GEOMETRY OF HOLOMORPHIC VECTOR FIELDS AND APPLICATIONS OF G -ACTIONS TO LINEAR ALGEBRAIC GROUPS m by ERSAN AKYILDIZ B.S., Middle East Technical University, 1973 ' A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY THE FACULTY OF GRADUATE STUDIES i n the Department of Mathematics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1977 J Ersan A k y i l d i z , 1977. In presenting t h i s thesis in 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 Library shall make i t freely a v a i l a b l e f o r reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or p u b l i c a t i o n of t h i s thesis 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. Department of MATHEMATICS The University of B r i t i s h Columbia 2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1W5 Date August 5.J977 ABSTRACT A generalization of a theorem of N.R. 0'Brian, zeroes of holomorphic vector f i e l d s and the Grothendieck residue, B u l l . London Math. S o c , 7 (1975) i s given. The theorem of Riemann-Roch and Hirzebruch for V-equivariant holomorphic vector bundles i s obtained, v i a holomorphic vector f i e l d s , i n the case a l l zeroes of the holomorphic vector f i e l d The Bruhat decomposition of action on G/B V are i s o l a t e d . i s obtained from the G - G/B . I t i s shown that a theorem of A. B i a l y n i c k i - B i r u l a , Some theorems on actions of algebraic groups, Ann. of Math. 98, 480-497 (1973) i s the generalization of the Bruhat decomposition on G/B , which was a conjecture of B. Iversen. The existence of a G -action on a point i s proved, where G i s a connected l i n e a r algebraic group defined over an a l g e b r a i c a l l y closed f i e l d and P G/P with only one fixed i s a parabolic subgroup of k of c h a r a c t e r i s t i c zero G . The following i s obtained P = N (P ) = {geG: Adg(P ) = P } G u u u where G i s a connected l i n e a r algebraic group, subgroup of G and P^ elements of P at the i d e n t i t y . P i s a parabolic i s the tangent space of the set of unipotent An elementary proof of P = N (P) o = {geG: gPg ~*~=P} i s given, iii where G i s a connected linear algebraic group and subgroup of G . P i s a parabolic iv TABLE OF CONTENTS INTRODUCTION CHAPTER I. 1 HOLOMORPHIC VECTOR FIELDS, EQUIVARIANT BUNDLES AND THE THEOREM OF RIEMANN—ROCH—HIRZEBRUCH 1. Hermitian D i f f e r e n t i a l Geometry 2. The Canonical Connection and Curvature of a Hermitian Holomorphic Vector Bundle 13 Holomorphic Vector Fields and Equivariant Vector Bundles 17 Grothendieck Residue and The Theorem of Riemann-RochHirzebruch 39 Chern Classes of Equivariant Bundles and Applications 62 3. 4. 5. 8 CHAPTER I I . APPLICATIONS OF G -ACTIONS TO LINEAR ALGEBRAIC GROUPS 1. Theorem of B i a l y n i c k i - B i r u l a 81 2. Bruhat Decomposition 92 3. A Vector F i e l d with One Zero on BIBLIOGRAPHY . . G/P . ... . . Ill 117 V \ To my grandparents Humak, Nuri Caliskan vi ACKNOWLEDGEMENTS I wish to thank my supervisors Professors James B. C a r r e l l and Larry Roberts for their help, advice and patience i n the production of this work. I also thank Professors Birger Iversen, David Lieberman and Ottmar Loos for several h e l p f u l discussions I had with them. I am s p e c i a l l y grateful to Gudrun Aubertin for l e t t i n g me play with her during my l o n e l i e s t days i n Vancouver. At this time I would also l i k e to mention my appreciation to my brother Yilmaz A k y i l d i z , my g i r l f r i e n d Elizabeth Charon,' and to my friends Marie and Bruce Aubertin and Maria Margaretta Klawe. F i n a l l y , l e t me thank Cat Stevens, who so many times has found the words to express my thoughts and feelings when I could not. vii ON THE ROAD TO FIND OUT Well, I l e f t my happy home to see what I could f i n d out. I l e f t my folk and friends with the aim to clear my mind out. Well I h i t the rowdy road, and many kinds I met there. So on and on I go, the seconds t i c k the time out, there's so much l e f t to know, and I am on the road to f i n d out. Well i n the end I ' l l know, but on the way I wander through descending snow, and through the f r o s t and thunder, I l i s t e n to the wind come howl., t e l l i n g me I have to hurry. I l i s t e n to the Robin's song saying not to worry. So on and on I go, the seconds t i c k the time out, there's so much l e f t to know, and I'm on the road to f i n d out. Then I found myself alone, hoping someone would miss me. Thinking about my home, and l a s t woman to kiss me, kiss me. But sometimes you have to moan when nothing seems to s u i t yer, but nevertheless you know you're locked towards to future. So on and on you go, the seconds t i c k the time out. There's so much l e f t to know, and I'm on the road to findout. Then I found my head one day when I wasn't even trying, and here I have to say, because there i s no use i n l y i n g , lying. Yes the answer l i e s within, so why not to take a look now, kick out the devil's s i n , and pick up, pick up a good book now. Yes the answer l i e s within, so why not to take a look now, kick out the devil's s i n , and pick up, pick up a good book now. Yes the answer l i e s within, so why not to take a look now, Kick out the devil's s i n , and pick up, pick up a good book now. CAT STEVENS INTRODUCTION We say (£ acts on a complex manifold az' <C x X ->• X holomorphic map homomorphism, where of X , and a ( ) x t = Aut(X) C -* Aut(X) , t -> a For an algebraic v a r i e t y an a l g e b r a i c a l l y closed f i e l d k , we say there exists a morphism x X G m Aut(X) , t -* <f> t cj>: G a holomorphic vector bundle C x ——y 0: E -* X = GL^(k) defined over acts on X ,i f of algebraic v a r i e t i e s such that (C x X X on a complex manifold X , i s said to be a-equivariant i f S : <C x E -> E there e x i s t s a C-action E X X i s a group homomorphism. For a given C-action (i) i s a group i s the group of holomorphic diffeomofphisms . o(t,x) such that X , i f there exists a on E such that the diagram E i s commutative, <E x X (ii) ° > X for t E (C and x e X , the map • e Let E -y X with the C - a c t i o n s a: <E x - E -»- E . * * ~ - l (a ) : 0 ( E ) -»• (a )^0 ( E ) , (a ) (s) = -a S CT 0 i s a l o c a l section of E . holomorphic vector f i e l d on X 0 This induces (by d i f f e r e n t i a t i o n ) f (resp. s) i s a l o c a l section of a: € x x -y X . V X t e C , we have an Then for each a C-module homomorphism. V: 0 ( E ) -> 0 ( E ) x ~x where i s C-linear. fc be a a-equivariant holomorphic vector bundle on 0 -module homomorphisms, where a (x) : E -y E , . t x a (x) such that V(f.s) = V(f).s+f.V(s) X x c (resp. E ) and V i s the induced (by d i f f e r e n t i a t i o n ) by i s c a l l e d the v - d e r v i a t i o n on E induced by For a given holomorphic vector f i e l d V on a complex manifold X , a holomorphic vector bundle E -y X a: C x E -> E i s said to be V-equivariant i f 2 there e x i s t s a C-module homomorphism V: 0 (E) -> 0 (E) ~x x such that - V(fs) = V ( f ) . s + fV(s) (resp. E). Such a where V f (resp. s) i s a l o c a l section of i s c a l l e d a V-derivation on X •*> C E . The main problem we are concerned with i n Chapter I, i s to obtain the theorem of Riemann-Roch and Hirzebruch v i a holomorphic vector fields. In sections 1 and 2 we begin by reviewing some of the basic differential-geometric concepts i n the context of holomorphic vector bundles. In section 3, i t i s shown that for a holomorphic vector bundle E X (i) (ii) on a compact complex manifold E i s a-equivariant E i s V-equivariant where induced by (iii) X , the following are equivalent: V i s the holomorphic vector f i e l d a . There exists a hermitian metric for some L e £(X, Horn(E,E)) associated with h and i where h 9 on E such that i (0,) = 8 (L) v. i s the canonical curvature matrix i s the contraction operator. As a c o r o l l a r y of t h i s , i t i s shown that P * AT (T (X)) (resp. U ,Q Jc f n ) Jc / n i s V-equivaricint f o r any holomorphic vector f i e l d V on X (resp. on Gr where U, i s the universal k-plane bundle on the Grassman manifold k,n Gr, k,n of k-planes i n C and 11 Q_ = Gr. x(C /U. . k,n k,n k,n n Some examples of V-equivariant bundles are given, and some computations are done. Let X , and l e t C-action a: C x x X 1 E X be a a-equi variant holomorphic vector bundle with the a: C x E •+ E . isomorphism ^-1 a : E be a C-action on a compact complex manifold Then, for each -1 * (° ) E . t t e C , we have a bundle By taking the pull-back r e l a t i v e to ) a : X -*• X , we have a natural geometric endomorphism t Let and l e t V V <f> : a E -* E t t be the holomorphic vector f i e l d on be the V-derivation on E induced by X a . induced by If expression of p V on of N(p) = Z a V , and i f Vi i=l i 'N(p) E by We also define the Todd class of X by z n E and N(p) of i s the l o c a l 9z. (t,z) e <E x N(p)-> q tv^z) Ch(E,z,t) = E e i=l n Td(z,t) = .'II i=l tw. (z) • tw.(z) 1-e i s the rank of a , then we define f o r the Chern character of q E . z = (z^ >. • • • i ) denotes holomorphic l o c a l coordinates on a small neighbourhood the i s o l a t e d zero of v ^ ( z ) , w_. (z) where 1 are defined by the following formal i d e n t i t i e s : q det(I +XV(z)) = II (1+Xv.(z)) = Y <I i=l n 1 det I +X n Then where Ch(E,z,t) W 9a.(z) I 9z. = D and n j=l Td(z,t) 1 (i+xw.:(z)) = c.(V(z))\ I c(x,z)x J 1 j 0 are holomorphic functions i n i s a s u f f i c i e n t l y small neighbourhood of c.(V(z)) (resp. c.(X,z)) :L W x N(p) 0 e C , since each i s an elementary symmetric function i n v, (z) k (resp. i n w^(z)), and also holomorphic. In section 4, the holomorphic Lefschetz fixed point formula for the i s o l a t e d fixed points, and an algorithm for c a l c u l a t i n g the Grothendieck Residue are stated. a theorem of N.R. ©'Brian [34]) The following formula (which generalizes 4 trace d> (z) dz., . .dz . _t 1 n Res P for — Res _n pi z -a (z) ,. . ,z -a (z) I t n t Ch(E,z,t)Td(z,t)dz.,..dz 1 n a ,...,a 1 n t e W - 0 , i s shown. The theorem of Riemann-Roch and Hirzebruch f o r V-equivariant bundles, i s obtained from the holomorphic Lefschetz fixed point formula and the formula i n [9], f o r holomorphic vector f i e l d s Let manifold X V T(Z,0 ) o z the sheaf of r i n g with i s o l a t e d zeroes. be a holomorphic vector f i e l d on a compact Kaehler with i s o l a t e d zeroes. f filtration V 0 F ^ n F I t i s shown i n [11] that there exists s n+1 ... , of the global section of = 0 / i (ft ) on the set of zeroes z x 1 Z of V , such that (*) Let ^ V on E . gr(T(Z,0 )) = © F E -> X be a V-equivariant bundle with the V-derivation In section 5, i t i s shown that represents the k-th Chern class isomorphism (*). * n H ( P :C) c (E) of i k A (—) c, (V(z)) e T(Z,0 ) 2TT k z E with respect to the (This r e s u l t i s obtained i n [ 1 2 ] , independently.). i s computed through the isomorphism (resp. c. (Q, )) l k,n in /F = H (X:C) -p -p+1 are computed i n T(Z,0 ) z [12] v i a a d i f f e r e n t approach). (*) . Also c.(U ) l Jc f n (this computation i s done Some calculations are done and motivation i s given f o r the concepts i n the e a r l i e r sections. <j>:G m In chapter II section 1, we show that f o r any G^-action x x - > X on a complete variety X , there exists an everywhere defined map <j): E> x X X 5 such that the diagram i s commutative and cj> ': P -> X G xx m <J> Cp) = <j>(P/X) i s a morphism for each x e X , where B? = G u m {0} u Although i t s proof i s not d i f f i c u l t , i t has many useful applications. For example, we are able to show that extensive use of this map cb gives an elementary proof of several well-known theorems i n the theory of Linear Algebraic Groups. I t i s also a useful tool i n understanding the structure theory of complete v a r i e t i e s with G -actions. Furthermore, m G by our methods i t i s simple to show that the fixed point scheme i s connected i f and only i f $ acts t r i v i a l l y on X . X This l a s t r e s u l t i s proved i n [3, p. 495], but only after much work. In chapter I I , we give some of the applications of the use of (j> . One of the important c o r o l l a r i e s to the existence of existence of enough fixed points. a projective variety or equal to one, then Namely, i f T i s the i s a torus, acting on X , and i f the dimension of T <j> X i s greater than fixes at least two points of X [22, p. 153]. By using t h i s , we are able to prove; (i) In section 2: P = N (P ) = {geG: G u and P = N (P ) = {geG: G u where G group of gP g u 1 = P } u Adg(P ) = P } , u u i s a connected linear algebraic group, G , P P i s a parabolic sub- i s the set of unipotent elements of P and P is {°°}. 6 the tangent space of P^ at the i d e n t i t y . These r e s u l t s give elementary proofs of the well-known theorems and P = N (P) = {geG: gPg =P} , P = N (P) = {geG: Adg(P) = P} , -1 G where (ii) P i s the Lie algebra of P . In section 3: The existence of a G -action on G/p. with only one fixed point, 3. where G i s a connected l i n e a r algebraic group defined over an algebra- i c a l l y closed f i e l d subgroup of G G k of c h a r a c t e r i s t i c zero, and i s established. p > 0 , the existence of a G -action on P i s a parabolic For a reductive linear algebraic group defined over an a l g e b r a i c a l l y closed f i e l d i s discussed, where P k of c h a r a c t e r i s t i c G / P with only one f i x e d point i s a parabolic supgroup of G . The structure theory for complete v a r i e t i e s with G^-action i n i t i a t e d by B i a l y n i c k i - B i r u l a i n [3], by the following theorem: Theorem ( B i a l y n i c k i - B i r u l a ) . d>: G x x-:->- X on a complete variety m For a given G -action G^ r X , let X = \_) Z. be 0 G was ' m the 1 decomposition of the fixed point scheme X into connected components. Then there exists a unique l o c a l l y closed G^-invariant decomposition of and morphisms y.: U. -> Z. for i= 0,1,..r so that X 7 (ii) (iii) y. 1 i s a G -fibration m f o r every closed point p e T (U. ) = T (Z. ) ® T (X) p i p i p i where T (X) p = {veT (X): d<t (p)v=t v t m p for some We c a l l the above decomposition B-B's metric proof of this theorem (using cb) D. m<0, f o r a l l teG }. m decomposition. A geo- was communicated to us by Lieberman. In section 1, with any G -action on m decomposition on X we give B-B's p and n decomposition on p n , and discuss the r e l a t i o n between on (P n associated B-B's under the G^-equivariant closed n immersion X D? In section 2, we have obtained the Brunat decomposition of from B-B's decomposition, which was a conjecture of B. Iversen [26]. G/B 8 CHAPTER I HOLOMORPHIC VECTOR FIELDS, EQUIVARIANT BUNDLES AND THE THEOREM OF RIEMANN-ROCH-HIRZEBRUCH 1. HERMITIAN DIFFERENTIAL GEOMETRY In t h i s section, we w i l l review some of the basic d i f f e r e n t i a l - geometric concepts i n the context of holomorphic vector bundles and, more generally, d i f f e r e n t i a b l e C-vector bundles. For.more d e t a i l s , the reader i s referred to R.O. Wells [40]. We s h a l l denote by the term vector bundle a d i f f e r e n t i a b l e C-vector bundle over a d i f f e r e n t i a b l e manifold, Let £(E) be a d i f f e r e n t i a b l e vector bundle over the set of d i f f e r e n t i a b l e sections of X , moreover we denote p-forms on £_( A^T*X) metric on E to each f i b r e E C, the function e h by E over an open subset £ ( U , E ) X" Let E •* X be a vector bundle. E X . A Hermitian i s an assignment of a hermitian inner product of U , the sheaf of d i f f e r e n t i a b l e X , £ ° : = £. the sheaf of d i f f e r e n t i a b l e functions on D e f i n i t i o n 1.1.1. n X , and l e t be the sheaf of d i f f e r e n t i a b l e sections of E . Then we denote by £(U,E) of E -> X E ->- X . <,> such that f o r any open set U C X and <£,ri>: U C <C,n>(x) = <£(x),n(x)> A vector bundle E given by is c OO equipped with a Hermitian metric h is c a l l e d a Hermitian vector bundle. If q „: U A U. -> GL (C) are the t r a n s i t i o n functions of E a,3 a 3 r Then any global smooth section by the following data; s = {s} s of E s on =9"a,3 3 ( X on i s uniquely determined U A U where 9 s : a U C r Let t f i s C°° . a E -> X CK be a Hermitian vector bundle on Ct = (e.,....,e ) a i r of f a r x r be a frame f o r E over X u , where a and l e t t f = transpose a (h ) = <e ,e"> , and l e t h F ((h ) ) a p,q p q a a py.q . We define a matrix of the C the hermitian metric h be the functions {(h) j . Thus h i s a positive a p,q a d e f i n i t e Hermitian symetric matrix, and i s a (local) representative f o r with respect to the frame the change of frame over h a = g„ h^g„ 3,a 3 3,a on U a . Now we have a f = g„ f„ ; i t i s easy to see that a 3,a 3 u rv U_ , a 0 A'U„ 3 E f i s the transformation'law representations of the Hermitian metric metric on fc h . i s given by the above data. for local Conversely any hermitian By a p a r t i t i o n of unity argument i t can be shown that every complex vector bundle can be given an Hermitian metric. Example 1.1.1. Let U, , -> IP be the t a u t o l o g i c a l l i n e l,n+l bundle on the projective n-space p , i . e . 0(U ,) = 0!(-l) . - l,n+l n n n Let (P ) n x a = {[x ,..x ,..x ] e u ot n t standard open a f f i n e covering of geneous coordinates of p n . p P : x n ot , where n We define h (fx.,..,x ]) = u ^ V ' V Now x g „([x ..x 1) = — a,$ O n x 1 x^,..is h : U a a I \\\ ^ 0} a = 0,..,n the homo- GL. (C) 1 by 2 0 v X 2 a i s the t r a n s i t i o n function of p be the U l,n+l on 10 ( P) r\ ( P ).. n , hence n x {h } a = 0,,.,n s a t i s f i e s the transformation a law f o r x l g „ = —••'} • "a,g x • l . J Therefore i t defines a Hermitian metric h on U, . , which i s c a l l e d the Fubini-Study metric. 