UBC Theses and Dissertations

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UBC Theses and Dissertations

Singular holomorphic foliations Sertöz, Ali Sinan 1984

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C . I SINGULAR HOLOMORPHIC FOLIATIONS by ALI SINAN SERTOZ B . S c . , M i d d l e Eas t T e c h n i c a l U n i v e r s i t y , 1 9 7 8 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MATHEMATICS We accept t h i s t h e s i s as conforming to the r e q u i r e d s t andard THE UNIVERSITY OF March © A l i S inan BRITISH COLUMBIA 1 984 S e r t o z , 1984 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the University of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I further agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Mathematics Department of • The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date A P r i l 25 L", 1984 /R-n i i T h e s i s S u p e r v i s o r ; P r o f e s s o r J a m e s B . C a r r e l l A B S T R A C T A g e n e r a l i z e d N a s h B l o w - u p M ' w i t h r e s p e c t t o c o h e r e n t s u b s h e a v e s o f l o c a l l y f r e e s h e a v e s i s d e f i n e d f o r c o m p l e x s p a c e s . I t i s shown t h a t M ' i s l o c a l l y i s o m o r p h i c t o a m o n o i d a l t r a n s f o r m a t i o n a n d h e n c e i s a n a l y t i c . E x a m p l e s o f M ' a r e g i v e n . A p p l i c a t i o n s a r e g i v e n t o S e r r e ' s e x t e n s i o n p r o b l e m a n d r e d u c t i v e g r o u p a c t i o n s . A C * a c t i o n on G r a s s m a n n i a n s a r e d e f i n e d , f i x e d p o i n t s e t s a n d B i a l y n i c k i - B i r u l a d e c o m p o s i t i o n i s d e s c r i b e d . T h i s a c t i o n i s g e n e r a l i z e d t o G r a s s m a n n b u n d l e s . T h e G r a s s m a n n g r a p h c o n s t r u c t i o n i s d e f i n e d f o r t h e a n a l y t i c c a s e a n d i t i s shown t h a t f o r a c o m p a c t K a e h l e r m a n i f o l d t h e c y c l e a t i n f i n i t y i s a n a n a l y t i c c y c l e . A c a l c u l a t i o n i n v o l v i n g t h e l o c a l i z e d c l a s s e s o f g r a p h c o n s t r u c t i o n i s g i v e n . N a s h r e s i d u e f o r s i n g u l a r h o l o m o r p h i c f o l i a t i o n s i s d e f i n e d a n d i t i s s h o w n t h a t t h e r e s i d u e o f B a u m - B o t t a n d t h e N a s h r e s i d u e d i f f e r by a t e r m t h a t c o m e s f r o m t h e G r a s s m a n n g r a p h c o n s t r u c t i o n o f t h e s i n g u l a r f o l i a t i o n . A s a n a p p l i c a t i o n c o n c l u s i o n s a r e d r a w n a b o u t t h e r a t i o n a l i t y c o n j e c t u r e o f B a u m - B o t t . P o n t r y a g i n c l a s s e s i n t h e c o h o m o l o g y o f t h e s p l i t t i n g m a n i f o l d a r e g i v e n w h i c h o b s t r u c t a n i m b e d d i n g o f a b u n d l e i n t o t h e t a n g e n t b u n d l e . i i i TABLE OF CONTENTS A b s t r a c t i i T a b l e of Content s i i i L i s t of F i g u r e s v Acknowledgement v i Chapter 0: INTRODUCTION 1 Chapter 1: NASH CONSTRUCTION 5 0 . I n t r o d u c t i o n 5 1 .Coherent Sheaves 7 2. M o n o i d a l T r a n s f o r m a t i o n s 12 3. The Nash C o n s t r u c t i o n 14 4 4. Examples 25 5 . S e r r e ' s E x t e n s i o n Problem 36 Chapter 2: GRASSMANN GRAPH CONSTRUCTION 43 0. 1 . t r o d u c t i o n 43 1. B i a l y n i c k i - B i r u l a d e c o m p o s i t i o n 44 2. C * - A c t i o n s on G ( k , n ) 45 3. Examples 53 4. C * - A c t i o n s on Grassmann Bundles 55 5. Graph of Complexes 59 6. Examples 62 Chapter 3: SINGULAR HOLOMORPHIC FOLIATIONS 70 0 . I n t r o d u c t i o n 70 i v 1. P r e l i m i n a r i e s 71 2. Baum-Bott Res idues 75 3.Suwa's Work 79 4 .Nash Res idue and R e d u c t i o n 83 Chapter 4: OBSTRUCTION CLASSES 100 0 . I n t r o d u c t i o n 100 1 . O b s t r u c t i o n C l a s s e s 101 2 . F u t u r e Research P r o j e c t s 109 B i b l i o g r a p h y 112 V L i s t of F i g u r e s F i g u r e 1 53 F i g u r e 2 54 v i Acknowledgement I would l i k e t o thank to my t h e s i s s u p e r v i s o r James B . C a r r e l l who sugges ted the use of Grassmann Graph t e c h n i q u e w i t h r e s i d u e s and made s e v e r a l v a l u a b l e comments. My thanks a l s o go to J . K i n g fo r h i s numerous s u g g e s t i o n s and t o L . G . Rober t s for h i s encouragement at most needed moments. The p r o d u c t i o n of t h i s t h e s i s on computer owes much t o T i i l i n who not o n l y p r o v i d e d me w i t h a workable environment but a l s o typed l o n g s e c t i o n s be fore the d e a d l i n e . S p e c i a l thanks are to W. Holzmann who c o n t r i b u t e d h i s programming a b i l i t i e s and to A . C a y f o r d fo r s u p p l y i n g computer funds . F i n a l l y I thank The S c i e n t i f i c and T e c h n o l o g i c a l Research C o u n c i l of Turkey (TUBITAK), P r o f e s s o r J . B . C a r r e l l , Mathemat ics Department and the F a c u l t y of Graduate S t u d i e s at the U n i v e r s i t y of B r i t i s h Columbia f o r the f i n a n c i a l support tha t they have p r o v i d e d throughout my s t u d i e s . T h e i r g e n e r o u s i t y i s g r a t e f u l l y acknowledged. A . S inan S e r t o z 1 CHAPTER 0 INTRODUCTION A s i n g u l a r ho lomorphic f o l i a t i o n i s d e f i n e d as an i n t e g r a b l e coherent subsheaf of the tangent sheaf of a complex m a n i f o l d . [see B B 2 ] . We f i n d i t p r o m i s i n g to s tudy coherent subsheaves of l o c a l l y f r e e sheaves i n o r d e r to get a b e t t e r u n d e r s t a n d i n g of s i n g u l a r ho lomorphic f o l i a t i o n s . T h i s p r o j e c t i s c a r r i e d out i n Chapter 1 where a g e n e r a l i z e d Nash blow-up i s d e f i n e d ; the c o n t e x t of t h i s work i s the complex a n a l y t i c c a t e g o r y . Le t M be a complex a n a l y t i c space w i t h G a complex a n a l y t i c l o c a l l y f r e e sheaf on M and F a complex a n a l y t i c coherent subsheaf of G. We w i l l d rop the phrase "complex a n a l y t i c " from now on u n l e s s we want t o emphasize i t . Assume wi thout l o s s of g e n e r a l i t y t h a t M i s connec ted and t h a t the support of F i s M. There i s a p roper c l o s e d s u b v a r i e t y S of M such tha t F i s l o c a l l y f r ee on M-S. Le t F have rank k on M-S and G have rank n on M. For every p o i n t xeM-S, F d e f i n e s a k-p lane i n G , hence a p o i n t of 2 the Grassmann m a n i f o l d G ( k , n ) of k -p l anes i n n- space . Le t G(k,t7) denote the Grassmann bundle of k -p l anes i n G over M. Thus we have a ho lomorphic map from M-S i n t o the Grassmann bundle G ( k , t7 ) . The t o p o l o g i c a l c l o s u r e of the image of t h i s map i n G(k ,G) i s denoted by M' and i s c a l l e d the g e n e r a l i z e d Nash blow up of M w i t h r e spec t to F and G. In the a l g e b r a i c c a t e g o r y i t i s c l e a r tha t M' c o i n c i d e s w i t h the Z a r i s k i c l o s u r e of the image and hence i s a l g e b r a i c . I t i s not however c l e a r tha t M' i s a complex a n a l y t i c space . I t i s shown i n c h a p t e r 1 t h a t M' i s a complex a n a l y t i c space and t h a t i n f a c t M' i s a monoida l t r a n s f o r m a t i o n of M whose c e n t r e need not c o i n c i d e w i t h S,, the s i n g u l a r set of F. T h i s g e n e r a l i z e s works of N o b i l e on Nash blow-up and R o s s i and Riemenschneider on b lowing up coherent sheaves u s i n g the s t r u c t u r e sheaf [ N ] , [ R o ] , [ R i ] . Counterexamples which y i e l d n o n - a n a l y t i c s e t s are g i v e n a l o n g w i t h examples to the theorem. C o n d i t i o n s on smoothness of M' are a l s o g i v e n . S e v e r a l a p p l i c a t i o n s are g iven to S e r r e ' s e x t e n s i o n problem and to r e d u c t i v e group a c t i o n s . One i n t e r e s t i n g a spec t of g e n e r a l i z e d Nash blow-up i s t h a t M' comes equipped w i t h a v e c t o r b u n d l e , the r e s t r i c t i o n of the t a u t o l o g i c a l bundle on G ( k , G ) , which agrees w i t h the p u l l - b a c k of F | M - S . We then want to measure i n terms of c h a r a c t e r i s t i c c l a s s e s how much the t a u t o l o g i c a l bundle 3 d i f f e r s from the p u l l - b a c k of F. For t h i s we employ the t e c h n i q u e of Grassmann graph c o n s t r u c t i o n of the a l g e b r a i c c a t e g o r y [BFM]. I t must then be shown that t h i s t e c h n i q u e i s v a l i d i n the complex a n a l y t i c c a t e g o r y . T h i s i s a c c o m p l i s h e d i n Chapter 2 which s t a r t s w i t h C * - a c t i o n s on Grasssmann m a n i f o l d s . We d e s c r i b e the B i a l y n i c k i - B i r u l a d e c o m p o s i t i o n of t h i s a c t i o n on Grassmann m a n i f o l d s and g i v e examples . C * - a c t i o n s are then used to d e s c r i b e a n a l y t i c a l l y the graph c o n s t r u c t i o n which produces l o c a l i z e d Chern c l a s s e s . T h i s l o c a l i z e d c l a s s i s computed for a s p e c i a l case at the end of the c h a p t e r as an example. These t e c h n i q u e s are c o l l a b o r a t e d i n Chapter 3 t o g i v e a c a l c u l a t i o n of Baum-Bott r e s i d u e s . F i r s t we d e f i n e a Nash r e s i d u e fo r s i n g u l a r ho lomorphic f o l i a t i o n s fo r which the Nash Blow-up i s smooth. Then we c o n s i d e r those s i n g u l a r ho lomorphic f o l i a t i o n s t h a t are i n t e g r a b l e images of v e c t o r bundles i n the tangent b u n d l e , i . e . l e t E be a v e c t o r b u n d l e , T be the tangent bundle on M and * : E >T be a ho lomorphic v e c t o r bundle map; * ( E ) d e f i n e s a s i n g u l a r ho lomorphic f o l i a t i o n i f i t i s i n t e g r a b l e . When the Nash Blow-up of t h i s s i n g u l a r ho lomorphic f o l i a t i o n i s smooth the Baum-Bott r e s i d u e i s shown to be equa l to the sum of the Nash r e s i d u e and a term tha t i s c a l c u l a t e d by u s i n g Grassmann Graph c o n s t r u c t i o n on the Nash B low-up . T h i s 4 r e s u l t a l l o w s us to conc lude that the R a t i o n a l i t y C o n j e c t u r e of Baum and B o t t i s t r u e i n t h i s set up. In c h a p t e r 4 we c o n t i n u e our i n v e s t i g a t i o n of s i n g u l a r ho lomorphic f o l i a t i o n s by a g a i n v i e w i n g them as i n t e g r a b l e images of v e c t o r bundles i n the tangent b u n d l e . We then ask i f any v e c t o r bundle E can be imbedded i n t o T , d r o p p i n g the i n t e g r a b i l i t y c o n d i t i o n on the image and r e q u i r i n g t h a t $ be i n j e c t i v e . S e v e r a l t o p o l o g i c a l o b s t r u c t i o n s can be found i n the l i t e r a t u r e . Here we look at the problem from a D i f f e r e n t i a l Geometr ic p o i n t of v i e w . The problem i s p u l l e d back to the s p l i t t i n g m a n i f o l d M g of T . I f E imbeds i n t o T then i t i s shown t h a t c e r t a i n P o n t r y a g i n c l a s s e s i n the cohomology r i n g of M g v a n i s h . For t h i s we need and prove a g e n e r a l s tatement of B o t t ' s v a n i s h i n g theorem [B2 ] . We conc lude by d i s c u s s i n g some fu ture r e s e a r c h p r o j e c t s t h a t f o l l o w from t h i s work on s i n g u l a r ho lomorphic f o l i a t i o n s . 5 CHAPTER 1 NASH CONSTRUCTION 0:.INTRODUCTION We w i l l be working i n the ca tegory of complex a n a l y t i c s p a c e s . A complex a n a l y t i c space i s l o c a l l y the v a r i e t y of an i d e a l of ho lomorphic f u n c t i o n s . We w i l l drop the phrases " c o m p l e x " , " a n a l y t i c " or even "complex a n a l y t i c " from "complex a n a l y t i c space" when the r e f e r e n c e i s c l e a r . A l l sheaves are complex a n a l y t i c and aga in we w i l l drop "complex a n a l y t i c " from t h e i r names when the o m i s s i o n causes no a m b i g u i t y . L e t F be a coherent complex a n a l y t i c subsheaf of a l o c a l l y f r ee complex a n a l y t i c sheaf G on a complex space M. O u t s i d e a c l o s e d proper s u b v a r i e t y S of M the sheaf F i s l o c a l l y f r e e . Assuming t h a t rankC?=n and rankF|M-S=k we can d e f i n e a ho lomorphic map F from M-S i n t o the Grassmann 6 bundle G ( k , G ) over M. Le t M' be the c l o s u r e of the image of t h i s map i n the Grassmann b u n d l e . Our c o n t r i b u t i o n i s to show t h a t the map F i s meromorphic i n the sense of Remmert, i . e . M' i s complex a n a l y t i c . T h i s i s a c c o m p l i s h e d i n s e c t i o n 3 by showing tha t M 1 i s l o c a l l y a monoidal t r a n s f o r m a t i o n . The f i r s t two s e c t i o n s ga ther toge ther some f a c t s on coherent sheaves and monoida l t r a n s f o r m a t i o n s and set up the n o t a t i o n t h a t i s go ing to be used throughout the c h a p t e r . The t h i r d s e c t i o n d e s c r i b e s the c o n s t r u c t i o n of M' and proves tha t i t i s a n a l y t i c . We then d i s c u s s i t s r e l a t i o n t o the l i t e r a t u r e , i n p a r t i c u l a r to the works of N o b i l e , R o s s i and R iemenschne ider . Smoothness of M' and c o n d i t i o n s fo r M' to be g l o b a l l y a monoidal t r a n s f o r m a t i o n are a l s o d i s c u s s e d . Examples and counterexamples are g i v e n t o demonstrate the theorem and r e d u c t i v e group a c t i o n s are d i s c u s s e d as an a p p l i c a t i o n . In the second p a r t of the c h a p t e r S e r r e ' s e x t e n s i o n problem i s s t a t e d and S i u ' s s o l u t i o n i s g i v e n a l o n g w i t h the neces sa ry t e r m i n o l o g y . We then g i v e a p p l i c a t i o n s to t h i s problem which f o l l o w from the a n a l y t i c i t y of M'. 7 1:COHERENT SHEAVES In t h i s s e c t i o n we c o l l e c t t oge ther some of the f a c t s on coherent sheaves i n the forms tha t we are go ing to use them i n the s e q u e l . A l l sheaves are go ing t o be complex a n a l y t i c ; i n p a r t i c u l a r " c o h e r e n t " w i l l mean "complex a n a l y t i c c o h e r e n t " . For f u r t h e r d e t a i l s on coherent sheaves t o g e t h e r w i t h the p r o o f s of the s ta tements of t h i s s e c t i o n the reader i s r e f e r r e d to [GH,p 695 f f ] , [ F , p 1-3 ,94-95] , [ C ] . Le t M be a complex space and l e t 0^ be i t s s t r u c t u r e shea f . A sheaf of 0 M -modules w i l l be c a l l e d a sheaf of modules , the s t r u c t u r e sheaf b e i n g u n d e r s t o o d . D e f i n i t i o n : A sheaf of modules F over M i s c a l l e d coherent i f f o r every xeM there i s an open neighbourhood U of x such tha t t h e r e e x i s t s an exact sequence °u ^° fo r some i n t e g e r s m and k. For any epimorphism of the form °u " ^ u * ° l e t J?y denote the k e r n e l shea f ; 0 >RU >FU >0 . Then by the above d e f i n i t i o n of c o h e r e n c e , F i s coherent i f 8 J?y i s f i n i t e l y genera ted as a Oy-module f o r a l l U i n M. I t i s a c l a s s i c a l r e s u l t of Oka t h a t i f F i s coherent then i s a l s o c o h e r e n t . The most common examples of coherent sheaves a r i s e as the sheaves of ho lomorphic s e c t i o n s of v e c t o r b u n d l e s . Such coherent sheaves are c a l l e d l o c a l l y f r e e . For a l o c a l l y f ree sheaf F l e t s i r . . . , s be g l o b a l holomorphic s e c t i o n s . Then s 1 , . . . , s r generate an a n a l y t i c subsheaf which i s c o h e r e n t . T h i s example can be g e n e r a l i z e d as f o l l o w s ; c a l l a sheaf of modules G of f i n i t e type i f for any xeM t h e r e i s a ne ighbourhood U of x and an epimorphism for some i n t e g e r m. Then for a coherent sheaf any subsheaf of f i n i t e type i s c o h e r e n t . I f G i s a coherent subsheaf of a coherent sheaf F then the q u o t i e n t sheaf F/G i s a l s o c o h e r e n t . In g e n e r a l i f there i s a shor t exact sequence of sheaves 0 >F i >F2 >F3 ^0 such tha t any two of them are coherent then the t h i r d one i s a l s o c o h e r e n t . Support of a sheaf F i s d e f i n e d as supp/ r={xeM|/ r ; c^0}. Assume t h a t F i s coherent and that suppF i s open i n M. 9 D e f i n e a f u n c t i o n rk on M as rk(x)=rank of F as an 0 -module . X X T h i s i s an upper s e m i - c o n t i n u o u s f u n c t i o n on M. To see t h i s l e t rk(p)=m for some peM. F i s coherent so i n p a r t i c u l a r i t i s l o c a l l y of f i n i t e t y p e . Hence t h e r e i s an open neighbourhood U of p such tha t t h e r e e x i s t s an epimorphism °u  > F u *° fo r some m. Then there are m s e c t i o n s s , , . . . , s _ , of FTT such 1 m U that they generate each s t a l k Fx f o r a l l xeU. C l e a r l y t h i s i m p l i e s t h a t rk(x)<m for a l l xeU. S i n c e rk takes on nonnegat ive numbers, i t a c h i e v e s a minimum. The minimum va lue of rk i s c a l l e d the rank of F and i s denoted by r a n k F . D e f i n e a subset S of M as S={xeM|rk(x)>rankF} . S i s c a l l e d the s i n g u l a r se t of F and i s a c l o s e d proper s u b v a r i e t y of M. O u t s i d e S the coherent sheaf F i s the sheaf of s e c t i o n s of a v e c t o r bundle of rank=n where n = r a n k F , i . e . F |M-S i s l o c a l l y f r e e . For t h i s reason rankF i s a l s o r e f e r r e d to as the g e n e r i c rank of F . The above d e f i n i t i o n of rank c o i n c i d e s w i t h the more u s u a l d e f i n i t i o n of rank which i s g i v e n as r ( x ) = d i m k ( x ) F x B 0 k ( x ) 10 w h e r e . k ( x ) = 0 /m , m b e i n g t h e m a x i m a l i d e a l o f O . The f a c t t h a t r ( x ) = r k ( x ) f o l l o w s f r o m N a k a y a m a ' s l e m m a : N a k a y a m a Lemma: L e t A be a f i n i t e l y g e n e r a t e d m o d u l e o v e r t h e r i n g o f c o n v e r g e n t p o w e r s e r i e s C { X 1 f . . . , X } a n d l e t m be t h e m a x i m a l i d e a l . T h e n : , a k g e n e r a t e A i f f a 1 f . . . , a ^ g e n e r a t e A/mA. i . . . , i F o r a p r o o f o f N a k a y a m a ' s lemma s e e [ G H , p p 6 8 0 - 6 8 l ] . F o r t h e e q u i v a l e n c e o f r a n d r k s e e [ H a , p 2 8 8 , ( 1 2 . 7 . 