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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 in the Department of Mathematics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1977 J Ersan Akyildiz, 1977. In presenting this thesis in partia l fulfilment of the requirements for 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 available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of MATHEMATICS  The University of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date August 5.J977 ABSTRACT A generalization of a theorem of N.R. 0'Brian, zeroes of  holomorphic vector fields and the Grothendieck residue, Bull. London Math. Soc, 7 (1975) i s given. The theorem of Riemann-Roch and Hirzebruch for V-equivariant holomorphic vector bundles is obtained, via holomorphic vector fields, in the case a l l zeroes of the holomorphic vector f i e l d V are isolated. The Bruhat decomposition of G/B is obtained from the G -action on G/B . It i s shown that a theorem of A. Bialynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. 98, 480-497 (1973) is the generalization of the Bruhat decomposition on G/B , which was a conjecture of B. Iversen. The existence of a G -action on G/P with only one fixed a point i s proved, where G is a connected linear algebraic group defined over an algebraically closed f i e l d k of characteristic zero and P is a parabolic subgroup of G . The following i s obtained P = N (P ) = {geG: Adg(P ) = P } G u u u where G i s a connected linear algebraic group, P i s a parabolic subgroup of G and P^ i s the tangent space of the set of unipotent elements of P at the identity. An elementary proof of P = No(P) = {geG: gPg ~*~=P} is given, i i i where G i s a connected linear algebraic group and P i s a parabolic subgroup of G . iv TABLE OF CONTENTS INTRODUCTION 1 CHAPTER I. HOLOMORPHIC VECTOR FIELDS, EQUIVARIANT BUNDLES AND THE THEOREM OF RIEMANN—ROCH—HIRZEBRUCH 1. Hermitian Differential Geometry 8 2. The Canonical Connection and Curvature of a Hermitian Holomorphic Vector Bundle 13 3. Holomorphic Vector Fields and Equivariant Vector Bundles 17 4. Grothendieck Residue and The Theorem of Riemann-Roch-Hirzebruch 39 5. Chern Classes of Equivariant Bundles and Applications 62 CHAPTER II. APPLICATIONS OF G -ACTIONS TO LINEAR ALGEBRAIC GROUPS 1. Theorem of Bialynicki-Birula 81 2. Bruhat Decomposition 92 3. A Vector Field with One Zero on G/P I l l BIBLIOGRAPHY . . . . . . . . 117 V \ To my grandparents Humak, Nuri Caliskan v i ACKNOWLEDGEMENTS I wish to thank my supervisors Professors James B. Carrell and Larry Roberts for their help, advice and patience in the production of this work. I also thank Professors Birger Iversen, David Lieberman and Ottmar Loos for several helpful discussions I had with them. I am specially grateful to Gudrun Aubertin for letting me play with her during my loneliest days in Vancouver. At this time I would also like to mention my appreciation to my brother Yilmaz Akyildiz, my girlfriend Elizabeth Charon,' and to my friends Marie and Bruce Aubertin and Maria Margaretta Klawe. Finally, let me thank Cat Stevens, who so many times has found the words to express my thoughts and feelings when I could not. v i i ON THE ROAD TO FIND OUT Well, I l e f t my happy home to see what I could find 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 tick the time out, there's so much l e f t to know, and I am on the road to find out. Well i n the end I ' l l know, but on the way I wander through descending snow, and through the frost and thunder, I listen to the wind come howl., t e l l i n g me I have to hurry. I listen to the Robin's song saying not to worry. So on and on I go, the seconds tick the time out, there's so much l e f t to know, and I'm on the road to find out. Then I found myself alone, hoping someone would miss me. Thinking about my home, and last woman to kiss me, kiss me. But sometimes you have to moan when nothing seems to suit yer, but nevertheless you know you're locked towards to future. So on and on you go, the seconds tick 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 in lying, lying. Yes the answer li e s within, so why not to take a look now, kick out the devil's sin, 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 sin, and pick up, pick up a good book now. Yes the answer li e s within, so why not to take a look now, Kick out the devil's sin, and pick up, pick up a good book now. CAT STEVENS I N T R O D U C T I O N We say (£ acts on a complex manifold X , i f there exists a holomorphic map az' <C x X ->• X such that C -* Aut(X) , t -> a i s a group homomorphism, where Aut(X) i s the group of holomorphic diffeomofphisms of X , and a t ( x ) = o(t,x) . For an algebraic variety X defined over an algebraically closed f i e l d k , we say = GL^(k) acts on X , i f there exists a morphism cj>: G x X X of algebraic varieties such that G Aut(X) , t -* <f> is a group homomorphism. m t For a given C-action 0: (C x X X on a complex manifold X , a holomorphic vector bundle E -* X i s said to be a-equivariant i f (i) there exists a C-action S : <C x E -> E on E such that the diagram C x E ——y E is commutative, <E x X ° > X (ii) for t E (C and x e X , the map a (x) : E -y E , . is C-linear. • e t x afc(x) Let E -y X be a a-equivariant holomorphic vector bundle on X with the C-action a: <E x-E -»- E . Then for each t e C , we have an * * ~ - l 0 -module homomorphisms, (a ) : 0 ( E ) -»• (a )^0 ( E ) , (a ) (s) = -a 0S 0CT where s i s a local section of E . This induces (by differentiation) a C-module homomorphism. V: 0 ( E ) -> 0 ( E ) such that V(f.s) = V(f).s+f.V(s) x ~x where f (resp. s) i s a local section of X x c (resp. E ) and V i s the holomorphic vector f i e l d on X induced (by differentiation) by a: € x x -y X . V i s called the v-derviation on E induced by a: C x E -> E For a given holomorphic vector f i e l d V on a complex manifold X , a holomorphic vector bundle E -y X i s said to be V-equivariant i f 2 there exists a C-module homomorphism V: 0 (E) -> 0 (E) such that ~x -x V(fs) = V(f).s + fV(s) where f (resp. s) i s a local section of X •*> C (resp. E). Such a V i s called a V-derivation on E . The main problem we are concerned with in Chapter I, is to obtain the theorem of Riemann-Roch and Hirzebruch via holomorphic vector fields. In sections 1 and 2 we begin by reviewing some of the basic differential-geometric concepts in the context of holomorphic vector bundles. In section 3, i t i s shown that for a holomorphic vector bundle E X on a compact complex manifold X , the following are equivalent: (i) E i s a-equivariant (ii) E i s V-equivariant where V i s the holomorphic vector f i e l d induced by a . ( i i i ) There exists a hermitian metric h on E such that i (0,) = 8 (L) v . for some L e £(X, Horn(E,E)) where 9 is the canonical curvature matrix associated with h and i i s the contraction operator. P * As a corollary of this, i t i s shown that AT (T (X)) (resp. U ,Q ) Jc f n Jc / n i s V-equivaricint for 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, of k-planes in C11 and Q_ = Gr. x(Cn/U. . Some examples k,n k,n k,n k,n of V-equivariant bundles are given, and some computations are done. Let a:1 C x x X be a C-action on a compact complex manifold X , and l e t E X be a a-equi variant holomorphic vector bundle with the C-action a: C x E •+ E . Then, for each t e C , we have a bundle ^-1 -1 * isomorphism a : E (° t ) E . By taking the pull-back relative to a : X -*• X , we have a natural geometric endomorphism <f> : a E -* E of E . t t t Let V be the holomorphic vector f i e l d on X induced by a and l e t V be the V-derivation on E induced by a . If z = (z^ >. • • • i zn) denotes holomorphic local coordinates on a small neighbourhood N(p) of the isolated zero p of V , and i f Vi 'N(p) = Z a i=l i 9z. i s the local expression of V on N(p) , then we define for (t,z) e <E x N(p)-> q t v ^ z ) the Chern character of E by Ch(E,z,t) = E e i=l n tw. (z) We also define the Todd class of X by Td(z,t) = .'II i=l • tw.