1, n+1 Let E -> X be a vector bundle. £. (E) = £( A T*X ® P Then we l e t E) • P c be the sheaf of E-valued d i f f e r e n t i a l forms of degree p . We have the natural isomorphism £( A T*X) 8 1(E) = t P We denote the image where P — <|> ® £ » £(E) = £ (E) — . P under t h i s isomorphism by <j> . E, e c, (E) , P cf> e fc , E, e £,(E) . P Definition connection D on 1.1.2. E Let E -> X be a vector bundle. Then a i s a C-linear mapping D: £(X,E) £, (X,E) , which s a t i s f i e s 1 D(<J>.5) = d(j> . K + tjj.DC where <J) e|(X) and 5 e fc(X,E) . Remark: In case E = X x C , the t r i v i a l l i n e bundle, we see 1 that we may take ordinary exterior d i f f e r e n t i a t i o n as a connection on E . d: £ (X) ->• £ (X) Thus a connection i s a generalization of exterior d i f f e r e n t i a t i o n to vector v a l u e d - d i f f e r e n t i a l forms. We now want to give a l o c a l description of a connection. t Let f = (e^,..,e > r be a frame over U f o r a vector bundle E •> X , 11 equipped with a connection D , where define the connection matrix rank of E is r . Then we 6(f) with respect to the frame , by setting. 0(f) = (6(f) ), 0(f) e ^ ( u ) , where p^q p#q - 0(f) P/q is defined by De = q p t ± T 0(f) .e q,P P The e f f e c t on the connection matrix under a change of the frame f i e l d can e a s i l y be found. In fact, l e t :f = g p new frame f i e l d , where g f 01r p be the 01 i s nonsingular rxr-matrix of C -functions, a, p we f i n d immediately This i s the equation f o r the change of connection matrix under a change of the frame f i e l d . Conversly. any connection D on E i s given by the above data. We s h a l l f i n a l l y define the curvature: bundle with a connection D , and l e t 0 ( f ) a matrix over a l o c a l frame ^ f . We define a 0(f ) = d0(f ) - 0(f ) A a a a i.e. If 0(f ) a p,q f. = g . f 3 a,3 a 0(f ) a Let E -»• X be a vector be the associated connection which i s an r x r matrix of 2-forms d0(f ) - T 0'"'(f ) /s, 0 ( f ) a p,q £ a p,k a k,q y i s the new frame f i e l d , we f i n d e a s i l y 12 0(f ) 3 = g D 0 ( f ) = d6 (f ) - 9 (f ) A 6 (f ) 3 3 3 3 0 ( f )g , where a a,3 1 a,3 Q Q Because o f t h e t r a n s f o r m a t i o n law { 0 ( f )} a d e f i n e s a g l o b a l element 2 0 e £, (X„ Horn (E,E)) the c o n n e c t i o n Let the m e t r i c t h e c u r v a t u r e form a s s o c i a t e d t o D . E h which i s c a l l e d be a H e r m i t i a n v e c t o r bundle on on E X . Then we c a n extend i n a n a t u r a l manner t o a c t on E - v a l u e d c o v e c t o r s . Namely, s e t <w»£, w'®n> = WAW'<5,TI> X w' e /?T*X , and x inner product for w e A?T*X , X £, n e E x X for x e X . Thus t h e e x t e n s i o n o f t h e t o d i f f e r e n t i a l forms i n d u c e s a mapping h: £ (X,E) ® £ (X,E) ->£. (X,E) . P D e f i n i t i o n ^ 1.1.3. q P+q A connection compatible w i t h the h e r m i t i a n m e t r i c h D on on E i s s a i d t o be E i f d<£,n> = <D£,n> + <£,Dn> . I t i s easy t o see t h a t a H e r m i t i a n v e c t o r b u n d l e E admits a c o n n e c t i o n w h i c h i s c o m p a t i b l e w i t h t h e h e r m i t i a n m e t r i c , and t h a t i t i s n o t unique i n g e n e r a l . B u t i n t h e h o l o m o r p h i c c a t e g o r y , we s h a l l o b t a i n a unique c o n n e c t i o n satisfying type o f 0 . an a d d i t i o n a l r e s t r i c t i o n on t h e 13 2. THE CANONICAL CONNECTION AND CURVATURE OF A HERMITIAN HOLOMORPHIC VECTOR BUNDLE. Suppose now that a complex manifold X . E •> X If E i s a holomorphic vector bundle over as a d i f f e r e n t i a b l e bundle i s equipped with a d i f f e r e n t i a b l e Hermitian metric h , we s h a l l r e f e r to i t as a Hermitian holomorphic vector bundle. Recall that since r P , q (E) = £ forms of type (p,q) P , i s a complex manifold, I £ ' ( E ) , wher p+q=r £ (E) = £ X P q € » q |(E) and £ P , q i s the sheaf of d i f f e r e n t i a l . Suppose then that we have a connection on E D: £(X,E)-* £ (X,E) = £ 1 into 1/0 ( X , E )ffi£°' (X,E) . Then 1 1,0 Theorem 1.2.1. vector bundle (a) E If h E -* X , then h and D": fc(X,E) - * £ 0,1 (X,E) . i s a Hermitian metric on a holomorphic induces canonically a connection which s a t i s f i e s , f o r U D (h) an open set i n X . I f £, n e £ ( U , E ) , d<£,n> = <D£,n> + <E ,Dn> ; i.e. (b) s p l i t s naturally D = D' + D" , where D' : £(X,E) ->& (X,E) on D D i s compatible with the metric i f £ e 0 (U,E) , i . e . , £ "X h . i s a holomorphic section of E , then D"£ = 0 . Proof. Let { f } a t be the holomorphic frames f o r E and l e t 14 g „: U r\ U„ -> GL (C) a,B a B r be the holomorphic over an open covering {U } of X . a ael representative f o r the hermitian metric t f f 3 . a = g Now . a,3 a ' f h Va,3 n " c e ( 8 h h 3 = 3 h 3 g a,3 a\,B' 1 ) g h a,3 0 = {9g g e .h + g D 9h a,3 a a,3 = Thus i s the l o c a l a with respect to the frame h h u t t h e a,3 U A U_ a 3 we have n + a , e g 3 a \ , B h ttt- ^" -1 -1 „3h g „ + g „h 9 g „} g „h g „g „ a,3 a a,3 ^a,3 a a,3 a,3 a a,3 a,3 - y }h ot _ 1 a 8„ = dg g + 3 a,3 a,3 1 0 0 unique global connection on (a) and { 8 g B E = 9g . + g .9h h " a,3 a,3 a a 1 = dg . + g .6 a,3 a,3 a i s holomorphic. a, p law (1). Assume -1 = 9h h ; f o r a frame change over a a a t„h g +g ^a,3 a a,3 ={9g Since 6 define t r a n s i t i o n functions of r g „ 8 g / a 3- a a,3 1 0 hence {8 } a defines a E , because i t s a t i s f i e s the transformation I t i s easy to see that, t h i s connection s a t i s f i e s the conditions (b) . Q.E.D. This theorem given a simple formula for the canonical connection i n terms of the metric h : 8(f) = 9h(f) h(f) Corollary. holomorphic 8(f) and Let vector bundle 0(f) 1 D , where fc f i s a holomorphic frame. be the cononical connection of a Hermitian E-^-X with the Hermitian metric h . Let be the connection and curvature matrices defined by with respect to the holomorphic frame f . Then D 15 (a) 0(f) i s of type (b) 0(f) = 99(f) Proof. (1,0) 0(f) i s of type By d e f i n i t i o n (1,1) 0(f) , 30(f) = 0 0(f) = 3h(f)h '''(f) 1 + h(f)3h (f) , - 1 _ 1 - 3h(f)h(f)_1A 30 (f) = 0 (f) A For Hence by and 30(f) = 0(f) A and which i s of type 30(f) = 3 ( 3 h ( f ) h ~ ( f ) ) = - 3h(f) A 3 h ( f ) , now = 3h(f)h(f) (1,0) _1 3h(f)h(f)_1A hence , but 3(h(f)h (f)) _ 1 3h(f)h(f) = - 1 s h ( f ) 3 h ( f ) = - 3h(f) A 3 h ( f ) = 0 (f) A 0(f) , hence - 1 _1 0 (f) . (b) we have by d e f i n i t i o n (a) we get 0(f) = d0(f) - 0(f) A 0(f) . 0(f) = d0(f) - 30(f) = (3+3)0(f) - 30(f) = 30(f) . which i s obviously type (1,1) and 30(f) = 330(f) = 0 . Q.E.D. We want to give one p r i n c i p a l example concerning of connection and curvatures. Example 1.2.1. bundle on p n Then by example the computation , let Let -> n + 1 (P n be the t a u t o l o g i c a l l i n e be the homogeneous coordinates for IP n x„,....,x O n 1.1.1 " , r 2 1=0 -i Iv[x„,...,x cx '0' ' nJ = L 2 J 1 defines a hermitian metric h on U a' ,.. . Hence the associated canonical l,n+l connection and curvature matrices are given by 0 = 3h h and a a a n 1 0 a = 30 a . Let z. i be the holomorphic l o c a l coordinates on .2, h (z.,...,z ) = 1 + Z |z.| . a 1 n 1 Let z = (z ,...,z ) , |z| 2 U a , then 2 £ |z | i=l n = , 16 then 0 (z) = 3 log h (z) = a a 2 1+l'zl / z.dz. i=l n 0 (z) = 36 (z) = 3 a ' a n z i=l hence i=l and z.dz. 1+| z| (1+1 z| ^)dz.«dz. - ,E, z.z.dz.^dz. l l 2j = l ± 3 3 i 1 1 TV (1+|z|) 17 3. HOLOMORPHIC VECTOR FIELDS AND EQUIVARIANT BUNDLES Let T(X) C = T(X) Since X X IR ® C be a complex manifold of dimension n , and l e t be the complexification of the tangent bundle IR T(X) iR T(X) i s a complex manifold, has a complex structure of i t s +. C C own, and so we may single but the subspaces T (X) and T (X) on + C which these structures agree and anti-agree. Moreover T (X) is in C d u a l i t y with the forms of type (1,0) while T (X) i s i n d u a l i t y with C + C C the forms of type (0,1) , and T(X) = T (X) ® T (X) A complex vector f i e l d each point x of X an element d i f f e r e n t i a b i l i t y condition. system over on X i s a map which assigns to <C V x of T(X) I f ^ ±^-i 1 with an obvious holomorphic l o c a l coordinate z 3/8z. , 3/3z. span U , say, then respectively f o r every V V 1 + C T (X) X and C T (X) X x e U . Hence we can express a complex vector f i e l d uniquely i n the form n v| = J a. 3/3z. + b. 3/3z. , 'U . , x l l I i-l where V a^, b^ e £(U) . We notice that IR e T (X) X f o r each X V x E T x ^ can write i s a r e a l vector f i e l d , i . e . , — x EX i f and only i f a. = b. 1 A complex vector, f i e l d + V V 1 for i = l,...,n . i s said to be of type (1,0) i f C a t V eac ^ P o i n t x • L e t v b e o f tyP e (IfO) . Then we l o c a l l y i n the form n V = y a. 3/3z. . i=l 1 If the components a^ of V are holomorphic functions of the holomorphic l o c a l coordinates z^ , then we c a l l V a holomorphic vector f i e l d on 18 X . Since on X TX = T is bundle simply TX V = V + V and i s X a complex v e c t o r (V) the:, map exits C -»- A u t ( X ) X X . a a holomorphic section , where We c a l l vector field of of the x . type Then vector holomorphic map C x a: : X -* X Aut(X) is or X -* X is group We define a real V vector x of a l l the o(t,x) = t of X , if associated is holomorphic group field, . such that ° ( ) 1-parameter a V . a group homo- diffeomorphisms automorphisms of . Let of X . V where For x f each d dt = C x x -> X a: is x * (a. t e X be , we a t=0 a holomorphic group of automorphisms define e T x x (x)) 1-parameter by x function V (f) = around dt x (f (a. . t (x)) t=0 Then V = {V } X a holomorphic from the compact, from meter vector 1-parameter it is well a 1-parameter If of field tangent on a complex m a n i f o l d given by the (1,0) associated to We s a y C - a c t s -> a a V each 1.3.1. t field t a holomorphic morphism of = V x x real Definition there bundle, . where called as a C - v e c t o r a global holomorphic of For (X) group V of is field, group known group called of a holomorphic generated by of a any holomorphic automorphisms automorphisms automorphisms holomorphic automorphisms that, of the a vector . V of field, We c a l l . X a vector X . vector is xeX field When field induced X is is induced . induced the from a 1-para- one parameter group 19 Contraction Operator and Koszul Complex Let X be a complex manifold of dimension n , and V be a n holomorphic vector f i e l d on X . I f ' f ^ holomorphic l o c a l z a r e i coordinates on an open subset U of i = 1 X , we define i I :£ P , q ( U ) -> £ P 1 , u by Y q l < i , < . <i <n l'"' p - 1 pK j <..<j <n - 1 q- v i i ; 3 . dz. A....A.dz. />. dz. A<....Adz. l'--' q *1 V i q 3 D 3 ; I l<i <..<i <n - 1 p- (-1) . dz ... kdz.A. . A ( J Z . A, dz . ^ l ' - ' V V ' ^ q 1 l j> 1 £=1 ± X A ..Adz : X 3 . q K j <-.<j <n - l qJ J where V Then, i where £, U k=l defines an £_-module homomorphism between P , q i s the sheaf of d i f f e r e n t i a b l e £ P , q (p,q)-forms.on and X . ^ 1 , q Moreover i t s a t i s f i e s the following i d e n t i t i e s on forms. (a) (b) (c) 2 i = i i = 0 v v v i 9 + 9i =0 v v i(<f>Ai{0 = i ( $ ) A i f i + v (-l) v d e g ( *^Ai W v Because of the property (b), i 0 -module ~X i s the sheaf of holomorphic v homomorphism p-forms on i : 9? -> X . ft P , where Qp Thus we get a complex induces an 9 20 i i v o -> fi — • fi n i v ->•• •-> fi —>• n p fi v •-»- s r — » • o p ~x which i s c a l l e d the Koszul Complex associated to i i s c a l l e d the contraction v Let £(E) t P , q i s an map E -> X V , and the £ 6(E) = & P , q be a holomorphic vector bundle on ( E ) , we get £, -module homomorphism. "X i : H (X,ft (E)) -> H ( X , n q P operator operator, be the sheaf of d i f f e r e n t i a b l e sections of ® -> o q i ^ := ± 8 1: £ v E P , q on X , and l e t X . Since (E)-» £ " ' (E) P 1 q which Moreover, contraction induces a natural P - 1 (E)) . Equivariant Vector Bundles Let X vector f i e l d on be a complex manifold, and l e t V be a holomorphic X . D e f i n i t i o n 1.3.2. A holomorphic vector bundle E -> X i s said A to be V-equivariant, i f there exists a C-linear morphism V: o (E)-*0 (E) such that V(fs) = V ( f ) s + fV(s) i s a l o c a l section of where X x C (resp. E). f (resp. s) V i s c a l l e d a V-derivation on E . A, To give the l o c a l expression of a V-derivation need to look at the holomorphic vector f i e l d on Since of automorphisms n Aut(P ) = PGL a of IP n _ = GL ,/C n+1 n+1 i s given by * .1 V on E , we" IP . n . , any 1-parameter group n+1 21 for some M e gSt, , . n+1 For a given M = (a. .) e gjl , , we define 1,3 n+1 V: <C[x .x. ,.. ,x ] -»- t[x ,x.,...,x ] u i n u i n as a C-derivation as follows; l e t n V(x.) = 1 for T a. .x. j=0 ^ 3 i = 0, l , . . . , n , and extend coordinate ring of (Pn . V {u.} Let P define extend n n be the natural open a f f i n e covering i=0 1 of as a C-derivation on the homogenous , i . e . , U. = ( P ) = { [x ,x, ,.. ,x ] e l x. 0 1 n l p : x. =1= 0} . l n R rt vl : 0 (U.) 'U. - n l I P vl 0 (U.) - n I IP 1 intersection x n 0 (U.) . { v l } - n 1 u. . „ P i i=0 U. n U. , to give a C-derivation of J derivation by V . Now f o r x e P i s a holomorphic function around define z. = x./x. 3 3 1 , V x ^) = 0 , we denote t h i s P ( f ) ( ) where f v x x ; hence the derivation naturally a holomorphic vector f i e l d coordinates v V on p n and 2 l patches i n the 1 1 Then we x,V(x..)-x^V(x,) v| (x./x.) 'U. 3 l as as a C-derivation of U. 1 V will give , and i n the holomorphic has the form x.V(x . )-x .V(x.) »l„. - ? 1 1 3 3 2 1 • x^ I t i s clear that t h i s i s the holomorphic vector f i e l d induced from the 1-parameter family a tM = 'e . Thus any holomorphic vector f i e l d on P gives a C-derivation of n 0 which i s induced from a C-derivation on the P homogeneous coordinate ring of P as above. We can e a s i l y generalize t h i s s i t u a t i o n as follows. Let R 22 be a reduced C-algebra and l e t V: R -* R i s C-linear" and V(rs) = V(r)s + rV(s) be a £-derivation, i . e . , V f o r r , s e R . Now given M = (b. .) e ql , (R) , we define a V-derivation i,D n+1 Let l U n i^i~0 ^ e t ^ ie n a t u r a x V of 9. n R as follows. p open a f f i n e covering of . - (P , then n {w. = Spec(R) x U.}._. i s an open a f f i n e covering of IP = Spec(R)x (P 1 i i—0 R Define v| : = V. : 0 (W.) -* 0 (W.) , as 'W. x l l l n n (P P R R x.V(x.)-x .V(x. ) V. (x./x.) = — ^—r-J — i j• i 2 x. l where J and extend V. x to be a C-derivation of n V(x ) = T b . x , r ,^„ r,k k k=0 0 P i n the i n t e r s e c t i o n W. r\ W. V. I = V and x' R n R (W.) . x to give a V-derivation n {V.}. „ .patches x x=0 V of 0 , and IP R conversely any V-derivation of 0 i s obtained i n t h i s way. What we have done geometrically i s the following: an open subset of C , and l e t V Let U be be a holomorphic vector f i e l d on U " n Then any holomorphxc vector fxeld V on U x tp such that a n d-rrCV) = V i s obtained by the procedure above, where TT : U -X P U , d-rr: T ( U x P ) + n Let X vector f i e l d on bundle on T(U) . be a complex manifold and l e t V X . I f E ->• X be a holomorphic i s a V-equivariant holomorphic vector X , then there exists a V-derivation V on E . Let { u } a 23 be the open a f f i n e covering of X Let V '= V : 0 (E) a map V :U a a y 0(E) X a a A such that V a of 0• •• where n U >< <P a V of 0 ~P(E) dir(v) = V 0 0 | X U a induces naturally a V : = v|„ a U a i s the derivation 1 induced from the holomorphic vector f i e l d of 0 "U x (P a n+1 © . Then the holomorphic T (U , Hom(E,E)) x -»• V (x) : E -> E a a x x V -derivation a V 0 (E)| = X rv V . The V^-derivations patches i n the i n t e r s e c t i o n to give a V-derivation n hence a holomorphic vector f i e l d where TT : P(E) -> X , since V V is a on P(E) . such that V -derivation. a From a the above procedure, we immediately see that the converse i s also true, namely i f V i s a holomorphic vector f i e l d on P(E) such that dir(V) = V , TT : P(E) •+ X then there exists a V-derivation W: 0 (E) ->• 0 (E) of E such that W = V . Thus we have the following X X Lemma. Lemma 1.3.1. A holomorphic vector bundle E ->• X i s V-equiva- r i a n t i f and only i f there exists a holomorphic vector f i e l d P(E) such that dir(w) = V Remark: where ir : P(E) ->• X . D. Lieberman has proved this Lemma i n [ 2 9 ] , Let us look at the s i t u a t i o n i n terms of 1-parameter of automorphisms. let f: X — X W on Let ^ : E -»• X group be a holomorphic vector bundle and, be an automorphism of X . I f there exists a bundle 24 automorphism f : E —*• E such that the diagram -> E commutes, X then we have f o r each f(x) a C-linear map takes lines through the o r i g i n i n E^ origin i n E f: x e X -> X Therefore f (x) P(E) ->• P(E) f f ( x ) : E —> E... , , hence x f(x) to the lines through the induces naturally an autormorphism such that the diagram P(E) -> P(E) commutes. X X Conversely any such f i s obtained i n this way, because Aut( P_) = PGL_,,. = G L n+1 n Let n+1 /Z (GL a: C x x ->• • X a complex manifold n+1 ) , ZfGL,^) = C .1 n+1 n+1' be a 1-parameter group of automorphisms of X . D e f i n i t i o n 1.3.3. A holomorphic vector bundle E —> X i s said to be a a-eguivariant bundle, i f (i) of there exists a 1-parameter group of automorphisms E such that a: £ x E -»- E 25 C x- E >E the diagram commutes. <E x X (ii) for t e C and >- x x e X , the map E E ,, o (x) i s C-linear. t We c a l l a a a-equivariant 1-parameter group of automorphisms of As we have seen above f o r each a : P(E) P(E) t e C , a^: E -* E E induces such that the diagram P(E) P(E) -+ X commutes. I t i s easy to see that of automorphism { } a fc a: C x P(E) -»- P (E) C x p(E) of defines a 1-parameter group P(E) ., such that the diagram -> P(E) commutes. C x Conversely any a x -> X i s obtained i n t h i s way, because Aut( P;) = PGL n+1 Let V be a holomorphic vector f i e l d induced from the 1-para- meter group of automorphisms a of X . If a i s a a-equivariant- 1-parameter group of automorphisms of a holomorphic vector bundle E * X , then f o r each t e C , we have an 0 -module homomorphism X 26 a : 0 (E) -* (a ) 0 (E) , given by t'* t -. X X a l o c a l section of E . a (s) = a V(fs) = - ^ ( c r dt t s Thus V of (fs)) E . dt ( t t=0 ( P(E) E . If = V ( f ) s +f V(s) t = 0 W i s the holomorphic induced from the one parameter family W a of i s the holomorphic vector f i e l d induced V , i.e., V = W . manifold X . V be a holomorphic vector f i e l d on a compact complex X , induced from the 1-parameter group of automorphisms X . a Then we have the following. Theorem 1.3.1. on t defines a V-derivation on Let of Then + P(E) , then i t i s clear that from is t=0 a; s) a;(f) A . a ; s ) } vector f i e l d on s = -£:(a. (£) cr.