2 ) ] . I f F i s a c o h e r e n t s h e a f , t h e n i t a d m i t s a l o c a l s y z y g y ; f o r a n y p o i n t peM t h e r e i s a n o p e n n e i g h b o u r h o o d U o f p s u c h t h a t 0 X 9 y >Om >Fy >0 i s e x a c t f o r some i n t e g e r s k a n d m. G l o b a l s y z y g i e s h o w e v e r n e e d n o t e x i s t f o r c o m p l e x a n a l y t i c c o h e r e n t s h e a v e s . T h i s d i f f i c u l t y i s c i r c u m v e n t e d by A t i y a h a n d H i r z e b r u c h by p a s s i n g t o t h e r e a l a n a l y t i c c a t e g o r y [ A H ] , V i e w M a s a r e a l a n a l y t i c s p a c e a n d l e t A be t h e r e a l s t r u c t u r e s h e a f o f M , i . e . A i s t h e s h e a f o f c o m p l e x v a l u e d r e a l a n a l y t i c f u n c t i o n s . A s h e a f G o f ^ - m o d u l e s on M i s c a l l e d a c o h e r e n t s h e a f o f / 4 - m o d u l e s i f f o r e v e r y xeM t h e r e i s a n o p e n n e i g h b o u r h o o d U o f x a n d a n e x a c t s e q u e n c e ^ g - >A% ^ ^0 f o r some i n t e g e r s p a n d q . F o r a n y c o h e r e n t s h e a f G o f 11 /4-modules and any compact subset X of the r e a l a n a l y t i c m a n i f o l d M, G^ has a r e s o l u t i o n on open subsets V of X by l o c a l l y f r e e ^-modules, [AH,p29,(2.6)],[BB2,p310,(6.30)3. I f M i s a compact m a n i f o l d then every coherent sheaf of /4-modules has a g l o b a l r e s o l u t i o n of l o c a l l y f r e e sheaves of ^-modules. For a coherent sheaf F of 0-modules FBQA i s a coherent sheaf of ^-modules. I f M i s compact then F®QA has a g l o b a l r e s o l u t i o n 0 >H >' • • *-H0 >FB>A >0 m where are l o c a l l y f r e e sheaves of ^-modules and m i s the r e a l dimension of M. Let H^ be the r e a l a n a l y t i c complex v e c t o r bundle whose sheaf of r e a l a n a l y t i c s e c t i o n s i s . The Chern c l a s s of F i s d e f i n e d i n terms of Chern c l a s s e s of H^s as f o l l o w s ; l e t H be the v i r t u a l bundle which i s d e f i n e d as the a l t e r n a t i n g sums of H^'s H = Z m = 0 ( - D 1 H i Then c(F)=c(H) =n m = 0c( H i)P ( i ) where p ( i ) i s +1 i f i i s even and -1 i f i i s odd. For f u r t h e r d e t a i l s on c h a r a c t e r i s t i c c l a s s e s of v i r t u a l bundles see [BB2]. For a proof t h a t c(F) depends only on F see [BS,p106,(lemmal1)]. 12 2:MONOIDAL TRANSFORMATIONS We w i l l f o l l o w the e x p o s i t i o n of H i r o n a k a and R o s s i [HR] to remind the reader of the t e r m i n o l o g y r e l a t e d to monoida l t r a n s f o r m a t i o n s , a l s o known as H i r o n a k a b low-ups . We adopt the d e f i n i t i o n of meromorphic map as i n t r o d u c e d by Remmert. Le t X and Y be complex a n a l y t i c spaces . A map f : X >-Y i s c a l l e d meromorphic i f t h e r e e x i s t s a proper s u b v a r i e t y V of X such t h a t i ) f | X - V >-Y i s ho lomorphic i i ) the c l o s u r e i n XxY of the graph of f | X - V i s a complex a n a l y t i c v a r i e t y . I t i s easy to see tha t t h i s d e f i n i t i o n reduces to the d e f i n i t i o n of a meromorphic f u n c t i o n when X=C, the complex numbers, and Y = P 1 , t h e p r o j e c t i v e l i n e . Le t f : X >Y be a morphism of complex spaces and l e t I be an i d e a l sheaf on Y . The p u l l back of / under f i s denoted by f * ( 7 ) , and the i d e a l genera ted by f*(7) i n t ? x i s denoted by f ~ 1 ( 7 ) . I f D i n Y i s the v a r i e t y d e f i n e d by / then the v a r i e t y tha t c o r r e s p o n d s to f ~ 1 ( 7 ) i n X i s denoted by f " 1 ( D ) . The p a i r f : X >Y and D i s c a l l e d the monoidal transformation of Y w i t h c e n t r e D i f i ) the i d e a l sheaf f ~ 1 ( 7 ) on X i s i n v e r t i b l e , i . e . l o c a l l y 13 f r e e of rank=1. i i ) i f g:Z >-Y i s any morphism of complex spaces having the p r o p e r t y ' ( i ) then there i s a unique morphism h:Z >X such that g=f.h. From the d e f i n i t i o n i t i s easy to see that Y-D i s isomorphic to X - f " ' ( D ) . I t can a l s o be shown that 'the monoidal t r a n s f o r m a t i o n i s determined by D and not X, see [HR], but we c a l l X the monoidal t r a n s f o r m a t i o n of Y. To c o n s t r u c t one such monoidal t r a n s f o r m a t i o n of Y l e t U be an open subset of Y which i s isomorphic to C m. Let ^0'**"'^n k e holomorphic f u n c t i o n s on U, not a l l i d e n t i c a l l y equal to zero, and l e t I be the i d e a l generated by the f ^ ' s . I f we denote the v a r i e t y of I as V then we can d e f i n e a holomorphic map F:U-V s-Pn where F ( x ) = [ f Q ( x ) : • • • : f n ( x ) ] . We wish to show that F:U >-Pn i s meromorphic. For t h i s l e t J be the i d e a l generated by the f u n c t i o n s ( x , [ X Q : • • • : X n ] ) ^ ( X ^ j ( x ) - X j f i ( x ) ) , 0^i,j<n, i * j over the s t r u c t u r e sheaf of UxP n where X^ are homogeneous c o o r d i n a t e s on P n. The v a r i e t y that i s d e f i n e d by J i n UxP n i s the c l o s u r e of the graph of F. Let U denote t h i s c l o s u r e . We have U=V(J), hence U i s a n a l y t i c and F i s meromorphic. 14 I t can be shown t h a t U i s a m o n o i d a l t r a n s f o r m a t i o n of U , see [ H R ] . The c e n t r e of t h i s t r a n s f o r m a t i o n i s V ( I ) i n U . T h i s p a r t i c u l a r d e s c r i p t i o n of a m o n o i d a l t r a n s f o r m a t i o n w i l l be u sed i n the n e x t s e c t i o n on the Nash b l o w - u p . 3:THE NASH CONSTRUCTION L e t F be an a n a l y t i c c o h e r e n t subshea f of a l o c a l l y f r e e shea f G on a complex a n a l y t i c space M . L e t rankF=k and rankt7=n. T h e r e i s a p r o p e r a n a l y t i c s u b v a r i e t y S o f M s u c h t h a t F | M - S i s l o c a l l y f r e e of rank k . C o n s i d e r t h e Grassmann b u n d l e G(k,c7) o v e r M . E a c h f i b r e o v e r xeM i s t h e space of k - p l a n e s i n Gx and i s t h e r e f o r e i s o m o r p h i c t o G ( k , n ) , t h e Grassmann space of k - p l a n e s i n n - s p a c e . D e f i n e a map F : M - S >MxG(k,G) by F ( x ) = ( x , [ F x ] ) . Here [ •] i s used t o denote t h e p o i n t t h a t " • " r e p r e s e n t s i n G ( k , n ) . L e t M ' be t h e t o p o l o g i c a l c l o s u r e o f F (M-S ) i n M x G ( k , G ) . L e t 7r :M' >M be t h e r e s t r i c t i o n of t h e n a t u r a l p r o j e c t i o n MxG(k,(?) >-M. Then we have t h e D e f i n i t i o n : w : M ' *-M i s t h e Nash blow-up o f M w i t h r e s p e c t t o F and G. We w i l l sometimes abuse l a n g u a g e and c a l l M ' t h e Nash b l o w - u p of M . 15 Le t T *-G(k,G) be the v e c t o r bundle on G(k,(7) which r e s t r i c t s to the t a u t o l o g i c a l bundle of G ( k , n ) on each s t a l k . P u l l back T to MxG(k,G) by the n a t u r a l p r o j e c t i o n MxG(k,G) ^G(k,(7) and r e s t r i c t i t to M ' . A g a i n use " T " to denote t h i s r e s t r i c t i o n and c a l l the bundle r >M' the t a u t o l o g i c a l bundle on M ' . I f U i s an open subset of M then l e t U ' denote -n~ 1 (U) where ?r:M' *-M i s the Nash blow-up of M. S ince the Nash Blow-up i s d e f i n e d as the c l o s u r e of the image of F , i t f o l l o w s tha t 7T 1 ( U ) i s the c l o s u r e of F ( U - U n S ) i n UxG(k,c7). THEOREM 1. M' i s l o c a l l y a monoida l t r a n s f o r m a t i o n of M. Consequent ly M' i s a complex a n a l y t i c space . P r o o f : Le t U be an open neighbourhood i n M such t h a t t7|U i s t r i v i a l and F\U i s f i n i t e l y g e n e r a t e d . Le t f 1 f . . . , f be ho lomorphic s e c t i o n s of F | U tha t generate i t , where r>k. I f U n S = 0 then U'=U by c o n s t r u c t i o n , so we c o n s i d e r the case when U n S ? £ 0 . S ince F\\J i s a subsheaf of the l o c a l l y f r ee sheaf G | U we can w r i t e each s e c t i o n f^ as f i = ( f i 1 f i n > 1 * U r where f^ j are ho lomorphic f u n c t i o n s on U . T h i s d e f i n e s an rxn m a t r i x A = ( f . j ) 1<i<r, 1<j<n. The row v e c t o r s of A generate a k-p lane i n n-space when A i s 16 e v a l u a t e d on U - U n S , s i n c e r a n k F ^ k for x e U - U n S , i . e . we have r a n k A | U - U n S = k and r a n k A | U n S ^ k . Let [A(x ) ] denote the p o i n t [F ] i n G ( k , n ) £ G ( k , G ) tha t i s r e p r e s e n t e d by the k-p l ane genera ted by the row v e c t o r s of A(x) , w i t h x e U - U n S. [A(x ) ]= [F ] e G ( k , n ) , x e U - U n S . In t roduce an i n d e x i n g set Bm={(N 1 , . . . , N k ) e Z k | I S N ^ . • .<Nk<m}. B w i l l be used to p i c k k rows of A and B w i l l be used to r ^ n p i c k k columns of A . I f ueB^ and 0eB n then d e f i n e A ^ = d e t (f . j ) ieu, je/3. Let A t f =( f_ j ) ieu, I S j S n . A^ i s the kxn-submatr ix of A formed by c h o o s i n g o n l y the k rows of A t h a t c o r r e s p o n d to u. For ueBr l e t I be the i d e a l genera ted by {A^^|/3eB n}, the d e t e r m i n a n t s of a l l k x k - s u b m a t r i c e s of A . S i n c e the rank of A on U - U n S i s not z e r o t h e r e e x i s t s a ueBr for which the c o r r e s p o n d i n g I i s not t r i v i a l . Choose and f i x t h i s u f o r the r e s t of the argument. v t 0 ! -R e c a l l i n g t h a t V(I ) denotes the proper s u b v a r i e t y of U on 17 which I v a n i s h e s , i t can be seen tha t V(I ) i s not n u n e c e s s a r i l y equa l to U n S . T h i s i s the reason why the c e n t r e of the monoida l t r a n s f o r m a t i o n need not c o i n c i d e w i t h the s i n g u l a r set S of F. On U - V ( I ) the rank of A i s k and t h e r e f o r e i t r e p r e s e n t s a k - p l a n e i n n-space which c o r r e s p o n d s to the same p lane d e f i n e d by Fx i n Gx; [A ( x ) ] = [ A ( x l ] = [ F x ] e G ( k , n ) , xeU-V(I ) . We d e f i n e a new map H : U - V ( I ) ^UxG(k,n) where H(x) = ( x , [ A (x) ] ) . N o t i c e tha t U - V ( I ^ ) and U - U n S are dense i n U . Le t 2 = V ( I M ) u ( U n S ) I n g e n e r a l Z need not be e q u a l to V ( I ^ ) but s i n c e F i s a subsheaf of a l o c a l l y f r ee sheaf we w i l l assume t h a t Z=V(I^) as i n [BB2, pp283-284] . F i r s t i t w i l l be shown t h a t H(U-V(I )) c o i n c i d e s w i t h F ( U - U n S ) . C l e a r l y F | U - V ( I )=H|U-V(I ) . U - U n S i s dense i n U and c o n t a i n s U-V(I ) . F | U - U n S extends F | U - V ( I ) . I t f o l l o w s from t h i s t h a t F ( U - V ( I ) = F ( U - U n S ) . S i n c e F and H agree on U-V(I ) we conc lude t h a t the c l o s u r e of the image of H c o i n c i d e s w i t h the c l o s u r e of the image of 18 F, i . e . HCU-VII ) ) = F(U-U nS) . One important aspect of t h i s equality i s that the right hand side does not depend on the choice of a. Therefore i f the l e f t hand side i s a monoidal transformation then t h i s w i l l also hold for F(U-U n S) and w i l l be independent of the choice of u. To show that H(U-V(I )) i s a monoidal transformation order B in some fixed manner n BQ,B} , . . . , 0 NeB n, N=CJJ-1. r e c a l l that a monoidal transformation i s defined as the closure of the image of T; T:U-V(I ) s-UxPN where T ( x ) . ( x , [ A ^ A ^ ] ) Then T(U-V(I D" = U i s the monoidal transformation of U with centre V(I ) and consequently U i s a complex analytic space as shown in section 2. It remains to show that U = H(U-V(I ) ) . For t h i s we use the Pliicker imbedding PI, which i s an imbedding of G(k,n) into P N where N=(£)-1. To define Pi l e t A be a kxn matrix representing a point yeG(k,n) and l e t A^(A) be the determinant of the kxk-submatrix of A determined by choosing a l l the rows and ^-columns of A for 19 A ^ ( A ) = d e t ( A i j ) 1<i<k, je/5. U s i n g the p r e v i o u s l y f i x e d o r d e r i n g of B n we d e f i n e P l ( y ) as P l ( y ) = P l ( [ A ] ) = [ A ( A ) : . . . : A - ( A ) ] . p 0 P N With the a i d of the P l u c k e r imbedding d e f i n e a new map ( i d , P I ) :UxG(k ,n ) KJxP N where ( i d , P l ) ( x , [ A ] ) = ( x , [ A „ ( A ) : . . . : A f l ( A ) ] ) . I f A=A then » 0 * N vA>"v*»,*4«",i 0 £ U N -T h e r e f o r e for xeU-V(I^) we have the f o l l o w i n g e q u a l i t i e s ; ( i d , P I ) . H ( x ) = ( i d , P l ) ( x , [ A (x ) ] ) = (x,[A„ (A ( x)):---:A F L (A ( x ) ) ] ) . = ( X , [ A m / 3 o ( X ) : . . . : A M / 3 N ( X ) ] ) =T(x) i . e . we have ( i d , P I ) . H = T . S i n c e P i i s an imbedding then ( i d , P i ) i s an isomorphism of H(U-V(I )) onto T ( U - V ( I ) ) . T h e r e f o r e M M ( i d , P I ) ( H ( U - V ( I ) ) ) = T ( U - V ( I ) ) . F i n a l l y we l i s t the s t r i n g of e q u a l i t i e s tha t we have p r o v e n ; here U i s the monoida l t r a n s f o r m a t i o n of U w i t h c e n t r e v d y ) and U ' i s the Nash Blow-up of U w i t h r e s p e c t to F and G. U = T ( U - V ( I )) 20 = F ( U - U n S ) = U ' . Hence U ' i s i somorphic t o a monoida l t r a n s f o r m a t i o n and i n p a r t i c u l a r U ' i s a n a l y t i c . I f V i s another open neighbourhood i n M such t h a t G | V i s f ree and F\V i s f i n i t e l y genera ted w i t h V n u V 0 , then U ' and V agree on U p V s i n c e F (U n V - U n V n S) depends o n l y on F and G. U and V are a n a l y t i c t h e r e f o r e U ' and V g lue toge ther t o g i v e a complex a n a l y t i c space . Hence M' i s a complex a n a l y t i c space . OED More can be s a i d about the nature of M' i f more data i s a v a i l a b l e . In chap te r 3 we w i l l be i n t e r e s t e d i n the case when M' i s smooth. For t h i s we f i r s t g i v e a d e f i n i t i o n : D e f i n i t i o n : A coherent sheaf F of rank k on M w i t h s i n g u l a r set S i s c a l l e d rich i f f o r any peS there e x i s t an open ne ighbourhood U of p and k s e c t i o n s s 1 , . . . , s k of F | U such tha t i ) s 1 f . . . , s k generate F | U - U n S , i i )s 1 , . . . , S j . are l i n e a r l y dependent on U n S . Examples of r i c h sheaves are easy t o f i n d . Complex a c t i o n s of r e d u c t i v e groups on complex m a n i f o l d s g i v e r i s e , to r i c h subsheaves of the tangent sheaf of M. F i n i t e l y genera ted subsheaves of l o c a l l y f r ee sheaves are r i c h . For a 21 l o c a l l y f ree sheaf subsheaves which are l o c a l l y of f i n i t e type are a l s o r i c h . Most f o l i a t i o n s tha t w i l l be c o n s i d e r e d w i l l be r i c h . The s i g n i f i c a n c e of r i c h n e s s becomes c l e a r i n the next c o r o l l a r y . Let the set up be as b e f o r e ; F i s a coherent subsheaf of a l o c a l l y f r ee sheaf G on a complex space M w i t h S be ing the s i n g u l a r set of F. Corollary 1: I f F i s r i c h then M' i s a monoida l t r a n s f o r m a t i o n of M w i t h c e n t r e S. P r o o f : The n o t a t i o n be ing as i n theorem 1 we have to e s t a b l i s h two f a c t s : ( i ) l o c a l l y the c e n t r e of the monoida l t r a n s f o r m a t i o n c o i n c i d e s w i t h S, and ( i i ) g l o b a l l y these l o c a l p i e c e s g lue t o g e t h e r . ( i ) T h i s f o l l o w s e a s i l y from the d e f i n i t i o n of r i c h n e s s . ( i i ) Le t U and V be open neighbourhoods i n M such t h a t U ' and V are monoida l t r a n s f o r m a t i o n s of U and V r e s p e c t i v e l y , w i t h c e n t r e s U n S and V n S r e s p e c t i v e l y . I f U n V ^ 0 , then from the uniqueness of monoida l t r a n s f o r m a t i o n s U 1 and V agree on U n V and hence g lue t o g e t h e r . S i n c e a monoida l t r a n s f o r m a t i o n i s u n i q u e l y determined by i t s c e n t r e , ( U U V ) ' i s the monoida l t r a n s f o r m a t i o n of ( U U V ) w i t h c e n t r e ( U U V ) n S . Q E D The above proo f showed t h a t i f F i s r i c h then the c e n t r e of the monoida l t r a n s f o r m a t i o n can be e x p l i c i t l y d e f i n e d . 22 C a r r y i n g on w i t h t h i s theme we can say more; l e t the n o t a t i o n be as b e f o r e , we have C o r o l l a r y 2 : I f F i s r i c h and S i s smooth then M' i s a complex m a n i f o l d . P r o o f : I f F i s r i c h then M' i s a monoidal t r a n s f o r m a t i o n of M w i t h c e n t r e S, by c o r o l l a r y 1. A monoida l t r a n s f o r m a t i o n w i t h a smooth c e n t r e i s smooth. QED Remmert has proven the f o l l o w i n g r e s u l t ; i f N i s an a n a l y t i c subset of an a n a l y t i c space X and i f Z i s an a n a l y t i c subset of X-N then the c l o s u r e of Z i n X i s an a n a l y t i c space i f d im(N)<dim(Z) , see [ R ] . I t i s i n t e r e s t i n g to r e l a t e t h i s r e s u l t w i t h theorem 1. Le t X be the t o t a l space of the Grassmann bundle 7r : G ( k , G ) >M and l e t N be i r 1 ( S ) which i s an a n a l y t i c subset of G ( k , G ) . Le t Z be F(M-S) which i s a l s o a n a l y t i c and i s i n G ( k , G ) - 7 r 1 (S) . I f d im(7T- 1 (S) )<dim(F(M-S) ) then Remmert's theorem a s sure s that M'=F(M-U) i s a n a l y t i c . In g e n e r a l the c o n d i t i o n of Remmert's theorem i s not s a t i s f i e d . For example take rankF=3, rankG=5, dim(S)=2 and dim(M)=7; then d im( i r - 1 (S) )=2+3^ (5-3) = 8 and dim(F(M-S) )=7. 2 3 As an a p p l i c a t i o n of theorem 1 we can r e t r i e v e N o b i l e ' s theorem on Nash b lowing-up [ N ] , For t h i s l e t us d e f i n e the u s u a l Nash c o n s t r u c t i o n as used by N o b i l e . L e t X be a s i n g u l a r s u b v a r i e t y of C n w i t h d imens ion k. Le t S be the set of s i n g u l a r p o i n t s of X . Then the tangent sheaf TX of X i s a coherent subsheaf of the tangent sheaf of C n . D e f i n e T?:X-S ^XxG(k ,n) where 7}(x) = ( x , [ ( 7 T ) v ] ) . The c l o s u r e TJ (X-S) i n XxG(k ,n) i s c a l l e d the Nash B lowing-up of X and i s denoted by X * , see [ N ] . Theorem ( N o b i l e ) : A Nash B lowing-up i s l o c a l l y a monoida l t r a n s f o r m (wi th c e n t r e a s u i t a b l e i d e a l ) . P r o o f : TX i s a coherent subsheaf of the tangent sheaf T of C n w i t h supp7T=X. Le t F=7T|X, G=T\K. Then the g e n e r a l i z e d Nash Blow-up of X w i t h r e s p e c t to F and G as i n theorem 1 i s l o c a l l y a monoida l t r a n s f o r m a t i o n . QED H . R o s s i has proven tha t fo r any coherent sheaf F on M w i t h s i n g u l a r set S, t h e r e e x i s t s an a n a l y t i c space N w i t h a 24 proper map «//: N >M such tha t ( i )\p:H-\l/~ 1 (S) : >-M-S i s b i h o l o m o r p h i c and (ii)\}/*F i s l o c a l l y f ree modulo t o r s i o n . . R o s s i c o n s t r u c t s t h i s N as f o l l o w s , see [ R o ] . Le t peS. S i n c e F i s coherent t h e r e e x i s t s an open neighbourhood U of p such t h a t there i s an exact sequence ° v — ^ 5 — — for some i n t e g e r s m and n , where 0 i s the s t r u c t u r e sheaf of M. Le t / be the image sheaf of the map „m _n °u * V Then I i s a coherent subsheaf of 0™ and i s l o c a l l y f r ee of rank k on U - U n S where k i s r a n k F . For each p o i n t x e U - U n S , 7 d e f i n e s an ( n - k ) - p l a n e i n the n-space of o " . D e f i n e a map A W r j : U - U n S HJxG(n-k ,n ) where 7?(x) = ( x , [ / x ] ) . Then R o s s i d e f i n e s N a s , see [Ro ] , N|U=??(U-U n S) i n U x G ( n - k , n ) . Theorem ( R o s s i ) : TJ i s a meromorphic map. P r o o f : We have to show t h a t N |U i s a complex a n a l y t i c space . N | U i s the g e n e r a l i z e d Nash blow-up of U w i t h r e s p e c t to 7 |U and 0^, hence i s a n a l y t i c by theorem 1. 25 QED That N | U ' s g lue t o g e t h e r to g i v e a complex a n a l y t i c space N f o l l o w s from the uniqueness of monoida l t r a n s f o r m a t i o n s . L a t e r 0. R iemenschneider has shown t h a t i f N , i s another a n a l y t i c space w i t h a p roper map x : N , >M such t h a t ( i ) x : N r x " ' ( S ) *-M-S i s b i h o l o m o r p h i c and ( i i ) x*F i s l o c a l l y f r e e modulo t o r s i o n , then t h e r e i s a unique ho lomorphic map u : N , such t h a t X=i//.a>, see [ R i ] . He a l s o showed t h a t i f F i s an i d e a l sheaf then N c o i n c i d e s w i t h a monoida l t r a n s f o r m a t i o n . Our Nash c o n s t r u c t i o n shows tha t M' i s a lways l o c a l l y a monoida l t r a n s f o r m a t i o n . The u n i v e r s a l p r o p e r t y of M' i s an i n t e r e s t i n g problem which we propose to s tudy e l s e w h e r e . In the next s e c t i o n we w i l l g i v e s e v e r a l examples of the Nash c o n s t r u c t i o n . 