(z) 1-e 1 where q i s the rank of E and v^(z), w_. (z) are defined by the following formal identities: q det(I +XV(z)) = II (1+Xv.(z)) = Y c.(V(z))\:L <I i=l 1 n 1 det I +X n 9a.(z) I 9z. D = n (i+xw.:(z)) = I c ( x , z ) x j j=l 0 Then Ch(E,z,t) and Td(z,t) are holomorphic functions in W x N(p) where W i s a sufficiently small neighbourhood of 0 e C , since each c.(V(z)) (resp. c.(X,z)) i s an elementary symmetric function in v, (z) 1 J k (resp. i n w^(z)), and also holomorphic. In section 4, the holomorphic Lefschetz fixed point formula for the isolated fixed points, and an algorithm for calculating the Grothendieck Residue are stated. The following formula (which generalizes a theorem of N.R. ©'Brian [34]) 4 Res P trace d> (z) dz., . .dz . _t 1 n z -a (z) ,. . ,z -a (z) I t n t — Res _n p i Ch(E,z,t)Td(z,t)dz.,..dz 1 n a 1,...,a n for t e W - 0 , is shown. The theorem of Riemann-Roch and Hirzebruch for V-equivariant bundles, i s obtained from the holomorphic Lefschetz fixed point formula and the formula in [9], for holomorphic vector fields V with isolated zeroes. Let V be a holomorphic vector f i e l d on a compact Kaehler manifold X with isolated zeroes. It i s shown in [11] that there exists f f i l t r a t i o n T(Z,0 ) o F ^ F s ... , of the global section of z n n+1 the sheaf of ring 0 = 0 / i (ft1) on the set of zeroes Z of V , such z x that (*) gr(T(Z,0 )) = © F /F = H (X:C) -p -p+1 Let E -> X be a V-equivariant bundle with the V-derivation ^ i k A V on E . In section 5, i t i s shown that (—) c, (V(z)) e T(Z,0 ) 2TT k -z represents the k-th Chern class c (E) of E with respect to the isomorphism (*). (This result i s obtained in [12], independently.). * n H ( P :C) i s computed through the isomorphism (*) . Also c.(U ) l Jc f n (resp. c. (Q, )) are computed in T(Z,0 ) (this computation i s done l k,n z in [12] via a different approach). Some calculations are done and motivation i s given for the concepts in the earlier sections. In chapter II section 1, we show that for any G^-action <j>:G xx->X on a complete variety X , there exists an everywhere m defined map <j): E> x X X 5 such that the diagram is 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 {0} u {°°}. m 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 in the theory of Linear Algebraic Groups. It i s also a useful tool i n understanding the structure theory of complete varieties with G -actions. Furthermore, Gm by our methods i t i s simple to show that the fixed point scheme X i s connected i f and only i f $ acts t r i v i a l l y on X . This last result i s proved in [3, p. 495], but only after much work. In chapter II, we give some of the applications of the use of (j> . One of the important corollaries to the existence of <j> i s the existence of enough fixed points. Namely, i f T i s a torus, acting on a projective variety X , and i f the dimension of X i s greater than or equal to one, then T fixes at least two points of X [22, p. 153]. By using this, we are able to prove; (i) In section 2: P = N (P ) = {geG: gP g 1 = P } G u u u and P = N (P ) = {geG: Adg(P ) = P } , G u u u where G i s a connected linear algebraic group, P i s a parabolic sub-group of G , P i s the set of unipotent elements of P and P i s 6 the tangent space of P^ at the identity. These results give elementary proofs of the well-known theorems P = N (P) = {geG: gPg-1=P} , and P = N (P) = {geG: Adg(P) = P} , G where P i s the Lie algebra of P . (ii) In section 3: The existence of a G -action on G/p. with only one fixed point, 3. where G is a connected linear algebraic group defined over an algebra-i c a l l y closed f i e l d k of characteristic zero, and P is a parabolic subgroup of G i s established. For a reductive linear algebraic group G defined over an algebraically closed f i e l d k of characteristic p > 0 , the existence of a G -action on G / P with only one fixed point i s discussed, where P i s a parabolic supgroup of G . The structure theory for complete varieties with G^-action was initiated by Bialynicki-Birula i n [3], by the following theorem: ' Theorem (Bialynicki-Birula). For a given G -action ^ m G r d>: G x x-:->- X on a complete variety X , l e t X = \_) Z. be the m 0 1 G decomposition of the fixed point scheme X into connected components. Then there exists a unique locally closed G^-invariant decomposition of X and morphisms y.: U. -> Z. for i= 0,1,..r so that 7 (ii) y. i s a G -fibration 1 m ( i i i ) for every closed point p e T (U. ) = T (Z. ) ® T (X) p i p i p i where T (X) = {veT (X): d<t (p)v=tmv for some m<0, for a l l teG }. p p t m We c a l l the above decomposition B-B's decomposition. A geo-metric proof of this theorem (using cb) was communicated to us by D. Lieberman. In section 1, we give B-B's decomposition on p n associated with any G -action on p n , and discuss the relation between B-B's m decomposition on X and on (Pn under the G^-equivariant closed n immersion X D? In section 2, we have obtained the Brunat decomposition of G/B from B-B's decomposition, which was a conjecture of B. Iversen [26]. 8 CHAPTER I HOLOMORPHIC VECTOR FIELDS, EQUIVARIANT BUNDLES AND THE THEOREM OF RIEMANN-ROCH-HIRZEBRUCH 1. HERMITIAN DIFFERENTIAL GEOMETRY In this section, we w i l l review some of the basic dif f e r e n t i a l -geometric concepts in the context of holomorphic vector bundles and, more generally, differentiable C-vector bundles. For.more details, the reader i s referred to R.O. Wells [40]. We shall denote by the term vector bundle a differentiable C-vector bundle over a differentiable manifold, E ->- X . Let E -> X be a differentiable vector bundle over X , and let £(E) be the sheaf of differentiable sections of E . Then we denote by £(U,E) the set of differentiable sections of E over an open subset U of X , moreover we denote £_( A^T*X) by , the sheaf of differentiable p-forms on X , £°: = £. the sheaf of differentiable functions on X . Definition 1.1.1. Let E •* X be a vector bundle. A Hermitian metric h on E i s an assignment of a hermitian inner product <,> to each fibre E of E such that for any open set U C X and X" C , n e £ ( U , E ) the function <£,ri>: U C given by OO <C,n>(x) = <£(x),n(x)> i s c A vector bundle E equipped with a Hermitian metric h i s called a Hermitian vector bundle. If q „: U A U. -> GL (C) are the transition functions of E a,3 a 3 r Then any global smooth section s of E on X i s uniquely determined by the following data; s = {s} s =9"( a,3 3 on U A U where 9 s : U C r i s C°° . a a Let E -> X be a Hermitian vector bundle on X and l e t t CK Ct t f = (e.,....,e ) be a frame for E over u , where f = transpose a i r a a of f . We define (h ) = <ea,e"> , and l e t h F ((h ) ) be the a a p,q p q a a py.q r x r matrix of the C functions {(h) j . Thus h i s a positive a p,q a definite Hermitian symetric matrix, and i s a (local) representative for the hermitian metric h with respect to the frame fcf . Now we have a the change of frame over u rv U_ , f = g„ f„ ; i t i s easy to see that a 0 a 3,a 3 h = g„ h^g„ on U A'U„ i s the transformation'law for local a 3,a 3 3,a a 3 representations of the Hermitian metric h . Conversely any hermitian metric on E i s given by the above data. By a partition 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, , -> IPn be the tautological line l,n+l bundle on the projective n-space p n , i.e. 0(Un ,) = 0!