(s)) I = , where Now we define d "* V(s) = - ( o . ( s ) ) dt t for a l o c a l section o s o a Let E •> X be a holomorphic vector bundle Then the following are equivalent. (i) (ii) (iii) E i s a a-equivariant bundle E i s V-equivariant There exists a hermitian metric 1^(0) = 3(L) , f o r some h on E , such that L e &(X, Horn(E,E)) , where i s the canonical curvature matrix associated to h . 0 27 Proof. there exists (i) => (ii). If E i s a ff-equivariant bundle, then a: C x E ->• E *• E <EXE a: commutes. C x x a: >• X we define v ( s ) = -£r(o\ <s)) dt t t=0 as we have seen above V i s a V-derivation of E , hence E is V-equivariant. ( i i ) => ( i ) . This i s the existence of the solution of the E-valued d i f f e r e n t i a l equation. Let V be the V-derivation of holomorphic vector f i e l d V group of automorphisms of because P(E) automorphisms I t can be seen as follows. on P(E) i s compact. i - t °^ a a a- equivariant bundle. E E , then P(E) . Let V induces a be the 1-parameter generated by V , a exists Then there exists a 1-parameter group of w n l c h induces a on In fact = V , dt t=0 so the correspondence i n (i) and ( i i ) b i j e c t i v e . P(E) , hence E is 28 ( i i ) => ( i i i ) Let V be a V-derivation on holomorphic vector f i e l d on P(E) . E , and P(E) and on I t i s easy to see that the associated derivation where of f V i s a V-derivation, i . e . ; i s a smooth function on X , and s X be respectively. V: £(E) -> £(E) of the V(fs) = V(f)s + f V(S) i s a smooth l o c a l section E . Let i : £ (E)—• £(E) from the smooth vector f i e l d : C i s the induced Let V = V + V , V = V + V the associated r e a l vector f i e l d s on r e a l vector f i e l d V 1 , 0 V be the contraction operator on X . (Namely; i = i induced + i_ v ( E )ffi£^'^"(E) -> fc(E) with respect to the natural s p l i t t i n g of £ (E)) . For any hermitian metric L = V_- i D : £(E) v V h on E , we define D > £(E) >• £ (E) V £(E) where D i s the canonical connection of E associated to h . Then L(fs) = V(fs) - i D ( f s ) = V(f)s + fV(s) - i (dfs+fDs) v v f(V(s) - iyDts)) = fL(s) where f i s a smooth function on X and s i s a smooth l o c a l section 29 of E . Thus L: £ . ( E ) -»- fc(E) i s an £-module homomorphism, hence naturally defines an element D . Let U such that v| - £(X, Horn ( E , E ) ) 0 where . i s the curvature == U x c U . Therefore q . Then i t follows from above that . Hence L | = "(V - i - L | = (V - i ^ D') | ^ y since D) , 'U 1 v 3(L| ) = 3VI ' U ' U since 3(L| ) = i (<^D')[ i s holomorphic. contraction i d e n t i t y . But then Since ^D'ly = but D = D' i D = (i^+i_) (D +D") = i D' +i D" = i D* , D" = 0 . Hence v V v v| associated be a holomorphic l o c a l coordinate neighborhood of X = V: 0 ( E ) -*• 0 ( E ) X X A on E| E 3(L) = i v ( 9 ) We claim to L - 3(i D')l = - 3i D , v ' U v 1 by the u ®lg » w e g ^ e t H y L = """v^^U ' i (0) = 9(L) . thus ( i i i ) => ( i i ) Let iyO) such that connection where V h D be the hermitian metric on where = of E V = L + i D: v V = L + iyD' on U , since = 3L + 8i D' = 3L - i . Moreover i f v V v such that D = D' on U ->&(E) i s a holomorphic l o c a l E | = U U . 3D' = 3L - i y 0 = 0 induces a V-derivation on E , namely x c q E i s V-equivariant. Remark: Since ' then Therefore on U , since V 0(E)' Hence £ ( E ) V ,is the r e a l vector f i e l d assoicated to V . I t i s clear that coordinate neighborhood of X Thus Hom(E,E)) i s the curvature of the canonical associated to h . Define i s a V-derivation of £ ( E ) 3V 0 , and L e £(X, E i v O ) = 3(L) . = V: 0 (E) -y 0 (E) X X Q.E.D. 3(0) = 0 , we then have a cohomology class 30 [0] e H (X,Hom(E,E) <3 Q ) — — > 1 H (X, Horn(E,E)) . 1 Thus the condition 1 ( i i i ) i n the theorem i s equivalent the vanishing of the cohomology class i ([0]) i n H^X,. Horn(E,E)) . Corollary 1.3.1. If E + X holomorphic vector bundles, then E i = 1,2 © E are V-equivariant * ,E^ E 2 1 ® E 2 , P AE E. and are 1 a l l V-equivariant. Proof. Let h. be the hermitian metrics on 0. be the canonical curvature matrices associated to h. . Then we have l L e &(X, Horn ( E ^ E j ) f o r i = 1,2, such that i ( © ) = JHLJ . Take v l L i ° E £(X, Horn(E ffiE , E ffiE )) , L = 2 0 2 L„ and l e t l ° 0 h. h h = be the hermitian metric on canonical connection of , h E © E on E 9 1 1 2 , then the curvature © E 2 0 of the l o c a l l y i n the form ° 0 = 0 0, Thus f i V (0) = i V 1 0 0 i \ 0 0.- V l 2 j 0 i 0; V 2 = J = 9L . 0 2 31 Hence ffi E^ For i s V-equivariant by Theorem 1.3.1. E , take L * = - L * and take the hermitian metric h * on E^ , where * _ * For X E^ ® E^ , l e t 0 generated by V . _1 t = (h^ ) * 0^ = - 0^ h^ , then If * 0^ hence E^ v i s V-equivariant by Theorem 1.3.1. be the 1-parameter group of automorphisms -1 of 1 * Therefore v * h * i s the canonical curvature matrix of i ( 0 ^ ) = 9(L^) . * e £(X, Horn (E^E.^) = £(X, HomfE^E.^ ) If -2 a^_, are the a-equi variant, 1-parameter group of automorphisms of E^ and E^ respectively, then a^= a^_ ® a^_ i s a a-equivariant 1-parameter group of automorphisms of E^ ® E^ . Hence by Theorem 1.3.1 For Aa P if A E P E^ ® E^ , take is V-equivariant. S^E^ -^E , i t i s clear that i s a a-equivariant 1-parameter group of automorphisms of -1 a i s f o r E^ . Hence by Theorem 1.3.1 p A E^ is ^ -^ E V-equivariant. Q.E.D. Now we w i l l give some examples of V-equivariant bundles f o r further use. Example 1.3.1. (i) TX , the holomorphic tangent bundle of variant f o r any holomorphic vector f i e l d Proof. generated by If V , then a i s V-equivariant. on is V-equi- X . i s the 1-parameter group of automorphisms of a = cla^: TX -> TX 1-parameter group of automorphisms of TX V X i s c l e a r l y a a-equivariant TX . Hence by Theorem 1.3.1 X 32 Let us compute the V-derivation V of 0 (TX) induced from ~X a t = d a : TX t open subset TX . I f {z.} ? , i s a l o c a l coordinate system over some 1 i=l 1 U of X . Then we have 9a^(z) da (z) = where 1 TX) t -y a : 0 (u) -> 0 (a (U)), a^(z.) = a. (z) and t t. t i t x Hence r(a (u), 3z, t x o^z) = da a C-linear map. -1 fc T(U,TX) z = (z „,z ....,z ) 1 2 i -1 -1 (z) = C3a (z) /3z_.) : T(U,TX) -»• r (a (u) ; TX) as fc fc Therefore V(z) = ^ (a; z))| ( 3a^(z) - A t = 0 -1 t=0 = -^!(3a^(z)/3z )} t = Q d i , . -rr a. z dt t 3z, = - t=0 3z. 3 where n vl = Y b. -r^— U . . l 3z. i=l l 1 L '3b. N dt t a ( z ) = - . dZ.' 1 t=0 Hence C-linear map given by the matrix derivation }• V(z): 0 (U,TX) ->• 0 (U,TX) X X 3b. l 3z. 3 J I 3z I 3- as a I t i s easy to see that L i e [V,*]: 0 (U,TX) •+ 0 (U,TX):W ->• [V,W], X X i s given by the same 33 matrix as a C-linear map. have Since both of them are V-derivations, we then V = [V,*]: 0 (TX) -> 0 (TX) X d i r e c t i o n of V . which i s the L i e derivation i n the X Similarily V: £(TX) -> £(TX) i s given by Lie derivation of smooth vector f i e l d s i n the d i r e c t i o n of P (ii) f i e l d on V V . * /f(TX) on [V=V+V, * ], X i s V-equivariant f o r any holomorphic vector by Corollary 1.3.1. ( i i i ) I t i s wellknown that any holomorphic vector f i e l d on Gr, the Grassmann manifold of k-planes i n C , i s induced from a k,n 1-parameter group of automorphisms a : Gr Gr , o ([W]')=[e . ] n tV for some \T e gH (C) , where W c C i s a k-plane and e .W i s the n 11 tV image of k-plane W under the isomorphism be the universal k-plane bundle on a : U, t k,n U, k,n I t i s clear that Gr, . :..k,n (cr (x), e t tV ^ J a i s <- a t U a t ~ e . Hence V on n Then the automorphism of Let where U w e W c C . 11 /-equivariant 1-parameter group U i s V-equivariant f o r any be the universal quotient bundle cr : Gr, xc -»• Gr, xc t k,n k,n 11 11 , a. (x,w) = t n U, *—> Gr, x C invariant f o r each k,n k,n leaves a . Gr, , we define f o r t e C , k,n Hence i t naturally induces an automorphism that n J3r^ ^ . Q, = Gr, xc /u, k,n k,n k,n .w) C n tV holomorphic vector f i e l d on t V as follows, a ([W],w) = (a [w] , e .w) t t of automorphisms of Let e ; C w a. of Q . t e C . I t i s clear {cJ =e }-equivariant 1-parameter group of automorphisms tV t Q Hence Q i s V-equivariant f o r any holomorphic vector f i e l d 34 V on Gr, k ,n (iv) Let G be a l i n e a r algebraic group over C , and P a * parabolic subgroup of P , we define g' = gp and G . Given \: V •+ £ L = GxC/~ , where X v' = x (P ) 1 v > f° r holomorphic character of (g v) ~ (g',v') i f and only i f r some p e P . Then L induces X naturally a holomorphic l i n e bundle on homogeneous l i n e bundle associated to For a given 1-parameter a: C x G/p -> G/p of G/P , G/p which i s called the x • subgroup a: C -> P of P , we get a (gP) = a(t)gP , 1-parameter group of automorphisms t . We define a : L -> L by a (gP,v) = (a (gP) , x(°(t) ) v ) . t X X t t It i s easy to see that or i s a-equivariant 1-parameter group of automorphisms of 1 . Hence L i s V-equivariant, where X X holomorphic vector f i e l d induced from a . (v) L If V i s the H (X,0 ) = 0 , then any l i n e bundle bundle L on X ~X 1 i s V-equivariant. Since Horn(L,L) = 0 , then we have H (X, Horn(L,L)) = X Hence by the remark following Theorem 1.3.1 L i s V-equivariant for 1 any holomorphic vector f i e l d Let V V on X . be a holomorphic vector f i e l d on a V-derivation of a holomorphic vector bundle V-derivation W on E , consider V - W: X , and l e t V E •+ X . , then element of X x C and E where respectively. H°(X, Horn(E,E)) . X (V-W) (fs) = V(fs) - W(fs) = V ( f ) s + fV(s) - V ( f ) s - fW(s) = f(V-W)(s) , sections of Then for any 0 (E) -*• 0 (E) , ~X (V-W)(s) = V(s) - W(s) be f Hence and V-W s are l o c a l ( defines an 35 0 T e H (X, Horn(E,E)) , we have the Conversely f o r a given natural induced 0 -module homomorphism X T: 0 (E) X 0 (E) , now consider ~X V + T: 0 (E) ->• 0 (E) defined by (V+T) (s) = V(s) + T(s) , where s i s X X a l o c a l section of E . Hence V + T : 0 (E) •+ 0 (E) defines obviously X X a V-derivation of E . I t i s clear that above correspondence i s b i j e c t i v e . We state t h i s f a c t as a Lemma for further use. Lemma 1.3.2. I f E"*"X i s a V-equivariant bundle, then the set of a l l V-derivations of E H°(X, i s a p r i n c i p a l homogeneous for the group Horn(E,E)). Let V be a holomorphic vector f i e l d on X , induced from the 1-parameter group of automorphisms a of X . I f E X i s a V- equivariant holomorphic vector bundle then by Theorem 1.3.1 we have a a-equivariant 1-parameter group of automorphisms Z be the zero set of V . Now given o p e Z , we have l e a s t for small values of | t | . Then which i s nothing else than 5 Ix) : E of E . Let CT t (P) = p at the 0 -module homomorphism X a : 0 (E) (o.).0 (E) induces naturally a C-linear map 5 (x) : E -*E .. t —. t *— t x —1 . . X X a (x) u then we get a representation a X -*• E °t . I f we take x= p , ;: 1 .. . ( X ) ) , t •+ a (p) : E ->• E , since p t p p i s a 1-parameter group of automorphisms of E . Therefore * tA a (p) = e for some t C—* GL(E A e g£(E ) , but we have p i s the V-derivation of E > Viz) induced from d * = -jHo. (z)) • •• - d t t "* tV (p) a = e t a , since A t=0 where " V 36 Hence A = - V(p) , namely ° (P) = -tV(p) e t We w i l l f i n i s h t h i s section with a very u s e f u l l example. Example 1.3.2. A. Let A = 0 °. 0 g £ \ n+l ' i ^ i A X ° f r + 1 * j Let V n tA the holomorphic vector f i e l d induced from vl . , ( IP" ) x = e . Then Y V(z.)3/3z. a where z. = x./x , but l l a V(z.) = A(x./x ) = I a I x A(x. )-x.A(x ) a I x a = (A.-A )z. . x a x Hence therefore e = [0,..1,..0] i s the only zero of V in ( ip ) , n a hence Z = {e : a=0, n} i s the set of a l l zeroes of a a : U„ , -»• U, t l,n+l l,n+l be given as i n Example 1.3.1 (iii). tA 0 t ( [ e a ' ] e a ) = ( a t ( [ e a ] ) ' e * a e V . tA ) ' H e n c e = 6 Let a Then be 37 Now i f V V(e a ) = - X a i s t h e V - d e r i v a t i o n induced f o r each Consider from a , then a = 0,...,n . the Fubini-Study metric h on U n n , l,n+l 2 I |x.| h a ( [ x O'- n X = ~ ] ) X as g i v e n i n Example 1.1.1. associated t o ( P ) X h . and on n Then we have (: p ) X A n a Let 0 n ( P ) X Therefore L e t z. = I 2 = h |z I , p p on (P ) x N a Hom(u\ _ ,U. .,,)) i J-,n+l l , n + l s g X hence r\ : ( P " ) x i v . 91og h ^ a + log5 ) + X • a Therefore g element £ ( P°) • B u t - i (0 ) = i 9 91og h = - 9 i 91og h = 9(L ) , v a v a v a a since — a hence i (91og h ) = i (91og h.) v a v 3 - X • a n N ct . on dz. + z, {L = - i 9 l o g h -X } d e f i n e s a unique a v a a cc=0,. ,n L e £(P , - h a = 9 log h " 3 3 D , then n g , Z„ = i (91og h„) + A 3 v 3 • 3 = i 9 l o g h_ + X v 3 3 h 3 1 2 s i n c e 9z„ = 9z„ = 0 . 3 3 a h = 1—1 a 'x ' a g = 9 log h + 9 log z 3 " 3 matrix 0 = 8 O h h "S = 9(91og h ) a a a a + log|z | ) = 9(log h ) + 9(log z o +' • ' Z„ be t h e c a n o n i c a l c u r v a t u r e ( P?) X„ 3 a be h o l o m o r p h i c c o o r d i n a t e s on 9(log h ) = 9(log h 2 a 9X = 0 a 38 Hence i ( Q ) = 3(L) . v L = V - i h , and o D , where V = V + V . By the proof of Theorem 1.3.1 D i s the canonical connection we see that associated 39 4. GROTHENDIECK RESIDUE AND THE THEOREM OF RIEMANN-ROCH-HIRZEBURCH Let us f i r s t formulate the Holomorphic Lefschetz Fixed Point Theorem of Atiyah-Bott. dimension Let X n , and l e t E X Then a holomorphic geometric be a compact complex manifold of be a holomorphic vector bundle on endomorphism of E consists a p a i r X . (f,<)>) * where f: X X i s a holomorphic map and cb: f E -»• E i s a holomorphic bundle homomorphism. Under these circumstances there are induced homomorphisms of cohomology groups * H (X,0 (E) ) — X k • H (X,0 (f*E) X k H k W > H (X,0 (E) ) X k where the f i r s t map i s the standard pull-back, and the second i s induced by f u n c t o r i a l i t y , by < f > . The composition gives a <C-linear endomorphism of the finite-dimensional complex vector space denote H (f,cf>) . Let z ,...,z I n n the isolated fixed point hood N(p) of p be l o c a l holomorphic coordinates centered at peX n i i (z.) = f (z) = f (z,,..,z ) x I n f (1) X . Then on a small neighbour- i s 0 = (0,..,0) , where 1 * of f : X i n X , the only common zeros of the functions z. - f ( z , . . , z ) i = l , . . , n l I n f H (X,0 (E)) which we ~X V ( (b,E) = Res P P are the l o c a l coordinates of } trace <j> (z) dz, dz 1 n 1 n z - f (z) , ,z - f (z) 1 n f ( z ) . Let 40 where R e S p i st n e Grothendieck residue symbol and representation of <f> on i s the matrix N(p) . Now i f the fixed point set X^ a l l the fixed points of cf> (z) f of f : X -*• X i s finite, i.e.; are i s o l a t e d , then the Holomorphic Lefschetz Fixed point formula can be stated as, , n (2) I (-D k-0 L(f,<j>): = trace H(f,<f>) = k k £ f peX v (<f>,E) P v see D. Toledo [36]. Algorithms For Calculating Res P There are two methods f o r c a l c u l a t i o n of Res , one i s purely p algebraic, and the other i s a n a l y t i c . We w i l l describe both of them f o r the l a t e r use. Let fu(z) = z^ - f ( z ) , i = l , . . , n , and l e t W(z) 1 holomorphic function defined on (i) be any N(p) . Algebraic Calculation of Res P m. There e x i s t integers by h_. (z) j = l,..,n . defined near z^ i s i n the i d e a l generated This follows from the f a c t that i s o l a t e d fixed point of B. . (z) iru > 1 , so p i s an f ( z ) . Hence we can f i n d holomorphic functions p such that n m. Z.l = T B. .(z) h.(z) j=l ^ 1 1 This granted, one then has 3 41 W(z).dz. .,. .dz 1 n h (z) ,. . .h (z) 1 n Res ,P m^-1 ~ l z^ ....z^ m n i s equal to the c o e f f i c i e n t of expansion of W(z) det(B. . (z)) , see P.F. Baum and R. Bott [ 2 ] . 1 (ii) i n the power series 'J Integral Representation of Res P Let S c N(p) be a 2 n - l sphere around p . Recall Y'.L.L. Tong [ 3 8 ] , that the Grothendieck residue can be expressed as an i n t e g r a l formula: W(z) dz, dz 1 n h (z),.. . ,h (z) 1 n Res =k f W(z) n where k |h(z)| J s n 2 n k i s constant and If p I ^ (-l) h(z) k + 1 h (z)dh ..dh, ..dh dz ..dz k 1 k n 1 n n = (Eh.(z) ' I 2 1 1 / 2 ) i s a transversal fixed point of f^(z),...,f (z) form holomorphic n where df(p) P S f: X -* X , namely l o c a l coordinates around then by the algorthim (i) i t can be seen e a s i l y Res . W(z)dz,....dz 1 n h (z),..,h (z) 1 n p e X , (or see D. Toledo [36] W(p) det(I-df (p)) n i s the d i f f e r e n t i a l of f at p ; df(p): T^X -> T^X . 42 In t h i s case we have x> (6 v) trace (ft (p) p ' - det(I-df(p)) (3) V ( ( f ) E ) Moreover i f a l l the fixed points of number L(f,cj>) X , '* T L(f f } _ v trace (ft (p) det(I-df(p)) - I peX i s the fixed points set of Let manifold are transversal, then the Lefschetz i s simply (4) where f V f : X -> X . be a holomorphic vector f i e l d on a compact complex X , induced from the 1-parameter group of automorphisms X , and l e t V: 0 (E) -> 0 (E) ~X ~X E -* X . of be the V-derivation induced from the e>-equivariant 1-parameter group of automorphisms vector bundle a Then f o r each a of a holomorphic t e C , we have a bundle isomorphism — 1 -1 * a : E -> (a ) E . By taking the pull-back a t i v e to a z X -> X , 1 *r e l * we obtain a bundle isomorphism <ft ( ) v • Hence we have t a 0 t a natural geometric endomorphism <|> E E t (cr^_, cf>^_) of E f o r each t , moreover i s compatible with the C-action. If p i s an isolated zero of fixed point of (5) a V , then at least f o r small values of v (<j> .E) P t y ^ Res P i p i s an i s o l a t e d | t | , hence trace (ft (z) dz,....dz _t 1_ n z -a. (z) , — ,z -a. (z) I t n t 43 makes sense at least f o r small values of |t| In t h i s section we w i l l compute a much more e x p l i c i t dependance at the case when p on . v^Ctb^E) , and w i l l obtain t . Before doing this l e t us look i s a simple isolated zero of V . • An i s o l a t e d zero p of V 1 9a. 2 det i s said to be simple, i f ? 