4:EXAMPLES 1 ) C o n s i d e r the C * a c t i o n on P N ; X . [ x 0 : . . . : x n ] = [ x 0 : X a ( l ) x i : . . . : X a ( n ) x n ] where X e C * , a ( i ) a re i n t e g e r s w i t h 0 < a ( 1 ) < . . . < a ( n ) , and [ x n : » » « : x ] are homogeneous c o o r d i n a t e s of P N . The f i x e d 26 p o i n t s of t h i s C * a c t i o n are [l:0:--«:0],[0:1:«--:0],[0:0 :---: l] . The o r b i t s of t h i s a c t i o n d e f i n e a s i n g u l a r ho lomorphic f o l i a t i o n whose set of s i n g u l a r i t i e s i s the f i x e d p o i n t set of the C * a c t i o n . T h i s f o l l o w s s i n c e the i s o t r o p y groups are f i n i t e . The Nash Blow-up of t h i s s i n g u l a r ho lomorphic f o l i a t i o n i s a smooth m a n i f o l d . To see t h i s l e t us choose E u c l i d e a n c o o r d i n a t e s around the f i x e d p o i n t [1:0: •••:0]; x 1 = x i / / x0 ' * * * f X n = x r/ x 0* The above C * a c t i o n has the f o l l o w i n g form w i t h r e s p e c t to these c o o r d i n a t e s ; x . ( x 1 , . . . , x n ) = ( x a ( 1 ) x 1 , . . , , x a ( n ) x n ) . In t h i s c o o r d i n a t e p a t c h the f i x e d p o i n t set i s the o r i g i n 0=(0,...,0). Any X = ( X 1 , . . . , X n ) e C n d e f i n e s a ho lomorphic curve x ^x-x=(x a ( 1 ) x 1 x a ( n ) x n ) . The d i r e c t i o n of t h i s curve at X , i . e . when X=1 , i s [ a ( 1 ) X 1 : • • • : a ( n ) X ] i n P n . T h i s i s the image of X under the Nash c o n s t r u c t i o n ; to see why, l e t P(T) be the p r o j e c t i v i z e d tangent bundle of C n , P ( T ) = C n x P n _ 1 . D e f i n e a s e c t i o n S of P(T) as S :C n -0 ^P(T) 2 7 where S ( X ) = ( X , [ a ( 1 ) X 1 : . . . : a ( n ) X n ] ) . Then the Nash Blow-up of t h i s c o o r d i n a t e p a t c h w i t h r e spec t t o the above C * a c t i o n i s the c l o s u r e of S ( C n - 0 ) i n P ( T ) . Le t M 0 denote t h i s c l o s u r e . To show t h a t M 0 i s a complex m a n i f o l d we c o n s t r u c t the f o l l o w i n g c o o r d i n a t e maps; l e t e = [ e 1 : • • • : e n ] be a p o i n t i n P n 1 and l e t t e C . D e f i n e a map A i : P n _ 1 x C >M0 where A i ( e , t ) = ( ( t e 1 / a ( 1 ) e i , . . . , t e n / a ( n ) e i ) , e ) fo r e ^ O and i = 1 , . . . , n . D i f f e r e n t A ^ ' s p a t c h toge ther to d e f i n e a ho lomorphic c o o r d i n a t e system. Hence M 0 i s a complex m a n i f o l d . Le t " TT:M0 >Cn be the n a t u r a l p r o j e c t i o n . Then the f i b r e above the s i n g u l a r set 0 of the s i n g u l a r f o l i a t i o n i s 7 r " 1 ( 0 ) = P n 1 . S i m i l a r l y we can c o n s t r u c t by b lowing up tha t c o o r d i n a t e p a t c h of P n w i t h {x^#0}. A l i these ' s a re smooth and they g lue toge ther to form a complex m a n i f o l d M. I t i s p o s s i b l e to g i v e d e f i n i n g e q u a t i o n s f o r . I f f o r example H x 1 , . . . f x ) , [ y 1 : • • • : y n ] ) are c o o r d i n a t e s f o r P ( T ) | U 0 then M 0 i s d e f i n e d by the e q u a t i o n s a ( J ) y _ x j = a ( i ) y , i , j = 1 , . . . , n , i # j . The i somorphism between the Nash Blow-up M 0 and the monoidal 28 t r a n s f o r m a t i o n of {XQ*0} w i t h c e n t r e [ 1 : 0 : :0] i s g iven by the map where T ( ( x 1 , . . . , x n ) , [ y 1 : T : P ( T ) - -=>-P(T) i y n » -: a ( n ) y N 3 ) . ( ( x 1 , — ,x ) , [ a ( 1 ) y . 2) In t h i s example we w i l l d e f i n e a rank 2 s i n g u l a r f o l i a t i o n on P" and c o n s t r u c t i t s Nash Blow-up. S t a r t w i t h two C * a c t i o n s on P* d e f i n e d as X- [ x 0 : • • • : x , ] = [ X C l 0 x 0 : • • • : X C 1 • x , ] and X « [ x 0 : where ' X e C * , [ x 0 : • • : x „ ] = [ X C 2 0 x 0 : :X L x j : x „ ] e P f t and c^^eZ, i = 1,2 j=0, 4. D e f i n e a 2x4 matr ix C ^ , for k = 0 , . . . , 4 , as f o l l o w s C k = ( c i j " c i k ) 1 ^ 2 , 0 < j < 4 , j * k Choose c . . such that 1) a l l the e n t r i e s of C, a re nonzero 1 _ K and 2) the de te rminant s of a l l the 2x2 minors of C k are n o n z e r o . To d e s c r i b e a rank 2 s i n g u l a r f o l i a t i o n on P" choose E u c l i d e a n c o o r d i n a t e pa tches U . - C x ^ O } i = 0 , . . . , 4 . In U Q l e t X,=Xi /x_, . . . ,X4=Xfl /xo. On UQ l e t a^j denote the e n t r i e s of the m a t r i x CQ C 0 = ( a . j ) 1<i<2,0<j<4. Note t h a t by the above c o n d i t i o n s on C K the e n t r i e s a^j a re 29 nonzero , a i j = c i j " c i O ' 1 - i - 2 » 0^j^4. On UQ the above C * a c t i o n s take the form X * ( X i r . . . , X ( j ) = (X 1 1 X i , . . . , X ^*X(j) and X * ( X , , . . . , X j l ) = (X 2 ' X ) , . . . , X 2 "Xj | ) At any p o i n t X = ( X , , . . . , X a ) f o r each C * a c t i o n the d i r e c t i o n of the o r b i t p a s s i n g through X d e f i n e s v e c t o r f i e l d s V , (X ) = (a , , X , , . . . , a , aXj|) and V 2 (X) — ( a 2 i X f , . . . , 3 2 4 X 4 ) . N o t i c e t h a t V I , i = l , 2 , i s d e f i n e d for a l l X e U n = C „ . V , ( X ) and V 2 ( X ) t o g e t h e r generate a v e c t o r subspace of the tangent space of UQ a t the p o i n t X CU Q . Le t V ( X ) = ( a . j X j ) 1<i<2, 1 £ j < 4 , and S n = { X e U n | r a n k V (X)<2} . On UQ-SQ, V , and V 2 span an i n t e g r a b l e 2-subbundle of the tangent b u n d l e . To c o n s t r u c t the Nash Blow-up d e f i n e a map F :UQ-SQ s-UnxG (2 ,4) where F ( X ) = ( X , [ V ( X ) ] ) . The c l o s u r e M 0 = F ( U Q - S Q ) o f ~ F ( U Q - S Q ) i n U QxG ( 2 , 4 ) i s the Nash Blow-up of U n w i t h r e spec t to the above C * a c t i o n s . 30 L e t ir:M 0 >-UN be t h e u s u a l p r o j e c t i o n . T h e n a s i n t h e p r e v i o u s e x a m p l e i t c a n be shown t h a t 1) M 0 - 7 r " 1 ( 0 ) i s a c o m p l e x m a n i f o l d . V , ( X ) a n d V 2 ( X ) g e n e r a t e a c o h e r e n t s u b s h e a f . F o f t h e t a n g e n t s h e a f TQ o f U Q - ( 0 ) . V , ( X ) a n d V 2 ( X ) a r e l i n e a r l y d e p e n d e n t a l o n g t h e c o o r d i n a t e a x e s a n d by t h e d e f i n i t i o n o f r i c h s h e a v e s F i s a r i c h c o h e r e n t s h e a f w i t h s i n g u l a r i t i e s a l o n g t h e a x e s . S i n c e t h e s i n g u l a r s e t i s s m o o t h , by c o r o l l a r y 2 o f s e c t i o n 3 t h e N a s h b l o w - u p , M 0 - 7 r ~ 1 ( 0 ) , i s a l s o s m o o t h . 2) 7 r:M 0 -7r~ 1 (SQ) S^UQ-SQ i s a n i s o m o r p h i s m . T h i s f o l l o w s f r o m t h e t h e a b o v e e x p l a n a t i o n s b y o b s e r v i n g t h a t SQ c o n s i s t s o f t h e c o o r d i n a t e a x e s i n U 0 . 3) F o r X e S p - O , 7 T 1 ( X ) = P 2 . 4) 7r" 1 ( 0 ) i s i s o m o r p h i c t o f o u r c o p i e s o f P i n G ( 2 , 4 ) , w h i c h t o u c h e a c h o t h e r a t a s i n g l e p o i n t . T h u s f o u r p l a n e s e a c h t o u c h i n g e a c h o f t h e o t h e r t h r e e a t a s i n g l e p o i n t g i v e s i x s i n g u l a r i t i e s o f M 0 T h e s e l a s t t w o a s s e r t i o n s f o l l o w i f a m a t r i x r e p r e s e n t a t i o n f o r e l e m e n t s o f UQXG(2,4) i s u s e d i n c a l c u l a t i n g t h e c l o s u r e o f F ( U Q - S Q ) . I n p a r t i c u l a r ?~ 1 ( 0 ) i s o b t a i n e d ' b y o b s e r v i n g t h e f i b r e s i n M 0 a b o v e t h e c o o r d i n a t e 2 a x e s ; t h e f o u r c o p i e s o f P i n n~1(0) a r e e a c h c o n t r i b u t e d by a c o o r d i n a t e a x i s a t t h e o r i g i n . T h e f a c t t h a t t h e y t o u c h 31 each o t h e r i s aga in found by u s i n g the mat r ix r e p r e s e n t a t i o n and c a l c u l a t i n g the l i m i t s . These c a l c u l a t i o n s are s t r a i g h t f o r w a r d and are o m i t t e d . We can s i m i l a r l y c o n s t r u c t M 1 r . . . , M 4 each of which w i l l s a t i s f y s i m i l a r c o n d i t i o n s 1-4 as above . They g lue t o g e t h e r to g i v e an a n a l y t i c space M w i t h 30 i s o l a t e d s i n g u l a r i t i e s . 3) An example used i n [CS1] can be adopted to d e s c r i b e a graph c o n s t r u c t i o n which g i v e s r i s e t o a n o n a n a l y t i c s e t . Le t Z ac t on C 2 ZxC 2 ^ C 2 where n « ( z , , z 2 ) = ( 2 n z , , 2 n z 2 ) . Le t H=C 2 - { (0 ,0 ) } /Z and d e f i n e 7T : H 1 as 7r(<z,w>) = [ z : w ] , where <z,w> denotes the e q u i v a l e n c e c l a s s of ( z , w ) e C 2 i n H . There i s a C * a c t i o n on H d e f i n e d as f o l l o w s C*xH • H where X « < z , w > = < X z , w > . S i m i l a r l y we have a C * a c t i o n on P 1 d e f i n e d as C * x P 1 >t?} where 32 X « [ x 0 : x 1 ] = [ X x 0 : x 1 ] . 7r i s e q u i v a r i a n t w i t h r e s p e c t t o t h i s C * a c t i o n . L e t T d e n o t e t h e p r e i m a g e o f [ 0 : 1 ] , i . e . . T=TT - 1 ( [ 0 : 1 ] ) . T i s t h e f i x e d p o i n t s e t o f t h e C * a c t i o n on H . H - T i s i s o m o r p h i c t o ( C * / Z ) x C u n d e r t h e map w h i c h s e n d s <z,w> t o ( < z , 0 > , w / z ) . C o n s i d e r t h e b u n d l e p r o j e c t i o n pr:H-T >C*/Z w h e r e p r ( < z , w > ) ->-<zf0>. L e t T d e n o t e t h e c l o s u r e o f t h e g r a p h o f pr i n H x ( C * / Z ) , l e t T n d e n o t e t h e p o i n t s o v e r < z , 0 > i n T , i . e . i f p 2 : H x ( C * / Z ) >C*/Z i s t h e n a t u r a l p r o j e c t i o n t h e n r < z Q > = p 2 " 1 ( < z , 0 > ) . I t c a n be shown t h a t T n = { ( < z , t > , < z , 0 > ) e H x ( C * / Z ) | t e C ] u { ( < 0 , X > , < z , 0 > ) | X e C * } I t i s c l e a r how t o o b t a i n t h e f i r s t c o m p o n e n t . T o o b t a i n t h e s e c o n d c o m p o n e n t c o n s i d e r l i m ( < z , 2 n X > , < z , 0 > ) = l i m ( < 2 ~ n z , X ) , < 2 ~ n z , 0 > ) B u t s i n c e <2 z , 0 > = < z , 0 > o n C * / Z , t h i s f i n a l l i m i t i s ( < 0 , X > , < z , 0 > ) w h i c h i s o u r s e c o n d c o m p o n e n t . I f T i s a s u b v a r i e t y t h e n i t m u s t be i r r e d u c i b l e . B u t t h e s e c o n d c o m p o n e n t { ( < 0 , X > , < z , 0 > ) e H x ( C * / Z ) | X e C * } i s a c l o s e d s u b v a r i e t y o f H x ( C * / Z ) w h o s e d i m e n s i o n i s e q u a l 33 to the d imens ion of T. T h i s c o m p o s i t i o n shows tha t T i s not a n a l y t i c . R e c a l l t h a t our Nash c o n s t r u c t i o n i s a l s o a graph c l o s u r e . In the above example the map pr can not be m e r o m o r p h i c a l l y extended a c r o s s T whereas the data tha t we supp ly fo r the Nash c o n s t r u c t i o n guarantees t h a t the map c o n s t r u c t e d t h e r e i s a lways meromorphic . Another such example u s i n g Hopf m a n i f o l d s i s g i v e n i n [ G , p 2 9 , e g 4 ] . 4) A s imple example of n o n a n a l y t i c graph c l o s u r e can be c o n s t r u c t e d by c o n s i d e r i n g a ho lomorphic f u n c t i o n f : C * >C w i t h an e s s e n t i a l s i n g u l a r i t y a t the o r i g i n . S i n c e by P i c a r d ' s theorem f a t t a i n s every v a l u e i n f i n i t e l y many t imes i n any ne ighbourhood of the s i n g u l a r i t y , the c l o s u r e r of the graph of f in CxC can not be a n a l y t i c . Otherwise we can i n t e r s e c t r w i t h Cx{y] f o r any yeC and we s h o u l d get an a n a l y t i c v a r i e t y as the i n t e r s e c t i o n . But by P i c a r d ' s theorem (Cx{7))«r w i l l have i n f i n i t e l y many i s o l a t e d p o i n t s , hence i s not an a n a l y t i c v a r i e t y . T h i s c o n t r a d i c t i o n shows t h a t r i s not a n a l y t i c . 5 ) Le t G be a connec ted compact L i e group and M a complex m a n i f o l d . Le t 34 * : GxM >M be a C°° a c t i o n of G by means of b i h o l o m o r p h i s m s . I f V i s a r e a l v e c t o r f i e l d induced by a 1-parameter subgroup of G and J i s the complex s t r u c t u r e t ensor of M, then V-iJV=W i s a ho lomorphic v e c t o r f i e l d and the f i x e d p o i n t set of * i s a complex submani fo ld which i s the set of common zeros of a l l such W, [CS1, page 50 ] , These W's generate a coherent i n t e g r a b l e subsheaf F of the tangent sheaf T of M, i . e . F i s c l o s e d under the bracket o p e r a t i o n . By R i c h a r d s o n ' s theory G has a p r i n c i p a l o r b i t t y p e , i . e . on an open dense subset U of M a l l o r b i t s have the same rank , see [ R c ] . The c l o s u r e of an o r b i t p i c k s up o r b i t s of s m a l l e r r a n k s . S i n g u l a r set of F c o n s i s t s of the un ion of a l l o r b i t s whose ranks are l e s s ' than the rank of the p r i n c i p a l o r b i t . Let G be the i n t e g r a b l e l o c a l l y f r ee subsheaf of r|U d e f i n e d by the p r i n c i p a l o r b i t s of G i n U. Then F|U=<7. Hence the Nash c o n s t r u c t i o n d e f i n e d by u s i n g the p r i n c i p a l o r b i t s c o i n c i d e s w i t h the Nash c o n s t r u c t i o n of M w i t h r e s p e c t to F and T, and c o n s e q u e n t l y i s a n a l y t i c . 6) C o n s i d e r a C * a c t i o n on C n as C * x C n * C n where X*(X.|,..., X^ ) = (X X^,...,X x^) 3 5 w i t h a ( i ) e Z . The Nash Blow-up of C n w i t h r e s p e c t to the o r b i t s of t h i s a c t i o n i s a smooth submani fo ld M of C n x P n 1 . L e t TT:M >-CN be the u s u a l p r o j e c t i o n . We can d e f i n e a C * a c t i o n on M which extends the C * a c t i o n on TT" M C n - 0 ) ; C*xM >M where fo r XeC* and pcM d e f i n e X « p as f o l l o w s : 1) I f p = ( ( X 1 , . . . , X n ) , [ a ( 1 ) X 1 : • • • : a ( n ) X n ] ) , then X . p = ( ( X a ( l ) X 1 , . . . , X a ( n ) X n ) , [ a ( D X a ( l ) X 1 : . . . : a ( n ) X a ( n ) X N 3 ) . C l e a r l y X « p e M . 2) I f p = ( 0 , [ X 1 , . . . , X n ] ) , then X . p = ( r j , [ x a ( l ) X 1 : . . . : X a ( n ) X n ] ) . T h i s a c t i o n i s the r e s t r i c t i o n to M of the C * a c t i o n of C n x P n 1 which i s d e f i n e d as f o l l o w s X • ( ( x 1 , . . . , x n ) , [ e 1 : • • • : e n ] ) = ( ( X a { l ) X l X a ( n ) x n ) , [ X a ( l ) e i : . . . : X a ( n ) e n ] ) where X e C * , ( x 1 , . . . , x n ) e C n and [ e 1 : • • • : e n ] e P n \ In t h i s example the C * a c t i o n on C n i s l i f t e d up t o the Nash Blow-up tha t i s d e f i n e d by the C * a c t i o n . I t can be c o n j e c t u r e d t h a t r e d u c t i v e group a c t i o n s can be l i f t e d to the Nash Blow-up that i s d e f i n e d i n the p r e v i o u s example. 7 ) Le t E and F be v e c t o r bundles on a complex space M. Let rankE=r and rankF=n. Le t 36 d> ; E >»F be a v e c t o r bundle map. There e x i s t s a proper s u b v a r i e t y S of M such t h a t rank<t>|M-S=k rank$|S<k where k i s some i n t e g e r not g r e a t e r than r . Then we can d e f i n e a meromorphic map F :M >MxG( k ,F) as F ( x ) = ( x , [ * ( x ) ] ) . The proof t h a t F i s meromorphic i s q u i t e ana logous to the proo f of theorem 1 t h e r e f o r e we do not repeat i t h e r e . I f M i s a complex m a n i f o l d and F i s i t s tangent bundle then $(E) d e f i n e s a d i s t r i b u t i o n . I f i n - a d d i t i o n to t h i s , $(E) i s i n t e g r a b l e then i t d e f i n e s a s i n g u l a r ho lomorphic f o l i a t i o n whose s i n g u l a r i t y set i s S, i . e . M-S i s f o l i a t e d w i t h k - d i m e n s i o n a l l eaves and fo r any xeM-S the tangent space of the l e a f t h a t passes through x i s * ( E ) . We w i l l r e t u r n to t h i s approach to s i n g u l a r f o l i a t i o n s i n Chapter 3 and 4. 5 :SERRE'S EXTENSION PROBLEM • In t h i s s e c t i o n we w i l l g i v e an a p p l i c a t i o n of our theorem to S e r r e ' s problem on e x t e n d i n g coherent sheaves . In 1966 Ser re posed h i s famous e x t e n s i o n prob lem, [ S ] . Le t M be 37 a complex a n a l y t i c space w i t h a c l o s e d a n a l y t i c subset Z and a coherent a n a l y t i c sheaf F d e f i n e d on M - Z . I f i : M - Z >VL i s the u s u a l i n c l u s i o n then l e t i^F denote the d i r e c t image of F, i . e . for any open subset of M the sheaf of s e c t i o n s of the sheaf i * F on U , r ( U , i * F ) , i s d e f i n e d t o be e q u a l to r ( U - Z , F ) . S e r r e ' s problem i s to determine i f i * F i s c o h e r e n t . T h i s problem has been s u c c e s s f u l l y a t t a c k e d by S i u , Trautmann and Thimm, see [ S T ] , [ T ] , S e r r e h i m s e l f proved the f o l l o w i n g , [ S ] ; Theorem ( S e r r e ) : I f M i s n o r m a l , F i f t o r s i o n f r ee and codim(Z)>2 then the f o l l o w i n g are e q u i v a l e n t : i ) i * F i s c o h e r e n t . i i ) There i s an a n a l y t i c coherent sheaf E on M which extends F. i i i ) For a l l peZ, there i s an open neighbourhood U of p such t h a t for a l l xeU-Z the image of T ( U - Z , F ) genera te s F^ as an O -module , where O i s the s t r u c t u r e sheaf of M. x ' O b v i o u s l y ( i ) i m p l i e s ( i i ) . I t i s s u r p r i s i n g however tha t ( i i ) does not imply ( i ) w i thout the as sumpt ions of the theorem; fo r example i*<? M _2 * s n o t c o h e r e n t , [ S ] . D e f i n i t i o n : I f i * F i s coherent then F i s c a l l e d extendable. The set up be ing as above l e t G be a l o c a l l y f r ee sheaf on M and l e t F be a coherent subsheaf of G on M - Z . Le t M' be the Nash Blow-up of M w i t h r e s p e c t to F and G. S i n c e F i s not 38 d e f i n e d everywhere theorem 1 of s e c t i o n 3 does not d i r e c t l y a p p l y . In t h i s case we have the f o l l o w i n g theorem. THEOREM 2. Assume that F i s t o r s i o n f r e e , M i s normal and codim(Z)>2. Then M' i s a n a l y t i c i f f F i s ex tendab le as a coherent subsheaf of G. P r o o f : I f F i s e x t e n d a b l e , i . e . i * F i s a coherent subsheaf of G, then M' i s the Nash Blow-up of M w i t h r e spec t t o i * F and G, and hence i s a n a l y t i c by theorem 1 of s e c t i o n 3. Assume then t h a t M' i s a n a l y t i c . Le t rankF=k and rankG !