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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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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. 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