0 9z. (P) where V N(p) or equivalently p Z c. X , where If Z p = I a i=l 9/9z , i s a non singular point of the closed subvariety i s the zero set of V . i s simple i s o l a t e d zero of V , then fixed point of a at least f o r small values of p i s a transversal | t | . In this case we get by (3) and (5) trace l (p) c ) t V But we have P ( ( | V da (p) = t d> (x) : E . . -> E t a (x) t e = det<l-da Ap)) E ) L ^ by section 3. Moreover by d e f i n i t i o n of which i s the p u l l back of t a r e l a t i v e to a , we get ^ ( p ) t t = Thus we have a.^x) : E •*• E t - l a ~-l ^ (p) t = a A * ^ (p) > hence t ( " < f > (p) = e t x ) tV(p) 44 (6) trace e V (<(>.,E) => P t tV(p) det(%-e t U p ) i where n 8a. L(p) = Let us take cotangant bundle of N(p) E = X V (P) 3z. = I /v (T X) , where and a. 3 z . i=l T X 0 < p < dim(X) i s the holomorphic Then 5 = ( A da P s i s a a-equivariant 1-parameter group of automorphisms of If a l l the zeros of called V 1 ) :E t s E = AT (T X) . are simple (such a holomorphic vector f i e l d i s nondegenerate) then we have the theorem of G. Lusztig [ 3 1 ] , * n . L (<)>., /?(T X)) = I (-1) dim t . „ i=0 . 1 for a l l t and for a l l 0 < p < n . independent of t e C . HNX/) Namely the Lefschetz number i s For a V-equivariant vector bundle E we have the b i j e c t i v e correspondence between a-equivariant group of automorphisms of Lemma 1.3.2. E and the set X , 1-parameter H°(X, Horn(E,E)) , given by So what G. Lusztig proved i n our language i s , that there e x i s t a a-equivariant 1-parameter group of automorphisms p * such that the Lefschetz number, L,{§_^, A (T X)) i f the holomorphic vector f i e l d induced from a a of p * A (T X) i s independent of i s nondegenerate. Thus we ask the natural question; i s t h i s the case f o r any V-equivariant holomorphic vector bundle? t , Answer of t h i s question i s no i n general, 45 A simple example i s the following: Let V be the holomorphic vector f i e l d on \ ' ° ' o a where V A ^ A^ . x x = e t Then we see immediately from the example 1.3.2 that i s a nondegenerate vector f i e l d with zeroes Let o : U s of ) :U U then of U {e = [ 1 , 0 ] , e =[0,1]} . be the natural extension of a , then * A-1 (a (P^ induced from * * -> U, „ i s a-equivariant 1-parameter group of automorphisms 1,2 1,2 A-1 - I f V i s the V-derivation of U induced from (a ) , 1,2 V(e.) = A . I U 1,2 t * S I i = 0,1 . i s i n the form But then by Lemma 1.3.2 any V-derivation V + a f o r some a e C . Hence f o r any * a-equivariant 1-parameter group of automorphisms of U , we get the Lefschetz number t(A +a) tfXj+a) Q * L(<l> ,U, J t'"l,2' by (2) and (6) . e , .. t ( A - A ) 1-e = 1 . e + t ( () 1-e W Hence L ( ( | ) . * t' l 2 U tA ta • 0 n ) = 6 ( e + S tA. 1. } * Therefore L(<j>,U^ ^ t i s not constant for any choice of a e C or equivalently f o r any a-equivariant 1-parameter group of automorphisms 46 a of 'U 1 . We note that 2 L(<J> ,U ) = e ' ( e +e a 0 1 2 i s Riemann-Roch f o r the i n v e r t i b l e sheaf 0(1) . ) = 2 which I t i s easy to see that, one can obtain Riemann-Roch f o r any i n v e r t i b l e sheaf on IP 1 holomorphic Lefschetz fixed point formula with t h i s method. from the We w i l l discuss the Theorem of Riemann-Roch and Hirzebruch l a t e r , actually our main concern i n computing v^((j) ,E) w i l l be to obtain t h i s theorem t through holomorphic vector f i e l d s . I t i s easy to see that A o L(,$^,XJ^ ^)= 0 ° 6 c t f o r any A, = e ( ^ Q ^ A ) - e q u i v a r i a n t 1-parameter group of automorphisms of U . In fact 1/2 L ( *t' l,n+l U ) = ° f o r a n y V a (A^=f=A_. t 0 = e i=j=j)-equivariant 1-parameter group of automorphisms of This observation raises the natural question; i f L(c|>,E) t of t U^ . i s independent f o r a fixed a-equivariant 1-parameter group of automorphisms of then i s i t true that L.(i/; ,E) i s independent of 1-parameter group of automorphisms of E . t f o r any a-equivariant Unfortunately t h i s i s not true either, and our simple example i s the following: Let V be the holomorphic vector f i e l d on E , P"*" induced from 47 *0 ° 0 0 * h a : tr „ U „ be the canonical extension of t 1z2 1j^ a , then t o Let * a ® a : ® U o t ® a * L( VT P * P . If t(A -A 1 i s a a-equi variant 1-parameter group V A ^ =i|= - A ^ , then . ,, ) t(A -A^) + , . "t( W 1 ) induced Hence we get — 2A,t e ( * 1 T (P i s the V-derivation of i=0, 1 . = Q 1-e Now take X i ->- T —2A„t e ° = } ' V(e.) =-2A. i i 1 P = e * 1 T • IP , then t t i = T of automorphisms of from 1- X " 6 1-e L(c)> ,T* IP1 ) i s not constant. t But we know * 1 by the theorem of G. Lusztig that L(^ ,T (P ) i s constant, i f tp^ i s * 1 -1 t * 1 * 1 the geometric endomorphism of T IP induced from (& ) : T P ->• T P s a s We can actually see this quite e a s i l y i n t h i s case. V-derivation of V(e ) = A Q - A T Q * P ( induced from , V(e ) = A 1 t L 1 \ ' T t ( ± p Q > = - A s t ( + i s the - I t ) # then we have by section 3. tIA^-AQ) V c W 1-e Remark: (^ If 6 Therefore V i' u A ^ i = - 1 1-e The main idea f o r asking whether L ( f ' ) ( or not i s to t r y to give a formula f o r dim H (X,0 (E)) q l E t i s constant i n terms of 48 eigevalues of 9a. i ~ ( (p)) and V(p) as j p runs through the zero set of V . C. Kosniowski gave such a formula i n [27] f o r E = P * his formula just from the fact that L($^ A T X ) r p * A. T X . He gets i s constant. We w i l l give some applications of his formula i n the next section. Now we can s t a r t computing v (<fc ,E) . Let V be a holomorphic P t vector f i e l d on a compact complex manifold X , induced from the 1-parameter group of automorphisms a of X , and l e t V: 0 (E) -* 0 (E) be the X ~X V-derivation induced from the a-equivariant 1-parameter group of automorphisms a of a holomorphic vector bundle E -> X . Then for each a natural geometric,' endomorphism of E , where (0^,$^) t e C , we have < j > = (a^ ) : a^E-^E. For an i s o l a t e d zero p of V , we have the holomorphic l o c a l coordinates z ,..,z on a small neighbourhood N(p) of p , such that 1 n trace d> (z) dz, dz _t 1 n z -a (z) , ,z -a (z) I t n t v (<l> ,E) = Res p t p T makes sense at least for small values of W of | t | , say i n a neighbourhood 0 e C . )a (z) i Let L(z) = 2fj:(da (z)) = t representation of the V-derivation of (da —i t * * ) : T X -> T X , where s v. (z) , w_. (z) T X v| , , = T 'N(p) ^ for 1 < i < n and be the transpose of the matrix 9z. on ^ a. I N(p) , induced from 9 . Now consider functions 9z^ 1_< j < n on N(p) , defined from 49 ,the following formal i d e n t i t i e s ; 3 , q det(I +AV(z)) = Y c.(V(z))A = n (1+Av.(z)) q . ~ i _• , i i=0 i=l det(I +AL(z) n where q = i s the rank of z = (z,,..,z ) . 1 n character of L(z) by V(z) Td(z,t) = n y i=0 c. ( L ( z ) ) \ . E , I n n j=l = 1 i s the Then we define for by 1 n tw.(z) II • . , tw. (z) i=l l 1-e (1+Aw . (z)) 3 k x k i d e n t i t y matrix, and (t,z) e C x N(p) q tv^z) £ e i=l Ch(E,z,t) = and . The Chern and The Todd class of where the functions on the right-hand 1 side should be regarded as standing for the corresponding power series expansion. W x N(p) Both , since functions i n the holomorphic Ch(E,z,t) and Td(z,t) are holomorphic c_.(V(z)) (resp. c_.(L(z))) v, (z) (resp. w„(z)) and k I functions. i s the elementary c.(V(z)) c.(L(z)) l j Then we have the following. , Theorem 1.4.1. .dz trace l ( ) dz . ( ) z t (1) v (*. P ,E) Res z -a ,(.z) , ....,z -a (z) L. t n t Ch(E,z,t)Td(z,t)dz — n for Res p t e W - 0 . i functions i n 1 a;(z),. . . ,a (z) 1 n dz n symetric are 50 . To prove t h i s theorem, we need some "elementary facts from Linear Algebra, Let A. whose proofs we w i l l include f o r completeness. M be any n x n e C i = l,..,n matrics over by the following equation 1 where x i s an indeterminate. A . C . We define n det(I +xM) = II (1+A.x) i=l i s c a l l e d a c h a r a c t e r i s t i c root of 1 M . We note that Lemma 1.4.1. X M , then where e l n Y A. 1=1 Trace (M) = X ,...,e n If A , . . , A 1 n n n and det(M) = 1 1 A . . i=l are the c h a r a c t e r i s t i c roots of M v M_ are the c h a r a c t e r i s t i c roots of e = I M — r=0 r! M e g£ (C) n Proof. By induction on n . The case n = 1 (n-1) x (n-1) Now we assume the Lemma holds for matrices. be any c h a r a c t e r i s t i c root of M , then we can f i n d that the matrix is trivial. Let A^ g e GL^JC) such g "'"Mg i s i n the form, l*' 0" 0 0 A g "'"Mg (1) for some N e g£ (C) . I f A „ , . . , A are the c h a r a c t e r i s t i c n-1 2 n n roots of N , then An , 1 g Mg . Hence 1 are the c h a r a c t e r i s t i c roots of A„,.-.,A 2 A , ^2'"*' n X n a r e t * ie c n a r a c t e r ; '- s t : '- c r o ° t s of M But 51 then, we have by (1) 2 n X Now by induction hence e , e e ,...,e X ,...,e are the c h a r a c t e r i s t i c roots of are the c h a r a c t e r i s t i c roots of n they are the c h a r a c t e r i s t i c roots of M e , since q Mq e -1 e^ e , ^ , so -1 M = g e g . Q.E.D. Corollary 1.4.1. If ^i'*-'^ tM of Me g£^(C) , then trace n ^ tM t(trace M) det e = e = Proof. then f o r any If t e C r e " II e i=l t = r e tx. 1 I e i=l ^ e c h a r a c t e r i s t i c roots _ = Ch(M,t) , and i . ^ , f o r any t e C X X , ..,X 1 n tX ,..,tX a n are the c h a r a c t e r i s t i c roots of M , are the c h a r a c t e r i s t i c roots of n tM , hence the claim follows from Lemma 1.4.1. Q.E.DJ Proof of Theorem 1.4.1 We assume f i r s t that Then left-hand side of (1) p i s simple isolated zero of V . 52 trace V (<j> ,E) p t = Res Z by (6) • But then by l ~ P a l l t e W . R.H.S = (1) compute using , d e t ( I - e l e tus = — t and ' Z n ~ Ch(V(p) t Now of t ^ a 0 C o r o l l a r y 1.4.1, V <VE) for d> • ( z ) ; d z , .. . d z t 1 n i p look Res n the algorithm Z we e tV(p) d e t ( I - e ^ t L ( p ) ) g e t t) t L ( p ) ) a t the right-hand side of (1) C h ( E , z , t ) T d ( z , t ) d z , ... d z 1 n •{ P t ^ trace a ( i ) . i ' " - ' Since p a n i s simple zero of n 'N(p) l o c a l = / a. 8/9z. . . i i i = l coordinates functions B. algorithm ( i ) R.H.S. i n .(z) = — _1_ . n , i t f o l l o w s N(p) such , and that . {constant that a.,..,a I n therefore z. term {Ch(E,p,t)Td(p,t) = of Z j=l there form e x i s t B. .(z)a.(z) i»D 3 Ch(E,z,t)Td(z,t) det(B. 113 .(p)} a holomorphic holomorphic . Hence det(B. , f o r t e W by the .(z)} . 53 since Ch(E,z,t)Td(z,t)det(B. . (z) ) i s a holomorphic function i n z e N(p) Ai3 ' -1 (C. . (z)) =' (B. .(z)) i,J »3 Let 1 3a. 3z Then -ip) = (C 1,3 .(p)) = (B. . ( p ) ) " 1,3 1 . a 3a. . = j=l I Hence C. .z. , ^ ( p ) = C. 3a. (p)) = (det ("^(p))) oz^ det(B. 1,3 Therefore the det(L(p)) „ R.H.S w Q n f n\ - - L Ch(E,p,t)Td(p,t) Of (1) —— T\ n det(L(p)) But by d e f i n i t i o n of Ch(E,z,t) Ch(E,p,t) =Ch(V(p),t) and Td(z,t) , we have and n tw. (p) Td(p,t) = tw. (p) 1=1 n 1-e 1 where det(I+AL(p)) = n n (1+Xw.(p)) i=l 1 Thus we get R.H.S. of (1) = , . . n v _1_ n Ch(V(p) ,t) det(L(p)) _1_ n Ch(V(p),t) det(L(p)) tw. (p) l *' tw (p) 1-e n > for t e W ± t n n i=l n det(L(p)) tw. (p) (1-e ) 1 . (p) , > for t e W -1 54 Ch(V(p) ,t) Ch(V(p) ft) n tw.(p) n (1-e ) i=l det(I-e 1 v (cL ,E) p t for t t L ( p ) for t e W - 0 ) W - 0 . E Hence we have the r e s u l t for simple i s o l a t e d zeros. Let (i.e. p be an i s o l a t e d zero of m = dim(0 P,X . (a..,..,a )0 / "p,X 1 V , with m u l t i p l i c i t y m ; ) , where n n vl = I a. 9/9z. 'N(p) i=l and V has only one zero z.,.., z 1 n p i n the neighbourhood holomorphic coordinates on N(p) a(z) = (a. (z) ,.. . ,z (z)) : N(p) -»- C . 1 n regular value r = (r^,..,r ) e C n n . N(p) of p ) , and We denote by By Sard's theorem we can f i n d a such that 1/2 n r = i I |r.| i s s u f f i c i e n t l y small and the function a(z) - r = a(r,z) only simple i s o l a t e d zeroes i n a small neighbourhood of loss of generality, we may zeroes i n the neighbourhood Let a^_(r,z) assume N(p) a(r,z) of say w i l l have p . Without w i l l have only simple p . be the flow generated by the vector f i e l d isolated 55 V(r,z) = £ i=l a.(r,z) 9/Sz. 1 , where 1 a.(r,z) = a.(z) - r. • 1 1 Since we 1 are i n a l o c a l coordinate, by the l o c a l study i n section 3, we can f i n d a^(r,z)equivariant 1-parameter group of automorphisms E| . , , such that N(p) r •+ 0 through regular values of V(r,z) -* V(z) as a (r,z) -* a Az) t t r -> 0 and a (r,z) converges uniformly i n a(z) for each C h ( E , z , r , t ) -> C h ( E , z , t ) through regular values of of t t . N(p) as Therefore converge uniformly i n a(z) for each N(p) t ; where i V(r,z) and i s the V ( r , z ) - d e r i v a t i o n of C h ( E , z , r , t ) i s the Chern character of <)> (r,z) -> f> ( ) < z t t values of converges uniformly i n a(z) , where (6^_(r,z) endomorphisms corresponding Now * i T X , . 'N(p) L(r,z) if L(r,z) induced from to and V(r,z) < f ^) ) z as th a r e and a e ( ) z t , Moreover r -> 0 t a^(r,z) . o~^(x,z) through regular geometric respectively. i s the transpose of the V ( r , z ) - d e r i v a t i o n of I t (da (r,z)) , then s , where i s o l a t e d , then we have f o r P ^ ' - ' ' P L(r,z) = L(z) , since a^(r,z) = a^(z) - r ^ . Td(r,z,t) Since a l l the zeroes t =)= 0) induced from N (p) depends only the derivatives of Td(z,r,t) = Td(z,t) of E|^ ^ ^ i s the Todd class of M °^ a(r,z) are Hence L(r,z) . simple t e W - 0 ; ( i . e . s u f f i c i e n t l y small values 1 56 trace ( ^ ( r j Z j d z ^ . . . . d z ^ m I j=l (2) Res z -0 ( r , z ) , . . , z -0 I t m I .n j=l n a(z) . (r,z) Ch(E,z,r,t).Td(z,r,t)dz Res a For each f i x of t (r,z), t e W - 0 , let dz n ,a (r,z) n r -* 0 through regular values Then we see that, by the continuity of Grothendieck residue, the r i g h t hand-side of (2) tends to the right-hand side of (1) and s i m i l a r i l y the l e f t hand-side of (2) tends to the left-hand side of (1) (a (r,z) t 0 (z) t Hence we have the as r 0 , since V(r,z) -> V(z) as r -> 0) . claim. Q.E.D. Remark: Theorem 1.4.1 generalizes the theorem of N.R. p [ 3 4 ] , where he a c t u a l l y proves this theorem for -.^ (A do^ t ID* O'Brian ID * ) : A^T X ->• A^T X We would l i k e to thank L. Roberts for simplifying our complicated looking Todd class for us. We wouldn't have had the following r e s u l t without his s i m p l i f i c a t i o n . We keep our notation as Proposition 1.4.1. Let before. A,,..,A I n be the c h a r a c t e r i s t i c roots 9a. of v(p) L(p) = . 9z. (P) , and l e t u,,..,u. 1 q be the c h a r a c t e r i s t i c roots of 57 We set X.(t) = 1 — — tX. i f X. f 0 i and X. (t) = 0 i otherwise, 1-e and tp. Y_. (t) = e 3 for 1 < i < n and i < j < q . Then f o r t e W - 0 , we have v U> ,E) = — p where P v (<b ,E) p t t i s a polynomial P(Y (t),...,Y (t),X (t),..,X (t),t) 1 g in n + q + 1 i s a meromorphic function of Proof. 1 n variables, i n p a r t i c u l a r i l y t in W . By theorem 1.4.1 we have f o r t e W - 0 Ch(E,z,t)Td(z,t)dz v (<f> ,E) P — t Res n p but then by the algorithm .(i) m i z n ....z a (z) , 1 1 dz^ ,a (z) n n v (<j> ,E) = l / t . p u n {the c o e f f i c i e n t of m i n the power series expansion of Ch(z,t)Td(z,t)det(B.i>3 .(z)} B. .(z) and f o r some H L > 1 k = l , . . , n . 1,3 K Therefore i t i s s u f f i c i e n t to prove f o r each multi-index (i ,..,i ) for some holomorphic functions n i +..+i , 1 n n ^ 1 „ n 9z, ...3z 1 n and i s a polynomial (Ch(E,z,t)Td(z,t)) i n t , X_^(t) z=0 Y_. (t) , 1 < i < n , 1 < j < q . But then by the product rule - of 58 d i f f e r e n t i a t i o n i t i s enough to show N 1 n (Ch(E,z,t) ) and ~ 1 ~ n dz, .. . 9z 1 n z=0 i,+..+i , 1 n ri (Td(z,t)) i dz, . . . dz 1 n For i , +..+i , 1 n i dz, 1 1 show ^ ...dz . 1 T are i n t h i s form. z=0 (Ch(E,z,t)) , by the chain rule i t i s enough to t=0 n n n — 9v, ...9v 1 n 1 n form, which i s obvious. Now l , +..+i , 1 n 1 q tv.