=n. Then l e t G ( k , G ) =>-M be the Grassmann bundle of k - p l a n e s i n G. Le t T >G(k,G) be t h a t v e c t o r bundle which r e s t r i c t s ' to the t a u t o l o g i c a l v e c t o r bundle of G(k-,n) on each f i b r e of G ( k , G ) . Use the same n o t a t i o n T >M' to denote the r e s t r i c t i o n of t h i s bundle to M ' . On M' we have the f o l l o w i n g shor t exact sequence of b u n d l e s : 0 >T >Cn K> ^0 (*) where Q i s the q u o t i e n t bundle C n / r . Le t 7 r : M ' >M be the u s u a l p r o j e c t i o n . On M-Z there i s a shor t exact sequence of sheaves : 0 >F >G >K no-where K i s the q u o t i e n t sheaf G/F. T h i s sequence p u l l s back by TT* t o the sequence of sheaves c o r r e s p o n d i n g to the sequence of bundles (*) on M ' - f M Z ) . Hence on M' the sheaf of s e c t i o n s of T , which we denote by r i s a c o h e r e n t sheaf 39 t h a t extends ir*F. T h i s e x t e n s i o n l i e s i n C n = 7 r * G . S i n c e it i s a p roper map TT^£ i s a coherent sheaf on M and s i n c e r_ l i e s i n 7r *G the coherent sheaf 7r*jr l i e s i n G. Then by S e r r e ' s theorem i^F i s c o h e r e n t . Hence F i s e x t e n d a b l e . Q E D T h i s r e s u l t not o n l y t e l l s us when M' i s a n a l y t i c but a l s o p r o v i d e s a s o l u t i o n to S e r r e ' s e x t e n s i o n problem when F i s t o r s i o n f r e e . In g e n e r a l i f F has t o r s i o n then we can ask i f t h e r e e x i s t s an a n a l y t i c coherent subsheaf E of G which extends F. In t h i s case E need not be equa l t o i * F . > C o r o l l a r y 3: Le t M be a complex a n a l y t i c space w i t h a proper a n a l y t i c subset Z. Le t G be a l o c a l l y f r ee sheaf on M and F >VL-Z be a coherent subsheaf of <7|M-Z. Then t h e r e e x i s t s an a n a l y t i c coherent subsheaf E >Vl which extends F i f f the Nash c o n s t r u c t i o n M' of M w i t h r e s p e c t to F and G i s a n a l y t i c . P r o o f : U s i n g the n o t a t i o n of the above theorem i f M' i s a n a l y t i c then E=-n1(T_ extends F. I f there e x i s t s a coherent E which extends F then the Nash Blow-up of M w i t h r e s p e c t to F and G i s p r e c i s e l y the Nash Blow-up of M w i t h r e s p e c t to E and G and i s t h e r e f o r e a n a l y t i c by theorem 1 of s e c t i o n 3. Q E D 40 Other s o l u t i o n s to S e r r e ' s problem p r o v i d e us w i t h f u r t h e r c r i t e r i a i n d e c i d i n g when M' i s a n a l y t i c . Le t us r e v i v e our set up . M i s an a n a l y t i c space w i t h the proper c l o s e d a n a l y t i c subset Z . There i s a l o c a l l y f r ee sheaf G of M and a coherent sheaf F of M - Z . Le t F be a subsheaf of G on M - Z . Le t rankF=k, rankG=n and S be the s i n g u l a r set of F. Then on each p o i n t x o f f Z U S , Fx d e f i n e s a k-p lane i n Gx and t h i s d e f i n e s an imbedding of M - Z U S i n t o G ( k , C ) . The c l o s u r e of the image of t h i s imbedding i s denoted by M ' , and 7r : M ' >M i s the u s u a l p r o j e c t i o n . We know, tha t M ' - i r _ 1 ( Z ) i s a n a l y t i c . However F need not extend c o h e r e n t l y over Z and we can not say much about M' in g e n e r a l . T h e r e f o r e we look for c o n d i t i o n s on F which make i t p o s s i b l e to extend i t -c o h e r e n t l y over Z. I f F extends c o h e r e n t l y then M' i s a n a l y t i c . The f o l l o w i n g theorem i s due to S i u and Trautmann. For the p r o o f see [ S T ] . Theorem ( S i u , T r a u t m a n n ) : Le t M be a complex space and D an open subset which i s s t r o n g l y r -concave at a p o i n t x 0 e M . Le t G be a coherent a n a l y t i c sheaf on M and F a coherent a n a l y t i c sheaf on D. Le t F be a subsheaf of C | D . I f the r - t h r e l a t i v e -gap sheaf of F i n G i s equa l to F, i . e . FV=F, then F can be extended c o h e r e n t l y to an open ne ighbourhood of x 0 as a subsheaf of G. 41 C o r o l l a r y 4: Le t M and D be, as above, G a l o c a l l y f r ee sheaf on M, F a coherent sheaf d e f i n e d on D. Le t F be a subsheaf of G | D . I f F f = F then M' i s a n a l y t i c . QED Let us b r i e f l y d e s c r i b e the terms u s e d . A twice d i f f e r e n t i a b l e r e a l v a l u e d f u n c t i o n f i s s a i d to be s t r o n g l y r -convex on a subset D of C n i f a t every p o i n t of D the h e r m i t i a n m a t r i x 0 2 f / 3 z . 3 2 j ) has at l e a s t n-r+1 p o s i t i v e e i g e n v a l u e s . An open subset U of a complex space M i s s t r o n g l y r -concave at a p o i n t x 0 e M i f t h e r e i s an open neighbourhood V of x 0 on which t h e r e i s a s t r o n g l y r -convex f u n c t i o n f such t h a t f ( x ) = 0 and U n V = { y e V | f (y)>0 }. For a s u b v a r i e t y A of M the gap sheaf F[h] of F i n G w i t h r e s p e c t to A i s the sheaf d e f i n e d by the pre shea f U ^ { s e r ( U , G ) | s | U - A e r ( U - A , F ) }. The r - t h r e l a t i v e gap sheaf , F of F i n G i s the sheaf d e f i n e d by the pre shea f U s- l im {r (U,F[A]) | A i s a s u b v a r i e t y of U , dimA<r} F i n a l l y be fore we c l o s e t h i s c h a p t e r we g i v e an a l t e r n a t e d e s c r i p t i o n of Nash c o n s t r u c t i o n , borrowing an idea of G i r a u d [ G r ] . Le t F be a coherent sheaf on M w i t h S be ing the s i n g u l a r set of F . Le t O be the s t r u c t u r e sheaf of 42 M. R e c a l l i n g tha t M s a sheaf of 0-modules c o n s t r u c t the sheaf P r o j ( A k F ) s-M On M-S there i s a n a t u r a l imbedding of M i n t o t h i s shea f . The c l o s u r e of the image of t h i s imbedding can be c a l l e d the i n t r i n s i c Nash c o n s t r u c t i o n on M. See a l s o Thimm [T] where he mentions P l u c k e r i a n c o o r d i n a t e s of F. 43 CHAPTER 2 GRASSMANN GRAPH CONSTRUCTION 0:INTRODUCTION T h i s c h a p t e r d e s c r i b e s the Grassmann Graph c o n s t r u c t i o n of MacPherson i n the a n a l y t i c c a t e g o r y u s i n g C * a c t i o n s . The d e t a i l s of the a l g e b r a i c case can be found i n [BFM]. In s e c t i o n 1 we summarize the decompos i t i on theorem of B i a l y n i c k i - B i r u l a i n the compact K a e h l e r c a s e , [ B B c ] , [CS1 ] . S e c t i o n 2 d e s c r i b e s a C * a c t i o n on Grassmann m a n i f o l d s and g i v e s the c o r r e s p o n d i n g B i a l y n i c k i - B i r u l a d e c o m p o s i t i o n . Examples are g i v e n i n the next s e c t i o n . In s e c t i o n 4 t h i s C * a c t i o n i s c a r r i e d on to Grassmann bundles and Z , the c y c l e at i n f i n i t y c o r r e s p o n d i n g to a bundle morphism i s d e f i n e d . I t i s shown t h a t i n the compact K a e h l e r case Zn i s an a n a l y t i c c y c l e . The graph c o n s t r u c t i o n i s f i n a l l y a c c o m p l i s h e d i n s e c t i o n 5. Examples are g i v e n i n s e c t i o n 6. 44 1;BIALYNICKI-BIRULA DECOMPOSITION The r e f e r e n c e s fo r t h i s s e c t i o n are [BBc] fo r the a l g e b r a i c case and [CS13 f o r the complex c a s e . There i s a l s o a c l e a r summary i n [ C S 2 , s e c t i o n I c ] . Le t M be a compact K a e h l e r m a n i f o l d w i t h a C * a c t i o n on i t . Le t t h i s C * a c t i o n have n o n t r i v i a l f i x e d p o i n t set B w i t h components B 1 f . . . , B . The components of the f i x e d p o i n t set are complex submani fo lds of M. For XeC* and pcM l e t X « p denote the a c t i o n of X on p . The C * a c t i o n extends to a meromorphic map P 1x{p} ?>M hence l i m X » p and l i m X « p e x i s t i n M. C l e a r l y these l i m i t s X>0 X > » are i n B. There a re two c a n o n i c a l d e c o m p o s i t i o n s of M i n t o i n v a r i a n t complex s u b m a n i f o l d s . D e f i n e MT={peM| l i m X « p e B . } ; X^O 1 fo r i = 1 , . . . , m . Each M^ i s a complex m a n i f o l d of M and M = U M T 1<iSm. T h i s i s c a l l e d the plus decomposition of M. S i m i l a r l y the minus decomposition i s d e f i n e d as M.={peM| l i m X « p e B . } X*-» f o r i = 1 , . . . , m . Each M^ i s a complex s u b m a n i f o l d and s i m i l a r l y M = U M 7 i< i^m. 45 There are two d i s t i n g u i s h e d components of the f i x e d p o i n t se t B, say B 1 and B m , which are de termined by the p r o p e r t y t h a t M * and M are open and dense i n M. B, i s c a l l e d the 1 m  r 1 source and B i s c a l l e d the sink. m 2 : C * - A C T I O N S ON G ( k , n ) In t h i s s e c t i o n we d e s c r i b e a p a r t i c u l a r C * a c t i o n on G ( k , n ) , the Grassmannian m a n i f o l d of k -p l anes i n n- space . F i x a c o o r d i n a t e system on C n . We w i l l use the r e p r e s e n t a t i o n of G ( k , n ) by m a t r i c e s . Any p o i n t peG(k ,n ) can be r e p r e s e n t e d by a kxn-matr ix A of rank k. Two such m a t r i c e s A and B r e p r e s e n t the same p o i n t i n G ( k , n ) i f there i s an i n v e r t i b l e kxk-mat r ix g c G L ( k , C ) such t h a t gA=B. For a kxn-matr ix A of rank k se t [A]=the row*space of A . G i v e n a kxn-matr ix A=(a^j) 1<i<k, 1<j<n d e f i n e two submat r i ce s A ^ C a ^ ) 1<i , j<k and A 2 = ( a . . ) 1<i<k, k+1<j<n. A , i s a kxk-mat r ix and A 2 i s a k x ( n - k ) - m a t r i x and A = ( A l f A 2 ) i s a p a r t i t i o n i n g of A . D e f i n e a C * a c t i o n on G ( k , n ) C * x G ( k , n ) >G(k,n) by 46 X - [ A ] = [ ( A , , X A 2 ) ] To d e s c r i b e the behav iour of t h i s a c t i o n d e f i n e a subset X^ j of G ( k , n ) as the set of a l l p i n G ( k , n ) which can be r e p r e s e n t e d by a kxn-matr ix A = ( A , , A 2 ) such tha t r ankA,= i and rankA 2 = j where k-min{k,n-k}<i<k and 0^ j <min{k ,n-k] . Le t B = ( B 1 f B 2 ) be another kxn-matr ix r e p r e s e n t i n g p . Then there i s an i n v e r t i b l e kxk-matr ix g such t h a t gA=B. gA,=B! and g A 2 = B 2 . Hence r a n k B 1 = r a n k ( g A 1 ) = r a n k A , = i and s i m i l a r l y r a n k B 2 = j , and the f o l l o w i n g d e f i n i t i o n of X^ j i s w e l l d e f i n e d : X i j = { [ A ] e G ( k , n ) | r a n k A , = i , r ankA 2 = j } where k-min{k,n-k}<i<k and 0 £ j ^ m i n { k , n - k } . N e c e s s a r i l y we have i+j>k; to see t h i s r e c a l l t h a t A r e p r e s e n t s a p o i n t i n G ( k , n ) hence has rank k and i f A , has rank i then A 2 must supp ly at l e a s t the r e m a i n i n g k - i r a n k s . To d e s c r i b e the behav iour of the C * a c t i o n tha t i s d e f i n e d above we prove the f o l l o w i n g lemmas. Lemma 1. X. , _ . a re the f i x e d p o i n t components of the C * a c t i o n , k-min{k,n-k}<i<k. P r o o f : Le t [A]eX. A = ( A , , A 2 ) . We f i r s t show tha t 1 K. 2. X « [ A ] = [ A ] . I f i=0 then A,=0 and i f i=k then A 2 = 0 . In both cases X « [ A ] = [ A ] . Assume 0<i<k. Then t h e r e e x i s t s an i n v e r t i b l e kxk-matr ix g such that gA i s of the form 47 where B , e G L ( i , C ) and B 2 e G L ( k - i , C ) . For XeC* d e f i n e h x to be the d i a g o n a l mat r ix [1 , , 1 , 1 / X , . . . , 1 / X ] where the number of 1/X's i s k - i . We then have the f o l l o w i n g sequence of e q u a l i t i e s ; . X . [A]=X-[gA] 48 B , B 2 = [ A ] . Thus we have proven t h a t X. . _ . i s a subset of the f i x e d p o i n t s e t . That i n f a c t t h e r e are no o ther f i x e d p o i n t s than U X ^ y^-^r k-min{k,n-k}<i<k f o l l o w s from the r e s u l t s of the f o l l o w i n g two lemmas. Q E D Lemma 2 . I f [ A ] e X . . then l i m X - [ A ] e X . . . , where IT >^r\ X K "~ X X^O k-min{k,n-k}<i<k, 0^ j^min{k ,n-k} , .i+j>k. I n p a r t i c u l a r X m Q i s the source where m=k-min{k,n-k} P r o o f : I f i=0 or i=k then X^ j i s a component of the f i x e d p o i n t set as i n lemma 1. Assume 0<i<k. There e x i s t s geGL(k ,C) such tha t / B , • ° gA= I *. B 2 \ 0 : B a where B ^ G L d . j C ) , B 3 e G L ( k - i , C ) and B 2 i s ( i + j - k ) x ( n - k ) - m a t r i x . . Le t h ^ be as i n lemma 1. Then 49 h^XgA= B and s i n c e l i m XB 2 = 0 we have X^O l i m X « [ A ] = l i m [h ,XgA] X>0 X>0 A 0 B 3 T h i s l a s t m a t r i x i s c l e a r l y i n X^ as c l a i m e d . QED Lemma 3 . I f [ A ] e X . . then l i m X « [ A ] e X , . . , where 1 3 X > » K~l 3 k-min{k,n-k}<i<k, O^j^min{k ,n-k} . In p a r t i c u l a r X k _ m i s the s i n k where m=miri{k,n-k}. P r o o f : I f i = 0 or i=k then X^ j i s a f i x e d p o i n t component. Assume 0<i<k. There e x i s t s geGL(k ,C) such t h a t gA= where B , e G L ( k - j , C ) , B 3 e G L ( j , C ) and B 2 i s a ( i + j - k ) x k - m a t r i x . Then 50 l i m X- [A]= l im [Xh.gA] X^oo X > » =lim 1 \ X - 1 B 2 T h i s l a s t mat r ix i s i n X k - j j as d e s i r e d . QED These l a s t two lemmas show tha t i k - i f o r k-min{n-k}<i<k are the o n l y f i x e d p o i n t components and thus complete the proof of lemma 1. We can app ly these lemmas t o examine the behav iour of Schubert c e l l s under the a c t i o n of C * on the Grassmann m a n i f o l d . We w i l l adopt the t e r m i n o l o g y of G r i f f i t h s and H a r r i s on Schubert c e l l s . For d e t a i l s r e f e r t o [GH,page 195-196]. Le t { e 1 f . . . , e } be the s t andard b a s i s fo r C n and V ^ = s p a n { e 1 , . . . , e ^ } . Then { V 1 , . . . , V n ) d e f i n e s a f l a g . For any n o n i n c r e a s i n g sequence of nonnegat ive i n t e g e r s between 0 and n-k d e f i n e a c e l l 51 W a ={[A]eG(k,n) | d i m ( A n V n _ k + i + a - ) = i }. The sequence of n o n i n c r e a s i n g i n t e g e r s a = ( a 1 , . . . , a ^ ) w i t h 0<a^<n-k i s c a l l e d a Schubert symbol . For [ A ] e G ( k , n ) l e t A be a kxn-matr ix such t h a t [A]= [A] , I f [A]eW for some Schubert symbol a = ( a 1 , . . . , a ^ ) then the rank of the f i r s t kx(n-k+i-a^) minor i s i and the rank of the l a s t kx(k-i+a^) minor i s k - i . The c l o s u r e of W vT={[A]eG(k,n) | d i m ( A n V n _ k + i _ a ) ^ i } i s c a l l e d a Schubert v a r i e t y . I f A i s a m a t r i x r e p r e s e n t i n g [A] as above , then [A] i s i n W~ i f f the rank of the f i r s t 3 kx(n-k+i-a^) minor of A i s a t l e a s t i and the rank of the l a s t kx(k- i+a^) minor of A i s at most k - i . I t i s w e l l known tha t W~~ i s an a n a l y t i c s u b v a r i e t y of G ( k , n ) and the homology c l a s s of W , denoted by o , i s independent of the f l a g used a a i n i t s d e f i n i t i o n , [ G H , p l 9 6 ] . a i s c a l l e d the Schubert a c y c l e c o r r e s p o n d i n g to a = ( a 1 , . . . , a ^ ) . Regard ing the behav iour of Schubert c y c l e s under the C * a c t i o n we g i v e the f o l l o w i n g c o r o l l a r y t o the above lemmas; COROLLARY 1. A l l Schubert c y c l e s of p o s i t i v e cod imens ion i n G ( k , 2 k ) l i e i n X . / s where j<k. In p a r t i c u l a r they do not flow to the s i n k , i . e . i f peW~ then l i m X « p i s a X>°° not i n the s i n k . P r o o f : The cod imens ion of VP fo r a = ( a 1 , . . . , a k ) i s £ a ^ , [GH,page 196] . I t s u f f i c e s to prove the c o r o l l a r y f o r 52 a = (1,0 , . . . , 0 ) . For [A]eW l e t A = ( A , , A 2 ) be a m a t r i x 3 r e p r e s e n t a t i o n where A i s a kxn-matr ix of rank k, and A , , A 2 are k x k - m a t r i c e s . The rank of the l a s t kxk minor of A i s of rank at most k-1. Hence i n p a r t i c u l a r the rank of A 2 i s not k, t h e r e f o r e [A] i s not i n X ^ . S i n c e the o n l y p o i n t s t h a t f low to the s ink be long to the components of the form X ^ , [A] does not flow to the s i n k . In g e n e r a l i f a=(a 1 , . . . , a ^ ) w i t h a^1 then the l a s t k x C k + a ^ l ) minor has rank at most k-1. S i n c e k+a,-1£k, the rank of A 2 can not be k. Hence W 1 a does not flow to the s i n k . I f a ^ O then a = ( 0 , . . . , 0 ) and W~~ 1 a does not have p o s i t i v e c o d i m e n s i o n . Q E D U s i n g the same n o t a t i o n as i n the p r e v i o u s c o r o l l a r y we can g e n e r a l i z e as f o l l o w s : C o r o l l a r y 2. Le t ViF, a=(a 1 , . . . , a^) , be a Schubert v a r i e t y i n G ( k , n ) where a,^n-2k+1. Then W~~ does not flow to 1 3 the s i n k i f n£2k. P r o o f : Le t A = ( A 1 f A 2 ) be a kxn-matr ix w i t h rank k r e p r e s e n t i n g a p o i n t [A] i n V P . A , i s a k x ( n - k ) - m a t r i x and [A] w i l l f low t o the s i n k i f rank A 2 i s maximal . S ince n£2k 53 m e a n s n - k £ k , t h e m a x i m a l r a n k o f A 2 i s k . T h e r a n k o f t h e l a s t k x ( k + a 1 ~ l ) m i n o r o f A i s a t m o s t k - 1 . By a s s u m p t i o n k + a ^ l ^ n - k , t h e r e f o r e t h e r a n k o f A 2 c a n n o t be k . H e n c e d o e s n o t f l o w t o t h e s i n k . QED 3 : E X A M P L E S I n e x a m p l e s 1 a n d 2 we a s s u m e t h a t t h e C * a c t i o n o f t h e p r e v i o u s s e c t i o n i s d e f i n e d o n t h e s p a c e s G ( 2 , 4 ) a n d G ( 4 , 9 ) . 1) G ( 2 , 4 ) I n G ( 2 , 4 ) we h a v e d e f i n e d t h e f o l l o w i n g s e t s ; ^ 2 0 r ^ 2 l ( X 2 2 i X t 1 , X 1 2 , X 0 2 . T h e f i r s t t h r e e s e t s a r e t h e f i x e d p o i n t s e t s . A s X >-0 t h e e l e m e n t s o f X21 a n d X- 2 2 f l o w t o t h e s o u r c e X 2 0 , a n d t h e e l e m e n t s o f X 1 2 f l o w t o X n . A s X t h e e l e m e n t s o f X 2 2 a n d X , 2 f l o w t o t h e s i n k X 0 2 f a n d t h e e l e m e n t s o f X 2 , f l o w t o X t 1 . F i g u r e 1 54 A l l v a r i e t i e s t h a t l i e i n X 2 1 f l o w t o X , , a s X S e e f i g u r e 1 f o r t h e d i r e c t i o n o f t h e s e f l o w s f o r e a c h X ^ j a s X 2) G ( 4 , 9 ) F o r t h e d i r e c t i o n o f f l o w a s X : — s e e f i g u r e 2 . F r o m t h e d e c o m p o s i t i o n o f G ( 4 , 9 ) i n t o X ^ j i t c a n be s e e n t h a t t h e p o i n t s t h a t l i e i n d o n o t f l o w t o t h e s i n k o r t h e s o u r c e u n d e r t h e a c t i o n o f C * . F i g u r e 2 55 4 :C* -ACTIONS ON GRASSMANN BUNDLES T h i s s e c t i o n d e f i n e s i n the compact Kaeh le r case the Grassmann Graph c o n s t r u c t i o n of [BFM,pp120-121] . Let E , F be v e c t o r bundles of ranks k and n r e s p e c t i v e l y on an a n a l y t i c space M. Let G(k,EffiF) s»-M denote the Grassmann bundle whose f i b r e at each xeM i s G ( k , E ©F ) , the X X Grassmanian of k -p l anes i n E x © F x . D e f i n e a C * a c t i o n on G ( k , E © F ) as the f i b r e w i s e C * a c t i o n . Le t 7T, :EffiF >E TT2 :E©F >F and 7 r : G ( k , E © F ) >M be the p r o j e c t i o n s . Any p e G ( k , E © F ) i s r e p r e s e n t e d by a k - p l a n e H i n E ©F where x = 7 r ( p ) . TT,(H) and 7 r 2 ( H ) a re l i n e a r X X subspaces of E x and F x r e s p e c t i v e l y . The t o t a l space G(k,E@F) can be decomposed i n t o C * - e q u i v a r i a n t subbundle s . X i j = { [ H ] e G ( k , E © F ) |dim7r, (H) = i , d i m 7 r 2 ( H ) = j} where k - m i n ( k , n ) ^ i ^ k , 0^ j^min(k ,n ) and i+j>k. I t i s easy to see t h a t X ^ S G d ^ J x G d . F ) i + j = k, which are the f i x e d p o i n t se t s of the C * a c t i o n . Le t 5 6 Hom(E,F) *-M be the bundle of morphisms from E to F and l e t j : H o m ( E , F ) > G ( k , E © F ) be the n a t u r a l i n c l u s i o n d e f i n e d f i b r e w i s e as j v($)=graph(<i>|E v) = { ( e , * ( e ) e E © F v } A X X X R e c a l l tha t C can be imbedded i n t o P 1 as C ^ P 1 X • [ 1 : X ] , [ B F M , p l 2 0 ] . D e f i n e a C * a c t i o n on G ( k , E © F ) x P 1 C*xG ( k , E©F) x P 1 >G ( k , E©F) x P 1 as ( X , p , [ X 0 : X, ]) ^ ( X - p , [ X 0 :XX , ]) where X « p i s the C * a c t i o n which i s d e f i n e d above . A l s o d e f i n e the C * a c t i o n on MxC, C*xMxC ^MxC as ( X , x , t ) M x . X t ) . Every $eHom(E,F) d e f i n e s an e q u i v a r i a n t imbedding s(<i>) of MxC i n t o G ( k,E8F ) x P ' , s(<i>):MxC ^ ( k ^ S F j x P 1 where s ( * ) ( x , X ) = ( [ j v ( X * v ) ] , [ 1 : X ] ) . A A s ( $ ) ( M , X ) i s the graph of X<i>. Now d e f i n e Z ^ l i m s ( * ) ( M , X ) . X>°° 57 Theorem 1. I f M i s a compact K a e h l e r m a n i f o l d , then fo r any <i>eHom(E,F) the c o r r e s p o n d i n g Z__ i s an a n a l y t i c c y c l e . P r o o f : Let p : C * x G ( k , E © F ) * - G ( k , E © F ) be the C * a c t i o n d e f i n e d above . C o n s i d e r M as a subspace of G ( k , E ( B F ) by the imbedding S ( # ) ( M , 1 ) ; i . e . i d e n t i f y M and the graph of D e f i n e a ho lomorphic map A : M x C * X 3 ( k , E © F ) as A(m, t ) = s(<i>) ( m , t ) , where meM and t e C * . T h i s map i s e q u i v a r i a n t w i t h r e s p e c t to p and the t r i v i a l a c t i o n of C * on M x C * , m u l t i p l i c a t i o n i n the second component; f o r i f XeC* then A ( m , X - t ) = s ( * ) ( m , X t ) = s (X* ) (m, t ) = X-s(4>) (m,t) = p(X,s(4>) (m,t ) ) =p (X ,A(m, t ) ) hence e q u i v a r i a n c e . But Sommese has shown t h a t i f i / / : Y x C * >X i s a ho lomorphic map e q u i v a r i e n t w i t h r e s p e c t to the t r i v i a l a c t i o n of C * on YxC* and the a c t i o n of C * on X w i t h f i x e d p o i n t s then \f/ extends m e r o m o r p h i c a l l y to Y x P 1 , [So ,p111 (lemma I I - B ) ] . Thus A extends m e r o m o r p h i c a l l y to A ' :MxP 1 M3 (k,E©F). 