(-l) . - l,n+l -Let ( P n) = {[x ,..x ,..x ] e P n: x ^ 0} a = 0,..,n be the x u ot n t ot a standard open affine covering of p n , where x^,..is the homo-geneous coordinates of p n . We define h : U GL. (C) by a a 1 h (fx.,..,x ]) = 1 0 I \\\2 u ^ V ' V v 2 X a x Now g „([x ..x 1) = — is the transition function of U on a,$ O n x l,n+l p 10 ( Pn) r\ ( Pn).. , hence {h } a = 0,,.,n satisfies the transformation x a x law for lg „ = —••'} . Therefore i t defines a Hermitian metric h on •l"a,g x • J U, . , which i s called the Fubini-Study metric. 1, n+1 Let E -> X be a vector bundle. Then we let £.P(E) = £( APT*X ® c E) • be the sheaf of E-valued di f f e r e n t i a l forms of degree p . We have the natural isomorphism £( APT*X) 8 1(E) = tP » £(E) = £ P(E) . — — We denote the image <|> ® £ under this isomorphism by <j> . E, e c,P(E) , where cf> e fcP , E, e £,(E) . Definition 1.1.2. Let E -> X be a vector bundle. Then a connection D on E is a C-linear mapping D: £(X,E) £,1(X,E) , which satisfies 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 line bundle, we see 1 that we may take ordinary exterior differentiation d: £ (X) ->• £ (X) as a connection on E . Thus a connection i s a generalization of exterior differentiation to vector valued-differential forms. We now want to give a local description of a connection. t Let f = (e^,..,er> be a frame over U for a vector bundle E •> X , 11 equipped with a connection D , where rank of E i s r . Then we define the connection matrix 6(f) with respect to the frame , by setting. 0(f) = (6(f) ), 0(f) e ^ ( u ) , where 0(f) i s p^q p#q - P/q defined by De = T 0(f) .e q pt± q,P P The effect on the connection matrix under a change of the frame f i e l d can easily be found. In fact, let : f = g f be the p 01 r p 01 new frame f i e l d , where g i s nonsingular rxr-matrix of C -functions, a, p we find immediately This i s the equation for 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 shall f i n a l l y define the curvature: Let E -»• X be a vector bundle with a connection D , and l e t 0 ( f ) be the associated connection a matrix over a local frame ^f . We define a 0(f ) = d0(f ) - 0(f ) A 0(f ) which is an rxr matrix of 2-forms a a a a i.e. 0(f ) d0(f ) - T 0'"'(f ) /s, 0 ( f ) a p,q a p,q £ ayp,k a k,q If f. = g . f i s the new frame f i e l d , we find easily 3 a,3 a 12 0 ( f D ) = g Q 0 ( f )g 1 Q , where 0 ( f ) = d6 (f ) - 9 (f ) A 6 (f ) 3 a,3 a a,3 3 3 3 3 Because of the trans f o r m a t i o n law {0(f )} de f i n e s a g l o b a l element a 2 0 e £, (X„ Horn (E,E)) which i s c a l l e d the curvature form a s s o c i a t e d to the connection D . Let E be a Hermitian v e c t o r bundle on X . Then we can extend the m e t r i c h on E i n a n a t u r a l manner to a c t on E-valued covectors. Namely, s e t <w»£, w'®n> = WAW'<5,TI> f o r w e A?T*X , X X X w' e /?T*X , and £, n e E f o r x e X . Thus the extension of the x x inner product t o d i f f e r e n t i a l forms induces a mapping h: £P(X,E) ® £q(X,E) ->£.P+q(X,E) . D e f i n i t i o n ^ 1.1.3. A connection D on E i s s a i d to be compatible w i t h the h e r m i t i a n m e t r i c h on E i f d<£,n> = <D£,n> + <£,Dn> . I t i s easy to see t h a t a Hermitian v e c t o r bundle E admits a connection which i s compatible w i t h the h e r m i t i a n m e t r i c , and t h a t i t i s not unique i n gen e r a l . But i n the holomorphic category, we s h a l l o b t a i n a unique connection s a t i s f y i n g an a d d i t i o n a l r e s t r i c t i o n on the type of 0 . 13 2. THE CANONICAL CONNECTION AND CURVATURE OF A HERMITIAN HOLOMORPHIC VECTOR BUNDLE. Suppose now that E •> X is a holomorphic vector bundle over a complex manifold X . If E as a differentiable bundle i s equipped with a differentiable Hermitian metric h , we shall refer to i t as a Hermitian holomorphic vector bundle. Recall that since X i s a complex manifold, £ r(E) = I £ P' q(E) , wher€ p+q=r £ P , q ( E ) = £ P , q » |(E) and £ P , q i s the sheaf of differential forms of type (p,q) . Suppose then that we have a connection on E D: £(X,E)-* £ 1(X,E) = £ 1 / 0(X,E) ffi £°'1(X,E) . Then D splits naturally into D = D' + D" , where D' : £(X,E) ->&1,0(X,E) and D": fc(X,E) -*£ 0 , 1(X,E) . Theorem 1.2.1. If h is a Hermitian metric on a holomorphic vector bundle E -* X , then h induces canonically a connection D (h) on E which satisfies, for U an open set in X . (a) If £, n e£(U,E), d<£,n> = <D£,n> + <E ,Dn> ; i.e. D i s compatible with the metric h . (b) i f £ e 0 (U,E) , i.e., £ i s a holomorphic section of E , then "X D"£ = 0 . Proof. Let {tf } be the holomorphic frames for E and l e t a 14 g „: U r\ U„ -> GL (C) be the holomorphic transition functions of E a,B a B r r over an open covering {U } of X . Assume h i s the local - a ael a representative for the hermitian metric h with respect to the frame t . -1 f . Now define 6 = 9h h ; for a frame change over U A U_ we have a a a a a 3 f 3 = ga,3 fa ' h e n c e h 3 = g a , 3 h a \ , B ' B u t t h e n V a , 3 " ( 8 h 3 h 3 1 ) g a , 3 = { 8 g a , 3 + g a , e 3 h a \ , B t- t- t- t- -^" -1 -1 ={9g „h g 0 + g „3h g „ + g „h 9 g „} g „h g „g „ ^a,3 a a,3 a,3 a ya,3 ^a,3 a a ,3 a ,3 a a ,3 a ,3 = {9g .h + g D9h }h _ 1 = 9g . + g .9h h" 1 = dg . + g .6 a,3 a a,3 ot a a,3 a,3 a a a,3 a,3 a Since g i s holomorphic. a, p Thus 8„ = dg 0 g 1 0 + g „ 8 g 1 0 / hence {8 } defines a 3 a,3 a,3 a 3- a a,3 a unique global connection on E , because i t satisfies the transformation law (1). It i s easy to see that, this connection satisfies the conditions (a) and (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) 1 , where fcf is a holomorphic frame. Corollary. Let D be the cononical connection of a Hermitian holomorphic vector bundle E-^ -X with the Hermitian metric h . Let 8(f) and 0(f) be the connection and curvature matrices defined by D with respect to the holomorphic frame f . Then 15 (a) 0(f) i s of type (1,0) and 30(f) = 0(f) A 0(f) (b) 0(f) = 99(f) and 0(f) i s of type (1,1) , 30(f) = 0 Proof. By definition 0(f) = 3h(f)h '''(f) which i s of type (1,0) , now 30(f) = 3(3h(f)h~ 1(f)) = - 3h(f) A 3h _ 1(f) , but 3(h(f)h _ 1(f)) = 3 h ( f ) h ( f ) - 1 + h(f)3h _ 1(f) , hence 3 h ( f ) h ( f ) _ 1 A 3 h ( f ) h ( f ) - 1 = s - 3 h ( f ) h ( f ) _ 1 A h(f)3h - 1(f) = - 3h(f) A 3h _ 1(f) = 0 (f) A 0(f) , hence 30 (f) = 0 (f) A 0 (f) . For (b) we have by definition 0(f) = d0(f) - 0(f) A 0(f) . Hence by (a) we get 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 principal example concerning the computation of connection and curvatures. Example 1.2.1. Let n + 1 -> (Pn be the tautological line bundle on p n , l e t x„,....,x be the homogeneous coordinates for IP O n Then by example 1.1.1 " 2 , r - i 1=0 Iv[x„,...,x J = n cxL'0' ' n J 2 1 a' defines a hermitian metric h on Un ,.. . Hence the associated canonical l,n+l connection and curvature matrices are given by 0 = 3h h 1 and a a a 0 = 30 . Let z. be the holomorphic local coordinates on U , then a a i a .2, 2 n 2 h (z.,...,z ) = 1 + Z |z.| . Let z = (z ,...,z ) , |z| = £ |z | , a 1 n 1 i=l 16 then 0 (z) = 3 log h (z) = -a a 2 1+l'zl / z.dz. i=l and 0 (z) = 36 (z) = 3 a ' a n z.dz. i=l 1+| z| n (1+1 z| ^ )dz.«dz. - ,E, z.z.dz.