( ) I e ) i=l Z (ChCE,z,t) = 1 z=0 for (Td(z,t)) 1 3z, . . .9z 1 n i s i n the desired 1 , i t i s again by the chain rule, t=0 n enough to obtain the desired form for > 1 1 1 9w, 1 tw. (z) n n X ...9w (Td(z,t) = n -i=1 n we only need to consider one dimensional tw. (z) 1-e , hence z=0 case where the r e s u l t may checked by d i r e c t d i f f e r e n t i a t i o n , considering the case separately. be A = 0, A =f 0 Q.E.D. 59 Remark: I t i s actually obvious by Theorem 1.4.1 and by the algorithm (i) that v^((j> ,E) i s a meromorphic function of t t with a pole at t = 0 . v^(<(>,E) in W has a unique analytic continuation as t a meromorphic function on the whole complex plane. n k L(<j> ,E) = £(-1) k=0 Now l e t us look at the fixed-point formula. trace H^fa ,<}> ) i s a holomorphic function of t i n the complex plane. On the other hand we have the meromorphic functions v^($^,E) complex plane such that t L(<j>,E) equals to E t Vp(cJ) ,E) on the for sufficiently peZ small [t| , by the fixed-point theorem, where which i s f i n i t e . Z i s the zero set of V , Hence by the uniqueness of analytic continuation the fixed point formula (1) I L(<|> ,E) = t V (<f> /E) t pez then holds f o r a l l t . Hence the singular parts of these series, must cancel out as we sum over p e Z , and the constant term must add up to * the l e f t member at t = 0 . But f o r t = 0 < | > = i d : a E -* E , hence the l e f t member of (1) reduces to L (<j) ,E) = -n 1 ( - l ) dim H (X,0 (E)): = X (0 (E) ) 0 X X k k By theorem 1.4.1 we have Ch(E ,z,t)Td(z,t)dz ..dz for t e W -0 v (d^jE) = — • Res p t' n p y t a l' n 60 Ch(E,z,t)Td(z,t)dz^....dz^ Since Res (E,t) = Res < P P holomorphic function of of t n > V t c W n (by the algorithm (i)) . \ i n the power series expansion of term of Res^(E,t) V (E,t) . But the c o e f f i c i e n t of P expansion of Res^(E,t) is a t n Then the c o e f f i c i e n t w i l l be the constant i n the power series equals Ch(E,z)Td(X,z)dz ...,dz t Res a a Ch(E,z)Td(X,z) = the c o e f f i c i e n t of of Ch(E,z,t)Td(z,t) residue). (2) where i'--" n t i n the power series expansion (by the l i n e a r i t y property of the Grothendieck Hence we get X ( 0 (E)) = I peZ Ch(E,z)Td(X,z)dz Res 1 1 .. dz n V Now R.H.S. of (2) can be viewed as the value of the global residue on a section of 0 . Z defined as follows: N(p) of p e Z Let w This operator, Res: T(Z,0 ) -> C , i s Z be a holomorphic function i n a neighbourhood representing the function Res(s) = £ Res peZ s e T(Z,0 ) Z w(z)dz„....dz 1 n ci^ / > • • • f 3. at p . Then 61 i s a well defined linear map. Moreover by [9] we have a commutative diagram r(z,o ) R e s ) C H (X,ft ) n I f we apply n X (2) to this commutative diagram, we get the theorem of i Riemann-Roch and Hirzebruch for V-equivariant holomorphic vector bundle E , namely (3) X(0 (E)) = X Remark: Ch(E)Td(X) We believe our concepts are j u s t i f i e d by this R. Bott has obtained (3) i n [5] for the t r i v i a l l i n e bundle. formula. I t i s this b e a u t i f u l work of R. Bott, who l e t us study this subject. H i s t o r i c a l l y , during the course of this work, the need f o r Theorem 1.4.1 came as follows; We wanted to prove Riemann-Roch and Hirzebruch formula for l i n e bundles by using Bott's method for the t r i v i a l l i n e bundle. l i n e bundle L We started with a projective variety on i t . We computed trace <|>(p): L^ t Blanchard equivariant imbedding theorem knew how to compute this trace i n case and Lemma 1.3.2). X X = X and a by using N -* (P , because we already N P (see Example 1.3.1 ( i i i ) To our suprise, we found exactly the expression which i s i n Theorem 1.4.1. This showed us how to proceed to prove Rieamann- Roch and Hirzebruch formula for general equivariant bundles. 62 5. CHERN CLASSES OF EQUIVARIANT BUNDLES AND APPLICATIONS. There are two spectral sequences which are p a r t i c u l a r l y useful i n analyzing the cohomology of a compact Kaehler manifold which has a vector f i e l d with i s o l a t e d zeros. studied further i n [ 1 3 ] . These were introduced i n [11] and We w i l l begin by r e c a l l i n g these spectral sequences and some of t h e i r interesting consequences. Let manifold X V be a holomorphic vector f i e l d on a compact complex of dimension n . Then we have the anti-commutative t diagram of sheaves. Q + n,0 ft r 4- 0 -y c p,0 -> £ Or v p-1,0 ,i 0 + 0 X Is 0 + 0 -y £ ° ' = £°' / i„(£ v Z to the presheaf 9 \. 1v c0,0 + v q y v P-1 0 -y tt where 9 q 1 , q ) £°' / i (£ q v 1 , q t i s the sheaf of ) f o r each £°' -module associated 0 q = 0 ,. .. ,n 63 F i r s t spectral sequence of V . Take the q-th cohomology of the p-th row diagram. £r + t i n the By the Dolbeault lemma, we have H (X,fi ) = ker{3: £ ' ( X ) -> E ' q_1 (X)) Then the above diagram gives the f i r s t spectral sequence of V , q I E P , q 1 P P = H (X,fi ) => H q P q P q (K,D) P where q+1 (X)} / 3(& ' P K i s the t o t a l complex associated n — v D = 3 + i , namely K = I K , k=-n r to the diagram with the derivation K r = I K' p+q=r P and q K' P = £" ' (X) . q P q Second spectral sequence of V. We now take the q-th cohomology of the p-th column on the global sections, namely " E ^ ' - * = k e r { i : £ ' ( X ) + e^'^X)} / i ( £ q P v v Then the diagram gives the second spectral sequence of 2 By [13] H zeros of Z q P q P aftd -z P V , 1 1 =0 "x q fi } q note that tf° = 0 ' (X) \ ' - = > H - (K,D) = ker{i : ft -> V q + 1 / i (fi q-1 v ^ ' ' ^ = H (X,H ) P / i (fi ) 1 v we have two spectral sequences. q q + 1 ) i s supported on the set of f o r each q = 0,1,...,n . i s the structure sheaf of We Z . Hence 64 \ I ] : If X ™ E 'P = H (X,fi ) => H - (K,D) q q P q = H ,(X,H ) => H P q P-q P (K,D) i s Kaehler, then a theorem of J.B. C a r r e l l and D. Lieberman [11] says that the f i r s t spectral sequence degenerates at E , namely the l i n e a r operator 1 i ^ : H (X,fi ) -»• H (X,fi ) q In p a r t i c u l a r , i f the dimension of p =|= q get X P q P i s zero, then i s zero. 1 H (X,fi ) = 0 for P (by comparing with the second spectral sequence) . q Therefore we a graded ring isomorphism. (1) 9 """E ' oo P q = ® H (X,fi ) = 9 H (X,fi ) = H*(X,C) = gr(H°(K,D)) I q P P P On the other hand from the second spectral sequence, we get i : C E '" P q = 0 for a l l p + 0 or q + 0 and 1 1 E^' 0 = H°(X,0 ) = T(Z,0 ) ' Z 2 Z Hence = r(Z,0 ) = gr © " E E ' " * = E°J° XI (H°(K,D> . Z 0 • Moreover the f i l t r a t i o n i s t r i v i a l , i . e . , 9 jj( r 0 • (K,D)) = H (K,D) , H hence the edge morphism defines a ring isomorphism 0 • e : T(Z,0 ) = H (K,D) Let s e T(Z,0 ) , and l e t p ,...,p _L z smooth functions f on JC which i s given as follows; denote the zeros of X , whose germ at p^ represents V . Choose s(p^) i n 65 0 ~Z . Then g k £ f. e £ i=l " = ' (X) r e p r e s e n t s the image o f s in 1 e°'°(X) / i ( £ 1 , 0 v (X)) . But then 3g = 0 Q in £ 0 , 1 (X) /i (£ L f l v S i n c e the bottom row i n the diagram i s a f i n e r e s o l u t i o n o f (X)) 0 by . [13] Z and H (X,0 ) = 0 for q (2) 0 •+ r ( Z , 0 ) + £°'*(X) / i z Therefore there e x i s t s i V 3g q > 0 , we g e t an e x a c t sequence n 1 = - 3i g V n 1 g., e £ 1 - = 0 , hence (X) 1 , 1 (^^(X)) such t h a t 3g_ e k e r i : £ 1 v - 1 , 2 . i (g_) = 3g^ . v 1 0 But then ( X ) + £°' (X) . 2 Since 0 -+ H 1 + ker i I v i 1 * / . i v (£ '*) 2 " i ' i s a fine resolution of then we i n t h i s manner, we 1 (3) 1 and 2 2 g^ e £_ ' (X) can f i n d g. e f 1' (X) i - H H^CXjh' ) = 0 such t h a t 3g^ = i^Q-^ are a b l e t o c o n s t r u c t a sequence q > 0 , ' Proceeding o f forms such t h a t 3g. = i (g.,,) 1 V 1 +1 for n But f o r every 1 (3) says t h a t the form G = ^(-l) 0 c o c y c l e f o r the t o t a l d i f f e r e n t i a l i = l,..,n - 1 i 1 g. e K 0 n = £ P # P ( ) X i s a p=0 1 D = 3 + i \ . I t i s easy t o check 66 that the cohomology class made. e(s) The edge morphism e : i s independent of the p a r t i c u l a r choices 0 • T(Z,0 ) ->- H (K,D) ~Z i s p r e c i s e l y the map = G , i t ' s inverse i s defined as follows; given an a r b i t r a r y cocycle S o l g. e K , then i=0 G = £°' (X)/i (£ 1 v s e F(Z,0 ) Z 1,1 (X)) . By the edge isomorphism n O F n + 1 0 • (H (K,D) ) = gr 1 = - ^9j^ > hence 3g x Q v Q i s zero g Q . Then G -+ s provides the inverse. from the f i r s t spectral sequence, we get the f i l t r a t i o n 0 • 0 • -n H (K,D) = H (K,D) D T(Z,0) = F - 8g By the exactness of (2) there exists a unique whose image i s Now Of G 0 • -n+1 H (K,D) D e : 0 • r(Z,0 ) = H (K,D) o ... O F n © p=0 ... O H 1 ^ F° ? 0 0 • -1 (K,D) 5 0 • 0 H (K,D) 3 0 , we get a f i l t r a t i o n such that 0 • -p 0 • -p+1 H (K,D) * / H (K,D) ^ n = gr(T(Z,0 ) = © Z p=0 F -P / F -P +1 But then by (1) we get (4) H*(X,C) = gr(T(Z,0 ) z The natural question i s now to understand the isomorphism (4). For example, how w i l l we define the Chern classes of holomorphic vector bundles i n gr(T(Z,0 ) ? ~Z This was the our o r i g i n a l question which lead 67 us to study equivariant bundles. We only know how to define the Chern classes of V-equivariant bundles i n gr(J(Z,0 )) as we w i l l discuss ~Z now. 0 • Let us f i r s t look at the f i l t r a t i o n -p 0• H (K,D) ^ which arises from the f i r s t spectral sequence. For each of H (K,D) p > 0 we have a f i l t r a t i o n F i=0 i „F - , o P e°'°(x) = = £ IW-P.I 0 , 1 ^ ^ ( x ) +...+ a ' ( x ) c P + (x) + £ " F 1 , 2 ( x ) + ..+ £ " P p , p + 1 I K° = (x) c t L , ± i=0 n-1 K1 = " W I e'i (x) +1 i=o • " Then from the general theory of spectral sequences, we get H°(K,D) P = the image of { Ker{D| I P 1 VP'P Let D| „ 'l -p,0 P G = q„ + g +..+ g e 0 1 p then F n 3g p y = 0 , hence g ^p P cohomology class P,P [g ] i n H (X,f2 ) P P P X P P and / H°(K,D)- '° / D( F P+1 be the cocycle f o r defines a cohomology class ^ H (X,ft ) . Now i f g^ e £. (X) P 1 = H (X,ft ) = H°(K,D)P Prl I n H°(K,D) = ker{D: K ° + K } / D(K ) , in in _ = F '° + F } p P' [g 1 ^P J i s a smooth form representing the then 9g^ = 0 . But then 68 3i v (g ) = - i 9g = 0 p v p represents there i exists g , e £ p-1 - (g ) e 8 " ' p-1 2 P _ 1 n then 3g = 0) , hence p the cohomology c l a s s P v (since i-(<3^ j) hence t h e r e P 1 , P ( X ) 1 ( X ) , since represents exists £ H (X,fi [i (g )] v i (g ) e fc v p - P p such t h a t P 2 1 , "*") = 0 . P ( X ) Therefore i (g ) = - 3(g ,) . v p P l Consider _ 3 i (g .) = - i 3g = i i (g ) = 0 , v p-1 v p-1 v v p [ i ^ ( g ^ ^)] e H the cohomology c l a s s g . e£ p-2 - P P ' P 2 (X) such t h a t P ^(X,n _ a sequence o f forms g^ e £ >•••' P (X) Tg. G= e such t h a t F I S P ' ° 2 3(g „) = - i (g .) . P 2 v p-1 P r o c e e d i n g i n t h i s manner we are a b l e t o c o n s t r u c t 1 , : L P 3g^ = - i ( 9 ^ v i s a cocycle + 1 ) f o r DI for i = 0 - l • But „ , hence t h e l i n e a r map V'° 1 p H°(K,D)~ ^ P H (X,fi ) P [G] P [g ] i s surjective p P Moreover i f Y g. e G = 0 and g = 3(f) P Then G^ = g i s a cocycle g Q f o r some + get hence (5) i s a cocycle P VP'° f e £ ' P P - 1 (X) . (i.e. e for DI •. 'lp-p+1,0 n V [G] e H ° ( K , D ) ~ P + 1 ker(i);) = H°(K,D) G ~ G n 1 • • . P V I t i s clear that + lp: H°(K,D) I t i s easy t o see t h a t Since 1 P . Therefore / H°(K,D) P + [G] —>- 0 e H ( X , f i ) ) , P , where ) f = 3f + i (f) ) , and V DI for 1 ...+ gp_2. + g ' ^ Vp+1'° + i (f) = Df = (3+i p F ' g ' ^ = P gp_1 - ±v(f) mod D (G - G, = 1 [ G je H°(K,D)~ 1 H°(K,D)~ P + 1 Q P + 1 we k e r \\i , i n d u c e s an isomorphism 1 = H (X,ft ) \\i i s w e l l d e f i n e d , P P and that" the edge isomorphism ) 69 P'P = H (X,ft ) = H°(K,D) I P E P P / H°(K,D) P + i s precisely 1 <ft . OO Let/ d>: g£ (C) x...x g£. (C) -»- C r • r g^g ) = < j > (A^,.. ,A^) be a k ^ l i n e a r invariant form (i.e. < f > (g A g , . E ->- X be a V-equi variant holomorphic vector bundle of rank 1 1 f o r any g e GL^_(C)) , and l e t by theorem 1.3.1, there exists a hermitian metric i (9) = 9 ( D f o r some h on 0 L e £(X, Hom(E,E)) , where E r - Then such that i s the canonical curvature matrix associated to h . Consider <fr^_^ k = ( J <j)(L,L,.,L, 0 ^ 0 ^ . , 0 ) 'i ' i = 0,...,k By the derivation property of i — ' k-i' 9 0 = 0 , we get (0: = 1) . Since k—i k i e £ ' (X) f o r 9<|> = 9 (<j> ( 0 , . . , 0 ) ) = 0 . and the symmetry property of < j > , we have v k-i and v = 9 k-i-1 (.) (f)(L,.L, ( ± + 1 ) 0,..0) -ftCLj^L, i+1 Since = ( J (k-i) < j ) ( L ^ L , i v ( 0 ) , 0 , . . , 0 ) , L1 0,..0) (i+D 1 ^(L^^L^L,©,..©) , •' 9(L) = i v ( 0 ) / we get i Therefore G = v (<(>. ) = 3(4 ,..,,.) k-1 k-(i+l) i I -k 0 £ .(-1) (JK e F ' i=0 for each i = 0 ,. .. ,k - 1 . i s a cocycle f o r l j _ D k 0 * B u t 70 iK[G]) = then (-l)k[<t>v] edge isomorphism 0 (E) = ]e (-l)k[4> ( 9 , • .0). 0 k * e: T ( z , 0 ) = H (K,D) . I f V Z associated to L (i.e., . Now consider the H (X,ftk) L = V - i D) i s the V-derivation of then LI = VI (i i s X zero operator on Z , i . e . L we get k (-1) e (((> (V,. . ,V)) (f)(0,..0) i s the smooth extension of v| ) . Hence = G e H (K) ^ . Therefore the cohomology k k e H (X,ft ) class can be represented by the element <|>(V,..,V) e F ( Z , 0 ) i n the isomorphism ~Z * (6) gr( ( Z , 0 ) = H n (X,C) = Z In p a r t i c u l a r i f © H (X,fiP) p=0 P : g£^ ( C ) x . . xg£^_ ( C ) -> C . i s the polarized invariant j - l i n e a r form associated to invariant homogeneous polynomial r c . : ql -> C defined by det(I+XA) = Y c . (A) X ( i . e . , c . (A) = $ . (A,.. .A)) , D r o ' s 3 1 3 3 1 1 c. (E) = (—- ) j then the j - t h Chern class can be represented by the element [$.(0,..0)] J of E J j 1 j (-1) (——r-) $.(V,...,V) e T ( z , 0 ) i n •27T1 the i i e H (X,ft ) : - z isomorphism ( 6 ) . Hence we have the following theorem which i s also proved i n [12] independently. Theorem 1.5.1. compact Kaehler manifold Let V X be a holomorphic vector f i e l d on a with i s o l a t e d zeros, and l e t E--»- X V-equivariant holomorphic vector bundle with V-derivation i (——) be a V . Then j c . (V) J e T(Z,0 ) represents.:the j - t h Chern class Z i n the isomorphism c . (E) J of E 71 equation gr(T(z,0 ) = H (X,C) Z where det(I+XV) = Y cAV)X j . c . (V) i s defined by the 3 3 3 Remark: This Lemma c l a r i f i e s our notation i n section 4. Before giving examples, l e t us mention the following r e s u l t i n [13] = ker{Res: T(Z,0 ) -> c} F and „ , Res r (z-,o ) y c (1) is H where 3 commutative (x,n i s the natural map induced from Example 1.5.1. Let V -0 • = H (K,p) e': T(Z,0 ~Z be the holomorphic vector f i e l d on P induced from 0 a Let V t --x X. + ^ • f o r l ' 3 = e i + j 1 be the V-derivation of 0(U, .,) = 0(-l) given i n example - l,n+l 1.3.2. Then Z = {e ...,e } i s the zero set of V and V(e.) = - X. . O n i i n Since V i s nondegenerate, we get T(Z,0 ) = © C e. (Ce. = C) . Hence ~Z 0 1 the function s n -1 e T(Z,0 ) , s (e.) = ^ - (-X.) = -1 j 2TT j Z 1 j = 0,...,n n 2TTI 72 represents the Chern class s, (e.) = k 3 each 2TT kA . J c (u, ) . 1 l,n+l Similarily n j = 0,..,n 0(k) f o r k =)= 0 e Z . i that the function s e T(Z,0 ) , s(e.) = "z 3 the n e T(Z,0 ) , Jc - represents the Chern class of . Let us compute the cohomology algebra of s s 1 e F -n n Res (s ) = s e F . We have seen represents 3 -1 c T(Z,0 ) . - Therefore z i s (e.) = (—) = r ( Z , 0 ) , and Z n A. j = 0,...,n 2 7 7 * 1 n 1 (U .,) e H ( P ,ft ) , hence l,n+l p n A., j = 0 , . . . , n . 3 Let us compute 3 s v I (z)dz,....dz 1 n v a (z),...,a^(z) Res i=0 By the algorithm (i) i n section 4, s we get Res e. 1 S (e.) (z)dz,...dz 1 n n 3_ 3a. • det( — (e.)) v 3 3 3 z since vl ' = n, ( IP )x . I Hence ( A . - A .)z.3/3z. ^ 1 3 1 1 1 3 Res (s n = (—) 2TT v ; A n y . 1-0 L n n — — n (A.-; .;,-i-xj> J i (—) 2TT V n j n A . - A . i 3 73 Let JXL be the n x n matrix obtained by deleting the i + l - t h row and n+l-th column of the matrix 1 0 M = We note that det(M) = 1 X . 1 1 X 0 X X n T 1 for i = 0,1,.. ,n n II X.-X. • • 3 i 3>i (-l) and n det(M) = J moreover Y (-1) X det(M.) • i i 1=0 1 n n det(M.) l I t i s easy to see that II (X.-X.) det(M.) = (-1) det(M) . •1 • 3 ' 1 But then 1 we have n > _ 1 = 0 n ( X . - X . ) 3 1 X (-l) det(M.) l i=0 det(M) n 1 (-1) det(M) det(M) = J .I i=0 (-1) X" det(M ) det(M) (-D n, 1 Res(s ) = (-^-) ( - l ) = (-T—r) n Therefore Hence s P n n represents a non-zero element i n p = l,..,n . But the dimension of F /F P P + 1 F /F P =(= 0 . f o r each can't be bigger than 1, 74 since of dimr(Z,0 ) = n+1, ~Z and we have T(Z,0 ) , we conclude that F /F gr(T(Z,0 ) = © Z p=0 of P P P + F~ /F~ P = C. s (s°=l) . P + 1 = C[s]/(s' ^") 1 n+ subspaces i n the f i l t r a t i o n Therefore P which i s the cohomology algebra . Remark: F~ /F P n+1 P + 1 the argument of the non-vanishing of i s similar to the argument that for the Kaehler class w , since w 11 s in =f= 0 i n H (X,ft ) p < dim X P P i s a volume element. In fact from the above computation and diagram 1, we get . n 1 n ResCs ) = (——) = ,I , (^-) ( c l ( U l,n+l ) ) ' h S n C 1 e P i f we take n = 1, c^(U^ then l,n+l P ^) = - 1 , namely i n the cohomology sequence -* H ( P ,0) 1 n 6(0(1)) = 1 H ( P ,0*) 1 n where H ( P ,z) 0 (U., , .,) l,n+l 2 n 0(1) The Universal Chern Classes Let V be the holomorphic vector f i e l d on Gr induced Jc f n tM from the 1-parameter group of automorphism M = \ ° 0 'A e gX (C) n A. =(= A . l ' j = e for , where i 4= j 1 75 Then > U. *1 e Gr, : det k,n x -> n + P x. -> nxk i s the standard open a f f i n e covering of where x^ •> denotes the vector i n C Gr^ f o n - """l """2 "" k - ' < <i n . Let -1 Z x. -> 3i l,l'-" l,k Z be the z = Z holomorphic X . p,kj J P l o c a l coordinates on 1 P J 3 + J , p = n - k . U. If A t _,..,t ] denotes the k x k rows of the n x k matrix which i s obtained by taking the matrix A . t^.-jt^-th Then l * x + X Vtz) = d ar tM ((e X) tM hence 3 1'"" p tM (e^X) ) tx. l tx. e x.. . ="l •tx. e p e J e V(z) = lim t->0 where X = t=0 x. •' D'-> P x.~ + • -z X x. -> V n nxk k 76 3 -X. ) t(X. t(X. -X. ) i i 3 x 1 \ J 1/1 -X. ) t(X. 3 e P l,k - z t(X. -X. ) " l z p,l D ••• e P X k ) Z 1 z p,k _ s lim t->0 (X . -X. ) z . . . =>1 X l 1 ' . But then (1) V(z)= (X . -X, ) z D ^ t,l D namely V(z r,s p i x e. 1 X k 1 , k 3 t X k ' fc k .(X . -X. )z D i, p •- k P,k ) = (X. -X. )z i i r,s r s e Therefore P,l l . (X . -X. )z^ t (X . -X. )z " i k A D 1 . = < l'-" k 1 S 0 l " k + > . (e„ = (0...1..0) and zero elsewhere) i s the only zero of V on U. 1 (V(Z at r,s )) r=l,..,p s=l,..,k e. . i ,..,i 1 e. i s equal to the maximal i d e a l ^ the m u l t i p l i c i t y of the zero k is a simple i s o l a t e d zero of vector f i e l d on Gr. k,n with the zeros V . e. i i n the i ^ . l ' - - ' \ - t h r o w , since (Z ) r,s r=l,..,p s=l,..,k 1 # . i s one, namely ..,i Hence k V i s a nondegenerate Z = {e. . 1 < i . < i <..<i, < l^,.. ,i ' - 1 2 k k 77 Then we get T(Z,0)= Z © Ce. K i <i <..<i<n l . • C e. k 1 Now l e t us compute the Chern classes of If cr a : U, -> U, t k,n k,n t (e. . ) : ,. . 1 k f(t) i s the natural extension of (U. ) k,n e. i , - . , i 1 e GL (C) k tl i n T(Z,0 k,n a, , then t is -»• (U, ) k,n e-. = c k given the matrix k such that e e. V t M - " 1 . * = e. . f(t) i'--' k 1 1 But tA. e tM e. 1' 'k e. 1' 'k A. 0 tA. -t Hence a (e. t i Therefore ± l ) k by section 3. k V(e. l'--' k 1 If = e .) = e . . , i V(e. """l'"' 1 -A. 0 0 -A. c.(V) s T(Z,0 ) ~Z 1 i s the function given by 78 c (V)(e. . ) = (-1) cr. (A ,A. ) , where elementary symmetric function i n k-variables. i (—) i 1 (-1) a. = (——) j a. i s the j - t h Then by Theorem 1.5.1. j 2TT J 2TTI a. e'.T(Z,0 ) J - represents the j - t h Chern class of k ,n If V a : u -*.U, t t,n k,n i s the V-derivation of (cr = (a ) ) , then s s A. 0 1. 