58 Le t T be the c l o s u r e of the graph of A i n M x P 1 x G ( k , E 0 F ) . By the d e f i n i t i o n of a meromorphic map, T i s an a n a l y t i c space . S ince Z a , = T n ( M x { ° ° } x G ( k , E © F ) ) , the i n t e r s e c t i o n of two a n a l y t i c spaces , then i s a n a l y t i c as d e s i r e d . QED Z B i s c a l l e d the c y c l e at i n f i n i t y c o r r e s p o n d i n g to the map N o t i c e tha t t h e r e i s an a l t e r n a t e d e f i n i t i o n of Z ^ , see [ B F M , p 1 2 l ] ; d e f i n e an imbedding of MxP 1 i n t o MxP 'xGfk jn ) i : M x P 1 ^ - M x P ' x G U ^ ) as i ( m , [ X 0 : X , ] ) = [ { ( m , [ X 0 : X 1 ] , ( e , f >) ^ © ^ ( E j | X 0*f = X,$(e) } ] Let W be the c l o s u r e of i ( M x P 1 ) in M x P 1 x G ( k , n ) . w= i ( M x P 1 ) Then Z o = W n ( M x { » } x G ( k , E © F ) ) . In the a l g e b r a i c c a t e g o r y W i s an a l g e b r a i c v a r i e t y but i n the a n a l y t i c c a t egory the o b s e r v a t i o n that W can be o b t a i n e d through a C * a c t i o n w i t h f i x e d p o i n t s on a compact K a e h l e r m a n i f o l d i s c r u c i a l i n c o n c l u d i n g t h a t i t i s a n a l y t i c . C l e a r l y {Z.=s($)(M,X)} d e f i n e s a f a m i l y of c y c l e s which 59 are a l g e b r a i c a l l y and hence h o m o l o g i c a l l y e q u i v a l e n t . 5:GRAPH OF COMPLEXES In t h i s s e c t i o n we d e f i n e the Grassmann Graph c o n s t r u c t i o n and the c y c l e a t i n f i n i t y a s s o c i a t e d t o a complex of v e c t o r b u n d l e s . T h i s c o n s t r u c t i o n was i n v e n t e d by MacPherson and used by Baum, F u l t o n and MacPherson to prove Riemann-Roch theorem for s i n g u l a r a l g e b r a i c v a r i e t i e s , [BFM] and [ M c ] . C o n s i d e r a complex of v e c t o r bundles on M, ( E . ) : 0 3»E_ >Em > >En >0 , m m-1 U Denote the.maps by 7 ^ , i . e . 7 . : E . >-E. , ' 1 1 1-1 where i = 1 , . . . , m . Assume tha t there i s a s u b v a r i e t y S of M such t h a t ( E . ) i s exact on M-S. Le t G^=G(rankE^,E^ffiE^_ 1) i = 1 , . . . , m . and l e t T ^ >G^ the t a u t o l o g i c a l b u n d l e , i = 1 , . . . , m . D e f i n e G = G 0 x M . . . x M G 1 , where x M denotes the bundle p r o d u c t on M. On G l e t T . denote 60 the p u l l back of by the p r o j e c t i o n p r ^ : G >G^ of the i - t h component, i = 0 , . . . , m . Le t r = r 0 - r 1 + . . . + ( - l ) m T m be the v i r t u a l t a u t o l o g i c a l bundle on G . R e c a l l i n g the d e f i n i t i o n of s from the p r e v i o u s s e c t i o n fo r any XeC d e f i n e an imbedding s\:M >G. as s ^ ( x ) = s ( 7 i ) ( x , X ) where i = 1 , . . . , m . Then d e f i n e fo r any XeC an imbedding by s ^ : M >G s ^ ( x ) = ( s ^ ( x ) , . . . , s ™ ( x ) ) Le t a g a i n denote s^(M) fo r XeC. Then we d e f i n e Z ^ l i m Z^ to be the c y c l e at i n f i n i t y c o r r e s p o n d i n g to the complex ( E . ) . Le t 7r :G >M be the n a t u r a l p r o j e c t i o n . R e c a l l i n g tha t S i s the set o f f which ( E . ) i s exact we have the f o l l o w i n g r e s u l t ; f o r p r o o f s see [ B F M , p ! 2 l ] . Theorem (Baum, F u l t o n , MacPher son) : The c y c l e Z^ has a unique d e c o m p o s i t i o n Z o o=Z^+MA, where 1) it maps M* m e r o m o r p h i c a l l y onto M, 61 2) mM^-iry(S) >M~S i s a b i h o l o m o r p h i s m . 3) u maps Z i n t o S. 4) T r e s t r i c t s on to the ze ro b u n d l e . REMARK: By theorem 1 of the p r e v i o u s s e c t i o n Za i s a p roduc t of a n a l y t i c c y c l e s i n the product bundle G hence t h i s theorem can be s t a t e d i n the a n a l y t i c c a t e g o r y as above . Any c y c l e can be w r i t t e n as a sum of i r r e d u c i b l e c y c l e s . The decompos i t i on of Z^ i s such a sum. For a p r o o f of (4) see [BFM,p122] . F i n a l l y we d e f i n e two r e s i d u e s on S. Le t E be the v i r t u a l bundle E Q - E ^ . . . + (-1 ) m E m on M. Then T | Z q i s i somorphic to E s i n c e ZQ=M. S i n c e Z Q and Z__ a re r a t i o n a l l y e q u i v a l e n t c ( E ) n [M]=c(r) n z n = c ( T ) where c ( « ) denotes the Chern c l a s s and n denotes the cap p r o d u c t . S i n c e Za decomposes c I < T ) n Z „ = c I ( T ) n < Z * + M * ) =C.(T ) n Z * + c _ ( T ) n M * » C . ( T ) n Z * where i>0 and the l a s t e q u a l i t y f o l l o w s s i n c e T|M*=0 by (4) of the above theorem. Def ine Cg ( E . ) = 7 r ^ ( c I ( T ) n Z * ) e H * ( S ; C ) . S i m i l a r l y l e t c h ( « ) denote the chern c h a r a c t e r , then 62 ch(E) n [ M ] = c h ( r ) n Z 0 = c h ( 7 ) n Z B =ch(r) n Z*+ch(r ) n M * =ch(r) n Z * S i m i l a r l y d e f i n e c h g ( E . ) = 7r * ( ch ( r ) n Z* ) eH* (S ;C ) . For b a s i c p r o p e r t i e s of ch ( E . ) i n the a l g e b r a i c c a t e g o r y see [BFM,pp121-126] . We w i l l use Cg (E ) fo r c a l c u l a t i n g the Baum-Bott r e s i d u e s of s i n g u l a r ho lomorphic f o l i a t i o n s i n the next c h a p t e r . 6:EXAMPLES 1) Le t M be a compact K a e h l e r m a n i f o l d of d imens ion n w i t h tangent bundle T . Le t L be a l i n e bundle on M and l e t aeHom(L* ,T) w i t h i s o l a t e d z e r o s Z . a i s c a l l e d a meromorphic vector field. Le t peZ . Choose an open ne igbourhood U of p such t h a t ( i ) t h e r e are c o o r d i n a t e f u n c t i o n s z^,...,z on U and ( i i ) t h e r e i s a l o c a l g e n e r a t o r 1* of L * on U and ( i i i ) U n Z = f p } . Then a ( l * ) i s a ho lomorphic v e c t o r f i e l d on U g i v e n as a ( l * ) | x = . Z 1 a i ( x ) g | - | x , xeU where a ^ ( « ) are ho lomorphic f u n c t i o n s on U . Any element c f L * | x i s of the form c « ( l * | x ) f o r some c e C , and 63 a ( c l * ) | x = i ? 1 c a i ( x ) g ^ — | x , xeU. T h i s d e f i n e s a p o i n t i n L * © T | x , ( c l * , a ( c l * ) I x ) — ( c _ r C 3 ^ (x) f • • • f ca ( x ) ) e L * © T | x , xeU. Hence for xeU-P, the graph r of a i n U x P ( L * © T ) = U x P n i s g i v e n as r = { ( x , [ 1 : a , ( x ) : . . . : a n ( x ) ] ) e U x P n D e f i n e a C * - a c t i o n on U x P ( L * © T ) = U x P n C * x U x P n >UxPn as ( X , x , [ y Q : . . . : y n ] ) = ( x , [ y Q : X y 1 : . . . : X y n ] ) } C o n s i d e r l i m X»r=Zaj. X>°° X-r={ ( x , [ 1 : X a 1 ( x ) : . . . : X a n ( x ) ] ) e U x P n } I f xeU-p, then l i m X - T = { ( x , [ 0 : a 1 ( x ) : . . . : a n ( x ) ] ) e U x P } and hence |U-p=U-p. To f i n d Z ^ J p , d e f i n e a ho lomorphic funct ion F : U *~Cn as F (x ) = ( a 1 ( x ) , . . . f a n ( x ) ) • Then F ( p ) = 0 and F(U) i s an open neighbourhood of the o r i g i n . For any p o i n t [ c 1 : . . . : c n ] e P n 1 , l e t D be the l i n e i n C n tha t 64 passes through ( c 1 f . . . , c ) and the o r i g i n . C o n s i d e r the set C={xeU|F(x) e U n D } . Then C i s a union of ho lomorphic curves {$ 1 , . . . ,$^1 p a s s i n g through p . We may assume wi thout l o s s of g e n e r a l i t y tha t these curve s do not i n t e r s e c t i n U - p . The number k w i l l be r e f e r r e d to as the degree of F at p . Le t $ be one of these c u r v e s . Let {Pm} e$ be a sequence of p o i n t s such t h a t l i m p =p. x>°° The graph of T on any one of these P m ' s can be w r i t t e n as n r I P m = t ( P m , [ l : a 1 ( p m ) : . . . : a n ( p m ) ] ) e U x P " } = { ( p m , [ l : c 1 : . . . : c n ] ) e U x P n } s i n c e F ( p )e$ . Then *m l i m l i m T i p =lim l i m {(p ) , [ 1 : X c X c ] ) e U x P n } « 1cm » r m i n m>® X > « m*-°° X>°° =lim { ( p m , [ O r e , : . . . : c n ] ) e U x P n } m > » = { ( p , [ 1 : c 1 : . . . : c n ] ) e U x P n } . On the o ther hand r | p = { ( p , [ l : 0 : . . . : 0 ] ) } and x.r|p=r| P T h e r e f o r e Z o o | p = k . p n _ 1 + { ( p , [ 1 : 0 : . . . : 0 ] ) } 65 as c y c l e s . can be decomposed u n i q u e l y i n t o two c y c l e s where i s bimeromorphic to U and Z* l i e s over p , i . e . i f nil • U OO i s the u s u a l p r o j e c t i o n induced by the n a t u r a l p r o j e c t i o n U x P n >U, then 7r (Z* )=p . Hence Z * = ( k - 1 ) P n 1 + {point} as c y c l e s . Let r ' be the t a u t o l o g i c a l l i n e bundle on P n and l e t T denote the p u l l b a c k bundle on U x P n . Le t w be a d u a l hyperp lane c l a s s i n H * ( P n 1 ; C ) . C(T|U * )=1-W s i n c e r r e s t r i c t s to the t a u t o l o g i c a l bundle on P n 1 , where c ( « ) i s the t o t a l Chern c l a s s . " L e t c h ( « ) denote the Chern c h a r a c t e r . Then c h ( r ) n z * = ( e " w n C (k+1 ) P n ~ 1 + ( p o i n t } ] ) = (e w n t ( k - D P n 1 ] ) + (e w n n p o i n t } ] ) n-1 ( - D 1 r / u _ , i » n - 1 = ( ( i S o i l w x ) n t ( k - 1 ) P " + 1 _ , ( k - 1 ) ( - 1 ) " " 1 + n + (k-1 ) ( - D n " 2 n-2 f p n - 1 l + + ( k _ 1 H P n " 1 l Le t i r * : H * ( Z * ; C ) = » H * ( p ; C ) be the map induced by ir on the homology c l a s s e s . N o t i c e t h a t 66 H * ( { p } ; C ) = C h e n c e TT^ i s z e r o o n p o s i t i v e d i m e n s i o n a l c y c l e s a n d maps o n l y t h e 0 - c y c l e s . D e f i n e a l o c a l r e s i d u e c h ( a ) = i r ^ ( c h ( r ) n 2 * ) T h e n rh f n > - 1 + ( k - 1 ) ( - 1 ) n " 1 m  c h p ( a ) " 1 + TTFTTl ( 1 ) D e f i n e t h e t o t a l r e s i d u e a s : c h ( a ) = p Z z c h p ( a ) ( 2 ) I t f o l l o w s t h a t ,n-1 c h ( o ) - Z z ( 1 + ( 7 F 7 T I } ( - 1 ) n _ 1 ! * z + 1 T ^ T T T ( p ? z k ( P > - p 5 2 1 ) ( - i ) n ~ \ _ ( - i ) n _ 1 = # z - A T n ^ # z + " 1 T T F T T T p i z k ( P ) _ r ( n - l ) ! - ( - l ) n " 1 1 f ; , + ( - 1 ) n " 1 y k ( , - [ nrm J # z + ( n - i ) j p ? z M p ) w h e r e k ( p ) i s t h e d e g r e e o f F a t p a s d e s c r i b e d a b o v e a n d #Z i s c a r d i n a l i t y o f Z , w i t h o u t c o u n t i n g m u l t i p l i c i t y . B u t i t i s shown i n [ G H , p a g e 6 6 3 - 6 6 6 ] t h a t w h e r e 6 7 A=(9a^/3Zj) and r e s { « } i s the G r o t h e n d i e c k r e s i d u e symbol . I t i s a l s o known tha t Z 7 r e s f d e t A } = c n ( T - L * ) . peZ a . . . . a „ n i n T h i s i s the meromorphic v e c t o r f i e l d theorem of Baum and B o t t , [BB1] . S ince n=dimM, we have c (T-L* )=c ( L » T ) . n n Hence ( n - 1 ) ! - ( - l ) n ~ \ _ „ _ ( - 1 ) n ~ 1 I f L * imbeds i n t o T by a , then Z* = 0 and Z=0. Hence c h ( a ) = 0 and #Z = 0. Then e q u a t i o n (3) reduces t o 0 = c n ( L » T ) which e x h i b i t s " B o t t ' s v a n i s h i n g theorem, [ B 2 ] . I f L i s t r i v i a l and a has o n l y nondegenerate ze roes then k(p)=1 fo r a l l peZ and from e q u a t i o n (1) we f i n d t h a t chp(a )=1 for a l l peZ . Hence by e q u a t i o n (2) ch ( a )=#Z . Then e q u a t i o n (3) becomes #Z=c n (T) which i s a consequence of the Hopf f o r m u l a . 2) Le t E , F be v e c t o r bundles on M and i / /eHom(E,F). Then the graph r(\p) of ^ g i v e s r i s e to a c y c l e a t i n f i n i t y Z ^ . We 68 l e t B o ' * * * ' B m ^ E FC^E C O M P O N E N T S ° f t n e f i x e d p o i n t set B of G ( k , E ® F ) . Then ( Z o e n B ) p = ( Z o o n B o ) p u ( Z c c n B ^ p for peM, where i=rank^p . I t i s of course p o s s i b l e tha t ( B Q> p i s empty a t t h a t p o i n t . T h i s i s because r(\jj) and Za i n t e r s e c t E i n the same s e t , namely the k e r n e l of \p. I f i i s the l a r g e s t i n t e g e r fo r any peM such tha t ( Z a n B ) p = ( n B Q ) p u ( Z a n B . ) p , then we say t h a t i n t e r s e c t s the f i x e d p o i n t set g e n e r i c a l l y a t i " . Then the g e n e r i c rank of ^ i s i . In p a r t i c u l a r l e t K be the c u r v a t u r e of E , then K e H o m ( E , A 2 T » E ) and we have i t s graph i n G ( r a n k E , E © A 2 T * » E ) . I f K i n t e r s e c t s the f i x e d p o i n t set g e n e r i c a l l y at i then C j ( E ) = 0 fo r j > i . C o n v e r s e l y i f C j ( E ) = 0 for j> i f o r some i then K i n t e r s e c t s the f i x e d p o i n t se t g e n e r i c a l l y at t f o r some t ^ i . T h i s i s because (Z t-iB) c o n t a i n s (B . ) i f f rank i// i s i and rank i s oo I I p i p T p lower s e m i c o n t i n u o u s . 3) We want to show tha t the Hi ronaka Blow-up at a p o i n t can be r e c o v e r e d as a Grassmann Graph c o n s t r u c t i o n . The problem i s l o c a l so l e t M be an open set i n C n . D e f i n e two t r i v i a l bundles L and F as L=MxC and F=MxC n . D e f i n e a morphism 0eHom(L,F) a s : 0 ( p , t ) = ( p , t p ) for p e C n , t e C . The c y c l e at i n f i n i t y Z^ c o r r e s p o n d i n g to 8 i n t e r s e c t s the s i n k of G ( 1 , L 6 F ) i n M* ' , i . e . Z ^ M ^ + Z * . i s the H i r o n a k a 69 Blow-up of M at the o r i g i n . We can see t h i s as f o l l o w s . L e t p = ( p 1 , . . . , P n ) c M = C n . We a l s o i d e n t i f y P ( L © F ) w i t h P n . There i s a C * a c t i o n C*xMxP n ^MxP n g i v e n as ( X f p , [ y Q : y 1 : • • • : y n ] ) = ( p f [ y Q r X y , : • • • : X y N 3 ) The graph of 6 has the form r ( 0 ) = { ( p , [ 1 : p ] : • • • : p n ] ) e M x P n } The C * a c t i o n moves r ( 0 ) as X - r ( 0 ) = { ( p , [ 1 : X p 1 : • • • : X p n ] ) e M x P n } C o n s i d e r the u s u a l imbedding of C * i n P 1 as X = [ 1 : X ] = [ X 0 : X , ] , where X = X 1 / X 0 . S ince X ><*> i f f X 0 >0 w i t h X ^ O , we have the f o l l o w i n g l i m i t Z ^ - l i m x-ne) X>°° = l im { ( p , [ X 0 : X , p . : • • • : X , p ] )eMxP n ) X o >0 . = { ( p , [ 0 : X , p 1 : • • • : X , p n ] ) e M x P n } C l e a r l y [ 0 : X 1 p 1 : • • • : X , p ] can be c o n s i d e r e d as a p o i n t [ x 1 : » * » : x ] i n P n 1 such tha t p j x i = p i x j ' ^- i >^- n -From here i t i s easy to see tha t the i n t e r s e c t i o n of w i t h the s i n k of the C * a c t i o n i s the Hi ronaka blow-up of M at the o r i g i n . 70 CHAPTER 3 SINGULAR HOLOMORPHIC FOLIATIONS 0:INTRODUCTION F o l i a t i o n s a r i s e n a t u r a l l y i n mathemat ics , such as i n submers ions , group a c t i o n s and d i f f e r e n t i a l e q u a t i o n s . For an i n t r o d u c t i o n to the s u b j e c t we r e f e r t o the e x p o s i t o r y a r t i c l e of Lawson on f o l i a t i o n s , [ L ] , Lawson c l a i m s : "One of the reasons tha t f o l i a t i o n s i n t e r e s t peop le i n geometry i s tha t they c o n s t i t u t e a c l a s s of s t r u c t u r e s on m a n i f o l d s which i s c o m p l i c a t e d enough to shed l i g h t on the g e n e r a l s i t u a t i o n but has c e r t a i n geometr ic a s p e c t s tha t make i t t r a c t a b l e " , [ L ] . In t h i s chapter we w i l l i n v e s t i g a t e r e s i d u e p r o p e r t i e s of s i n g u l a r ho lomorphic f o l i a t i o n s . In S e c t i o n 1 we summarize some of the b a s i c i d e a s . In S e c t i o n 2 we d e f i n e Baum-Eott r e s i d u e s , see [BB2] . S e c t i o n 3 g i v e s Suwa's r ecent c o n t r i b u t i o n , see [ S u ] . The main theme of r e s i d u e s i s g i v e n 71 i n S e c t i o n 4 where we c a l c u l a t e Baum-Bott r e s i d u e s u s i n g Nash Blow-up and Grassmann Graph c o n s t r u c t i o n . 1:PRELIMINARIES A ho lomorphic f o l i a t i o n L of rank k on a complex m a n i f o l d M of d imens ion n i s a decompos i t ion of M i n t o d i s j o i n t connec ted s e t s L={L f l } ,acA w i t h a i n some i n d e x i n g set A , s a t i s f y i n g the f o l l o w i n g c o n d i t i o n ; for every p o i n t peM t h e r e e x i s t s an open neighbourhood U of p w i t h a ho lomorphic c o o r d i n a t e map x = (x 1 , . . . ,x ) : U >Cn such that f o r any aeA, e i t h e r L a n U = 0 or L a n U = { q e U | x i ( q ) = t " , k+1<i<n} where ( t j + 1 , . . . , tjj) e C n _ k depends on a and U . Each L i s c a l l e d a l e a f of the f o l i a t i o n . A rank k a f o l i a t i o n i n a complex m a n i f o l d of d imens ion n i s sometimes r e f e r r e d to as a cod imens ion n-k f o l i a t i o n . The l o c a l behav iour of a rank k f o l i a t i o n can be v i s u a l i z e d as the f i b r e s of a p r o j e c t i o n p r : C n ^ C n _ k where C n i s c o n s i d e r e d as C k 6 C n k and pr i s the p r o j e c t i o n n~ k — ] on the second component. Then f o r any ceC , pr (c) i s a l e a f of a rank k f o l i a t i o n {L }. Any f o l i a t i o n of rank k i s a l o c a l l y i somorph ic to { ^ a l . The i somorphism i s e s t a b l i s h e d 72 t h r o u g h t h e l o c a l c o o r d i n a t e s y s t e m ( U , x ) w h i c h i s d e s c r i b e d a b o v e . S u c h c o o r d i n a t e s y s t e m s a r e c a l l e d distinguished. L e t ( U , x ) a n d ( U , y ) , y = ( y 1 , . . . , y n ) be t w o d i s t i n g u i s h e d c o o r d i n a t e s on U f o r t h e f o l i a t i o n {L }. L e t g be t h e a x y t r a n s i t i o n f u n c t i o n b e t w e e n x a n d y x = ( x 1 x n ) = 9 X y - y = ( 9 x y - y - - " 3 x y ^ ) -T h e n Og__ y /9y-j)=0 f o r k + 1 < i < n , i < j < k . i . e . g^y / • • w y n ) = 9 x y ( y k + r • • • r Y n ) ^or i = k + 1 , . . . , n . T h i s p r o p e r t y o f f o l i a t i o n s l i e s a t t h e h e a r t o f B o t t ' s v a n i s h i n g t h e o r e m w h i c h we w i l l m e n t i o n n e x t a n d i t s g e n e r a l i z a t i o n w h i c h we w i l l g i v e i n t h e n e x t c h a p t e r . T h e r e i s a l s o a v e c t o r b u n d l e a p p r o a c h t o f o l i a t i o n s . L e t T be t h e t a n g e n t b u n d l e o f M , a n d l e t E be a s u b b u n d l e o f T w i t h r a n k k . I n t h e c l a s s i c a l t e r m i n o l o g y C°° s u b b u n d l e s o f T a r e c a l l e d s m o o t h d i s t r i b u t i o n s . E i s c a l l e d i n t e g r a b l e i f a t e a c h p o i n t p e M , t h e r e e x i s t s a s u b m a n i f o l d w h o s e t a n g e n t s p a c e a t p i s E p . E a c h s u c h s u b m a n i f o l d i s p a r t o f a l e a f o f a f o l i a t i o n on M . I t i s e a s y t o s e e t h a t i n t h i s c a s e E i s c l o s e d u n d e r t h e u s u a l b r a c k e t o p e r a t i o n . A s u b b u n d l e o f t h e t a n g e n t b u n d l e i s c a l l e d i n v o l u t i v e i f i t i s c l o s e d u n d e r t h e b r a c k e t o p e r a t i o n . I n t e g r a b l e a n d i n v o l u t i v e b u n d l e s a r e r e l a t e d t o e a c h o t h e r by t h e 73 f o l l o w i n g c l a s s i c a l t h e o r e m o f F r o b e n i u s . F r o b e n i u s T h e o r e m : A s u b b u n d l e o f t h e t a n g e n t b u n d l e i s i n t e g r a b l e i f f i t i s i n v o l u t i v e . One o f t h e q u e s t i o n s a s k e d a b o u t f o l i a t i o n s w a s , w h e t h e r g i v e n a r a n k k s u b b u n d l e E o f t h e t a n g e n t b u n d l e T , t h e r e i s an i n t e g r a b l e s u b b u n d l e F o f T s u c h t h a t E c a n be d e f o r m e d t o F . A n e c e s s a r y c o n d i t i o n i s g i v e n by B o t t ' s V a n i s h i n g T h e o r e m [B2]: I f a s u b b u n d l e E o f t h e t a n g e n t b u n d l e T i s i n t e g r a b l e , t h e n t h e r e a l P o n t r y a g i n c l a s s e s o f T / E g e n e r a t e a g r a d e d r i n g P o n t * ( T / E ) s u c h t h a t P o n t 1 ( T / E ) = 0 i f i > 2 - r a n k R ( T / E ) . C o m p l e x c a s e : I f a h o l o m o r p h i c s u b b u n d l e E o f t h e h o l o m o r p h i c t a n g e n t b u n d l e T i s i n t e g r a b l e t h e n t h e c h e r n c l a s s e s o f T / E g e n e r a t e a g r a d e d r i n g C h e r n * ( T / E ) s u c h t h a t C h e r n 1 ( T / E ) = 0 i f i > r a n k c ( T / E ) . T h e p r o o f o f t h i s t h e o r e m c a n be s e e n a s a n e l e g a n t e x p l o i t a t i o n o f d i s t i n g u i s h e d c o o r d i n a t e s , s e e [ B 2 ] . I n t h e n e x t c h a p t e r we w i l l g i v e a c o r o l l a r y t o t h i s t h e o r e m . I n g e n e r a l M may n o t a d m i t a n y f o l i a t i o n b u t o n e may f i n d a c l o s e d s u b s e t S o f M s u c h t h a t M - S a d m i t s a f o l i a t i o n . R . Thorn d i s c u s s e s p o s s i b l e w a y s o f c h o o s i n g s u c h S i n [ T h ] . O b v i o u s l y i f t h e c h o i c e o f S i s n o t f o r c e d b y t h e 7 4 f o l i a t i o n on M-S then t h i s does not lead to an int e r e s t i n g mathematical concept. If S in some i n t r i n s i c way depends on the f o l i a t i o n on M-S then i t i s natural to expect that S w i l l r e f l e c t some properties of the f o l i a t i o n . To t h i s end we adopt the d e f i n i t i o n of Baum and Bott for a singular holomorphic f o l i a t i o n ; Definition [ BB2 ]: A singular holomorphic foliation i s a coherent integrable complex analytic subsheaf of the tangent bundle. If F i s a singular holomorphic f o l i a t i o n then S i s the singular set of F as a coherent sheaf. In the next section we w i l l describe a residue on S coming from the f o l i a t i o n as given by Baum and Bott, [BB2]. 