^dz. 1 1 l l j=l ± 3 3 i z TV2 i=l (1+|z|) hence 17 3. HOLOMORPHIC VECTOR FIELDS AND EQUIVARIANT BUNDLES Let X be a complex manifold of dimension n , and let C IR IR T(X) = T(X) ® C be the complexification of the tangent bundle T(X) iR Since X i s a complex manifold, T(X) 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) i s in C duality with the forms of type (1,0) while T (X) i s i n duality 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 V on X is a map which assigns to <C each point x of X an element V of T(X) with an obvious x diff e r e n t i a b i l i t y condition. If ^z±^-i holomorphic local coordinate - + C - C system over U , say, then 3/8z. , 3/3z. span T (X) and T (X) 1 1 X X respectively for every x e U . Hence we can express a complex vector f i e l d V uniquely in the form n v| = J a. 3/3z. + b. 3/3z. , 'U . , x l l I i - l where a^, b^ e £(U) . We notice that V i s a real vector f i e l d , i.e., IR — V e T (X) for each x E X i f and only i f a. = b. for i = l,...,n . X X 1 1 A complex vector, f i e l d V i s said to be of type (1,0) i f + C Vx E T x ^ a t eac^ P o i n t x • L e t v b e o f tyP e (IfO) . Then we can write V locally in the form n V = y a. 3/3z. . i=l 1 If the components a^ of V are holomorphic functions of the holomorphic local coordinates z^ , then we c a l l V a holomorphic vector f i e l d on 1 8 X . S i n c e T X = T (X) a s a C - v e c t o r b u n d l e , a h o l o m o r p h i c v e c t o r f i e l d o n X i s s i m p l y a g l o b a l h o l o m o r p h i c s e c t i o n o f t h e h o l o m o r p h i c t a n g e n t b u n d l e TX o f X . F o r a c o m p l e x v e c t o r f i e l d V o f t y p e ( 1 , 0 ) . We d e f i n e V = V + V w h e r e (V) = V a t e a c h x . T h e n V i s a r e a l v e c t o r f i e l d , x x a n d i s c a l l e d t h e : , r e a l v e c t o r f i e l d a s s o c i a t e d t o V . D e f i n i t i o n 1 . 3 . 1 . We s a y C - a c t s o n a c o m p l e x m a n i f o l d X , i f t h e r e e x i t s a h o l o m o r p h i c map a: C x X -* X s u c h t h a t t h e a s s o c i a t e d m a p C -»- A u t ( X ) , t -> a : X -* X g i v e n b y ° t ( x ) = o ( t , x ) i s a g r o u p h o m o -morphism w h e r e A u t ( X ) i s t h e g r o u p o f a l l h o l o m o r p h i c d i f f e o m o r p h i s m s o f X . We c a l l a o r a 1 - p a r a m e t e r g r o u p o f a u t o m o r p h i s m s o f X . L e t a : C x x -> X b e a 1 - p a r a m e t e r g r o u p o f a u t o m o r p h i s m s o f X . F o r e a c h x e X , we d e f i n e d * V = ( a . (x)) x d t t e T x b y V ( f ) = ( f ( a . (x)) x x d t t t = 0 t = 0 w h e r e f i s a h o l o m o r p h i c f u n c t i o n a r o u n d x . T h e n V = {V } i s X x e X a h o l o m o r p h i c v e c t o r f i e l d , c a l l e d t h e h o l o m o r p h i c v e c t o r f i e l d i n d u c e d f r o m t h e 1 - p a r a m e t e r g r o u p o f a u t o m o r p h i s m s a o f X . W h e n X i s c o m p a c t , i t i s w e l l k n o w n t h a t , a n y h o l o m o r p h i c v e c t o r f i e l d i s i n d u c e d f r o m a 1 - p a r a m e t e r g r o u p o f a u t o m o r p h i s m s o f X . I f V i s a h o l o m o r p h i c v e c t o r f i e l d , i n d u c e d f r o m a 1 - p a r a -m e t e r g r o u p o f a u t o m o r p h i s m s a . We c a l l a t h e o n e p a r a m e t e r g r o u p o f a u t o m o r p h i s m s g e n e r a t e d b y V . 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 . If ' f z i ^ i = 1 a r e holomorphic local coordinates on an open subset U of X , we define i I : £ P , q(U) -> £ P 1 , 9 u by v Y q . dz. A....A.dz. />. dz. A<....Adz. l < i , < . <i <n i l ' " ' i p ; 3 l ' - - ' 3 q *1 V D i 3 q - 1 p-K j <..<j <n; - 1 q-l<i <..<i <n - 1 p-K j <-.<j <n - J l Jq-I (-1) . dz ... £=1 ^ l ' - ' V V ' ^ q ±1 kdz.A. .A(JZ. A, dz . A ..Adz . Xl Xj> :1 3 q where V U k=l Then, i defines an £_-module homomorphism between £ P , q and ^ 1 , q where £, P , q i s the sheaf of differentiable (p,q)-forms.on X . Moreover i t satisfies the following identities on forms. 2 (a) i = i i = 0 v v v (b) i 9 + 9i =0 v v (c) iv(<f>Ai{0 = i v ( $ ) A i f i + ( - l ) d e g ( * ^ A i v W Because of the property (b), i induces an 0 -module v ~X homomorphism i : 9? -> ftP , where Qp i s the sheaf of holomorphic p-forms on X . Thus we get a complex 20 i i i v v v o -> fin — • fin ->•• •-> fip —>• fip •-»- s r —»• o -> o ~ x which i s called the Koszul Complex associated to V , and the operator i i s called the contraction operator, v Let E -> X be a holomorphic vector bundle on X , and l e t £(E) be the sheaf of differentiable sections of E on X . Since t P , q ® £ 6(E) = & P , q(E) , we get i ^ : = ± v 8 1: £ P , q(E)-» £ P" 1' q(E) which is an £, -module homomorphism. Moreover, contraction induces a natural "X map i : H q(X,ft P(E)) -> H q (X,n P - 1 (E)) . Equivariant Vector Bundles Let X be a complex manifold, and l e t V be a holomorphic vector f i e l d on X . Definition 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) where f (resp. s) i s a local section of X x C (resp. E). V i s called a V-derivation on E . A, To give the local expression of a V-derivation V on E , we" need to look at the holomorphic vector f i e l d on IPn . n * Since Aut(P ) = PGL _ = GL ,/C .1 . , any 1-parameter group n+1 n+1 n+1 of automorphisms a of IPn i s given by 21 for some M e gSt, , . For a given M = (a. .) e gjl , , we define n+1 1,3 n+1 V: <C[x .x. ,.. ,x ] -»- t[x n,x.,...,x ] as a C-derivation as follows; l e t u i n u i n V(x.) = T a. .x. 1 j=0 ^ 3 for i = 0, l,...,n , and extend V as a C-derivation on the homogenous coordinate ring of ( P n . Let { u . } n be the natural open affine covering 1 i=0 of P n , i.e., U. = ( P n) = { [xrt ,x, ,.. ,x ] e p R: x. =1= 0} . Then we l x. 0 1 n l 1 l x,V(x..)-x^V(x,) define v l : 0 (U.) 0 (U.) as v | (x./x.) 'U. - n l - n I 'U. 3 l 2 I P IP 1 x l and n extend v l as a C-derivation of 0 (U.) . { v l } patches in the U. - n 1 1 u . . „ 1 P i i=0 intersection U. n U. , to give a C-derivation of 0 , we denote this J P derivation by V . Now for x e P define v x ^ ) = v ( f ) (x) where f is a holomorphic function around x ; hence the derivation V w i l l give naturally a holomorphic vector f i e l d V on p n , and i n the holomorphic coordinates z. = x./x. , V has the form 3 3 1 x.V(x . )-x .V(x.) »l„. - ? 1 3 2 1 • 1 3 x^ It i s clear that this i s the holomorphic vector f i e l d induced from the tM n 1-parameter family a = 'e . Thus any holomorphic vector f i e l d on P gives a C-derivation of 0 which is induced from a C-derivation on the P homogeneous coordinate ring of P as above. We can easily generalize this situation as follows. Let R 22 be a reduced C-algebra and l e t V: R -* R be a £-derivation, i.e., V is C-linear" and V(rs) = V(r)s + rV(s) for r , s e R . Now given M = (b. .) e ql , (R) , we define a V-derivation V of 9. as follows. i,D n+1 p n R . -n Let l U i ^ i ~ 0 ^ e t^ i e n a t u r a x open affine covering of (P , then n {w. = Spec(R) x U.}._. i s an open affine covering of IP = Spec(R)x (P 1 i i—0 R Define v| : = V. : 0 (W.) -* 0 (W.) , as V. I = V and 'W. x - l - l x' l n n R (P P R R x.V(x.)-x .V(x. ) n V. (x./x.) = — ^—r-J — where V(x ) = T b . x , i j• i 2 r ,^„ r,k k J x. k=0 l n and extend V. to be a C-derivation of 0 (W.) . {V.