0 '"-A. 1 Hence U, k,n V (e. associated to . ) = - V(e. . ) l^,..,!^ , . . , 1 ^ I i 3 (—) a. e T(Z,0 ) 2* 1 - represents the j - t h Chern z * class of U, k ,n 1 s = (-r^r) cr. e T(Z,0 ) , 3 By the similar reasoning we get e a s i l y 1 7 1 where cr. (e ) = j t h elementary symmetric function i n : t ,..,t 1 k 1 < r <...<r < n - 1 p- and {r ..r t,..t,} = {l,..n} l p l k Chern class of the universal quotient bundle Z D A ,..,A r x r p represents the j - t h Q K/n Remark: The same computations are done i n [12] by a d i f f e r e n t technique. Let us f i n a l l y compute of Grassmann manifold Gr, . k,n vector f i e l d on a compact Kahler b If (Gr V ) the 2p-th B e t t i number i s a nondegenerate holomorphic manifold X with zeros, then by the vanishing theorem of J.B. C a r r e l l and D. Lieberman [11], we have b 2 k + 1 ( X ) = 0 , and by the theorem of C. Kosniowski [27], we have , 79 b 2p (X) = {the number of zeros Real part of i J 0 (x) > 0 1 < j < n = dim X - let V of V 9a. (j^— j such that the for exactly , where - eigenvalues of Now, x e Z p 0 . (x) i s the 3 ' VIN(X) = I (x)) indices n a %/dzA i 1 be the holomorphic vector f i e l d on Gr induced K. f from oi tn 2. e 0 '-n . Then we have Z = {e. V and by (1) l . : 1 < i < i_<..<i, < n} , 2 kk {i - i } , ' are the eigenvalues of t s s=l,. . ,k t=l,-. ,P e. . r ..,r r .., n V at the zero Hence we get k b 2p (Gr ) = the number of k,n- there are exactly t Therefore p e. . 1 < i <..<i. < n i -.i - 1 k 1 such that k p o s i t i v e integers i n the set s s '1/ • • /k t=i,..,p _ b„ (Gr, ) = the number of 2p :.k,n there are exactly p 1 < i. - 1 < i<..<L 2 p o s i t i v e integers i n the set < n k - such that {j - i } _-, , t=l,..p where {j 1 j p , i ,. . i , } = {1,2, ..n} 1 k n Let us count this number: exist i^ . n.- i elements i n the set Hence there exists exactly elements i n the set for and p = n - k . 1 < i {l,2,..n} < i <..<i which are greater than n - i ^ - (k-1) {1,2,..n} \ {i^,..which < n , there = n - i ^ + k + l are greater than i ^ . 80 S i m i l a r l y there e x i s t exactly which are greater than elements i n n - elements i n the set , hence {l,2,..n}\ n - i ^ - (k-2) = n - i -fi- , - - - r4-^} 3 for {l,2,..n} \ {i 2 ~ k + 2 i.^ which are greater than proceeding i n t h i s manner, there e x i s t exactly i n the set {l,2,..n} , i , »-.i, } n - i g - k + s By elements which are greater than 0 . i , s = 1,..,k . But the eigenvalues of { A . - A . } 3 t 1 , l - {A. : n -A. t X 2 V } ^ e. . , , are {A . - A . 3 n t=l,..p n-i -k+1 at t \ t=l..p positives n - i -k+2 } ^ . t=l..p positives n _ 1 v positives Therefore we get b„ (Gr, ) = {# 1 < i <- -<i, < n such that Y (n-i -k+s) 2p k,n l k , s * s=l = p} But k r ) n - i i - k + s = kn - k 2 k k v v . , ,> k(k+1) + ) s - ) i = k(n-k) + i i s k v • > l i 2 s Hence (2) b 2p 0 (Gr ) = {# 1 < i < . -<i X K/Il In p a r t i c u l a r for K < n: k (n-k) + p = 1 , b (Gr 2 = n - k + 2,...i k = n Z k r \ i ^ = p} S ) = 1 ; since i rc / n i k (k+1) = n - k X i s the only one which s a t i s f i e s (2) . 81 CHAPTER II APPLICATIONS OF G -ACTIONS TO LINEAR ALGEBRAIC GROUPS m 1. THEOREM OF BIALYNICKI-BIRULA A structure theory f o r complete v a r i e t i e s with G -actions was G i n i t i a t e d by B i a l y n i c k i - B i r u l a i n [3]. I f G^ acts on X and x e X m i s a fixed point then one can f i n d coordinates f o r the tangent space T (X) so that the induced action of G on T (X) i s of the form x m x m m d(f>.:(x) (v. ,. . . ,v ) = (t v., . . . , t v ) , where dA (x): T (X) + T (X) t i n 1 n t x x n i s the d i f f e r e n t i a l of <j>, : X'-»- X at x , and < J > : G xX -> X i s the t m given G -action. Let T (X) be the invariant G submodule spanned m x m 3 by those U v e T (X) such that x d<J>, (x)v = t v f o r some t m m < 0 . Let and W be the complete v a r i e t i e s with G -actions. Define a G m m equivariant morphism f : U -> W to be a G^-fibration i f there exists a representation that f "''(W^) = factor. In this case the dimension of V the G - f i b r a t i o n m Let closed f i e l d G X a: G -> GL(V) and an open cover {w.} of W such m l x V with the action given by a on the second r = KJ i=0 X f: U W . be a complete variety defined over an a l g e b r a i c a l l y k . For a given G -action m Z. i s said to be the rank of cb: G x X m X , let be the decomposition of the fixed point scheme into 1 connected components. G For p e X , we define d(p) = dim p(X) T G By [25] d i s a l o c a l l y constant function on X for any p , q e Z^ , since Z^ i s connected. . Hence d(p) = d(q) We denote t h i s common 82 value by d(Z.O . 1 Theorem 2.1.1 ( B i a l y n i c k i - B i r u l a [3]). Let X be a complete variety defined over an algebraically closed f i e l d k . For a given G r G -action <f>: G * X X , l e t X = (j Z. be the decomposition of m m . _ I i=0 m the fixed point scheme into connected components. Then there exists a unique l o c a l l y closed G^-invariant decomposition of X = [J U. i=0 (called the X (-) decomposition) 1 and morphisms v.: U. -> Z. i i I for i = 0,...,r so that G (i) (UJ (ii) y. : U. -»• Z. - ' i i i m = Z ( i i i ) for any ± i s a G - f i b r a t i o n of rank m d(Z.) l p e Z. , T (U.) = T (Z. ) © T (X) i P i P i P We c a l l the above decomposition X = U. i=0 B-B's decomposition. 1 F i r s t of a l l , we would l i k e to discuss the r e l a t i o n between B-B's decomposition on immersion ij;: X -*• X . everywhere defined map and p n under the G^-equivariant closed For this we need to know the existence of an $: 1 P x P 1 (b X G x X m x X ->- X > X such that the diagram 83 i s commutative, where r e s u l t on P 1^ = G u{0} u {«>} . We w i l l f i r s t show this m IP , following [22] . n Let eft: G x p m n p be a given n G -action on p m defines a l i n e a r algebraic group homomorphism t < f t: P +p t n . Hence n T since <ft: G G m xp m i s a torus. -> p n for some <ft (G ) m Therefore Q Q x ep n , consider the o r b i t map cb : G u {0} + 1 m X X- Tl where X I X =,<(> cb. 1 H , f o r i = 1, 2 and Q m X "evaluation at t = 0" n Q n m -£ m cb (°°) = [t x ,...,t X k = min: P X m -k m -k cj>(0) = [t x ,...,t x ]| where n + 1 n+1 up to the change of bases of k* cb : G -»- G .x ci (t) = cb(t,x) e p " . Define m m cb : G u {°°} -> P 2- m n l i e s i n a maximal torus of PGL rn+1 e 2 . Now for a given X Aut( (P ) = P G L m m cb (t, [x , . . . ,x^]) = [t x , .. ., t x^] i s given by n <j>: G . Then < f > n t = Q -l x ] | ^ n "evaluation at t = ~" n {m.: x.4=0, i=0,...,n}, I = max{m.: x.fO i=0,...,n} and x = [x , ...,x ] . I t i s easy to see that the <JK are well defined for i = 1, 2 (<j>. |„ n for i = 1, 2) and that they patch i n the i n t e r s e c t i o n to give a morphism <j>: P X 1 -> P R XI 1 such that =§ o X m $ | = cj) . x X m Hence {<j)} xe P X Let defines the required cft:G x x - > X m map eft: p"*" x p n -+ p n b e a G -action on a projective v a r i e t y m X Then i t i s well known that such a G -action i s induced from a G -action m m 84 on P (see [35]). Namely, there exists a closed immersion N Tp: X -»- P , such that the diagram G x X m G x p >p m commutes, where For x e X we have <J> , V m ^ - (x) f Hence is ( X ) <p d) i s a G -action on m G m Since yX • (P) = ^ - a c t o r s ~x {<b } commutative. 1 ( through . 4*(x) X ) G ...x > m .. CU X > G .i> (x) c P m (G ) C ^ m >- ( x ) m X , say <j) : X defines the required (G ) = m P 1 N (cb (G ) ) c i|) (x ) = m X -> X , such that ip o <$*• = (jS^ ^ ~ 1 map <j>: P x x-> X . (j) : P xx yX (j) : P N x p N •p * (x ) , 5 Moreover 85 Proposition 2.1.1. ^ complete variety eft: P is x x •> X 1 Let <J>: G x x + X m b e a G -action on a m X . Then there exists an everywhere defined map such that the diagram P1 x x G x m x < f t —-—> X commutative. Proof. By the equivariant Chow Lemma, there exists a surjective b i r a t i o n a l morphism TT: Y •+ X , and a G ^ a c t i o n eft on Y such that the diagram commutes exists have X (ft y where y e Y Y *" V = TT o eft : P ~x {eft '}" xeX T 1 -> Y G x x m -* X i s a projective v a r i e t y . such that ~y 1 tft : IP -> Y - G xy m TT (y) = x , since which extends + X For a given TT i s s u r j e c t i v e . Then we y < f t : G -> G .y c Y . Hence - m m X i s the extension of eft : G define the required x e X , there G .x C X . Hence m m 1 map eft: P x x -*• X . Moreover the diagram 86 is commutative. Q.E.D. Now, l e t us f i n d B-B's decomposition of the G -action m n n 0 d>: G x P IP , d>(t, [x-,...,x]) = [t x . . . . , t x ] , m. :e Z . m U n :u n 1 m m n Without los Let of generality, we may assume m^ > m^ > m^ > ... > s„, s,,...,s be the p a r t i t i o n of n 0 1 r such that . m„ = m. = ... m 0 1 s ^0 s m = n m 0 s +1 = ••• = 0 m s +s - 1 1 0 l m = £+l ' m ••• n ^ = = m . s r Consider the following f i l t r a t i o n on X P . lP = X„ oX,3 — 0 1 n i= 1 and X i-1 1 OX n AV 1 ( X £ , where r +s. - l 1-2 i - '•••' l K n i-2 X l } w h e r (D.Lieberman s 1 = V(x ,.. . ,x s 1 o o i-2 N A-2 e 1 ^ s k=0 = 2 K filtration) .),..., for i > 2 V ( f (x , . . . , x ) , . . . , f (XQ,...,x )) i s the set of common zeros of n the homogeneous polynomials f ,...,f in (P . . Then the X^ are n •t G - i n v a r i a n t , i r r e d u c i b l e , closed subschemes of IP m <f> I = d>. : X. i l y| y G m x x. -> X. I I f o r i = 0, l , . . , r . I t i s easy to see that 1 i-1 h i = 0,...,r m V ( V % + j f ° r I \ - j < I\ 5 are the connected components of the fixed point scheme G r Z = ( IP ) = [J Z n = . Let G , and (X ^ X check e a s i l y that the morphism ) m i + = z i i = 0,...,r. y. = (<b.) : X. v. X. ,. ->• Z. , l I °° l l+l i One can 87 l y. (x) = A. (°° ,x) 1 1 and T p 1 p defines a G - f i b r a t i o n of rank m ,) = T (Z.)ffiT l+l p i p ( X . - X . l x X . -> X . i i d>. : G associated to • for ( X ) ~ I m p £ Z. l x X . decomposition. X . subset of One notes that (d>.) such that 1 Now, I IP X „ 2 = V(x .x ) , . 0 1 . . , X . = l case D. Lieberman's B-B's + 1 m n I Then by the uniqueness r = I ) • ~ 1=0 ( X . s X . ,) i i+l . Then X , = V(x ) , n 1 0 = e = [0,...,1] . In t h i s n n > ...> . , X m p r = Uu., 0 1 s x = U i=0 respectively. scheme i s the G (P ) G m If x. and n p n . n I|J : X -> pn i s the i s the maximal open decomposition i s the natural c e l l u l a r decomposition of Let Y s, ,„ k k=0 L i s the map d>. f i l t r a t i o n i s the natural f i l t r a t i o n of p) - i s a morphism. > m, > m„ 0 1 2 v(x ,...,x. , ) , . . 0 i i (Bruhat decomposition of pn X ^- X_^ 1 °° i f we take where X . . I of the B-B decomposition we conclude that B-B d(Z.)=n+l l be a G^-equivariant closed immersion, and l e t P be the B-B decomposition of and X 1 r Z = \^_J i=0 = Z Z. i s the decomposition of the f i x e d point 1 i n t o connected components, and f i b r a t i o n on y^ = cb^: -<t>: G m -> Z^ x (p -s- p U. l as given above, G invariant decomposition of n . n _ X = I j ij> (U.J iT0 r we have a l o c a l l y closed Moreover the morphism tft: ij> "*"(IL) IL factors through Then IJJ ~^( Z ^) n • Z. l Namely we get a morphism X^: ± (V^) -> ift ^( j_) Z such that the diagram 88 ip """(U. ) — ^ — y U. 1 1 4> (|) (ip (x)) £ ip ^ (Z) i s commutative, since ip (Z. ) = W u ... u W 1 1 K. . L f o r any x e X . Now l e t be the decomposition of ip "*"(Z.) 1 1 components and l e t V. = X (W.) 1 D f o r j = 1, — 3 0 0 from Lemma 4.2 i n [3] that Then i t follows 1 V. = X. j 3 ,k.. into connected f o r some 1 ip "''(U.) = LJ X^ . In p a r t i c u l a r j j 0 < i . < s . Hence " ~ 1 ip "'"(U.) = <J> i f and only i f 1 tp "'"(Z^) = cp . This observation e s s e n t i a l l y shows how to prove B-B's theorem from the decomposition of f a c t the morphisms on xT y. : X. 1.1. 3 IP 11 f o r projective v a r i e t i e s . In W. i s the r e s t r i c t i o n of X : ip "'"(U.)-Mp "*"(Z "J o o l r 3 T • [30]. j Let d>: G X - > X m beaG -action on a complete variety with m r i s o l a t e d fixed points {p ,...,p^} . I f X = IL i s B-B's i=0 d(p ) decomposition then U. = /A i = 0,...,r . Let b„ = Card{ie{0,..,r} I 2p such that d(p.) = p>0} . We w i l l now show that these numbers b are X Q ± the B e t t i numbers of X . Let X d>: G x X m r X beaG m with i s o l a t e d zeros -action on a compact Kaehler manifold {p ,...,p } . Then f o r each p. we can f i n d o r l coordinates f o r the tangent space T (X)' so that the induced action of i P 89 i n dd> (p. ) (v, ,... ,v ) = (t v, , . . . , t v ) t i l n 1 n m G m on T p . m. e 2 l (X) i s of the form 1 since G m i s diagonizable and G m G . But then the induced X-action on m T (X) v i a p. l i s of the form m^z m z (z,(v.,... ,v )) = (e v ,...,e v ) 1 n 1 n (*) Now consider the C-action i.e.; dd> (p.): G + GL(T (X)) i s a t i m p. I representation of exp: C m a: C x X -* X induced from a(z,x) = <b(e ,x) . Then the representation <b v i a exp: C + C + GL(T Z P z -* da (p.) i s given by z^ i J i m 9z. I 0 V o m 2 A -1 A e GL^ where v| N p *'m n + dim T (X) p. eigenvalues of the matrix M m (X): deb (p.)v = t v p. t I m. , i = l , . . . , n . and i = I a i=l 9 ± a . . = dimlveT m > 0} = the number of p o s i t i v e dim T (X) p. i z z (e ) = e . Therefore we get n f o r some i s the vector f i e l d induced from Hence d — dz (*) , since (X)) , i B "*"MB f o r some Since the are the .same, we get = the number of eigenvalues of p o s i t i v e r e a l part = the number of p o s i t i v e (- 9z. 3 (p.)) which have I rru i = l , . . , n . But then by the theorem of Kosniowski (see Chapter 1 section 5) the 2m-th B e t t i number of X i s given by b (X) = Card{ie{0,...,r} 9 a exists exactly m such that there k eigenvalues of (. (p^)) with p o s i t i v e r e a l part}. j 90 But then from the above we get b (X) = Card{ie{0,...,r}: dim T (X) = , 2m p. 1 d'(p.) = m} . Hence l b^ (X) = b' = Card{ie{0,. . . ,r} 2m 2m such that v dim T ( X ) = m} . By [3] we have p. + I b' = b„ . Hence 2m 2m b„ 2m i s the 2m-th B e t t i number of X . This observation proves that the number of fixed points of on a compact Kaehler manifold X i s greater than the dimension of X , since b = b (X) > 1 f o r each p . This i s a special case of 2p Zp — a theorem of Rosenlicht [3], i f G acts on a complete variety X , m then the number of fixed points of G^ on X i s greater than the dimension of X . We w i l l need the following lemma i n the next section. Lemma 2.1.1 [22, p. 153]. If T i s a torus, acting on a projective variety X , and i f the dimension of X equal to one, then T Proof. and i s greater than or fixes a t least two points of X . I f X(T) = Hom(T,G ) i s the character group of T m Y(T) = Horn(G^,T) , then the composite of X e Y(T) x e ( ) with X T y i e l d s a morphism of algebraic groups ' i - e . , an element of X(G ) = Z . This allows one to define a natural pairing X(T) x Y(T) -* m G G m denoted <x,A>•, under which m X(T) and Y(T) becomes dual Z-modules. Therefore there exists X e Y(T) such that <X-/^ + X • ^ J k where { x - } i s a basis f o r the free abelian group X(T) . I t i s i=0 > 1 K f o r 1 3 1 easy to see that the fixed points of T G = X(G ) m m and the fixed points of coincide. Hence we can assume T = G . Now i f the m G dimension of X > 1 , there i s nothing to prove. Otherwise we can G ~ f i n d x e X - X , since dim X > 1 . But then <t>(°°,x) =j= <|>(0,x) and m 91 they are the fixed points of G . m Hence we have the claim. Q.E.D, 92 2. BRUHAT DECOMPOSITION Let G be a connected reductive l i n e a r algebraic group defined over an a l g e b r a i c a l l y closed f i e l d subgroup of 0 B be a Borel G , T a maximal torus contained i n B . Then T n B where u be the set of roots of G semidirect product Let k , and l e t be the corresponding B is a B i s the unipotent r a d i c a l of B . u with respect to T and l e t W = N (T)/T G Weyl group We s h a l l denote by the same symbol an element of W and a representative i n N (T) when this can be done without ambiguity. G Let A be the set of simple roots of associated to B , and l e t $ $ for the ordering be the set of p o s i t i v e roots. + Then we have g = t © 1L g . a ae$ t, g, b, b^ b=tffib u ,b u = © g , dim g ^+ a ae$ are the Lie algebras of T, G, B = 1 , where and B^ respectively. For the basic facts about algebraic groups the reader i s referred to [22]. Now we have the following. Lemma 2.2.1. B = N (B ) = N (b ) , where G u G u N (B ) = {geG: gB g G u u = B } u 1 and N (b ) = {geG: Adg(b ) = b } . G u u u Proof. We have obviously • i s a normal subgroup of B . Now G/B and T B c N (B ) c N (b ) G u G u N^(b^)/B because B u i s a closed subvariety of acts on i t v i a l e f t m u l t i p l c i a t i o n . Since T i s connected, 93 i t s t a b i l i z e s each i r r e d u c i b l e component of N (b )/B . Let Z be any G u i r r e d u c i b l e component of N (b^.)/B . Then there e x i s t fixed points of G T on Z cr B e Z .. and they are of the form i s a fixed point of T , then cr e N (b ) . But t h i s implies G u the i d e n t i t y . hence aB , for some Therefore Z = {[B]} a a e $ a £ W . Now, i f Ad (a) (b ) = b , because u u for a l l a e $ + + , hence {[B]} i s the only fixed point of T by Lemma 2 . 1 . 1 . Since Z a is in Z , i s an arbitrary irreducible component of N (b )/B , we get N (b )/B = {[B]} , which means G u G u Since B c N„(B ) c N_(b ) we have the claim. G u G u N,_ (b ) = B . G u Q.E.D. This Lemma gives an elementary proof of the well known theorem (C. Chevalley) N (B) = B G Corollary 2 . 2 . 