75 2: BAUM-BOTT RESIDUES T h i s s e c t i o n p r e s e n t s a summary of Baum and B o t t ' s work on s i n g u l a r f o l i a t i o n r e s i d u e s , [BB2] . At the end of the s e c t i o n a r e s i d u e fo r v e c t o r bundles i s d e f i n e d which i s denoted by BRes and c a l l e d the g e n e r a l i z e d Baum-Bott r e s i d u e . For any holomorphic v e c t o r bundle E we w i l l use the n o t a t i o n tha t E '=sheaf of ho lomorphic s e c t i o n s of E . Then E ' w i l l denote the germs of ho lomorphic s e c t i o n s of E at x . Le t T be the ho lomorphic tangent bundle of a complex m a n i f o l d M, and l e t T be the tangent shea f . With the above n o t a t i o n T=T'. Let { be a s i n g u l a r holomorphic f o l i a t i o n of rank k. Then there e x i s t s a c l o s e d s u b v a r i e t y S of M such t h a t £ | M - S i s l o c a l l y f r e e , hence there i s a v e c t o r bundle E >M-S of rank k such tha t E ' = £ | M - S . To a v o i d any a r t i f i c i a l s i n g u l a r i t i e s for t h i s f o l i a t i o n i t i s assumed tha t £ i s f u l l , i . e . for every open U i n M and for every 7 e r ( U , T ) , i f 7 ( x ) e E fo r every x e U - U n S , then the germ of the ho lomorphic v e c t o r f i e l d 7 at x i s i n £ for every x e U n S , [ BB2, p282] . 76 Baum and B o t t compute t h e c h e r n p o l y n o m i a l s 4>(T/£) i n t e rms of l o c a l i n f o r m a t i o n a t S, where $ e C [ X 1 , . . . , X n ] i s a s y m m e t r i c homogeneous p o l y n o m i a l of degree d>n-k . L e t Z be a c o n n e c t e d component of S , i : Z >M be t h e i n c l u s i o n , i * : H * ( Z ; C ) ^ H * ( M ; C ) be the n a t u r a l map i n d u c e d by i and l e t PD' : H * ( M ; C ) >H*(M;C) be t h e P o i n c a r e d u a l i t y map. F o r any s y m m e t r i c homogeneous p o l y n o m i a l $ e C [ X 1 , . . . , X ] t h e r e e x i s t s a u n i q u e p o l y n o m i a l $ s u c h t h a t * ( a 1 ( X 1 , . . . , X n ) , . . . , a n ( X 1 f . . . , X n ) ) = * ( X r . . . , X n ) where a r e t h e e l e m e n t a r y s y m m e t r i c p o l y n o m i a l s . L e t Q=T/£ t h e n t h e c h e r n p o l y n o m i a l i s d e f i n e d by * ( Q ) = * ( c 1 ( Q ) , . . . , c n ( Q ) ) where t h e c ^ ( - ) i s the i - t h c h e r n c l a s s . Then we have Theorem (Baum, Bott): Assume t h a t Z i s c o m p a c t . Then t h e r e e x i s t s R e s # ( ^ , Z ) e H n _ d ( Z ; C ) s u c h t h a t i ) R e s . ( £ , Z ) depends o n l y on $ and t h e b e h a v i o u r of £ near Z . 77 i i ) i f M i s compact t h e n I P D ' • i * (Res^,( £ , Z) )=#(Q) where t h e summation i s o v e r a l l t h e c o n n e c t e d components Z o f S, [ B B 2 , p p 3 1 2 - 3 1 3 ] . L e t E be a v e c t o r b u n d l e of rank k on a complex m a n i f o l d m of d i m e n s i o n n . L e t U be an open s u b s e t of M . L e t D be a c o n n e c t i o n o f E | U and K a c u r v a t u r e m a t r i x f o r D . D e f i n e a 1 ( K ) , . . . , a (K) by t h e e q u a t i o n d e t ( I + t K ) = 1 + t a 1 ( K ) + . . . + t n a n ( K ) . E a c h a^(K) i s a 2 i - f o r m on U . I t i s w e l l known t h a t each a^(K) i s a c l o s e d form and d e f i n e s a u n i q u e cohomology c l a s s [ ^ ( K ) ] i n H 2 l ( M ; C ) . By t h e C h e r n - W e i l T h e o r y [ o ^ K ) ]=(2TT/V-1 ) 1 c i ( E ) , i=1 , . . . , n . F o r a s y m m e t r i c homogeneous p o l y n o m i a l fceCtX^,...,X ] d e f i n e * ( K ) - * ( i(7.j (K) , . . . , inon(K) ) where i= (2ir / \ / -1 ) " 1 . Then [ * ( K ) ] = * ( E ) . Assume t h a t M i s compact and t h e r e e x i s t s a c l o s e d s u b s e t S o f M s u c h t h a t E | M - S has a c o n n e c t i o n D 1 w i t h t h e p r o p e r t y t h a t f o r any s y m m e t r i c homogeneous p o l y n o m i a l * ( X ) e C [ X 1 , . . . , X ] w i t h deg *>d 0 f o r some d o >0 we have Z ( K 1 ) = 0 on M-S where K 1 i s t h e c u r v a t u r e m a t r i x o f D 1 . L e t 2 be a c l o s e d s u b s e t o f M s u c h t h a t S i s c o n t a i n e d i n t h e i n t e r i o r of E and l e t D- be a c o n n e c t i o n f o r E | I . Then t h e r e e x i s t s a 78 c o n n e c t i o n D f o r E on M s u c h t h a t D a g r e e s w i t h D 1 on M - Z . To c o n s t r u c t such a c o n n e c t i o n l e t f be a r e a l v a l u e d C°° f u n c t i o n on M such t h a t f v a n i s h e s on a n e i g h b o u r h o o d ' o f S and f=1 on M - Z . Then D i s d e f i n e d as D = f D 1 + ( 1 - f ) D 2 , w h i c h e x t e n d s D 1 , [BB2 , p 3 0 0 , Lemma 4 . 4 1 ] L e t K be a c u r v a t u r e m a t r i x f o r D . Then $(K)=0 on M - Z . Hence <t>(K) i s a d i f f e r e n t i a l form w i t h compact s u p p o r t and d e f i n e s an e lement i n t h e cohomology w i t h compact s u p p o r t s , [ * ( K ) ] « * ( E ) e H j ( M ; C ) . L e t Z be a c o n n e c t e d component of S and U an open n e i g h b o u r h o o d of Z w h i c h d e f o r m a t i o n r e t r a c t s t o i t . D e f i n e a r e s i d u e B R e s $ ( E , Z ) = ( i ^ ) - 1 . P D - ( $ ( E ) ) We w i l l c a l l BRes t h e generalized Baum-Bott residue and use BRes i n S e c t i o n 4 t o d e s c r i b e t h e Nash r e s i d u e . A l s o no te t h a t i f F i s a c o h e r e n t sheaf and * ( K ) i s a d i f f e r e n t i a l fo rm s u c h t h a t [ * (K) ]= * (F ) and $ ( K ) has compact s u p p o r t i n U as a b o v e , t h e n BRes i s d e f i n e d f o r F. I n p a r t i c u l a r i f £ i s an i n t e g r a b l e c o h e r e n t s u b s h e a f of 7" t h e n s u c h a d i f f e r e n t i a l form e x i s t s f o r T/%, [ B B 2 , p 3 ! 3 , ( 7 . 1 1 ) ] . Then no te t h a t 79 R e s ^ ( « , Z ) = B R e s ^ ( T / U . R e g a r d i n g t h e c a l c u l a t i o n o f R e s ^ , Baum and B o t t g i v e t h e f o l l o w i n g c o n j e c t u r e : R a t i o n a l i t y C o n j e c t u r e [BB2 , page 2 8 7 ] : I f $ e Q [ X 1 , . . . , X ] and deg <t>>n-k+1 , t h e n R e s ( J > ( ? , Z ) € H ^ ( Z ; Q ) . I n the n e x t s e c t i o n we w i l l summarize Suwa ' s r e c e n t c o n t r i b u t i o n t o t h i s c o n j e c t u r e and we w i l l r e l a t e h i s r e s u l t t o B R e s . In s e c t i o n 4 we w i l l g i v e a c a l c u l a t i o n f o r R e s ^ when M i s a compact K a e h l e r m a n i f o l d . 3 :SUWA'S WORK L e t 0 denote t h e c o t a n g e n t shea f o f M and l e t 4/ be a subshea f of R . D e f i n e £={6eT\ u)(6) = 0 f o r a l l u e ^ }. Then £ i s an i n t e g r a b l e subshea f o f t h e t a n g e n t shea f i f f \p i s c l o s e d under t h e e x t e r i o r d i f f e r e n t i a t i o n . Note 0 >l >T >Q - > 0 (1 ) where Q i s d e f i n e d as T/£ . L e t fi^ be d e f i n e d by t h e e x a c t sequence 0 >\p >Sl >to , >0. (2) T a k i n g t h e d u a l , H o m ^ f •,0), o f (2) we o b t a i n 0 - ^ H o m 6 > ( n ^ , O ) - ^ H o m o ( J 2 , C > ) - ^ H o m c > ( ^ , O ) -^T? ^ 0 ( 3 ) 80 where r)=Ext Q(SI^,0) . I n t h i s sequence Hom 0 ( J2 ,0 )=r Hom^(v//,0)=<//*=dual of i//. U s i n g t h e s e i d e n t i f i c a t i o n s (3) can be r e w r i t t e n as 0 >T >\p* >rj >0 (3 ' ) The k e r n e l of t h e map T i s £ by t h e e x a c t n e s s of ( 3 ' ) . From ( 1 ) T/£=Q, hence t h e s e g i v e an e x a c t sequence 0 K> M ' * >V >0 (4) Assume t h a t \p i s l o c a l l y f r e e of rank n - k , where k = r a n k £ and n = r a n k 7 \ t h e n yp i s c a l l e d a foliation of complete intersection type. I n t h i s c a s e t h e sequence (4) c an be i n t e r p r e t e d as r e s o l v i n g t h e c o h e r e n t "sheaf Q i n t o a d i f f e r e n c e of a v e c t o r b u n d l e \p* and a s k y s c r a p e r shea f TJ w h i c h has s u p p o r t i n S, t h e s i n g u l a r s e t of £ . L e t a ( i ; X 1 , . . . , X ) be t h e i - t h e l e m e n t a r y s y m m e t r i c p o l y n o m i a l on X 1 , . . . , X n . F o r a c o h e r e n t shea f E l e t o(i;E) denote t h e i - t h e l e m e n t a r y symmetr i c p o l y n o m i a l on t h e C h e r n c h a r a c t e r s of E. 81 Theorem (Suwa) : L e t $ be a f o l i a t i o n o f c o m p l e t e i n t e r s e c t i o n t y p e on M w i t h rank =n-k. L e t Z be a c o n n e c t e d component o f S, t h e s i n g u l a r s e t , and U an open n e i g h b o u r h o o d of Z w h i c h d e f o r m a t i o n r e t r a c t s t o Z . L e t 4 )e[X 1 , . . . , X n ] be a homogeneous symmetr i c p o l y n o m i a l w i t h degree m such t h a t * ( X 1 , . . . , X n ) = o ( J 1 ; X 1 , . . . , X n ) . . . o ( J r ; X 1 , . . . , X n ) w i t h + ' ' - + J r =ni and j^>n-k f o r some v. (*) Then R e s d ) ( £ , Z ) = ( i * ) - 1 . P D ( c . • • - c . U * - T ? ) ) . J 1 where i * and PD a r e as d e f i n e d i n s e c t i o n 2 . P r o o f : F o r d e t a i l s of t h e p r o o f we r e f e r t o [ S u ] , Here we w i l l c o n c e n t r a t e on t h e ma in a r g u m e n t . L e t us r e s t r i c t a l l t h e above sequences t o U . From t h e sequence (4) we h a v e , as v i r t u a l b u n d l e s Q=^*-r? D e f i n e 1+dj + d n =( 1+o( 1 ; T J ) + ff(n;r?) ) ~ 1 . Then ( 1 + O ( 1 ; Q ) + « • - + o ( n ; Q ) ) = ( 1+a( 1 ;\//*) + - • .+a(n-k,•!/>*)) ( 1 + d ^ - • - + d n ) . -82 From t h i s i t f o l l o w s t h a t f o r j = 1 , . . . , n we have o(j;Q)=a(j;^*)+o(j-1;^*)d1+-•1;^*)dj_1+d^ (5) and hence * ( Q ) = o ( ; Q ) • • - a ( J r ; Q ) = o( z*'/ '*) • • • o ( J r ; \ / / * ) + P - (6) where P i s a p o l y n o m i a l i n a ( 1 ; ^ * ) , . . . , a ( n - k ; ^ * ) , d 1 , . . . , d such t h a t each monomial of P has a t l e a s t one d . as a n o n t r i v i a l f a c t o r f o r l < i < n . S i n c e j^>n-k f o r some v by c o n d i t i o n ( * ) , we have o( j v,4>*) =0 and hence 4>(Q)=P. By a p p l y i n g ( i * ) ~ 1 . P D t o b o t h s i d e s of t h i s e q u a t i o n we ge t t h e r e q u i r e d r e s u l t . QED I n t h i s p r o o f s i n c e each d^ a l r e a d y r e p r e s e n t s c l a s s e s i n H * ( Z ; C ) , t h e n P i s a l s o i n H * ( Z ; C ) . Hence t h e r a t i o n a l i t y c o n j e c t u r e d i r e c t l y f o l l o w s i n t h i s ca se because one d i d no t have t o c a l c u l a t e r e s i d u e s ; R e S g ( £ , Z ) i s g i v e n by t h e image of # ( ^ * - T J ) i n H * ( U ; Q ) see [ S u , c o r o l l a r y 3 . 8 ] . REMARK: The s e t up b e i n g as i n Suwa' s theorem d r o p t h e c o n d i t i o n ( * ) , t h e n R e s ^ ( £ , Z ) = B R e s 4 ) ( \ / / * , Z ) + ( i * ) - 1 . P D ( P ) where P i s as i n e q u a t i o n ( 6 ) . T h i s can be seen as f o l l o w s ; s i n c e $ i s d e f i n e d as 83 * ( X 1 , . . . , X n ) = a ( j 1 ; X 1 , . . . , X n ) • • - a ( J r ; X 1 r . . . , X n ) t h e f i r s t t e r m on t h e l e f t hand s i d e of e q u a t i o n ( 6 ) i s • H ^ * ) . A p p l y i n g ( i * ) _ 1 . P D t o b o t h s i d e s o f e q u a t i o n ( 6 ) g i v e s ( i * ) - 1 . P D ( * ( Q ) ) = ( i * ) - 1 . P D ( $ U * ) ) + ( i * ) - 1 . P D ( P ) where by d e f i n i t i o n ( i * ) - 1 . P D ( * ( Q ) ) = R e s 4 > ( £ , Z ) and ( i * ) " 1 -PD(<I>U* )). =BRes$(<//*, Z ) . 4 :NASH RESIDUE AND REDUCTION T h i s s e c t i o n c u l l s t h e r e s u l t s of t h e p r e v i o u s c h a p t e r s t o g i v e a new a p p r o a c h t o t h e c a l c u l a t i o n of Baum-Bott r e s i d u e s f o r s i n g u l a r h o l o m o r p h i c f o l i a t i o n s on compact K a e h l e r m a n i f o l d s , where t h e Nash B l o w - u p g i v e s a smooth m a n i f o l d . I t w i l l be shown t h a t i f i n a d d i t i o n t h e f o l i a t i o n i s d e f i n e d by a b u n d l e morphism t h e n t h e R a t i o n a l i t y C o n j e c t u r e of Baum and B o t t h o l d s . F i r s t t h e Nash R e s i d u e i s d e f i n e d f o r g e n e r a l s i n g u l a r h o l o m o r p h i c f o l i a t i o n s . L e t M be a compact K a e h l e r m a n i f o l d of d i m e n s i o n n w i t h t a n g e n t shea f T. L e t F be an i n t e g r a b l e f u l l c o h e r e n t subshea f of T w i t h r a n k / ^ k , s u c h t h a t t h e Nash B l o w - u p N of M w i t h r e s p e c t t o F and T i s s m o o t h . On N t h e r e i s a s h o r t e x a c t sequence of v e c t o r b u n d l e s 0 >r >C n ^0 84 w h e r e T , t h e t a u t o l o g i c a l b u n d l e , r e s t r i c t s t o t h e t a u t o l o g i c a l b u n d l e o f e a c h f i b r e it'1 ( x ) = G ( k , n ) , x e M . C n i s t h e t r i v i a l n - b u n d l e a n d W i s t h e q u o t i e n t b u n d l e ' w h i c h r e s t r i c t s t o t h e u n i v e r s a l q u o t i e n t b u n d l e o f e a c h f i b r e . T h i s e x a c t s e q u e n c e r e s t r i c t s t o t h e f o l l o w i n g s h o r t e x a c t s e q u e n c e on N 0 >T >ir*T >VI J-0 w h e r e T i s t h e t a n g e n t b u n d l e on M a n d TT*T d e n o t e s t h e p u l l b a c k b u n d l e on N . We u s e t h e same n o t a t i o n T a n d W f o r t h e r e s t r i c t i o n o f t h e s e b u n d l e s t o N s i n c e we w i l l be w o r k i n g on N f r o m now on a n d t h e r e w i l l be no a m b i g u i t y a b o u t t h e b a s e s p a c e s . L e t S be t h e s i n g u l a r s e t o f F. On M - S t h e r e i s a u n i q u e h o l o m o r p h i c v e c t o r b u n d l e Y s u c h t h a t Y'=Q | M - S . By B o t t ' s v a n i s h i n g t h e o r e m Y h a s a c o n n e c t i o n D , s u c h t h a t i f K , i s t h e c o r r e s p o n d i n g c u r v a t u r e m a t r i x a n d • e C [ X 1 , . . . , X ] i s a s y m m e t r i c h o m o g e n e o u s p o l y n o m i a l w i t h d e g $ > n - k t h e n $ ( K ! ) = 0 o n M - S . S i n c e TT * (Y ) a n d W a g r e e on N - i r - 1 ( s ) , t h e c o n n e c t i o n D , o f Y p u l l s b a c k t o a c o n n e c t i o n 7 r * D , o f W | N - 7 r " 1 ( S ) . T h e r e e x i s t a c o n n e c t i o n D o f W on N a n d a c o m p a c t s u b s e t Z o f N w h i c h c o n t a i n s JT'MS ) i n i t s i n t e r i o r s u c h t h a t D | N - Z = i r * D 1 | N - Z , [ B B 2 , p 3 0 0 ] , L e t K u be t h e c o r r e s p o n d i n g c u r v a t u r e m a t r i x . 8 5 Then as above $(KW)=0 on N - I , i . e . * ( R W ) i s a d i f f e r e n t i a l form on N w i t h compact s u p p o r t . L e t Z be a c o n n e c t e d component o f S, U be an open n e i g h b o u h o o d of Z t h a t d e f o r m a t i o n r e t r a c t s t o Z , 6 : U >Z, and choose 2 such t h a t t h e c o n n e c t e d component of 2 t h a t c o n t a i n s i r ' M Z ) i n i t s i n t e r i o r i s c o n t a i n e d i n 7 r ~ 1 ( U ) . Then 4>(KW| TT" 1 (U) ) i s a d i f f e r e n t i a l form on TT" 1 (U) w i t h compact s u p p o r t . Hence t h e c h e r n p o l y n o m i a l <I>(W| TT~ 1 (U)) i s a cohomology c l a s s of U w i t h compact s u p p o r t . D e f i n i t i o n : The Nash R e s i d u e o f F on Z i s d e f i n e d ' a s N R e s 4 , ( F , Z ) = 8 * . 7 r A , P D 7 r . , ( u ) * ( W | TT" 1 (U)) R e m a r k : L e t V be an open n e i g h b o u r h o o d of M' s u c h t h a t V d e f o r m a t i o n r e t r a c t s t o i r " 1 ( Z ) , p:V >ir- 1 ( Z ) . Then choose U t o be TT(V) and choose 5 s u c h t h a t 6(x)=7rp7r 1 (x) i f xeU-S 6 ( x ) = x i f x e S . Then 6 i s a d e f o r m a t i o n r e t r a c t i o n and 6 . ff(y ) = 7r. p ( y ) f o r y e 7 r ~ 1 ( U ) = V . A t t h e homology l e v e l t h i s i m p l i e s t h a t Then we have 86 NRes ) J ) (/ : - ,Z) = 5 ^ . 