}. „ .patches x - x x x=0 n PR in the intersection W. r\ W. to give a V-derivation V of 0 , and IP R conversely any V-derivation of 0 i s obtained i n this way. What we have done geometrically i s the following: Let U be an open subset of C , and l e t V 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(UxP n)+ T(U) . Let X be a complex manifold and l e t V be a holomorphic vector f i e l d on X . If E ->• X i s a V-equivariant holomorphic vector bundle on X , then there exists a V-derivation V on E . Let { u } a 23 be the open affine covering of X such that 0 (E)| = © 0 X rv X n+1 | U a Let V '= V : 0 (E) y 0(E) . Then the holomorphic a A a X a map V : U T (U , Hom(E,E)) x -»• V (x) : E -> E induces naturally a a a a a x x V -derivation V of 0• • where V : = v|„ i s the derivation a a - n a 1U U >< <P a a induced from the holomorphic vector f i e l d V . The V^-derivations V of 0 patches in the intersection to give a V-derivation "U x (Pn a V of 0 hence a holomorphic vector f i e l d V on P(E) . such that ~P(E) dir(v) = V where TT : P(E) -> X , since V i s a V -derivation. From a 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-riant i f and only i f there exists a holomorphic vector f i e l d W on P(E) such that dir(w) = V where ir : P(E) ->• X . Remark: D. Lieberman has proved this Lemma i n [29], Let us look at the situation in terms of 1-parameter group of automorphisms. Let ^ : E -»• X be a holomorphic vector bundle and, let f: X — X be an automorphism of X . If there exists a bundle 24 automorphism f: E —*• E such that the diagram -> E commutes, X -> X then we have for each x e X a C-linear map f(x): E —> E... , , hence x f(x) f(x) takes lines through the origin in E^ to the lines through the origin i n E f (x) Therefore f induces naturally an autormorphism f: P(E) ->• P(E) such that the diagram P(E) -> P(E) X X commutes. Conversely any such f i s obtained in this way, because Aut( P_) = PGL_,,. = GL n + 1/Z (GL n + 1) , ZfGL,^) = C .1 n n+1 n+1' n+1 Let a: C x x ->• • X be a 1-parameter group of automorphisms of a complex manifold X . Definition 1.3.3. A holomorphic vector bundle E —> X i s said to be a a-eguivariant bundle, i f (i) there exists a 1-parameter group of automorphisms a: £ x E -»- E of E such that 25 the diagram C x- E > E <E x X >- x commutes. (ii) for t e C and x e X , the map E E , , i s C-linear. o t(x) We c a l l a a a-equivariant 1-parameter group of automorphisms of E As we have seen above for each t e C , a^: E -* E induces a : P(E) P(E) such that the diagram P(E) P(E) -+ X commutes. It i s easy to see that {afc} defines a 1-parameter group of automorphism a: C x P(E) -»- P (E) of P(E) ., such that the diagram C x p(E) -> P(E) C x x -> X commutes. Conversely any a is obtained i n this 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 is a a-equivariant-1-parameter group of automorphisms of a holomorphic vector bundle E * X , then for each t e C , we have an 0 -module homomorphism X 26 a : 0 (E) -* (a ) 0 (E) , given by a (s) = a o s o a , where s i s t -. t'* X X a local section of E . Now we define d "* V(s) = - ( o . ( s ) ) dt t t=0 for a local section s of E . Then V(fs) = - ^ ( c r (fs)) = -£:(a. (£) cr.(s)) dt t I dt t t t=0 = a; (s) +a;(f) A . a ; ( s ) } t = 0 = V(f)s +f V(s) Thus V defines a V-derivation on E . If W i s the holomorphic vector f i e l d on P(E) induced from the one parameter family a of P(E) , then i t i s clear that W i s the holomorphic vector f i e l d induced from V , i.e., V = W . Let V be a holomorphic vector f i e l d on a compact complex manifold X , induced from the 1-parameter group of automorphisms a of X . Then we have the following. Theorem 1.3.1. Let E •> X be a holomorphic vector bundle on X . Then the following are equivalent. (i) E i s a a-equivariant bundle (ii) E i s V-equivariant ( i i i ) There exists a hermitian metric h on E , such that 1^(0) = 3(L) , for some L e &(X, Horn(E,E)) , where 0 is the canonical curvature matrix associated to h . 27 Proof. (i) => ( i i ) . If E is a ff-equivariant bundle, then there exists a: C x E ->• E a: <EXE *• E a: C x x >• X commutes. 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 i s V-equivariant. (ii) => ( i ) . This i s the existence of the solution of the E-valued differential equation. It can be seen as follows. Let V be the V-derivation of E , then V induces a holomorphic vector f i e l d V on P(E) . Let be the 1-parameter group of automorphisms of P(E) generated by V , a exists because P(E) i s compact. Then there exists a 1-parameter group of automorphisms i a - t °^ E w n l c h induces a on P(E) , hence E i s a a- equivariant bundle. In fact dt = V , t=0 so the correspondence i n (i) and (ii) bijective. 28 (ii) => ( i i i ) Let V be a V-derivation on E , and V i s the induced holomorphic vector f i e l d on P(E) . Let V = V + V , V = V + V be the associated real vector fields on P(E) and on X respectively. It i s easy to see that the associated derivation V: £(E) -> £(E) of the real vector f i e l d V i s a V-derivation, i.e.; V(fs) = V(f)s + f V(S) where f i s a smooth function on X , and s i s a smooth local section of E . Let i : £ (E)—• £(E) be the contraction operator induced from the smooth vector f i e l d V on X . (Namely; i = i + i _ v : C 1 , 0(E) ffi £^'^"(E) -> fc(E) with respect to the natural splitting of £ (E)) . For any hermitian metric h on E , we define V D L = V_- i vD: £(E) > £(E) >• £ (E) V £(E) where D i s the canonical connection of E associated to h . Then L(fs) = V(fs) - i vD(fs) = V(f)s + fV(s) - i v(dfs+fDs) f(V(s) - iyDts)) = fL(s) where f is a smooth function on X and s i s a smooth local section 29 of E . Thus L: £ . ( E ) -»- fc(E) i s an £-module homomorphism, hence naturally defines an element L E £(X, Horn ( E , E ) ) . We claim 3(L) = i v ( 9 ) where 0 is the curvature associated to D . Let U be a holomorphic local coordinate neighborhood of X such that E | == U x c q . Then i t follows from above that v| = V: 0 ( E ) -*• 0 ( E ) . Hence L | = "(V - i D) , but D = D' - X X - 'U A on U . Therefore L| y = (V - i ^ D') | ^  since i vD = (i^+i_) (D1+D") = i D' +i D" = i D* , D" = 0 . Hence 3(L| ) = 3VI - 3(i D')l = - 3i D1, v - V ' U ' U v ' U v v since v| i s holomorphic. But then 3(L| u) = i (<^D')[ by the contraction identity. Since ^D'ly = ®lg » w e g e t ^ L H y = """v^^U ' thus i (0) = 9(L) . ( i i i ) => (ii) Let h be the hermitian metric on E , and L e £(X, Hom(E,E)) such that i y O ) = where 0 i s the curvature of the canonical connection D of E associated to h . Define V = L + i vD: £ ( E ) - > & ( E ) where V ,is the real vector f i e l d assoicated to V . It i s clear that V i s a V-derivation of £ ( E ) . Moreover i f U i s a holomorphic local coordinate neighborhood of X such that E | = U x c q then V = L + iyD' on U , since D = D' on U . Therefore 3V = 3L + 8ivD' = 3L - i v 3D' = 3L - i y 0 = 0 on U , since i v O ) = 3(L) . Thus V induces a V-derivation on E , namely V = V: 0 (E) -y 0 (E) 0(E)' X X Hence E is V-equivariant. ' Q.E.D. Remark: Since 3(0) = 0 , we then have a cohomology class 30 [0] e H1(X,Hom(E,E) <3 Q1) — — > H1(X, Horn(E,E)) . Thus the condition ( i i i ) i n the theorem i s equivalent the vanishing of the cohomology class i ([0]) in H^X,. Horn(E,E)) . Corollary 1.3.1. If E + X i = 1,2 are V-equivariant * P holomorphic vector bundles, then E © E 2 , E ^ E 1 ® E 2 , A E 1 are a l l V-equivariant. Proof. Let h. be the hermitian metrics on E. and 0 . be the canonical curvature matrices associated to h. . Then we have l L e &(X, Horn ( E ^ E j ) for i = 1,2, such that i v(© i) = JHLJ . Take L = L l ° 0 L„ E £(X, Horn(E ffiE2, E ffiE2)) , and let h = h l ° 0 h. be the hermitian metric on E © E 2 , then the curvature 0 of the canonical connection of , h on E 1 © E 2 locally i n the form 0 = 91 ° 0 0, Thus i (0) = i V V 0 1 0 2j i 0.-V l 0 i 0 ; V 2 f \ 0 = 0 J 2 = 9L . 31 Hence ffi E^ is V-equivariant by Theorem 1.3.1. * * For E , take L = - L e £(X, Horn (E^E.^) = £(X, HomfE^E.^ ) * * * _1 t * and take the hermitian metric h on E^ , where h 1 = (h^ ) If 0^ * * i s the canonical curvature matrix of h^ , then 0^ = - 0^ hence * _ * * i v ( 0 ^ ) = 9(L^) . Therefore E^ vis V-equivariant by Theorem 1.3.1. For E^ ® E^ , l e t 0 be the 1-parameter group of automorphisms -1 -2 of X generated by V . If 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 E^ ® E^ i s V-equivariant. For A PE , take S^E^ -^E , i t i s clear that A Pa i s a a-equivariant 1-parameter group of automorphisms of ^ E-^ -1 p i f a i s for E^ . Hence by Theorem 1.3.1 A E^ i s V-equivariant. Q.E.D. Now we w i l l give some examples of V-equivariant bundles for further use. Example 1.3.1. (i) TX , the holomorphic tangent bundle of X i s V-equi-variant for any holomorphic vector f i e l d V on X . Proof. If a i s the 1-parameter group of automorphisms of X generated by V , then a = cla^: TX -> TX i s clearly a a-equivariant 1-parameter group of automorphisms of TX . Hence by Theorem 1.3.1 TX is V-equivariant. 