1 . as follows. Let G be a connected l i n e a r algebraic group defined over an a l g e b r a i c a l l y closed f i e l d k and l e t B be a Borel subgroup with the unipotent r a d i c a l B^ . Then B = ( ) Q^ ^ where b i s the Lie algebra of B . In p a r t i c u l a r B = N_(B) . u u G N B G Proof. G/R (G) u U i s a reductive connected l i n e a r algebraic group. i s a Borel subgroup of G/R^G) [ 2 2 , p. N Let R (G) be the unipotent r a d i c a l of G , then u natural epimorphism of algebraic groups <j)(B) = U 1 3 6 ] . cb: G We have the G/R^(G) ~. Hence cb (B) with the corresponding unipotent, r a d i c a l For g e N (b ) we have G U <j> (g) E N_. / t j r r M (L(d>(B ))) VW. u u where L (d> (B ) ) = dd> (b ) i s the Lie algebra of d> (B ) . By Lemma 2 . 2 . 1 u u u we get <j> (g) £ cb (B) . This means g = br for some b £ B and r £ R^ (G) . U Since that R (G) c B u i t follows that B c N„(B ) c N (b ) G u G u implies Cv K g £ B . This, together with the fact B = N (B ) = N (b ) . G u G u B c N (B) c N (B ) , we have i n p a r t i c u l a r G G u Since N (B) = B G Q.E.D. 94 Let G be a connected reductive l i n e a r algebraic, group defined over an a l g e b r a i c a l l y closed f i e l d k , and l e t $ = {a.} 1i-i ^ x g = kv i=l,..,d -a. -a. x x morphism where that of Ad: G ->- GL(g) m and t = © k w. . Then by Lemma 2.2.1 the •, 3=1 GL(g)/P GL(g)/P = Gr "d,2d+m 3 induces a one to one morphism P = St(b ) = {-LeGL (g) u Ad: G/B g = kv a. a. x 1 such that L(b ) = b } . I t i s easy to see u u i s a closed immersion. d -* P ( A g ) Ad: G/B -> GL(g)/P By the Plucker imbedding , the morphism d <j>: G/B > - P(Ag) , tf) (gB) = [Adg v ....*Adg v ] e P ( A g ) i s a closed imersion. The l d A a a l e f t action of torus T on G/B l i f t s naturally to P ( A g ) so that ( j > becomes T-equivariant. This can be done as follows; the action of T on d g = t © b_ © b_ , b_ = © g_^ i s given on the basis {v_ ,w_.} by u u u . . -a. 3 i=l I +a. x. X Ad(t)w. = w. 3 J J 3 Ad(t)v = + a.(t) v . Hence t h i s induces naturally x +a. +a. x x an action of T see that on A^g which descends to d > i s T-equi variant, since P(A^g) . I t i s easy to Ad: G ->- GL(g) i s a morphism of algebraic groups. Hence any G -action on G/B which i s induced by T m d d l i f t s to a G - action on P ( A g ) so that d>: G/B -> P ( A g ) i s G -equivariant. Therefore to a G -action on G/B m decomposition on the B-B decomposition of G/B , associated induced by T P(Ag) can be computed from the B-B . Now we w i l l give a s p e c i a l G^-action on G/B induced by T which w i l l give us the Bruhat decomposition of G/B . The Bruhat decomposition i s the d i s j o i n t union G/B = u B^.OXQ (x =BeG/B) Q i s the unipotent r a d i c a l of the opposite Borel subgroup B where of G 95 and B . ax„ U 0 i s the o r b i t of ax„ 0 under the l e f t action of To f i n d the r i g h t G -action, we need to compute m If a e W W independent of the choice of (OX) on d(Z.) . l n . (t) = x(n t e T , x e X(T) tn) < a e G , X e Y(T) m a e W , x <a\,oX> = <X/^> for the d e f i n i t i o n of l^(gB) G/B . n e N (T) , then the following G X(T) = Horn (T,G ) and Y(T) = Horn(G ,T), m m -1 (aX) (a) = nX (a) n Let on has a coset representative formulas y i e l d an action of Moreover B u X'(T) , X e Y(T) e (see section 1 >: X(T) x Y(.T) -> Z) . b e B , and consider i , : G •+ G i , (g) = bgb^ b b = bgB . Then the commutative diagram 1 and l^-.G/B -> G/B D G G/B where TT (g) = gB -> G/B induces the commutative diagram on the tangent spaces Ad(b) g dTT (e) T x where T (G) = g , e g —.—y dir (e) (G/B) o T (G/B) x 7r(e) = x„. 0 o Hence we have 96 Ad(b): g/b g/b (1) diT d£: b T (G/B) x T and IT i s separable, since ker (dTT ) = b e For any regular 1-parameter subgroup such that <a,X> =f 0 for any a e $ ; mu l t i p l i c a t i o n , we get a G -action m m G/B on (G/B) V Q such that (G/B) X of T such X: G (i.e., X e Y(T) X. • exist) , v i a l e f t x G/B -> G/B •. X. (gB) = X(t)gB t = (G/B) = {cfx : aew} . Now l e t us compute Q d(ax ) = dim{veT(G/B): dX (ax,Jv = t v for some 0 Q (1) we have m m<0} . By the diagram aX d(x ) Q Since any 1 = dim{weg/b: g g/b = b t e G m d(x ) Q AdX(t)w = t w , and AdX"(t)x for some k < 0} = a(X(t))x = t < a , X > x for and x e g , we get a a = the number of a e $ such that <-a,X> = - <a,X> < 0 Therefore d(x ) (2) Q Let = Card{ae$ : <a,X> > 0} + n e N (T) be the representative for G a. e W , then we have 97 the commutative diagram G/B -> G/B A G/B for each A(t)XB G/B t e G , £ (xB) = m n nxB ,a H (xB) = n A(t)nxB, A (xB) = t 1 t . Hence we get a commutative diagram on the tangent spaces d£ (x ) T (G/B) — - — x„ 0 i y T (G/B) crx„ 0 da-\(x ) dA (ax ) 0 d£ (x ) (G/B) — T x t T o Q (G/B) o CTX ~1 rn d(ax„) = dim{veT (G/B): da A (x„)(v)=t v 0 x t O Therefore f o r some m<0} , Q .since d£ (x^) , dA (ox) n 0 t o spaces. (3) and da "*"A (x^) are isomorphisms of vector t o Hence we have (by taking a\ X instead of A i n (2)) d(ax ) = Card{ae$ : <a, a A > > 0} + 1 Q = Card{ae$ : <aa, A>>0} + since <a ^aa, a A> = <aa,A> 1 By [22, p. 166] we can pick a regular 1-parameter subgroup such that <a,A> < 0 (3) gives us i f and only i f a s $ + . A e Y(T) In t h i s case the formula 98 d(cpO = Card{ae$ : <a,a '\»0} = Card{ae$ : <aa,A>>0} + = Card{ae$ : aa<0} = £(a), "the length of a " . + Now i f we take k = C , then by section 1 we get that the 2p-th B e t t i number of G/B , b„ (G/B) = the number of a e W 2p such that £(a) = p which i s a c l a s s i c a l theorem of Borel. G / p where We have the s i m i l a r s i t u a t i o n for P i s a parabolic subgroup of G . ' To do t h i s we need the following important lemma. Lemma 2.2.2. Let H be a connected solvable linear algebraic group acting on the complete v a r i e t i e s X / TT: X -> Y i s surjective and H-equivariant, Let x e X Proof. h.Tr(x) = Tr(h.x) = IT. (x) H y e Y ; then . Since and Y . I f the morphism H H then TT ( X ) = Y TT i s H-equi variant we have for a l l h e H , hence —1 i IT (y) f <j> H Y c TT(X IT: G/B + G / P P o B , TT((G/B) ) (G/p) T = Tr(gB) = gP ( G / P ) T be such that p = b =j= cb , which means ' ' i s surjective and T-equivariant = {crP: aeW(P)} = {aP: aeW(P)} H Q . E . D . be a parabolic subgroup of G . Then the morphism v i a l e f t m u l t i p l i c a t i o n on G/B, G / P ) . T (TT ^ ( y ) ) H ' ) , and hence we have the claim. Let . Now l e t closed subvariety of X . i s a H-invariant By the Borel Fixed Point Theorem we have TT(X) e Y where Hence by Lemma 2.2.2 W(P) c W i s such that + + Let $ (P) C $ contains d i s t i n c t points. © g ae$ (P) " + , where (T acts p i s the L i e algebra of P . a For any regular 1-parameter subgroup A of T , the morphism 99 TT: G/B -> G/P where i s G -equivariant. m i X: G x G/P -> G/P m Hence we have i s given by (G/p) = {aP: aeW(P)} m X (gP) = X(t)gP . t By a similar argument above, i t i s easy to see that d(aP) = Card{ae$ (P): <a,a X> > 0} + 1 = Card{ae$ (P): <oa,X> > 0} + for a e W(P) . I f we choose X such that <a,X> < 0 i f and only i f a £ Then we have d(aP) = Card{ae<I> (P) : aa<0} = I (a), "the reduced length $ (P). + of a e W(P)" () . In this case b ^P (G/P) i s given by b„ (G/P) = Card{aeW(P): £ 4 Example 2.2.1. + Let * B = < • 1 * * * V C r} = {a. .:j>i} , $ GL n , T = 0 0 * C B > then. $ = {a. .: K i , j < n , i=)=j} i, J + (a) = p } $ (P) 2 P where = {a. .: i>j} . a. *1 if. 0 The map ° t = t.t: 1 100 a-.1 • -> 1 ,2,....,n o•• ,a ,. .. ,a 1 2 n y 5 • " -* n , = a e N ( T ) , where a j_ = e i defines an isomorphism between the symmetric group group a C T j . Now, i f we take t ( 0 , ... . , 1 , . . , f i and the Weyl W . I t i s easy to see that the action of W a.a. . = a i,3 i' = a < on $ i s given by A: G -> T by m n 0 . n-1 A(t) = then <a. . ,A> = j - i (i.e. < a, A > > 0 i s and only i f a e $ ) . Hence + we get d(aB) = Card{a. .e $ : <aa. . ,A> = <a , A > > 0} 1,3 1,3 °i j + : ,a = C a r d { ( i , j ) : n>j>i>l, a^>o^} • Therefore that b„ (GL /B) = Card{aeS : there exists exactly 2p n n < a_.}. p Ki<j<n - such Similarily- i t can be shown by formula (4) that b„ (GL /P) = C a r d { K i <...<i <n: k(n-k) + -±= '- - \ i = p} , where P = A * 0 * AeGL, ^ c GL k n , Kk<n-1 -- Before obtaining the Bruhat decomposition l e t us mention a Corollary of Lemma 2.2.2. Corollary 2.2.2. Let G be a connected linear algebraic group 101 defined over an a l g e b r a i c a l l y closed f i e l d subgroup of G . Then N G ^ P U ^ elements of P and N (P ) G u N (P) = P . G particular Proof. and B u Let B = • p w n e r e k , and P p 1 the set of unipotent S u i s the normalizer of P u be a Borel subgroup of G be the unipotent be a parabolic r a d i c a l of B . Now i n G . In contained i n B u P, acts v i a l e f t TT: G/B ->- G/p , Tr(gB) = gP B i s B -equivariant. By Corollary 2.2.1 we have (G/B) = {[B]} . But B then Lemma 2.2.2 gives (G/P) = { [P]} . Since B^ C • have m u l t i p l i c a t i o n on G/B and on G/B so that U U U p w e u P (G/P) U Since B c (G/P) U P , hence (G/P) = {[p ]} which proves Q N (P ) = P . G U N (P) = P . G P C N (P) c N (P ) we have i n p a r t i c u l a r G G u Q.E.D. Now we can s t a r t to obtain the Bruhat decomposition of G/B via the B-B decomposition. By [22, p. 166] we can choose a regular 1-parameter subgroup X: G^ + T for each i = l , . . . , d where the G -action m m action X: G x G/B d X: G x p ( A g ) + m cb(gB) = [Adgv A...AAdgv ot _ 1 in L y : v + such that <X,ct^> = iru > 0 = {ct^: i=l,...,d}. We have seen that G/B , X(t,gB) = X(t)gB d P ( A g ) so that ] ot -i d i a basis $ of T cb: G/B l i f t s to a G m d +P(Ag) becomes G -equivariant. e P(y\q) in Take d k=0,...,N} of A g the exterior products of d {v , w.: i - l , . . . , d ; j=l,...,m} — j +ct. where elements g = kv ,g = kv , a. a. -a. -a. 1 1 1 1 I t = m ® j = 1 kw. . Then the action of X D on Y k i s given by X, X(t,Y ) = t Y k y 102 for some <a.,X> m. X(t,v ) = a.(X(t))v = t v = t v , a. i a. a. a. 1i i i I, e Z , since k <-a.,X> -m. X(t,v ) = - a. (X(t))v = t v = t . Let -a. i -a. -a. -a. X(t,w.) = w. i i 1 I I {Y ,Y , ..,Y } be the ordered bases of \J -L. A g f o r which 0 = SL > I \J JN . d (£. i I 7 > I -L >...>J N & d m. > . , l 1=1 L Y = v A N -a. J- Jt . . . . A V > 1 - £„ > > 2 - I - . N-1 ) . Let x. -a l d I = N - Y . ^, m. , I i=l Y„ 0 = A . . . A V v a, 1 a, d be the homogeneous coordinate of corresponding to the basis vector , d , „ N (P ( Ag) = P > Y^ P N i n the natural isomorphism . Let P N .. = i_J U. i=0 N P r be the B-B decomposition of induced 1 by X , and G ( U j = Z^ . Then m Z Q = {e •= [1,0,...,0]} i s the unique Q G connected component of (P ) such that d(e^) = N , and moreover Q U = (P ) N Q Therefore x = {[1,*, <j) ^(UQ) ,*]}^P . i- a s that B <J> "^( Q) U < j > i s (<J)" (U )) 1 = {x = [B] e G/B} . m 0 Q member of the B-B decomposition (by section 1). -1 Let us compute Now N d n (U ) = {gBeG/B: A(Adg).Y ±i a Q = Y £ o^°^ " i=0 invariant under the l e f t action of B^ on G/B Q a W i s the unipotent r a d i c a l of the opposite Borel subgroup U For have d u e B , i f A(Adu ).Y = y u k ^ B c e l a i where of G . N a. , (u ) = 0 a , (u ) = 1 k ,k where -1 gB e <j) (U ) , then Q a. , (u.) Y. , then by [18, p. 9] we i,k l i f z ( i ) > z(k) or i f z ( i ) = z(k) , i =t k z(s) = Card{ae$ -.such that d A(Adg).Y v a n d A Y = 0} . Now i f a s N Q = £ a j_ Y i a o ^ ° " T n e r e f o r e f o r a n m Y 103 u - B £ u since , A d ( A d u g).Y •? = I a. Q z ( 0 ) = d > z (i) namely d> ^(U„) A for any invariant under 0 d <Adu ).Y. = a ^ i = 0 B N . u group on an a f f i n e v a r i e t y i s closed a ^ + . + a'Y . . ,+a'^ 2 + Hence ugB § "*"( Q) e U Since the o r b i t of a unipotent p. [22, we get 115] b U * X Q $ = ) * Because they have the same dimension, d , and ^^ 0^^ (as a member of the B-B d) i s a closed immersion U decomposition or, since S a n a ^ ^ i ^" n e v a r i e ty N and U i s open a f f i n e subvariety of Q IP ) . We w i l l proceed i n the same manner to obtain the other members of the B-B decomposition. i n an order, so that Let us put the connected components of N = d(Z ) Q > dfZ^ > d(Z )>...>d(Z ) = 0 . 2 r ( IP ) We will show step by step that <j> i s invariant under the l e f t action of B on G/B . We have already shown that i> "*~(LI ) i s B -invariant. For u 0 u m -1 (U ) , i t i s easy to see that (d> (U ) ) = d> hence <j» (U ) = d> -1 m Let us assume that (6. (n ) ) = {a,x-, . . . ,cr x.} •. Then U„ i s i n the 2 1 0 k 0 2 form { [ 0 , ,0,1,*, ,*]} u { [ 0 , . . . _ . , 0 , 1 , * , ...,*]} W -1 G -1 d> G S o+1 s +k U { [ 0 , ,0,1,*, w o s 4>(a.xJ e n l T ,*]} = O „ . + s i + k - for . • Let us assume that 2,j 1 i = 1, ,k ; 1 < j - 2,3^ 0 U j = 1 < s l + k . - x d> (U Note that s +k U = K^J 2 , 2 1 = l , , U i s a d i s j o i n t union, and 3 ,k . Now we w i l l show that <j> ( U 1 . ) '3 2_ . ) = X '- i 1 is i for each 0 B -invariant for 104 each i = l,....,k ; i n p a r t i c u l a r example, l e t us show cb (U„ . ) 2,: cb "^(U^) w i l l be B^-invariant. For i s B -invariant. 1 In t h i s case k Then vv f,-1... (0 2 1 . r ) = {gBeG/B: d. A(Adg) .Y = \ i=s _! a. Y ,a ± gB i s an element of < j > ( U ^ . ) , then A(Adg) .Y„ = a Y ± ' ' \ . 2 0 ~ a^/0 and Y A - . - A V =v ° W k l a 1 b„Y„ + b Y...+b Y +...+b 0 0 11 y y s z(s +-i -1) > z(s) 0 . _ 0 + J k F o u r 0 — e a B + . .+ y a s +j _!0 d , y N ' k "~ / \ (Adu g) . Y 2 s 0 + j _ k Y + a^Y +...+a'Y , since 0 SQ+J.^! YN 2 . s o j -i k d i n the expansion of assumption. Hence g),Y /\(Adu u gBgfcb' (U stays as .) 0 < y < s„ +-! -2 such that b o k y b =0 f o r 0 < y < s„ + \- _ o y • ~ 0 k a which i s not zero by Q i f and only i f there exists =1= 0 . We w i l l show now that a l l J J u gB e cb "*"(UQ) '> hence (u ) X . F i r s t of a l l u gB = gB e cb "*"(U ) b 0 = 0 , otherwise since cb ^(UQ) i s B -invariant, which contradicts the choice of gB e cb ^ (U u . 2 , 3 b 0 = 0 . Now i f b M ^ 0 which can not happen since for s < 0- y < s - ^ + i 0 k happen, since J - 2 1 < y < s — 0 then cb "*"(U^) = cb . S i m i l a r i l y > otherwise . ) . Hence k u gB e cb (U ) I 1 b^ = 0 u gB e cb (U ) which can't 2 1 cb ^ ( ^ J does not contain such an element, (otherwise [0,....,0,1,0,....,0,1,0, f f o r some K. ,0] = ° U d f o r any s = 0,....,N , the c o e f f i c i e n t of Y " k • °k a =f= 0} . - 0 Jr 0 + j k If g + i s a fixed point of A on 105 G/B , but there i s no such f i x e d point, since the fixed points of in p correspond to N e. = [0,.. .-,0,1,0,... ,0] for some G/B 0 < i < N) i+1 Therefore namely .b d> ^ (U„ B .a x„ u 0 -i •k for a l l 0 < y < s_ + -1J, - -2- 0 . ) k then (j) =0 u u gB e <j) ^ (U B . But i s a closed subvariety of the a f f i n e variety kV d(a (u . ) = A , since the o r b i t of a unipotent group on an Card{ae$ : such that k = d> (U„ • ) = B-.cf x„ . 2, u k 0 -1 a x Q But the dimension of a, a > 0} = d(a x ) . + k Hence i s invariant under the l e f t action of a f f i n e variety i s closed. * X • k B .a, x_ u 0 i s equal to Therefore k 0 S i m i l a r i l y we have X Dk = cf> (U„ 1 0.x Q . ) 2, . D = B .a.x„ . u i 0 Proceeding i n t h i s manner, we see that each <j) ^(U. , ) is 1 ,K B~ u (|)~(U.) = L J invariant where <!)~(U. . ) , i,k 1 U. , ) = X i,k 1 I Then by comparing the dimensions of the closed subvariety the a f f i n e variety X , we get TX decomposition of G/B , G/B = L J G/B X TX Q induced by X Q = B .TX . U 0 TXQ of B^.TX^ Hence the B-B i s the Bruhat decomposition of B .crx . u 0 0 aeW Remarks: (i) The only place we have used B-B's proof i s to compute the dimension of d> ''"(U. 1 theorem i n the above ) = X ,K . 0 X Q Now we will 106 compute the dimension of cb (u. ) without using B-B's theorem. This can be done as follows: Consider the morphism cb "'"(U. ) U. X ,k 1 zi . o {ax > — y I t induces on the tangent spaces a commutative diagram T <Jx„ 0 (cb ?"(ui .,k ,)) — ^ — y <H T . . (u.) ) 0x I Q dy. + Since dy. : T , . I , (U. ) = T . i $(ax ) Q T cb(crx ) ( 0 V (Z. ) © T . <K ) ax I Q ,( P ) + T, N <f>(ax ) frl <b(Ox ) 0 0 ,(Z.) I i s the projection to the f i r s t factor (see section 1) we get dci(T cr Q 0 T„ Therefore dim T <JXQ i,k )) C T n cr (G/B)' X o dim(B . x ) < dim <b ( U . , ) < dim T„ CT u (G/B)~ = <b (u.i,k , ) _1 (cp ( U . 1 Ox„ dim Hence (cj, ^ (U. , ) ) ) C. T . ( P )' i,k <('( x ) N Ox (cb(u. , )) < 1 0 d(Ox^) 0 - i,k T . Since = dim(B~.axJ u 0 - crx 1 d(cbc) = dim(B~.Ox = d(c o 0 x i/k Q u 0 ) , we get ). With t h i s argument,, the above proof should be considered as an elementary proof of the Bruhat decomposition. 107 (ii) information The about proof of the the Bruhat coefficients decomposition a /?(Adu ) . Y , . . . . ,a I = g i v e s more i n the e x p r e s s i o n a Y i=0 (iii) It i s clear B-B's is Let P through a parabolic subgroup of G/P , associated decomposition of G/P G containing IT: G/B G - e q u i v a r i a n t morphism decomposition the Bruhat be to G , i.e. m G/P , Tr(gB) -> G/P -action A: = G T m < A, a i s a regular > > 0 for a l l Let us f = E 12 0 = A: G is g i v e n by a m 3 21 E -> T * L Adg E Hence GL /B = 2 Take t 0 0 1 = n = E> [a ,a X l X 2 X 3 X 4 1 2 2 [x^ = X l X 2 X 3 X 4 , g o" 1 0 - x e GL x 1 {a} 3 ' i 3 ' B -> [x x G/P m .ax„ -* , G/P where u such that X ,x example . l = w 0 Then and f 0 1 ' 2.2.1 0 0 0 <|>: GL^/B Adg E 1 1 +IP(Ag!l ) 2 = a^E^ 2 \ 0 + a . ^ 2 = P + a ^ , then 2 U\ x T f where 3 = _ 1 i n the 0 = 2 g = g K 2 , a ,a ] Q of = gP example. + cj> (gB) <j> (gB) P = subgroup . . + , V = E = ' -a 21 A(t) t 12 2.2.2. 0 , 6 e $ 1 0 a a parameter give a simple Example v one G B aeW(P) A: B 2 ~ 3^ X ] . E12- • x x w 1 3 i Consider Via this + X i x w the natural 3 2 - natural x ^ } isomorphism isomorphism + 108 cb: P 1 3 -> P 2 ^([x^x.^]) = [x^, i s given by "Twisted Veronese mapping". - x x 1 3 ' [tx X l = (t, [x ,x ]) = [ t x ^ x ^ , and on ,x ,x , t -1 2 V ( X 0 ) ° 2 X = x^] V ( X . Then D-L's 0' 1' 2 X X X P 1 , <b (X ) = e 1 ' P = f i l t r a t i o n on P if. = [0,1] . P _1 1 y^ X 1 P( A g ^ ) 3 GL^/B (t, [ x ^ x ^ x ^ x . ^ ] i s given by are the homogeneous coordinates = (P) 1 1 Y Bruhat decomposition of X ° * ' ) d> ^~ (X ) = V(y ) c P of 2 ~ 3^ i 3' The corresponding G^-action on fl?"