7 r ^ . P D 7 r . , ( u ) # ( W | i r 1 (U) ) = 7 r ^ . p ^ P D 7 r . , ( u ) f ' ( W | 7 r - 1 ( U ) ) where p ^ . P D ^ . , ( u ) * ( W | w - 1 ( U ) ) e H * ( i r 1 ( Z ) ; C ) i s t h e BRes of W (see S e c t i o n 2 ) . T h e r e f o r e N R e s $ ( / r , Z ) = T r A - B R e s ^ ( W r 7 r - 1 (Z) ) . L e t us summarize the s e t u p : L e t M be a compact K a e h l e r m a n i f o l d of d i m e n s i o n n . L e t E be a v e c t o r b u n d l e of rank k and * : E vT be a b u n d l e morphism of max imal r a n k . Suppose t h a t * ( E ) g e n e r a t e s an i n t e g r a b l e c o h e r e n t subshea f F of t h e t a n g e n t shea f of M . C o n s t r u c t t h e Nash B l o w - u p 7T: N *-M c o r r e s p o n d i n g t o F. Assume t h a t N i s s m o o t h . L e t S be t h e s i n g u l a r s e t o f F and Z a c o n n e c t e d component o f S . Lemma 1: L e t :E >T be t h e map i n d u c e d by * : E >T a t the shea f l e v e l . Then i s i n j e c t i v e . P r o o f : L e t U be an open s u b s e t of M such t h a t E and T a r e t r i v i a l on U . L e t e . , . . . , e , and t . , . . . f t be l o c a l 87 g e n e r a t o r s f o r E and T r e s p e c t i v e l y . Then * can be d e f i n e d i n t e rms of t h e s e b a s i s e l e m e n t s a s : * ( e . ) = f i l V . . . + f i n t n i - 1 k where t^^eO^ a r e h o l o m o r p h i c f u n c t i o n s on U . I f h i s a s e c t i o n of E\U t h e n H i e i V k = ( h 1 , . . . , h k > where h^ a r e h o l o m o r p h i c f u n c t i o n s on U . ¥ ( h ) can t h e n be d e f i n e d by * ( h ) = h 1 * ( e 1 ) + • • - + h k * ( e k ) = ( h 1 f 1 1 + " * + h k f k 1 h l f 1 n + " • + V k n ) ' I f * (h )=0 t h e n i n p a r t i c u l a r * x ( h ( x ) ) = 0 , i . e . h 1 ( x ) f 1 i ( x ) + - • « + h k ( x ) f k i ( x ) = 0 f o r xeU and i = 1 , . . . , k . S i n c e * i s i n j e c t i v e as a b u n d l e morphism on U - U n S t h e n h ( x ) = 0 f o r x e U - U n S , i . e . h ^ ( x ) = 0 f o r x e U - U n S and i = 1 , . . . , k . Then h^ a r e h o l o m o r p h i c f u n c t i o n s on U v a n i s h i n g on the open s e t U - U n S and c o n s e q u e n t l y a r e i d e n t i c a l l y z e r o on U . T h i s p r o v e s t h a t i s i n j e c t i v e . QED 88 Theorem 1: R e s ^ ( E , Z)=NRes^(E , Z)+7r^(3 where /3eH*(ic' 1 (Z) ; C ) and i s c a l c u l a t e d by a Grassmann G r a p h c o n s t r u c t i o n . M o r e o v e r N R e s ^ ( E , Z ) and ir+B a r e r a t i o n a l , hence Res^( 'E ,Z) i s r a t i o n a l . P r o o f : On M t h e r e i s t h e e x a c t sequence of sheaves 0 >E >T >Q *-0 (1 ) where E >T i s i n d u c e d by $ . T h i s sequence p u l l s back t o a sequence of sheaves on N 0 s -7r*£ >ir*T >ic*Q >0. (2) On N t h e r e i s " a l s o t h e sequence o f v e c t o r b u n d l e s 0 5-r >i:*T s-W >0 ( 3 ) where T i s t h e t a u t o l o g i c a l b u n d l e and W i s t h e u n i v e r s a l q u o t i e n t b u n d l e . L e t X = 7 r " 1 ( S ) and W'=the sheaf of h o l o m o r p h i c s e c t i o n s of W. On N-X t h e sheaves ir*Q and W a r e e q u a l . Hence t h e sequence of sheaves on N 0 >ic*E >ic*T—: •() (4) i s e x a c t on N - X . The u n d e r l y i n g v e c t o r b u n d l e s o f t h i s sequence g i v e a complex of b u n d l e s on N 0 s~7r*E s-7r*T >VI 5-0 (5) w h i c h i s e x a c t on N - X . 89 C o n s i d e r on N t h e v i r t u a l b u n d l e 7 7=7r *T -7r*E-W. The c h e r n c l a s s of t h i s v i r t u a l b u n d l e i s C(7)=C(7T *T-7T*E-W) =c(7r * (T-E)-W) = C ( T T * ( T - E ) ) / c(W) . (6) From t h e e x a c t n e s s of sequence (2) t h e c h e r n c l a s s of TT*Q i s c ( i r * Q ) = c U * ( T - E ) ) = C ( T T * ( T - E ) ) (7) C o m b i n i n g t h e r e s u l t s of (6) and (7) one g e t s C ( 7 ) = C ( T T * Q ) / C ( W ) . (8) T h i s e q u a t i o n w i l l be u sed w i t h t h e Graph c o n s t r u c t i o n on ( 5 ) ; c o n s t r u c t t h e Grassman G r a p h c o r r e s p o n d i n g t o t h e complex o f v e c t o r b u n d l e s (5) on N , p : G ••—>N. (9) L e t i >G be t h e v i r t u a l t a u t o l o g i c a l b u n d l e on G and l e t ^ X ^ X e P 1 ^ e t n e f a m i l y o f r a t i o n a l l y e q u i v a l e n t c y c l e s i n G o b t a i n e d by t h e G r a p h c o n s t r u c t i o n . Then £ | Z 0 = p * ( 7 ) (10) s i n c e Z 0 i s i s o m o r p h i c t o N . The c y c l e a t i n f i n i t y Z B decomposes as Z_=N*+Z* ( 1 1 ) 00 where i s b i m e r o m o r p h i c t o N and Z * i s t h e f i b r e above S. I t i s known t h a t £ | N * = 0 (12) 90 [ B F M , p 1 2 2 ] , [ F u , p p 3 4 0 - 3 4 1 ] . S i n c e Z 0 i s r a t i o n a l l y e q u i v a l e n t t o Z B we ge t t h e f o l l o w i n g e q u a l i t i e s c i ( U n Z o = c i ( n n Z 0 6 = c i ( £ ) n N * + c i ( U n Z * = c i ( £ ) n Z * , i>0 (13) where t h e l a s t e q u a t i o n f o l l o w s f rom (12) and c_U) n z * e H * ( Z * ; C ) . U s i n g (10) and (13) g i v e s c i ( 7 ) n [ N ] = p * ( c i ( £ ) n Z 0 ) = P * ( c . U ) n Z * > (14) where p * ( c ^ ( £ ) n Z * ) e H * ( X ; C ) . D e f i n e a l o c a l i z e d c h e r n c l a s s c £ e H * ( X ; C ) as c'^=p y t(-c i(Un 2*)» i > 0 - <15> E q u a t i o n (14) can be r e w r i t t e n i n t h i s n o t a t i o n c i ( 7 ) n [ N ] = c J , i > 0 . ( 1 4 ' ) The t o t a l c h e r n c l a s s of 7 i s t h e n g i v e n by c ( 7 ) n [ N 3 = ( 1 + c 1 ( 7 ) + - " + c n ( 7 ) ) n [ N ] = [ N ] + c ^ + - • - + c £ (16) where t h e l a s t e q u a t i o n i s w r i t t e n u s i n g ( 1 4 ' ) and [N] i s t h e f u n d a m e n t a l c y c l e o f N. S u b s t i t u t e e q u a t i o n (8) t o t h e LHS o f e q u a t i o n (16) n {c(ir*Q)/c(W)} n [N] = [N] + i | 1 c ^ . ( 1 6 ' ) Cap b o t h s i d e s o f ( 1 6 ' ) by c(W) n c(w) n ( { c ( i * Q ) / c ( w ) ) n [N])=C(W) N [N] + .Z I C(W ) n c _ ; . (17) 91 The LHS of (17) can be w r i t t e n : (c(W) u { c (7r*Q ) / c (W)} ) n [N] (18) [ S p , p254 , ( 1 8 ) ] . I n t h i s e x p r e s s i o n t h e c ( W ) ' s c a n c e l e a c h o t h e r s i n c e t h e t o t a l c h e r n c l a s s of a v e c t o r b u n d l e i s i n v e r t i b l e . Hence (17) t a k e s t h e form n cU*Q) n [N]=c(w) n [N] + i L 1 c ( w ) n c _ _ . (19) To s i m p l i f y t h e n o t a t i o n d e f i n e e H 2 n - 2 i ^ X ; C ^ a s /3. + . . . + / 3 n = i Z 1 c ( W ) n c ^ . (20) Then from (19) and (20) t h e d u a l of t h e i - t h c h e r n c l a s s of 7r*Q becomes: c . U * Q ) n [N]=c. (W) n [N] + 0 . f i = l , . . . , n . (21) T h i s w i l l be used t o c a l c u l a t e ${n*Q) w h i c h i s d e f i n e d as 4>(7r*Q)=*(c 1 U*Q) , . . . , c n ( i r*Q)) where $ was d e f i n e d b e f o r e . To c a l c u l a t e i(ir*Q) f i r s t assume t h a t * U * Q ) = c . U * Q ) C j U * Q ) (22) f o r some i , j , 0 < i , j < n . Cap b o t h s i d e s of (22) by [N] <I>(7r*Q) n [N] = {c i (7r*Q )c ; j (7r*Q)} n [ N ] . (23) The RHS can be w r i t t e n as {c i (7r*Q )c : J (7r*Q)} n [N] = ( c . (ir*Q) n [N] ) • (c ^ ( TT*Q) n [N]) (24) where • i s t h e c y c l e i n t e r s e c t i o n , [ G H , p 5 9 ] . The RHS of (24) can be r e w r i t t e n u s i n g (21) RHS(24) = ( c i ( W ) n [ N l + P j ) • ( C j ( W ) n [N] + /3j) = (c.(w) N [ N ] ) - ( C J ( W ) n [N]) + ( c . (w) N [ N ] ) . j 5 j + ^ i * ( C j ( W ) N [ N ] ) + j 5 i - P j . (25) 92 To s h o r t e n t h e n o t a t i o n d e f i n e / J ^ e H ^ ( X ; C ) as / 3 i j = ( c I ( W ) n [ N ] ) . / 3 j + / 3 i . ( c J ( W ) n [N])+/3 .-^j. (26) T h a t /J^j i s a c y c l e i n X f o l l o w s from t h e f a c t t h a t each i s a c y c l e i n X , see t h e d e f i n i t i o n of i n ( 2 0 ) . Then (25) c a n be r e w r i t t e n as RHS(24) = ( C I ( W ) n CN]) • ( C j ( W ) n [N]) + j3. j . ( 2 5 ' ) . Use t h e same c o m p u t a t i o n as i n (24) f o r t h e f i r s t t e rm on t h e RHS of ( 2 5 ' ) ; ( c . ( W ) n [ N ] ) . ( C j ( W ) n [ N ] ) = { c . ( W ) C j ( W ) } n [ N ] (27) U s i n g t h e a s s u m p t i o n o f (22) t h a t * ( * ) = C . ( * ) C j ( * ) t h e RHS of (27) can be w r i t t e n • { c . ( W ) C j ( W ) } n [ N ] = * ( W ) n [ N ] . (28) These c a l c u l a t i o n s can be put t o g e t h e r as f o l l o w s ; * ( T T * Q ) n [ N ] = { c i ( 7 r * Q ) - C j ( 7 r * Q ) } n [N] ( f rom 23) = (ci(ir*Q) n [N]) • ( C J ( T T * Q ) n [N]) ( f rom 24) = ( c I ( W ) n [N]) • ( C j ( w ) n [ N ] ) + / 3 i : j ( f rom 2 5 ' ) = { c I ( W ) C j ( W ) } n [ N ] + 0 . j ( f rom 27) = * ( W ) N [ N ] + ^ I j . ( f rom t h e d e f i n i t i o n of * ) Hence by i n d u c t i o n on t h e s i z e of $ we o b t a i n f o r g e n e r a l $ * ( T T * Q ) n [ N ] = * ( w ) n [ N ] + /3 (29) where / 3 e H * ( X ; C ) . A p p l y Tr* t o b o t h s i d e s of ( 2 9 ) , T T * ( * ( T T * Q ) n [N] ) = TT * ($ (W ) n [N ] ) + T T * / 3 . (30) I n t h e f o l l o w i n g t h r e e s t e p s t h e te rms of t h i s e q u a t i o n w i l l be examined and w i l l be shown t o be r e l a t e d t o Baum-Bot t and 93 Nash R e s i d u e s . 1 ) N o t i c e t h a t * ( 7 r * Q ) = 7 r * * ( Q ) . (31) S i n c e each t e r m i n <t>(jr*Q) i s a p r o d u c t o f c h e r n c l a s s e s o f 7r*Q and as i s w e l l known c ( 7 r * Q ) = 7 r * c ( Q ) (32) Then LHS o f (30) can be s i m p l i f i e d ; 7 r * ( $ ( 7 r * Q ) n [N] ) = * r * U * * ( Q ) n [N] ) =*(Q) n ^ * f N ] =*(Q) n (degjr) [M] =4»(Q) i-l [ M 3 (33) where t h e f i r s t e q u a t i o n f o l l o w s f rom ( 3 1 ) , t h e s e c o n d e q u a t i o n i s a p r o p e r t y of c ap p r o d u c t s , [ Sp , p 2 5 4 , ( 1 6 > ] . The t h i r d e q u a t i o n h o l d s by d e f i n i t i o n s i n c e [N] and [M] a r e f u n d a m e n t a l c y c l e s . The l a s t e q u a t i o n f o l l o w s s i n c e deg7r=1. U s i n g t h e Baum-Bott c o n s t r u c t i o n more can be s a i d about $ ( Q ) . L e t Z 1 , . . . , Z be t h e c o n n e c t e d components o f S . F o r each choose an open n e i g h b o u r h o o d U^ of Z^ such t h a t U^ d e f o r m a t i o n r e t r a c t s t o 6 i : U i > 2 i ( 3 4 ) and U ^ n u j = 0 ' i » j = 1 r • • • rm. In e a c h U^ choose a compact s e t 1^ such t h a t Z . i s c o n t a i n e d i n t h e i n t e r i o r o f £ • , i = 1 , . . . , m . L e t u • • • u ^ m * (35) There e x i s t s a c l o s e d d i f f e r e n t i a l fo rm w on M w i t h s u p p o r t 94 on Z such t h a t [u]=*(Q) (36) where [•] d e n o t e s t h e cohomology e l ement d e f i n e d by t h a t f o r m , [ BB2, p31 2-31 3 ] . L e t CJ^ be d e f i n e d as f o l l o w s : o ; i | U i = c j | U i and U j i M - U . - O , i = 1 , . . . , m . (37) Then u>=u> 1 + . . . + < J M and [CJ] = [a^ ] +. . . + [ ^ m ] , hence [ w ] n [ M ] = [ w 1 ] n [ M ] + . . . + [ w m ] n [ M ] , (38) But s i n c e i s a d i f f e r e n t i a l form whose s u p p o r t i s compact and i s i n t h e n [w j ] n [ M M u j ] n [ U j ] , i = 1 , . . . m . (39) Then t h e LHS of (30) can be w r i t t e n , u s i n g ( 3 3 ) , ( 3 6 ) and (38) as 7T*(4>(7r*Q) n [ N ] ) - * ( Q ) n tM] ( f rom 33) = [ w ] n [M] ( f rom 36) ] n t M ] + . . . + [ " m ] n [ M 3 • (40) S u b s t i t u t i n g (39) i n t o (40) g i v e s 7r * ( * ( f f * Q ) n [N] ) = [w 1 ] n tu1 ] + . . . + ["„,] n [um3. (41 ) Hence the g l o b a l e x p r e s s i o n on t h e l e f t s p l i t s up as t h e sum of l o c a l e x p r e s s i o n s . 2) To c a l c u a l a t e t h e f i r s t t e r m on t h e RHS o f (30) f i r s t r e c a l l t h a t T / * ( E ) i s a v e c t o r b u n d l e on M-S and s i n c e • ( E ) | M - S i s i n t e g r a b l e T / * ( E ) has a b a s i c c o n n e c t i o n , i . e . i f K=K(D) i s t h e c u r v a t u r e m a t r i x f o r t h i s c o n n e c t i o n on M - S , t h e n t h e d i f f e r e n t i a l form #(K)=0 on M-S f o r deg *>n-k , [ B B 2 , p 2 9 5 ] . 95 S i n c e W | N - X = T T * ( T / * ( E ) ) | N - X , p u l l t h e c o n n e c t i o n D by n* t o a c o n n e c t i o n on W | N - X . Then t h e r e e x i s t s a c o n n e c t i o n D f o r W on N such t h a t D a g r e e s w i t h 7r*D on N - 7 r ~ 1 ( I ) , [ B B 2 , p 3 3 0 , ( 4 . 4 1 ) ] . I f K = K ( D ) i s t h e c u r v a t u r e m a t r i x f o r D, t h e n * ( K ) = 0 on N - 7 T " 1 ( I ) (42) s i n c e D and D agree on N -7r " 1 ( I ) . From (42) i t f o l l o w s t h a t we c a n d e f i n e c l o s e d forms $ j on N by and Then . | TT- 1 ( U . ) - i * ( K ) | TT- 1 ( U . ) , where i= (1 / 2 m / - 1 ) u c y ^ (43a) * . | N - 7 T " 1 ( U ^ O , i = 1 , . . . , m . (43b) i * ( K ) = # . + . . ' (44) 1 m where i i s as i n ( 4 3 a ) . By d e f i n i t i o n i [ * ( K ) ] - * ( W ) (45) U s i n g (44) and (45) t o g e t h e r g i v e s * (W) = i ' [ * ( K ) ] ] + . . . + [ * m ] (46) Cap b o t h s i d e s of (46) by [ N ] , * ( w ) N [ N ] * [ * 1 ] N [ N ] + . . . + [ * M ] N [ N ] . (47) E a c h i s a c l o s e d form w i t h compact s u p p o r t , whose s u p p o r t l i e s i n 7 r ~ 1 ( U ^ ) = V \ , i = 1 , . . . , m . Hence [ * _ ] n [ N ] * [ * _ ] n t v i ] , i = i , . . . , m . ( 4 8 ) Then ( 4 7 ) can be w r i t t e n as 96 * (W) n [ N ] = [ * 1 ] n [ V 1 ] + . . . + [ # m ] n [ V i n ] . (49) N o t i c e t h a t D | V \ i s a c o n n e c t i o n f o r W|v\ and K | V ^ i s a c u r v a t u r e m a t r i x f o r W | V ^ . I f deg$>n-k, t h e n $ ( K | V ^ ) has compact s u p p o r t i n V\ and i [ # ( K | V i ) ] - * ( W | V . ) . (50) U s i n g ( 5 0 ) , (43a) and (43b) g i v e s * ( W | V . ) = [ * . ] . (51) E a c h [*^] i s a cohomology c l a s s w i t h s u p p o r t i n T " ' ( L ) ( hence has compact s u p p o r t and i s t h e c h e r n c l a s s o f a v e c t o r b u n d l e , W | V ^ , i = 1 , . . . , n . Then [<i>^ ] i s i n t h e image of cohomology w i t h r a t i o n a l c o e f f i c i e n t s on . U s i n g (49) and (51) g i v e s * ( w ) n [ N ] » * ( w | v 1 ) n [ v 1 ] + • • - + * ( w | v m ) n [ v m l . (52) A p p l y 7T* t o b o t h s i d e s o f ( 5 2 ) , m » r * ( * (W) n [ N ] ) = i 5 1 i r # ( * ( W | V i ) n l V j ] ) (53) where TT* ( * ( W | V . ) n [ V i ]) eH* ( u . ;C ) f o r i = 1 , . . . , m . 3) The l a s t e l ement on t h e r i g h t hand s i d e o f (30) i s *r*/3. From t h e c o n s t r u c t i o n o f p1 i t f o l l o w s t h a t 0 n a t u r a l l y s p l i t s as /5=51 + ...+J in (54) where 0^eH* (X ;C) f o r i = 1 , . . . , m . S i n c e 0^  i s o b t a i n e d by c a p p i n g c h e r n c l a s s e s o f W w i t h t h e r e s i d u e o b t a i n e d from Grassmann G r a p h and by i n t e r s e c t i o n of t h e s e , t h e n i t f o l l o w s t h a t i s i n t h e image of homology w i t h r a t i o n a l 97 c o e f f i c i e n t s i n X, see e q u a t i o n s (20 ) , (26 ) and (29 ) . Apply 7T* t o both s i d e s of (54) ^P"=**V"- +^ m (55) where J T * / ^ eH* ( Z^ ;C) , i = 1 , . . . , m . T h i s completes the examina t ion of the terms of (30 ) . P u t t i n g e q u a t i o n s (41) , (53) and (55) i n t o (30) g i v e s m m m ~ i g 1 ^ i ] n [ U i ] = i Z 1 7 r * ( $ ( W | V i ) ) n [ V . ] ) + i Z 1 7 r * / 3 i . (56) From (56) we can now w r i t e [ « . ] n [ U i ] = 7 r * ( * ( W | V . ) ) n [ V . D + i r J i l . , (57) s i n c e each summand i n (56) i s i n H * ( t L ; C ) , i = 1 , . . . , m . The d e f o r m a t i o n r e t r a c t i o n 6^ of (34) induces an i somorphism 5 i * : H * ( U i ? C ) ^ H * ( Z i ; C ) (58) fo r i = 1 , . . . , m . App ly 6^* to both s i d e s of (57) * i * ( [ " - i ] n [ O i ] ) - f i i * * * ( * ( w | v . ) + (59) f o r i = 1 , . . . , m . For the f i r s t term on the LHS of (59) we have e . ^ d w ^ n f U j l J ^ R e s ^ t E ^ j ) , i + 1 , . . . , m (60) [BB2 ,p3 l3 ( 7 . 1 4 ) ] . The f i r s t term on the RHS of (59) i s the Nash Res idue by d e f i n i t i o n ^ • ^ ( • ( w I V . J n t V j l - N R e s ^ E f Z ^ , i = i , . . . , m . (61) S i n c e 7 r * p ^ i s a l r e a d y i n H * ( Z ^ ; C ) by (55 ) , 6^* does not change i t ; 5. * 7 r * / J , =ir*/3. , i = 1 , . . . , m . (62) l l i 98 S u b s t i t u t i n g ( 6 0 ) , (61) and (62) i n t o (59) g i v e s R e s $ ( E , Z i ) = N R e s ^ ( E , Z i ) + 7 r * / 3 i (63) f o r i = 1 , . . . , m as r e q u i r e d . By a p p e a l i n g t o t h e d i s c u s s i o n s t h a t f o l l o w e q u a t i o n s (51) and (54) we c o n c l u d e t h a t t h e RHS of (63) i s r a t i o n a l , and hence t h e LHS i s r a t i o n a l , Res^iE,!^) eH+iZ^jQ), i = 1 , . . . , m . QED REMARKS: 1) L e t M be a compact complex m a n i f o l d w i t h a p o s i t i v e l i n e b u n d l e . Then M i s a l g e b r a i c by K o d a i r a embedding t h e o r e m , [ G H , p l 8 l ] , Hence N b e i n g a s u b v a r i e t y of M x G ( k , T ) i s a l s o a l g e b r a i c . On a l g e b r a i c m a n i f o l d s c o h e r e n t sheaves have g l o b a l s y z y g i e s , [ G H , p 7 0 l ] . Then tr^iF) w i l l have g l o b a l s y z y g i e s on N a s suming t h a t t h e Nash b l o w - u p N i s s m o o t h . Thus theorem 1 w i l l h o l d f o r r i c h f o l i a t i o n s on a l g e b r a i c m a n i f o l d s f o r w h i c h t h e Nash B l o w - u p i s smooth w i t h o u t t h e f u r t h e r a s s u m p t i o n t h a t F be g e n e r a t e d by a b u n d l e m o r p h i s m . 2) I f F i s g e n e r a t e d by a b u n d l e morphism 4»:E s-T but r a n k E > r a n k F , t h e n * ' i s no t i n j e c t i v e . To make theorem 1 work i n t h i s ca se some r e s t r i c t i o n s must be imposed on t h e k e r n e l of * . F o r example f o r e v e r y c o n n e c t e d component Z of S assume t h a t t h e r e e x i s t an open n e i g h b o u r h o o d U of Z and a 9 9 v e c t o r b u n d l e H on U w i t h a b u n d l e morphism T?:H|U-Z s~E|U-Z s u c h t h a t TJ i s i n j e c t i v e . Then theorem 1 h o l d s f o r F. Note t h a t TJ need not be d e f i n e d on a l l of U bu t H s h o u l d be d e f i n e d on U t o e n s u r e t h e c o n s t r u c t i o n of Grassmann G r a p h . 3) N o t i c e t h a t theorem 1 h o l d s f o r a s u b c l a s s of r i c h f o l i a t i o n s . C a l l a r i c h f o l i a t i o n very rich i f f o r e v e r y c o n n e c t e d component Z o f S t h e r e e x i s t s an open n e i g h b o u r h o o d U of Z s u c h t h a t t o F|U t h e r e e x i s t s a complex of 0 -module s on U w h i c h g i v e a l o c a l l y f r e e r e s o l u t i o n on U-Z. Then theorem 1 h o l d s f o r v e r y r i c h f o l i a t i o n s f o r w h i c h t h e Nash B l o w - u p i s smooth . I t i s n a t u r a l t o c o n j e c t u r e t h a t a l l r i c h f o l i a t i o n s a r e v e r y r i c h . 100 CHAPTER 4 OBSTRUCTION CLASSES 0:INTRODUCTION T h i s f i n a l c h a p t e r p u r s u e s a p r o b l e m t h a t a r i s e s when s i n g u l a r h o l o m o r p h i c f o l i a t i o n s a r e c o n s i d e r e d as i n t e g r a b l e images o f b u n d l e m o r p h i s m s . S e c t i o n 1 d e f i n e s o b s t r u c t i o n c l a s s e s i n terms o f P o n t r y a g i n c l a s s e s w h i c h o b s t r u c t t h e i m b e d d i n g of a v e c t o r b u n d l e i n t o t h e t a n g e n t b u n d l e . O t h e r t o p o l o g i c a l o b s t r u c t i o n s i n t e rms o f S t i e f e l - W h i t n e y c l a s s e s can be found i n t h e l i t e r a t u r e , i n p a r t i c u l a r see Sundararaman [ S r ] . S e c t i o n 2 v e r y b r i e f l y summarizes immedia te f u t u r e r e s e a r c h p r o j e c t s t o w h i c h t h i s work l e a d s . We p r o p o s e t o s t u d y t h e p r o b l e m of R iemann-Roch as e x p l a i n e d i n s e c t i o n 2 as a consequence of t h i s w o r k . 101 1-.OBSTRUCTION CLASSES Most w e l l known examples o f s i n g u l a r h o l o m o r p h i c f o l i a t i o n s a r e meromorphic v e c t o r f i e l d s . A meromorphic v e c t o r f i e l d i s d e f i n e d as f o l l o w s : L e t M be a complex m a n i f o l d o f d i m e n s i o n n w i t h t a n g e n t b u n d l e T and l e t L be a l i n e b u n d l e on M . Assume t h a t t h e r e i s a b u n d l e morphism • * : L ->T. * i s c a l l e d a meromorphic v e c t o r f i e l d . ¥ ( L ) g e n e r a t e s a 1 d i m e n s i o n a l subshea f of t h e t a n g e n t sheaf T. By d i m e n s i o n c o n s i d e r a t i o n s t h i s c o h e r e n t subshea f i s i n t e g r a b l e , t h e r e f o r e i t d e f i n e s a s i n g u l a r f o l i a t i o n , t h e s i n g u l a r i t y s e t b e i n g S, where S={xeM| * =0 } To g e n e r a l i z e t h i s c o n c e p t l e t E be a v e c t o r b u n d l e of rank k on M and assume t h a t t h e r e i s a morphism B: E >T. I f /3(E) i s i n t e g r a b l e t h e n i t d e f i n e s a s i n g u l a r f o l i a t i o n whose s i n g u l a r i t y s e t i s S={xeM| rank (0 )<max.rank(0) } I f S=0 t h e n 0(E) d e f i n e s a f o l i a t i o n i f i t i s i n t e g r a b l e . I f m a x . r a n k ( p ' ) = r a n k ( E ) and S=0 t h e n B i s an i m b e d d i n g of E i n t o T . F o r an a r b i t r a r y E c l e a r l y no such B e x i s t s . T h i s s e c t i o n answers a n a t u r a l q u e s t i o n : " A r e t h e r e d i f f e r e n t i a l g e o m e t r i c o b s t r u c t i o n c l a s s e s f o r the e x i s t e n c e o f an 102 i m b e d d i n g B:E s-T?" F o r an answer see theorem 2 . F i r s t i n t h e f o l l o w i n g theorem we c o l l e c t a few f a c t s about s p l i t t i n g m a n i f o l d s , [ H , 1 . 