32 Let us compute the V-derivation V of 0 ( T X ) induced from ~ X a = d a : TX TX . If {z.}1? , i s a local coordinate system t t 1 i=l open subset U of X . Then we have over some da t(z) = 9a^(z) 3z, r ( a t 1 ( u ) , T X ) -y T ( U , T X ) where a : 0 (u) -> 0 (a (U)), a^(z.) = a. (z) and z = (z „,z ....,z ) t - x - x t . t i t 1 2 -1 i -1 -1 Hence o^z) = dafc (z) = C3afc (z) /3z_.) : T(U,TX) -»• r (afc (u) ; TX) as a C-linear map. Therefore V(z) = ^  ( a ; ( z ) ) | t = 0 - A 3a^(z) -1 t=0 = - ^ ! ( 3 a^(z ) / 3 z )} t = Q = -3z, d i , . -rr a. z dt t t=0 3z. 3 dt a t ( z ) t=0 }• = - dZ.' 1 '3b.N . I 3z I 3-n where vl = Y b. -r^— 1U . L. l 3z. i=l l Hence V(z): 0 (U,TX) ->• 0 (U,TX) as a X X C-linear map given by the matrix 3b. l 3z. 3 J It i s easy to see that Lie derivation [V,*]: 0 (U,TX) •+ 0 (U,TX):W ->• [V,W], i s given by the same X X 33 matrix as a C-linear map. Since both of them are V-derivations, we then have V = [V,*]: 0 (TX) -> 0 (TX) which i s the Lie derivation in the X X direction of V . Similarily V: £(TX) -> £(TX) i s given by [V=V+V, * ], Lie derivation of smooth vector fields in the direction of V . P * (ii) /f(TX) i s V-equivariant for any holomorphic vector f i e l d on V on X by Corollary 1.3.1. ( i i i ) It i s wellknown that any holomorphic vector f i e l d on Gr, the Grassmann manifold of k-planes in C11 , i s induced from a k,n 1-parameter group of automorphisms a : Gr Gr , o ([W]')=[etV.w] n tV for some \T e gH (C) , where W c C i s a k-plane and e .W i s the n image of k-plane W under the isomorphism e t V; C n C n . Let U be the universal k-plane bundle on Gr, , we define for t e C , k,n a : U, U, as follows, a ([W],w) = (a [w] , etV.w) where w e W c C11. t k,n k,n t t It i s clear that ^atJ i s a <-at ~ e /-equivariant 1-parameter group of automorphisms of U . Hence U i s V-equivariant for any holomorphic vector f i e l d V on J3r^ ^  . Let Q, = Gr, xcn/u, be the universal quotient bundle k,n k,n k,n on Gr, . Then the automorphism cr : Gr, xc 1 1 -»• Gr, xc 1 1 , a. (x,w) = :..k,n t k,n k,n t tV n (cr (x), e .w) leaves U, *—> Gr, x C invariant for each t e C . t k,n k,n Hence i t naturally induces an automorphism a. of Q . It i s clear that a {cJ t=e t V}-equivariant 1-parameter group of automorphisms of Q Hence Q i s V-equivariant for any holomorphic vector f i e l d 34 V on Gr, k ,n (iv) Let G be a linear algebraic group over C , and P a * parabolic subgroup of G . Given \: V •+ £ holomorphic character of P , we define L = GxC/~ , where (grv) ~ (g',v') i f and only i f X g' = gp and v' = x (P 1 ) v > f° r some p e P . Then L induces X naturally a holomorphic line bundle on G/p which i s called the homogeneous line bundle associated to x • For a given 1-parameter subgroup a: C -> P of P , we get a: C x G/p -> G/p , at(gP) = a(t)gP , 1-parameter group of automorphisms of G/P . We define a : L -> L by a (gP,v) = (a (gP) , x(°(t) 1)v) . t X X t t It i s easy to see that or i s a-equivariant 1-parameter group of automorphisms of L . Hence L i s V-equivariant, where V i s the X X holomorphic vector f i e l d induced from a . (v) If H1(X,0 ) = 0 , then any line 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 any holomorphic vector f i e l d V on X . Let V be a holomorphic vector f i e l d on X , and l e t V be a V-derivation of a holomorphic vector bundle E •+ X . Then for any V-derivation W on E , consider V - W: 0 (E) -*• 0 (E) , ~X X (V-W)(s) = V(s) - W(s) , then (V-W) (fs) = V(fs) - W(fs) = V(f)s + fV(s) - V(f)s - fW(s) = f(V-W)(s) , where f and s are local ( sections of X x C and E respectively. Hence V-W defines an element of H°(X, Horn(E,E)) . 35 0 Conversely for a given T e H (X, Horn(E,E)) , we have the natural induced 0 -module homomorphism T: 0 (E) 0 (E) , now consider X X ~X V + T: 0 (E) ->• 0 (E) defined by (V+T) (s) = V(s) + T(s) , where s i s X X a local section of E . Hence V + T : 0 (E) •+ 0 (E) defines obviously X X a V-derivation of E . It i s clear that above correspondence i s bijective. We state this fact as a Lemma for further use. Lemma 1.3.2. If E"*"X i s a V-equivariant bundle, then the set of a l l V-derivations of E i s a principal homogeneous for the group H°(X, 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 . If E X is a V-equivariant holomorphic vector bundle then by Theorem 1.3.1 we have a a-equivariant 1-parameter group of automorphisms o of E . Let Z be the zero set of V . Now given p e Z , we have CTt(P) = p at least for small values of |t| . Then 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) which i s nothing else than 5 Ix) : E -*• E . If we take x; := p , u X 1 . . . °t ( X ) then we get a representation C—* GL(E ) , t •+ a (p) : E ->• E , since p t p p a i s a 1-parameter group of automorphisms of E . Therefore * tA "* tV (p) " a (p) = e for some A e g£(E ) , but we have a = e where V t p t is the V-derivation of E induced from a , since > d A * Viz) = -jHo. (z)) • • - dt t t=0 36 Hence A = - V(p) , namely ° t(P) = e -tV(p) We w i l l finish this section with a very usefull example. Example 1.3.2. A. Let A = °. 0 0 \ n g £n+l ' A i ^  Xi f°r 1 + j * Let V be tA the holomorphic vector f i e l d induced from e . Then vl . , = Y V(z.)3/3z. ( IP" ) x a where z. = x./x , but l l a V(z.) = A(x./x ) = I I a x A(x. )-x.A(x ) a I x a Hence = (A.-A )z. . x a x therefore e = [0,..1,..0] i s the only zero of V in ( ipn) , a hence Z = {e : a=0, n} i s the set of a l l zeroes of V . Let a a a : U„ , -»• U, be given as in Example 1.3.1 ( i i i ) . Then t l,n+l l,n+l tA tA 0 t ( [ e a ] ' e a ) = ( a t ( [ e a ] ) ' e * e a ) ' H e n c e = 6 37 Now i f V i s the V - d e r i v a t i o n induced from a , then V(e ) = - X f o r each a = 0,...,n . a a Consider the Fubini-Study m e t r i c h on U n n , l,n+l 2 I |x.| h a ( [ x O ' - X n ] ) = ~ 2 X a as given i n Example 1.1.1. Let 0 be the c a n o n i c a l curvature m a t r i x a s s o c i a t e d t o h . Then we have 0 = 8 Oh h "S = 9(91og h ) on a a a a ( P n) and on (: p n) A ( P?) h = 1—1 h n . Let z. = — X X X„ a 'x 1 3 I X a a 3 ' a a n 2 be holomorphic coordinates on ( P ) , then h = h |z I , hence X ct p p a 9 ( l o g h o) = 9 ( l o g h + l o g | z g | 2 ) = 9 ( l o g h g) + 9(log z g + l o g 5 ) dz. = 9 l o g h + 9 log z = 9 l o g h + 3 " 3 " 3 z, s i n c e 9z„ = 9z„ = 0 . Therefore i (91og h ) = i (91og h.) 3 3 v a v 3 +' • ' a , Z„ = i (91og h„) + A - X hence i 91og h + X Z„ 3 v 3 3 • 3 • a v ^ a • a = i 9log h_ + X D on ( P N ) r\ :( P " ) . Therefore v 3 3 x a - x g {L =-i 9log h -X } n d e f i n e s a unique element a v a a cc=0,. ,n L e £ ( P N , Hom(u\ _ ,U. .,,)) s £ ( P°) • But - i J-,n+l l,n+l -i (0 ) = i 9 91og h = - 9 i 91og h = 9(L ) , sin c e 9X = 0 v a v a v a a a 38 Hence i v(Q) = 3(L) . By the proof of Theorem 1.3.1 we see that L = V - i o D , where D i s the canonical connection associated h , and V = V + V . 