*" = 3 i s simply x GL^/B = 0 u (e } = G u{°°) i s the a P"*" . In f a c t for any G -action on p n , if Z = I j 0 Z. i s the 1 i r r e d u c i b l e decomposition of the fixed point scheme, then the B-B. decomposition i s of the form P r = [J B .z. , where u i i=0 Borel subgroup B of GL^ ^ + B u i s the unipotent r a d i c a l of some (see section 1). Moreover one can use the T-equivariant morphism <p: G/B «-»- P(A^g) and the above method, to show that the B-B decomposition associated to any G -action on G/B (resp. on G/P) induced by T m s the form G/B = i I - „ , „ ~ > • — . \- B .Z. (resp. G/P = VJ B .W.) . i=0 u l i=0 u l i s of r J VT rs I am very g r a t e f u l indeed to D. Lieberman f o r the several h e l p f u l discussions I had with.him while t h i s work was i n progress at IHES. I am also g r a t e f u l to B. Iversen for mentioning this problem to'me. 109 Let us f i n a l l y give one more a p p l i c a t i o n of Lemma 2.1.1, Lemma 2.2.2 and Corollary 2.2.2 to appreciate the use of Lemma 2.1.1. Lemma 2.2.3. Let G be a connected l i n e a r algebraic group defined over an a l g e b r a i c a l l y closed f i e l d k , and P be a parabolic subgroup of G . Then P = N (P ) = {geG: Adg(P ) = P } , where G u u u P = T (P ) i s the tangent space of P at the i d e n t i t y e e G . In u e u u particular P = N (P) = {geG: Adg(P) = P} G i algebra of P . Proof. where We w i l l f i r s t prove Lemma when this case, l e t B P = L(P) G i s the L i e i s reductive. In be a Borel subgroup of G contained i n P , and l e t $ be the set of roots of G with respect to a torus T c B , and W = N (T)/T be the corresponding Weyl group. Now the torus T acts G on the closed subvariety N (P )/P v i a l e f t m u l t i p l i c a t i o n . By Lemma G u 2.2.2 the fixed points of T on N (P )/p are of the form oP f o r G U some a e W . Now i f oP e N (P )/P i s a fixed point of T , then G u Ada(P ) = P . But P = ©. g for some $ o ip o $ , and Ada (g ) = g u u u aeip a a aa + If i s the unique connected T-stable subgroup of G algebra g' [22, p. 161] for each a representative f o r a e W , then I(J = {a a^} d>: U x . . . . xrj a , a, 1 k T a e $ having L i e and n e N (T) i s a G nU n = U for each a e $ . Let a aa and consider the morphism of v a r i e t i e s 1 -> P : <j> (x, ,.. . ,x, ) = x,'. . x, u 1 k l k . I t i s an i n j e c t i v e morphism. Since they have the same dimension, the constructible set i> (U • 01, 1 contains a dense open subset of P , say u ' i •n (V=P ) c i (V) c <t(U u•- v " a X...XTJ a )= p u fc X . . . X T J OL ) k V . But then where -1 i (g) = gng . n Therefore 110 i n (P ) = P , namely u u Hence n e N_(P ) . By Corollary 2.2.2 we get n e P . G u {[P]} i s the only fixed point of T x 2.1.1 on N CP )/P . By Lemma G u we get N (P ) = P . G u Now, l e t G be any connected l i n e a r algebraic group, and l e t R ( ) Q U be the unipotent r a d i c a l of G . We have natural morphism of algebraic groups of TT: G -»• G' , G' = G/R^CG) . Hence G' , and cb (P ) = (<b(P)) u ' u <b(P) <J> (P) i s a parabolic subgroup i s the set of a l l unipotent elements of . For g e N (P ) we have cf) (g) e N G u G dir (e) (T (P )) = T ((<b(P)) ) . Since e u e u G' (T ((cb(P)) ) where e u i s reductive, we get cb (g) e cb (P) . and r E R (G) . Since R (G) l i e s u u i n every Borel subgroup of G , i n p a r t i c u l a r ( G ) C P , i t follows that This means g = pr for some p e P R U q e P . This, together with the f a c t that 3 Since P c N (P ) G u P c N (P) C N (P ) we have i n p a r t i c u l a r G G U implies P = N (P ) G u P = N CP). G Q.E.D. Remark: N CP) = P G i s a well known r e s u l t . In fact one can use Lemma 2.2.3, and obtain a T-equivariant closed immersion cb: G/P *-»- P (A g) where the dimension of P = r. O.Loos informed us that the following are also true : T C P ; ) - P and N (R (P)) = P ,where R (P) i s the unipotent r a d i c a l of P. Ill 3. A VECTOR FIELD WITH ONE ZERO ON G/P By Lemma 2.1.1, we know that there exists no G -action on a m complete variety X with only one fixed point i f the dimension of X i s greater than or equal to one. existence of a G -action on G/P a field k i s nice. In this section we w i l l show the with only one fixed point i f the ground This w i l l be suprisingly an application of Lemma 2.1.1. I am very g r a t e f u l indeed to 0. Loos for several h e l p f u l discussions I had with him while t h i s work was i n progress. Let G be a connected reductive l i n e a r algebraic group defined over an a l g e b r a i c a l l y closed f i e l d of k , and l e t G , T a maximal torus contained i n product T x> B^ where B^ be the set of roots of G B B . Then be a Borel subgroup B i s a semidirect i s the unipotent r a d i c a l of B . Let $ with respect to T and l e t W = N (T)/T G be. the corresponding Weyl group. We s h a l l denote by the same symbol an element of W and a representative i n N (T) when this can be done without ambiguity. G Let A be the set.of simple roots of $ for the ordering associated with have B , and l e t § + g = tffiJJL g , b = © • a u .+ aes> ae$ be the set of p o s i t i v e roots. g , dim g = 1 a a are the Lie algebras of T , G , and where respectively. Then we t ,g , b u Now we have the following important lemma. Lemma 2.3.1 (E.Y. A k y i l d i z ) . closed subvariety of G/B , where Proof. For any n e b , X /B u n X = {geG: Adg(n) e b } . n u I f n = 0 , there i s nothing to prove. i s 'a Assume n =(= 0 . 112 Let d = dim.b u = dim b , m = dim(t) , where u g = t © b © b . Consider u u x © g - Then .+ • -a ae$ Ad: G -> GL(g) C g&(g) and define . • P = (TegA(g): T(b ) c b } , Y u " " u n and h = u = {Teg£(g): T(ri)eb } . Let P = P A GL(g) u Y = Y f\ GL(g) , we have c l e a r l y n n X = Ad ( Y ) n n 1 and B = Ad~ (P) 1 because of Lemma 2.2.1. We claim see t h i s extend of g b ,b u u Y .P = { T O S : TeY , SeP} i s closed i n g£(g) . To n n n 4= 0 to an ordered bases {Ofn=v ,. ,v,,v. ,v~,.. , v , t ,. 1 a 1 2. d 1 such that {v,,..,v,} ,•{v.,..,v.J , { t . , . . , t } are bases of I d I d 1 m , and t respectively. With respect to t h i s ordered basis P n n i s the set of matrices of the form d+m and Y n * * 0 * £ qSL 2d+m i s the set of matrices of the form 2 d+m d+m Let Z be the set of matrices of the form e gZ , 2d+m 3 d+m { with rank (C) < d n j 113 Then Z = Y .P . n B A. For i f •e Y A = A. , B = n 0 Because the f i r s t column of rank (A ) < d . 3 Hence A.B 2 e P , then A.B = A B, * 3 1 B, 4, A^ * * B 1 i s zero, and rank ( A B 3 1 ) < e Z . Conversely, given e Z v i a column operations on Q C^ we can f i n d an i n v e r t i b l e d x d matrix such that, Now CQ 3 take -1 B = Q 0 0 Id e P , and A = e Y n A .B = Therefore 2d+m (Y .P) A GL (g) = (Y A GL(g)).(P (\ GL (g)) = Y .P n n n i n closed i n GL(g) Ad: G/B -y GL(g)/P GL(g)/P . The morphism Ad: G + GL(g) Y /P n induces a morphism which, as easy to see, i s a closed immersion. has the quotient topology and follows that i s closed i s closed i n Y P GL(g)/P . i s closed i n But then GL(g) Since ,it ~ -1 X /B = Ad (Y /P) n n 114 i s closed i n G/B , which completes the proof. Q.E.D. Let let of G cGL^(k) n c g c. gl^ G . be a connected l i n e a r algebraic group, and be any nilpotent element, where Then, i f the c h a r a c t e r i s t i c of k g i s the Lie algebra i s zero, there i s a well defined algebraic group morphism e: G •+ G such that de(l) = n , where de: k -»- g i s the cl d i f f e r e n t i a l of e at 0 e G , and 1 i s the unit of k . a Theorem 2.3.1. Let G be a connected reductive l i n e a r algebraic group defined over an a l g e b r a i c a l l y closed f i e l d zero, and l e t G -action on B be a Borel subgroup of G/B , induced by G . k of c h a r a c t e r i s t i c Then there e x i s t s a G , which has exactly one fixed point. cl Proof. Let n = £ X , where S i s any subset of $ + cteS containing A , and 0 =)= X e algebraic group homomorphism g for each e: G -> G aeS . such that Si induces, v i a l e f t m u l t i p l i c a t i o n , a G -action on cl {[B]} i s the only fixed point of this action. Now we have an ds(1) = n . G/B . e We claim that To see t h i s , consider X /B i n G/B which i s a closed subn variety of G/B by Lemma 2.3.1. The torus T acts on X /B v i a n l e f t m u l t i p l i c a t i o n , hence i t acts on each' i r r e d u c i b l e component of X /B n . Let Z be any of them. form aB , for some then Ada(n) = Ada( we must have aae$ a e W . Now, Fixed points of if aB e Z Y x ) = Y x e b aeS„ « aeSr. act for a l l aeS . Since S on Z are of the i s a fixed point of because u + T a e X n . T , But then contains the simple roots 115 A ,a T must be the i d e n t i t y . Hence {[B]} i s the only fixed point of i n Z . By Lemma 2.1.1, we get Z = {[B]} . Since Z i s an a r b i t r a r y i r r e d u c i b l e component of X /B , we have X /B = {[B]} ; i . e . , X = B . n n n G Now, i f gB e (G/B) i s a f i x e d point of G , then cL g 1 e (x)g e B This implies for a l l x e G g 1 e X n . Hence = B , therefore Adg ( d e ( l ) ) = Adg (n)eb 1 1 g e B , and {[B]} i s the only fixed point of this action. -.Q.E.D. The nilpotent element n , i n the form above, was suggested to us by 0. Loos. Remark: Since any nilpotent element i n g i s conjugate to an element i n b^ , i t i s clear from the proof of this theorem that any G -action, induced by a G , on G/B with only one fixed point, i s obtained.in this way. Corollary 2.3.1. Let G be a connected linear algebraic group defined over an a l g e b r a i c a l l y closed f i e l d zero, and l e t B be a Borel subgroup of G . Then there exists a G -action on G/B , induced by a Proof. G' = G/R^CG) of k of c h a r a c t e r i s t i c G , having only one f i x e d point, Let R (G) be the unipotent r a d i c a l of G . Then u i s a ,reductive group, and B' = <j> (B) i s a Borel subgroup G' . The natural epimorphism of algebraic groups induces an isomorphism <> J: G/B ^ G'/B' , since b i j e c t i v e and separable. subset of $ + containing Let n 1 = £ aeS A , and 0 =]= X x <j>: G R^(G) c B , e g ' where e g ' for each G' d > is S i s any aeS . Then 116 there exists, a nilpotent element since R (G) l i e s i n B u u dtp: g -* g' morphism G of g such that f o r a l l Borel subgroups i s surjective. e: G n dtp (n) = n' , B of G and Hence we have an algebraic group homo- 'such that de(l) = n . Let £' = cb o E : G a Then ->• G' a e(resp. e ) induces v i a l e f t m u l t i p l i c a t i o n , a G -action on 1 3. G/B (resp. G'/B') so that <j): G/B »- G'/B' i s G -equivariant. cl By Theorem 2.3.1 the G -action on G'/B induced by e' has only one a fixed point. Therefore the G -action on G/B induced by e has only 1 cl one fixed point {[B]} , since cb i s G -equivariant isomorphism. 3. Q.E.D. Theorem 2.3.2. Let G be a connected l i n e a r algebraic group defined over an a l g e b r a i c a l l y closed f i e l d and l e t P be a parabolic subgroup of G . Then there e x i s t s a G -action on a G/P , induced by Proof. and l e t k of c h a r a c t e r i s t i c zero, e: G Let B G , having only one fixed point, be a Borel subgroup of G * G/B -> G/B contained in P , be the given action i n Corollary 2.3.1. Then ci e induces a G -action on G/P so that the natural map TT : G/B •> G/P , 3. Tr(gB) = gP i s G -equivariant. By Corollary 2.3.1 and Lemma 2.2.2, we cl conclude. (G/P) = {[P]} . Q.E.D. Comments: The proofs of Theorem 2.3.1 and Theorem 2.3.2 f o r reductive groups would go through f o r c h a r a c t e r i s t i c p > 0 also, i f we knew the existence of an algebraic group morphism de(y) = such £ X aeS f o r some E does not e x i s t . y E k .and f o r some e: G -> G a with s D A . But, generally A. Borel.and T.A. Springer give a s u f f i c i e n t condition f o r the existence of e i n [4, p. 495]. 117 BIBLIOGRAPHY M.F. Atiyah and R. Bott, A Lefschetz fixed-point formula for e l l i p t i c complexes: I I , Annals of Math., 88(1968), 451-491. P.F. Baum and R. Bott, On the zeroes of meromorphic vector f i e l d s , i n Essays on Topology and Related Topics, ed. A. Haefliger and R. Narasimhan, Springer-Verlag (1970), 29-47. B i a l y n i c k i - B i r u l a , A., Some theorems on actions of algebraic groups, Ann. of Math. 98, 480-497 (1973). A. Borel and T. A. Springer, R a t i o n a l i t y Properties of Linear Algebraic groups IT, Tohoku Math. J . V o l . 20, 443-497 (1968). R. Bott, Vector f i e l d s and c h a r a c t e r i s t i c numbers. Mich. Math. J . 14, 231-244 (1967) . R. Bott, A residue formula.for holomorphic vector f i e l d s . Differential Geometry, V o l . 1(1967), 311-330. N. Caliskan, private communications (1976). J.B. C a r r e l l , Holomorphically i n j e c t i v e complex t o r a l actions, Proc. Conf. on Compact Transformation Groups, Springer Lecture Notes 299, 205-236. J.B. C a r r e l l , A remark on the Grothendieck Residue map, preprint. J.B. C a r r e l l , A. Howard, C. Kosniowski, Holomorphic vector f i e l d s on complex surfaces, Math. Ann. 204(1973), 73-82. J.B. C a r r e l l and D. Lieberman, Holomorphic vector f i e l d s and Kaehler manifolds Invent. Math. 21(1973), 303-309. J.B. C a r r e l l and.D. Lieberman, Vector f i e l d s and Chern numbers, Math. Ann. 225, 263-273 (1977). \ 118 13. J.B. C a r r e l l and D. Lieberman, Meromorphic vector f i e l d s and Residues, unpublished manuscript. * 14. J.B. C a r r e l l and A.J. Sommese, 15. S. S. Chern, Complex manifolds without p o t e n t i a l theory, Van Nostrand Notes 16. C actions, preprint. 1967. S.S. Chern, Meromorphic vector f i e l d s and c h a r a c t e r i s t i c numbers, Scripto Math. XXIX (1973), 243-252. 17. S.S. Chern, Geometry of c h a r a c t e r i s t i c classes, Proc. 13th Biennial seminar 18. (1972), C. Chevalley, Certains schemas, de groupes semi simples, Sem. Bourbaki (1960-61), Exp. 19. 1-40. 219. R. Hartshorne, Residues and Duality, Lecture notes i n Mathematics 20, Springer-Verlag (1966). 20. F. Hirzebruch, Topological Methods i n Algebraic Geometry, SpringerVerlag (1966). 21. G. Horrocks, Fixed point schemes of additive group actions, Topology, V o l . 8(1969), 233-242. 22. J.E. Humphreys, Linear Algebraic Groups. 23. B. Iversen, Cohomology and torus actions, preprint. 24. B. Iversen, A fixed-point formula for action of t o r i on algebraic v a r i e t i e s , Invent. Math. 16(1972), 229-236. 25. B. Iversen and H.A. Nielsen, Chern numbers and diagonalizable groups, J . London Math. Soc. (2) (1975), 223-232. 26. B. Iversen, Private communications at IHES (1976). 27. C. Kosniowski, Applications of the holomorphic Lefschetz formula, B u l l . London Math. S o c , 2(1970), 43-48. 119 28. D. Lieberman, Holomorphic vector f i e l d s on projective v a r i e t i e s , Proceedings of Symposia i n Pure Math. 30(1976), 271-274. 29. D. Lieberman, Holomorphic vector f i e l d s and r a t i o n a l i t y , unpublished manuscript, 1973. 30. D. Lieberman, Private communications at IHES (1976) 31. G. Lusztig, Remarks on the holomorphic Lefschetz numbers, Universite de Montreal Publ. Sem. Math. Sup. 42, Analyse Global, 193-204(1971). 32. D. Mumford, Introduction to A l g . Geometry. 33. E. Charon, private communications at Paris i n the Spring of 1976. 34. N.R. O'Brian, Zeroes of holomorphic vector f i e l d s and the Grothendieck residue, B u l l . London Math. S o c , 7(1975). 35. M. Rosenlicht, Tori operating on projective v a r i e t i e s , Rend d i Mat. 25(1966), 129-138. 36. D. Toledo, On the Atiyah-Bott formula f o r i s o l a t e d fixed points, J . D i f f e r e n t i a l Geometry. 8(1973), 401-436. 37. Y.L.L. Tong, De Rham's integrals and Lefschetz fixed point formula for d"-cohomology, 38. B u l l . Amer. Math. S o c , 78(1972) 420-422. Y.L.L. Tong, Integral representation formulae and Grothendieck residue symbol, American Journal of Math., 95(1973), 904-917. * 39. P. Wagreich, V a r i e t i e s with C action, Proceedings of Symposia i n Pure Mathematics, V o l . XXIX (1975). 40. R.O. Wells, D i f f e r e n t i a l Analysis on Complex Manifolds, Prentice-Hall Inc. 1973.
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Geometry of holomorphic vector fields and applications of Gm-actions to linear algebraic groups Akyildiz, Ersan 1977
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Title | Geometry of holomorphic vector fields and applications of Gm-actions to linear algebraic groups |
Creator |
Akyildiz, Ersan |
Date Issued | 1977 |
Description | A generalization of a theorem of N.R. 0'Brian, zeroes of holomorphic vector fields and the Grothendieck residue, Bull. London Math. Soc, 7 (1975) is given. The theorem of Riemann-Roch and Hirzebruch for V-equivariant holomorphic vector bundles is obtained, via holomorphic vector fields, in the case all zeroes of the holomorphic vector field V are isolated. The Bruhat decomposition of G/B is obtained from the G -action on G/B . It is shown that a theorem of A. Bialynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. 98, 480-497 (1973) is the generalization of the Bruhat decomposition on G/B , which was a conjecture of B. Iversen. The existence of a G -action on G/P with only one fixed a point is proved, where G is a connected linear algebraic group defined over an algebraically closed field k of characteristic zero and P is a parabolic subgroup of G . The following is obtained P = N[sub G](Pu) = {geG: Adg(Pu) = Pu} where G is a connected linear algebraic group, P is a parabolic subgroup of G and P^ is the tangent space of the set of unipotent elements of P at the identity. An elementary proof of P = N[sub G](P) = {geG: gPg ⁻¹=P} is given, where G is a connected linear algebraic group and P is a parabolic subgroup of G . |
Subject |
Linear algebraic groups Vector bundles |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-02-21 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080138 |
URI | http://hdl.handle.net/2429/20644 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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