4 . 2 , 1 1 1 . 1 3 . 2 . 1 ] . Theorem ( H i r z e b r u c h ) : L e t E be a v e c t o r b u n d l e o f rank k on a complex m a n i f o l d M w i t h t a n g e n t b u n d l e T . There e x i s t s a complex m a n i f o l d M g and a h o l o m o r p h i c map * : M s-M s w i t h t h e f o l l o w i n g p r o p e r t i e s : i ) M i s a f i b r e b u n d l e o v e r M w i t h t h e f l a g m a n i f o l d 5 F ( k ) = G L ( k , C ) / A ( k , C ) as f i b r e , where A ( k , C ) i s t h e subgroup of G L ( k r C ) c o n s i s t i n g of t r i a n g u l a r m a t r i c e s . i i ) $ *E s p l i t s as a sum of l i n e b u n d l e s on M g . i i i ) $ *T i s a q u o t i e n t b u n d l e of T , t h e t a n g e n t b u n d l e o f T = ( $ * T ) 6 E A s where E A i s t h e b u n d l e a l o n g t h e f i b r e s . Here t h e d i r e c t sum need not be h o l o m o r p h i c , i t i s i n g e n e r a l a C°° d i r e c t sum. i v ) The map i n d u c e d by $ $>* :H*(M;C) s - H * ( M s ; C ) i s a monomorphism. Remark: To c o n s t r u c t M g , c o n s i d e r t h e complex a n a l y t i c p r i n c i p a l b u n d l e 103 a s s o c i a t e d t o E . Then M s = F / A ( k f C ) . The p r o o f of theorem 2 w i l l need a g e n e r a l i z e d v e r s i o n of B o t t ' s v a n i s h i n g t h e o r e m , w h i c h i s g i v e n n e x t ; assume t h a t M i s a complex m a n i f o l d of d i m e n s i o n n+m w i t h t a n g e n t b u n d l e T . Theorem l : L e t T=A®B and E be a s u b b u n d l e of A . I f E and B a r e i n t e g r a b l e , t h e n t h e g r a d e d C h e r n r i n g C h e r n * ( A / E ) v a n i s h e s beyond t h e c o r a n k of E i n A , i . e . C h e r n 1 ( A / E ) = 0 i f i > r a n k A - r a n k E . P r o o f : L e t r ankA=n, rankB=m and rankE=k . I t s u f f i c e s t o show t h a t i f P i s a s y m m e t r i c , homogeneous a d - i n v a r i a n t p o l y n o m i a l on G L ( k , C ) o f degree g r e a t e r t h a n n - k , t h e n P a p p l i e d t o a c u r v a t u r e m a t r i x of ( A / E ) * i s z e r o . L e t {U} be an open c o v e r i n g of M s u c h t h a t a l l t h e above b u n d l e s a r e t r i v i a l on each U , and t h e r e i s a p a r t i t i o n o f u n i t y {Xy} c o r r e s p o n d i n g t o t h i s c o v e r i n g . On U l e t U U U U x 1 , . . . , x n , y 1 , . . . , y m be l o c a l c o o r d i n a t e s s u c h t h a t A * i s g e n e r a t e d by d x ^ , . . . , d x ^ B * i s g e n e r a t e d by d y 1 ^ , . . . , d y j j j . L e t V be a n o t h e r e l e m e n t o f {U} . Then s i m i l a r l y t h e r e a r e 104 c o o r d i n a t e s V V V V r on V s u c h t h a t x i ' * * * , x n ' y 1 ' * * * ' Y m A * i s g e n e r a t e d by d x y , . . . , d x ^ B * i s g e n e r a t e d by d y ^ , . . . , d y ^ . I f U n V # 0 t h e n t h e r e i s a t r a n s i t i o n f u n c t i o n h ^ f o r A * s u c h t h a t ( d x 1 , . . . , d x n ) = h u v ( d x 1 , . . . , d x n , d y 1 , . . . , d y m ) . B i s i n t e g r a b l e so by F r o b e n i u s vA , , V , V , V , V» , A / , V , V> h u v ( d x 1 , . . . , d x n , d y 1 , . . . , d y [ n ) = h u v ( d x 1 , . . . , d x n ) . S i n c e E i s i n t e g r a b l e we may assume t h a t t h e c o v e r i n g i s f i n e enough so t h a t E * i s g e n e r a t e d by and by , U , U d x n _ k + 1 , . . . , d x n on U d x n - k + 1 d x n o n V * Then ( A / E ) * i s g e n e r a t e d by and by d x , , . . . ,dx_. , on U 1 n-k d x Y , . . . , d x ^ _ , on V . J. il K I f UnV*0 t h e n t h e r e i s a t r a n s i t i o n f u n c t i o n g ^ such t h a t ( d x ^ , . . . , d x J J _ k ) = g u v ( d x y , . . . , d x ^ ) . E i s i n t e g r a b l e so by F r o b e n i u s 9 r j V ( d V ' * ' ' d x n ) = g U V ( d x T ' ' ' * ' d x n - k } L e t d x U = ( d x y , . . . , d x ^ _ k ) and d x V = ( d x ^ , . . . , d x ^ _ k ) . 105 From h e r e on the p r o o f o f B o t t ' s v a n i s h i n g theorem a p p l i e s , see [ B 1 ] . F o r c o m p l e t e n e s s we i n c l u d e t h e main p a r t s o f t h e p r o o f . L e t Dy be a c o n n e c t i o n f o r ( A / E ) * on U d e f i n e d as D y d x ^ O , i = 1 , . . . , n - k . Then D = V u D u i s a c o n n e c t i o n f o r ( A / E ) * . The a s s o c i a t e d c o n n e c t i o n m a t r i x 6y on U i s c a l c u l a t e d as f o l l o w s ; u = D d x U = Z v X v D v d x U = z v x v D v ( 9 u v d x V ) = Z v X v ( d g u v d x V + g u v D v d x V ) = 2 v x v d g u v d x V = Z v X v d g u v g v u d x U . We t h e n have t o i n v e s t i g a t e t h e n a t u r e o f d g ^ t o f i n d where t h e c u r v a t u r e m a t r i x dd^-d^AO^ l i e s . D i f f e r e n t i a t i n g b o t h s i d e s of g i v e s d x U = 9 r j V d x V 0 = d g u v d x V w h i c h i m p l i e s t h a t d g T T V l i e s i n t h e i d e a l g e n e r a t e d by 'UV d x ^ , . • • , d x ^ _ ^ . 106 C o n s e q u e n t l y t h e c u r v a t u r e m a t r i x l i e s i n t h e same i d e a l . Any i - f o l d p r o d u c t of t h i s i d e a l w i t h i>n-k i s c l e a r l y z e r o . Hence t h e t h e o r e m . QED N o t i c e t h a t when B=0 t h i s theorem r e d u c e s t o B o t t ' s v a n i s h i n g t h e o r e m . A l s o n o t e t h a t t h e above p r o o f shows t h a t i f P o n t * ( A / E ) i s t h e g r a d e d P o n t r y a g i n r i n g of A / E g e n e r a t e d by t h e r e a l P o n t r y a g i n c l a s s e s of A / E t h e n P o n t 1 ( A / E ) = 0 i f i > 2 ( r a n k R A - r a n k R E ) B e f o r e d e f i n i n g o b s t r u c t i o n c l a s s e s l e t us d e v e l o p some n o t a t i o n . L e t a=(a j / . . . , a ^ ) be an n - t u p l e of n o n n e g a t i v e i n t e g e r s and d e f i n e Ia I =a ,+2a~+•• •+na„ . 1 1 1 2 n F o r any v e c t o r b u n d l e E , d e f i n e c a ( E ) = ( c 1 ( E ) ) ( a l ) . . . ( c n ( E ) ) ( a n ) and p 3 ( E ) = ( p 1 ( E ) ) ( 2 a i ) . . . ( p n ( E ) ) ( 2 a n ) where c ^ ( « ) i s t h e i - t h C h e r n c l a s s i n H X ( M ; C ) and p ^ ( • ) i s t h e i - t h P o n t r y a g i n c l a s s i n H 2 l ( M ; C ) . A l s o r e c a l l t h a t f o r any two v e c t o r b u n d l e s E and F , t h e C h e r n c l a s s of t h e v i r t u a l b u n d l e E-F i s d e f i n e d as c ( E - F ) = ( c ( E ) / c ( F ) ) . 107 Theorem 2: L e t E be a v e c t o r b u n d l e o f rank k on a complex m a n i f o l d M of d i m e n s i o n n w i t h t a n g e n t b u n d l e T . I f E can be imbedded i n t o T t h e n t h e f o l l o w i n g o b s t r u c t i o n c l a s s e s i n t h e cohomology of t h e s p l i t t i n g m a n i f o l d M g a r e z e r o : p a ( * * T - L i ) = 0 , i = 1 , . . . , k . , | a | = 2 n , where # : M g i s t h e n a t u r a l p r o j e c t i o n , and L^ a r e l i n e b u n d l e s s u c h t h a t * * E = L 1 © ' • '©Lj. on M g . P r o o f : R e c a l l t h a t T = $ * T © E A where E A i s t h e b u n d l e s a l o n g t h e f i b r e s on M g and hence i s i n t e g r a b l e . I f E can be imbedded i n t o T t h e n * * E = L 1 © - • -©Lj. can be imbedded i n t o <I>*T and hence each L^ can be imbedded i n t o $ * T . By d i m e n s i o n c o n s i d e r a t i o n s each L j i s i n t e g r a b l e i n T . By t h e o r e m 1 t h e P o n t r y a g i n r i n g P o n t * ( $ * T / L ^ ) v a n i s h above t w i c e t h e c o r a n k of L^ i n $ * T , P o n t I ( * * T / L i ) = 0 i f J > 2 ( n - 1 ) . T h i s t h e n c o m p l e t e s t h e p r o o f . QED I n p a r t i c u l a r o b s e r v e t h a t i f t h e r e a r e c l a s s e s 7 ^ e H * ( M ; C ) s u c h t h a t * * ( 7 j ) = p 1 ( L ^ ) , t h e n t h e g r a d e d r i n g P * g e n e r a t e d 108 i n H * ( M ; C ) by { 7 ^ . . . , 7 ^ } and t h e P o n t r y a g i n c l a s s e s of T v a n i s h i n t h e t o p d i m e n s i o n . T h i s i s because $ * i s a monomorphism a t t h e cohomology l e v e l . E x a m p l e : L e t M be a complex m a n i f o l d of d i m e n s i o n n . L e t V=MxC, and 7r :V =»-M be t h e p r o j e c t i o n on t h e f i r s t component . L e t L be a l i n e b u n d l e on M . D e f i n e two l i n e b u n d l e s on V as L , = 7r * L L 2 = t h e l i n e b u n d l e a l o n g t h e f i b r e s . Then L 2 i s the t r i v i a l l i n e b u n d l e . L e t £ be a c o h e r e n t sheaf on M d e f i n e d by t h e p r e s h e a f r ( £ , u ) = r ( L 1 e L 2 , 7 r - 1 ( u ) ) f o r U open i n M . L e t T be t h e t a n g e n t sheaf of M . I f E c an be imbedded i n t o T t h e n c n ( D - c n _ 1 ( r ) C l (£•) + ..-+ (-1 ) n C l ( E ) n = 0 . To see t h i s n o t e t h a t n*E s p l i t s on V and c 1 ( i r * £ ) = c 1 ( L 1 « L 2 ) = c 1 ( L , )+c 1 ( L 2 )=c 1 ( L , ) f and c n ( i r * T - L 2 ) = 0 by t h e meromorphic v e c t o r f i e l d theorem of Baum and B o t t , 109 see [ B B 2 ] , o r see theorem 1 a b o v e . 2:FUTURE RESEARCH PROJECTS The r e s u l t s o f t h i s work n a t u r a l l y l e a d t o new p o s s i b i l i t i e s w h i c h a r e b r i e f l y m e n t i o n e d h e r e . i) MacPher son has d e f i n e d C h e r n c l a s s e s f o r s i n g u l a r v a r i e t i e s u s i n g C h e r n - M a t h e r t y p e of c h a r a c t e r i s t i c c l a s s e s w i t h c o r r e c t i o n f a c t o r s , see [ M c ] . One i n t e r e s t i n g p r o b l e m i s t o d e f i n e a l o c a l E u l e r o b s t r u c t i o n f o r a c o h e r e n t subshea f F o f a l o c a l l y f r e e shea f G u s i n g t h e a s s o c i a t e d Nash B l o w - u p as MacPher son d e f i n e s a l o c a l E u l e r o b s t r u c t i o n u s i n g t h e Nash c o n s t r u c t i o n c o r r e s p o n d i n g t o t h e t a n g e n t s h e a f of a s i n g u l a r v a r i e t y . T h i s w i l l d e f i n e h o m o l o g i c a l C h e r n c l a s s e s f o r F and i t w i l l be i n t e r e s t i n g t o check i f t h e s e c l a s s e s c o r r e s p o n d t o t h e u s u a l C h e r n c l a s s e s of F o b t a i n e d t h r o u g h a r e s o l u t i o n by l o c a l l y f r e e r e a l a n a l y t i c s h e a v e s . ii) I t was c o n j e c t u r e d f o r some t i m e t h a t t h e M e r o m o r p h i c V e c t o r F i e l d theorem of Baum-Bott w o u l d i m p l y t h e R iemann-Roch theorem as t h e H o l o m o r p h i c V e c t o r F i e l d theorem of B o t t d i d , see [BB1] and [ B 2 ] . I t w i l l be i n t e r e s t i n g t o see how f a r t h e Grassmann Graph c o n s t r u c t i o n can be used 110 t o w a r d s a s e t t l e m e n t of t h i s c o n j e c t u r e . I n t h e a l g e b r a i c c a s e B a u m - F u l t o n - M a c P h e r s o n used t h i s c o n s t r u c t i o n t o p r o v e a R iemann-Roch theorem f o r s i n g u l a r v a r i e t i e s , see [ B F M ] . U s i n g t h e g r a p h o f a meromorphic v e c t o r f i e l d i n t h e compact K a e h l e r ca se p r o m i s e s t o be t h e r i g h t way t o a t t a c k t h e above c o n j e c t u r e . iii) I t w i l l be an i n t e r e s t i n g p r o b l e m t o c o n c e n t r a t e on c a l c u l a t i n g Baum-Bott r e s i d u e s u s i n g t h e d e g e n e r a c y c y c l e s of t h e u n i v e r s a l q u o t i e n t b u n d l e W on t h e Nash B l o w - u p N . I t i s n a t u r a l t o c o n j e c t u r e t h a t t h e i n t e r s e c t i o n c y c l e c o r r e s p o n d i n g t o * ( c 1 ( W ) , . . . , c ^(W)) w i l l be homologous t o t h e sum of some r a t i o n a l c y c l e s t h a t l i e i n i r " 1 ( S ) . T h i s w i l l t h e n s o l v e t h e R a t i o n a l i t y c o n j e c t u r e i n t h e compact K a e h l e r c a s e . i v ) U s i n g t h e knowledge t h a t the Nash B l o w - u p c o r r e s p o n d i n g t o c o h e r e n t subsheaves of a l o c a l l y f r e e sheaf i s a n a l y t i c one can a p p r o a c h t h e work of A z n a r who i n t h e a l g e b r a i c c a t e g o r y g e n e r a l i z e s M a c P h e r s o n ' s l o c a l E u l e r o b s t r u c t i o n , see [ A z ] , A f u t u r e p r o j e c t i s t o s t u d y A z n a r ' s g e n e r a l i z a t i o n i n t h e a n a l y t i c c a se i n te rms of Segre c l a s s e s as m e n t i o n e d by F u l t o n i n h i s 1983 R e g i o n a l C o n f e r e n c e . v) The o b s t r u c t i o n c l a s s e s d e f i n e d i n t h i s c h a p t e r a r e open f o r f u r t h e r i n v e s t i g a t i o n . One p a r t i c u l a r d i r e c t i o n t o c o n t i n u e i s t o r e c o v e r t h e o b s t r u c t i o n c l a s s e s i n t e rms of t h e C h e r n c l a s s e s of E and T . F o r t h i s i t w i l l be n e c e s s a r y t o c l a s s i f y t h o s e c a s e s where on M t h e t a n g e n t b u n d l e T a c c e p t s <t>*T as a h o l o m o r p h i c f a c t o r i n t h e d i r e c t sum T = * * T 8 E A . 112 BIBLIOGRAPHY [AH] M . F . A t i y a h & F . H i r z e b r u c h , A n a l y t i c C y c l e s on Complex M a n i f o l d s , T o p o l o g y 1 (1961) 2 5 - 4 5 . [ A z ] V . N a v a r r o A z n a r , Sur l e s M u l t i p l i c i t e s de S c h u b e r t L o c a l e s des F a i s c e a u x A l g e b r i q u e s C o h e r e n t s , C o m p o s i t i o M a t h e m a t i c a 48 (1983) 311-326 . [B1] R. B o t t , On a T o p o l o g i c a l O b s t r u c t i o n t o I n t e g r a b i l i t y , P r o c e e d i n g s of Sympos ia i n P u r e M a t h e m a t i c s , A m e r i c a n M a t h e m a t i c a l S o c i e t y , 16 (1970) 127-131 . [B2] R. B o t t , V e c t o r F i e l d s and C h a r a c t e r i s t i c Numbers , M i c h i g a n M a t h e m a t i c a l J o u r n a l , 14 (1967) 231-244 . [BB1] P . F . Baum & R. B o t t , On t h e Z e r o e s of M e r o m o r p h i c V e c t o r F i e l d s , E s s a y s on T o p o l o g y and R e l a t e d T o p i c s , Memoire s d e d i e s a Georges de Rham, S p r i n g e r - V e r l a g (1970) 2 9 - 4 7 . [BB2] P . F . Baum & R. B o t t , S i n g u l a r i t i e s of H o l o m o r p h i c F o l i a t i o n s , J o u r n a l o f D i f f e r e n t i a l G e o m e t r y , 7 (1972) 2 7 9 - 3 4 2 . [BBc] A . B i a l y n i c k i - B i r u l a , Some Theorems on A c t i o n s of A l g e b r a i c G r o u p s , A n n a l s of M a t h e m a t i c s , 98 (1973) 4 8 0 - 4 9 7 . [BFM] P . F . Baum, W. F u l t o n & R. M a c P h e r s o n , R iemann-Roch f o r S i n g u l a r V a r i e t i e s , P u b l i c a t i o n s M a t h e m a t i q u e s I H E S , 45 (1975) 101-145 . 113 [BS] A . B o r e l & J . B S e r r e , Le theoreme de R i e m a n n - R o c h , B u l l e t i n de l a S o c i e t e M a t h e m a t i q u e de F r a n c e , 86 (1958) 9 7 - 1 3 6 . [C] H . C a r t a n , F a i s c e a u x A n a l y t i c C o h e r e n t s , C e n t r o I n t e r n a z i o n a l e M a t h e m a t i c o E s t i v o , V a r e n n a (1963) 1-88. [CS1] J . B . C a r r e l l & A . J . Sommese, C * - A c t i o n s , M a t h e m a t i c a S c a n d i n a v i c a , 43 (1978) 4 9 - 5 9 . [CS2] J . B . C a r r e l l & A . Sommese, Some T o p o l o g i c a l A s p e c t s o f C * - A c t i o n s on Compact K a e h l e r M a n i f o l d s , C o m m e n t a r i i M a t h e m a t i c i H e l v e t i c i , 54 (1979) 5 6 7 - 5 8 2 . [F ] G . F i s c h e r , Complex A n a l y t i c G e o m e t r y , L e c t u r e N o t e s i n M a t h e m a t i c s , Volume 538, S p r i n g e r - V e r l a g , (1976) [G] P . A . G r i f f i t h s , Two Theorems on E x t e n s i o n s o f H o l o m o r p h i c M a p p i n g s , I n v e n t i o n e s M a t h e m a t i c a e , 14 (1971) 2 7 - 6 2 . * [GH] P . A . G r i f f i t h s & J . H a r r i s , P r i n c i p l e s o f A l g e b r a i c G e o m e t r y , John W i l e y and S o n s , ( 1 9 7 8 ) . [ G r ] J . G i r a u d , M a t h e m a t i c a l R e v i e w s , 53#13217 (1977) 1846. [H] F . H i r z e b r u c h , T o p o l o g i c a l Methods i n A l g e b r a i c G e o m e t r y , 3 r d E d i t i o n , S p r i n g e r - V e r l a g ( 1 9 6 6 ) . [Ha] R. H a r t s h o r n e , A l g e b r a i c G e o m e t r y , G r a d u a t e T e x t s i n M a t h e m a t i c s 5 2 , S p r i n g e r - V e r l a g ( 1 9 7 7 ) . 1 14 [HR] H . H i r o n a k a & H . R o s s i , On t h e E q u i v a l e n c e of Imbeddings of E x c e p t i o n a l Complex S p a c e s , M a t h e m a t i s c h e A n n a l e n , 156 (1964) 3 1 3 - 3 3 3 . [ L ] H . B . Lawson J r . , F o l i a t i o n s , B u l l e t i n o f t h e A m e r i c a n M a t h e m a t i c a l S o c i e t y , 80 (1974) 3 6 9 - 4 1 8 . [Mc] R. D . M a c P h e r s o n , C h e r n C l a s s e s f o r S i n g u l a r A l g e b r a i c V a r i e t i e s , A n n a l s of M a t h e m a t i c s , 100 (1974) 4 2 3 - 4 3 2 . [N] A . N o b i l e , Some P r o p e r t i e s of t h e Nash B l o w i n g - u p , P a c i f i c J o u r n a l of M a t h e m a t i c s , 60 (1975) 2 9 7 - 3 0 5 . [R] R. Remmert, P r o j e k t i o n e n A n a l y t i s c h e Mengen, M a t h e m a t i s c h e A n n a l e n , 130 (1956) 4 1 0 - 4 4 1 . [Rc ] R. W. R i c h a r d s o n , J r . , P r i n c i p a l O r b i t Types f o r R e d u c t i v e Groups A c t i n g on S t e i n M a n i f o l d s , M a t h e m a t i s c h e A n n a l e n , 208 (1974) 3 2 3 - 3 3 1 . [ R i ] 0 . R i e m e n s c h n e i d e r , C h a r a c t e r i s i n g M o i s e z o n Spaces by A l m o s t P o s i t i v e C o h e r e n t A n a l y t i c S h e a v e s , M a t h e m a t i s c h e Z e i t s c h r i f t , 123 (1971) 2 6 3 - 2 8 4 . [Ro] H . R o s s i , P i c a r d V a r i e t y of an I s o l a t e d S i n g u l a r P o i n t , R i c e U n i v e r s i t y S t u d i e s , 54 (1968) 6 3 - 7 3 . [S ] J . P . S e r r e , P r o l o n g e m e n t de F a i s c e a u A n a l y t i q u e C o h e r e n t , A n n a l e s de L ' I n s t i t u F o u r i e r , G r e n o b l e , 16 (1966) 3 6 3 - 3 7 4 . 115 [So] A . J . Sommese, E x t e n s i o n Theorems f o r R e d u c t i v e Group A c t i o n s on Compact K a e h l e r M a n i f o l d s , M a t h e m a t i s c h e A n n a l e n , 218 (1975) 107-116 . [ S r ] D . Sundararaman , M o d u l i , D e f o r m a t i o n s and C l a s s i f i c a t i o n s o f Compact Complex M a n i f o l d s , P i t m a n P u b l i s h i n g L t d . , L o n d o n , (1980) [ST] Y . S i u & G . T r a u t m a n n , Gap Sheaves and E x t e n s i o n o f C o h e r e n t A n a l y t i c S u b s h e a v e s , L e c t u r e N o t e s i n M a t h e m a t i c s , Volume 172, S p r i n g e r - V e r l a g , ( 1 9 7 1 ) . [Su] T . Suwa, R e s i d u e s Complex A n a l y t i c F o l i a t i o n S i n g u l a r i t i e s and The Riemann-Roch Theorem f o r Embedd ings , p r e p r i n t . [T] W. Thimm, E x t e n s i o n of C o h e r e n t A n a l y t i c S u b s h e a v e s , S e v e r a l Complex V a r i a b l e s 1, M a r y l a n d , L e c t u r e N o t e s i n ' M a t h e m a t i c s volume 155 (1970) 191-202 . [Th] R. Thorn, On S i n g u l a r i t i e s o f F o l i a t i o n s , M a n i f o l d s , T o k y o , (1973) 171-173 . 

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