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. Let X be a compact complex manifold of dimension n , and let E X be a holomorphic vector bundle on X . Then a holomorphic geometric endomorphism of E consists a pair (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 * k Hk(X,0 (E) ) — • Hk(X,0 (f*E) H W > Hk(X,0 (E) ) X X X where the f i r s t map i s the standard pull-back, and the second i s induced by functoriality, by <f> . The composition gives a <C-linear endomorphism of the finite-dimensional complex vector space H (X,0 (E)) which we ~X denote H (f,cf>) . Let z n,...,z be local holomorphic coordinates centered at I n the isolated fixed point p e X of f: X X . Then on a small neighbour-hood N(p) of p i n X , the only common zeros of the functions z. - f 1 ( z n , . . , z ) i = l,..,n i s 0 = (0,..,0) , where l I n * i i f (z.) = f (z) = f (z,,..,z ) are the local coordinates of f(z) . Let x I n f } (1) V ( (b,E) = Res P P trace <j> (z) dz, dz 1 n 1 n z -f (z) , ,z -f (z) 1 n 40 where R e S p i s t n e Grothendieck residue symbol and cf> (z) i s the matrix representation of <f> on N(p) . Now i f the fixed point set X^ of f: X -*• X i s f i n i t e , i.e.; a l l the fixed points of f are isolated, then the Holomorphic Lefschetz Fixed point formula can be stated as, , n (2) L(f,<j>): = I (-Dk trace Hk(f,<f>) = £ v (<f>,E) k-0 v f P peX see D. Toledo [36]. Algorithms For Calculating Res P There are two methods for calculation of Res , one i s purely p algebraic, and the other i s analytic. We w i l l describe both of them for the later use. Let fu(z) = z^ - f 1(z) , i = l,..,n , and l e t W(z) be any holomorphic function defined on N(p) . (i) Algebraic Calculation of Res P m. There exist integers iru > 1 , so z^ is in the ideal generated by h_. (z) j = l,..,n . This follows from the fact that p i s an isolated fixed point of f(z) . Hence we can find holomorphic functions B. . (z) defined near p such that n m. l Z.1 = T B. .(z) h.(z) 1 j=l ^ 3 This granted, one then has 41 Res , P W(z).dz. .,. .dz 1 n h (z) ,. . .h (z) 1 n m^ -1 m n ~ l i s equal to the coefficient of z^ ....z^ in the power series expansion of W(z) det(B. . (z)) , see P.F. Baum and R. Bott [ 2 ] . 1' J (ii) Integral Representation of Res P Let S c N(p) be a 2n-l sphere around p . Recall Y'.L.L. Tong [38], that the Grothendieck residue can be expressed as an integral formula: Res W(z) dz, dz 1 n h (z),.. . ,h (z) 1 n =k f W(z) |h(z)| 2 n I ( - l ) k + 1 h (z)dh ..dh, ..dh dzn..dz n J s k ^ k 1 k n 1 n 2 1 / 2 . where k i s constant and h ( z ) = (Eh.(z) ) n ' I 1 If p i s a transversal fixed point of f: X -* X , namely f^(z),...,f n(z) form holomorphic local coordinates around p e X , then by the algorthim (i) i t can be seen easily (or see D. Toledo [36] Res S P W(z)dz,....dz 1 n h n (z),..,h (z) 1 n W(p) det(I-df (p)) where df(p) i s the di f f e r e n t i a l of f at p ; df(p): T^X -> T^X . 42 In this case we have (3) x> (6 v) trace (ft (p)  Vp ( ( f )' E ) - det(I-df(p)) Moreover i f a l l the fixed points of f are transversal, then the Lefschetz number L(f,cj>) i s simply (4) T , f _ v trace (ft (p)  L(f'*} - I det(I-df(p)) peX where X i s the fixed points set of f: X -> X . Let V be a holomorphic 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 V-derivation induced from the ~X ~X e>-equivariant 1-parameter group of automorphisms a of a holomorphic vector bundle E -* X . Then for each t e C , we have a bundle isomorphism — 1 -1 * a t : E -> (a ) E . By taking the pull-back relative to a z X -> X , we obtain a bundle isomorphism <ft 1 * * ( a t ) v 0 t E E • Hence we have a natural geometric endomorphism (cr^ _, cf>^_) of E for each t , moreover <|> i s compatible with the C-action. ^ If p i s an isolated zero of V , then p i s an isolated fixed point of a at least for small values of |t| , hence (5) v (<j> .E) P y t Res i P trace (ft (z) dz,....dz _t 1_ n z -a. (z) , — ,z -a. (z) I t n t 43 makes sense at least for small values of |t| . In this section we w i l l compute v^Ctb^E) , and w i l l obtain a much more explicit dependance on t . Before doing this l e t us look at the case when p i s a simple isolated zero of V . • An isolated zero p of V i s said to be simple, i f det 1 9a. 2 9z. (P) ? 0 where V = I a 9/9z , N(p) i=l or equivalently p i s a non singular point of the closed subvariety Z c. X , where Z i s the zero set of V . If p i s simple isolated zero of V , then p i s a transversal fixed point of a at least for small values of |t| . In this case we get by (3) and (5) trace cl ) t(p) V P ( ( | V E ) = det<l-da Ap)) But we have da t(p) = e t L ^ by section 3. Moreover by definition of d> (x) : E . . -> E which i s the pul l back of a.^x) : E •*• E t at(x) t a - l ( x ) ~ - l A * " tV(p) relative to a , we get ^(p) = a ^ (p) = a ^ (p) > hence <f> (p) = e t t t t t Thus we have 44 (6) V (<(>.,E) => P t trace e tV(p) d e t ( % - e t U p ) i where L(p) = 8a. 3z. (P) n V N(p) = I a. i=l 3z. Let us take E = /v (T X) , where T X i s the holomorphic cotangant bundle of X and 0 < p < dim(X) is a a-equivariant 1-parameter group of automorphisms of E = AT (T X) . If a l l the zeros of V are simple (such a holomorphic vector f i e l d i s called nondegenerate) then we have the theorem of G. Lusztig [31], Then 5 = ( A Pda 1 ) t : E s s * n . . L (<)>., /?(T X)) = I (-1)1 dim HNX/) t . „ i=0 for a l l t and for a l l 0 < p < n . Namely the Lefschetz number i s independent of t e C . For a V-equivariant vector bundle E X , we have the bijective correspondence between a-equivariant 1-parameter group of automorphisms of E and the set H°(X, Horn(E,E)) , given by Lemma 1.3.2. So what G. Lusztig proved in our language i s , that there p * exist a a-equivariant 1-parameter group of automorphisms a of A (T X) p * such that the Lefschetz number, L,{§_^, A (T X)) i s independent of t , i f the holomorphic vector f i e l d induced from a i s nondegenerate. Thus we ask the natural question; i s this the case for any V-equivariant holomorphic vector bundle? Answer of this question i s no in general, 45 A simple example i s the following: Let V be the holomorphic vector f i e l d on (P^  induced from \ ' ° ' o x x a t = e where A ^ A^ . Then we see immediately from the example 1.3.2 that V i s a nondegenerate vector f i e l d with zeroes {e =[1,0], e =[0,1]} . Let o : U U be the natural extension of a , then A - 1 * * (a ) : U -> U, „ is a-equivariant 1-parameter group of automorphisms s 1,2 1,2 * * A - 1 t of U - If V i s the V-derivation of U induced from (a ) , 1,2 S then V(e.) =A. i = 0,1 . But then by Lemma 1.3.2 any V-derivation I I of U i s in the form V + a for some a e C . Hence for any 1,2 * a-equivariant 1-parameter group of automorphisms of U , we get the Lefschetz number * t(A Q+a) tfXj+a) e . e L(<l> ,U, J = , + t ' " l , 2 ' .. t(A 1-A ( )) t ( W 1-e 1-e by (2) and (6) . Hence . t A n tA. * t a • 0 1. L ( ( | ) t ' U l 2) = 6 ( e + S } * Therefore L(<j>t,U^  ^ i s not constant for any choice of a e C or equivalently for any a-equivariant 1-parameter group of automorphisms 46 a of 'U1 2 . We note that L(<J>0,U1 2) = e ' a ( e +e ) = 2 which is Riemann-Roch for the invertible sheaf 0(1) . It i s easy to see that, one can obtain Riemann-Roch for any invertible sheaf on IP1 from the holomorphic Lefschetz fixed point formula with this method. We w i l l discuss the Theorem of Riemann-Roch and Hirzebruch later, actually our main concern in computing v^((j)t,E) w i l l be to obtain this theorem through holomorphic vector fi e l d s . It i s easy to see that L(,$^,XJ^ ^) = 0 for any Ao ° 6 A , c t = e (^Q^A)-equivariant 1-parameter group of automorphisms of U . In fact 1/2 L ( * t ' U l , n + l ) = ° f o r a n y V 0 a t = e (A^ =f=A_. i=j=j)-equivariant 1-parameter group of automorphisms of U^ . This observation raises the natural question; i f L(c|>t,E) i s independent of t for a fixed a-equivariant 1-parameter group of automorphisms of E , then i s i t true that L.(i/; ,E) i s independent of t for any a-equivariant 1-parameter group of automorphisms of E . Unfortunately this i s not true either, and our simple example i s the following: Let V be the holomorphic vector f i e l d on P"*" induced from 47 *0 ° 0 X 1 -o t = e ' X0 * h Let a : tr „ U „ be the canonical extension of a , then t 1 z 2 1 j ^  t * i * i a ® a : ® U = T P ->- T P i s a a-equi variant 1-parameter group * 1 * 1 of automorphisms of T • IP . If V i s the V-derivation of T (P induced from o ® a , then V(e.) =-2A. i = 0 , 1 . Hence we get t t i i —2A„t — 2A,t . ,, , . * 1 e ° e 1 " t ( W L (VT P } = t ( A 1 - A ( ) ) + t ( A Q - A ^ ) = " 6 1-e 1-e i * 1 Now take A ^ =|= - A ^ , then L(c)>t,T IP ) i s not constant. But we know * 1 by the theorem of G. Lusztig that L(^ s,T (P ) i s constant, i f tp^  i s * 1 -1 t * 1 * 1 the geometric endomorphism of T IP induced from (&a s ) : T P ->• T P We can actually see this quite easily in this case. If V is the * 1 - I t V-derivation of T P induced from (^cs ) # then we have V(e Q) = A - A Q , V(e 1) = A Q - A by section 3. Therefore t ± t ( W t ( V Ai' L ( \ ' T p > = t I A ^ - A Q ) + 6 u ^ i = - 1 1-e 1-e Remark: The main idea for asking whether L ( ( f l t ' E) i s constant or not i s to try to give a formula for dim Hq(X,0 (E)) in terms of 48 9a. i ~ eigevalues of ( (p)) and V(p) as p runs through the zero set of j p * V . C. Kosniowski gave such a formula in [27] for E = A. T X . He gets P * his formula just from the fact that L($^r A T X ) i s constant. We w i l l give some applications of his formula i n the next section. Now we can start 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 let 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 t e C , we have a natural geometric,' endomorphism (0^,$^) of E , where <j> = (a^ ) : a^ E-^ E. For an isolated zero p of V , we have the holomorphic local coordinates z ,..,z on a small neighbourhood N(p) of p , such that 1 n v (<l> ,E) = Res p T t p trace d> (z) dz, dz _t 1 n z -a (z) , ,z -a (z) I t n t makes sense at least for small values of |t| , say in a neighbourhood W of 0 e C . Let L(z) = 2fj:(dat(z)) = )a i(z) 9z. be the transpose of the matrix representation of the V-derivation of T X on N(p) , induced from — i t * * ^ 9 (da ) : T X -> T X , where v| , , = T a. . Now consider functions s 'N(p) ^ I 9z^ v. (z) , w_. (z) for 1 < i < n and 1_< j < n on N(p) , defined from 49 ,the following formal identities; 3 , q det(I +AV(z)) = Y c.(V(z))A = n (1+Av.(z)) and q . ~ i _• , i i=0 i=l det(I +AL(z) n n . n = y c. (L(z))\ 1 = n (1+Aw . (z)) i=0 1 j=l 3 where q i s the rank of E , I i s the k x k identity matrix, and z = (z,,..,z ) . Then we define for (t,z) e C x N(p) . The Chern 1 n q t v ^ z ) character of V(z) by Ch(E,z,t) = £ e and The Todd class of i=l n tw.(z) L(z) by Td(z,t) = II • 1 . , tw. (z) i=l l 1-e where the functions on the right-hand side should be regarded as standing for the corresponding power series expansion. Both Ch(E,z,t) and Td(z,t) are holomorphic functions i n W x N(p) , since c_.(V(z)) (resp. c_.(L(z))) i s the elementary symetric functions in the v, (z) (resp. w„(z)) and c.(V(z)) c.(L(z)) are k I l j holomorphic functions. Then we have the following. , Theorem 1.4.1. (1) v (*. ,E) P L. Res trace ( l ) t ( z ) dz . .dz z -at,(.z) , .. ..,z n-a t(z) — Res i n p Ch(E,z,t)Td(z,t)dz 1 dz a;(z),. . . ,a (z) 1 n n for t e W - 0 . 50 . To prove this theorem, we need some "elementary facts from Linear Algebra, whose proofs we w i l l include for completeness. Let M be any n x n matrics over C . We define n A . e C i = l,..,n by the following equation det(I +xM) = II (1+A.x) 1 i=l where x i s an indeterminate. A . i s called a characteristic root of 1 n n M . We note that Trace (M) = Y A . and det(M) = 1 1 A . . 1=1 i=l Lemma 1.4.1. If A n , . . , A are the characteristic roots of 1 n X l Xn M v M M , then e ,...,e are the characteristic roots of e = I — r=0 M_ r! where M e g£ (C) n Proof. By induction on n . The case n = 1 i s t r i v i a l . Now we assume the Lemma holds for (n-1) x (n-1) matrices. Let A^ be any characteristic root of M , then we can find g e GL^JC) such that the matrix g "'"Mg is in the form, (1) g "'"Mg A l * ' 0" 0 0 for some N e g£ n (C) . If A „ , . . , A are the characteristic n-1 2 n roots of N , then A n , A „ , . - . , A are the characteristic roots of 1 2 n g 1Mg . Hence A , ^2'"*'Xn a r e t* i e c n a r a c t e r ;'- s t :'- c r o°ts of M But 51 then, we have by (1) X2 Xn Now by induction e ,...,e are the characteristic roots of e , hence e , e ,...,e n are the characteristic roots of e^ ^ , so M q-1Mq -1 M they are the characteristic roots of e , since e = g e g . Q.E.D. Corollary 1.4.1. If ^ i ' * - ' ^ n a r e ^ e characteristic roots tx. t M r 1 _ of Me g£^(C) , then trace e = I e = Ch(M,t) , and i=l n ^ tM t(trace M) " t X i . ^ det e = e = II e , for any t e C i=l Proof. If X , ..,X are the characteristic roots of M , 1 n then for any t e C tX ,..,tX n are the characteristic roots of 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 p i s simple isolated zero of V . Then left-hand side of (1) 52 V p(<j> t,E) = R e s p i t r a c e d> • ( z ) ; d z , .. . d z t 1 n Z l ~ a t ^ ' , Z n ~ 0 t ^ Z ^ t r a c e e t V ( p ) d e t ( I - e t L ( p ) ) b y ( 6 ) • B u t t h e n b y C o r o l l a r y 1 . 4 . 1 , w e g e t V <VE) = Ch(V(p) t) P t d e t ( I - e t L ( p ) ) f o r a l l t e W . N o w l e t u s l o o k a t t h e r i g h t - h a n d s i d e o f ( 1 ) R.H.S o f ( 1 ) = — R e s •{ t n P C h ( E , z , t ) T d ( z , t ) d z , ... d z 1 n a i ' " - ' a n a n d c o m p u t e u s i n g t h e a l g o r i t h m ( i ) . S i n c e p i s s i m p l e z e r o o f ' N ( p ) n = / a . 8 / 9 z . , i t f o l l o w s t h a t a . , . . , a f o r m a h o l o m o r p h i c . . i i I n i = l l o c a l c o o r d i n a t e s i n N ( p ) , a n d t h e r e f o r e t h e r e e x i s t h o l o m o r p h i c f u n c t i o n s B . . ( z ) s u c h t h a t z . = Z B . . ( z ) a . ( z ) . H e n c e b y t h e j = l i » D 3 a l g o r i t h m ( i ) R . H . S . = — . { c o n s t a n t t e r m o f C h ( E , z , t ) T d ( z , t ) d e t ( B . . ( z ) } _1_ . n { C h ( E , p , t ) T d ( p , t ) d e t ( B . . ( p ) } , f o r t e W . 113 53 since Ch(E,z,t)Td(z,t)det(B. . (z) ) i s a holomorphic function in z e N(p) A i 3 ' Let (C. . (z)) =' (B. .(z)) i,J 1»3 -1 Then a j=l 3a. . = I C. .z. , ^ ( p ) = C. . (p) , 3a. 3z -ip) 3a. -1 = (C .(p)) = (B. .(p))" 1 . Hence det(B. (p)) = (det ("^(p))) 1,3 1,3 1,3 oz^ det(L(p)) Therefore the w „ Q n f n\ - -L Ch(E,p,t)Td(p,t) R.H.S Of (1) — — T\ n det(L(p)) But by definition of Ch(E,z,t) and Td(z,t) , we have Ch(E,p,t) =Ch(V(p),t) and Td(p,t) = n tw. (p) tw. (p) 1=1 n 1 1-e where n det(I+AL(p)) = n (1+Xw.(p)) i=l 1 Thus we get R.H.S. of (1) = _1_ n , . . v n tw. (p) Ch(V(p) ,t) n l *' det(L(p)) tw±(p) 1-e > for t e W _1_ n Ch(V(p),t) t n det(L(p)) - n tw. (p) det(L(p)) n (1-e 1 ) i=l > for t e W 54 Ch(V(p) ft) n tw.(p) n (1-e 1 ) i=l Ch(V(p) ,t) d e t ( I - e t L ( p ) ) for t e W - 0 v (cL ,E) for t E W - 0 . p t Hence we have the result for simple isolated zeros. Let p be an isolated zero of V , with multiplicity m ; (i.e. m = dim(0 . (a..,..,a )0 ) , where P , X / 1 n "p,X n vl = I a. 9/9z. 'N(p) i=l and V has only one zero p i n the neighbourhood N(p) of p ) , and z.,.., z holomorphic coordinates on N(p) . We denote by 1 n a(z) = (a. (z) ,.. . ,z (z)) : N(p) -»- C . By Sard's theorem we can find a 1 n regular value r = (r^,..,r n) e C n such that 1/2 r = i n I |r.| i s sufficiently small and the function a(z) - r = a(r,z) say w i l l have only simple isolated zeroes in a small neighbourhood of p . Without loss of generality, we may assume a(r,z) w i l l have only simple isolated zeroes i n the neighbourhood N(p) of p . Let a^_(r,z) be the flow generated by the vector f i e l d 55 V(r,z) = £ a.(r,z) 9/Sz. , where a.(r,z) = a.(z) - r. • Since we i=l 1 1 1 1 1 are in a local coordinate, by the local study i n section 3, we can find a^(r,z)equivariant 1-parameter group of automorphisms a t(r,z) of E | . , , such that a (r,z) -* a Az) converges uniformly in N(p) as N(p) t t r •+ 0 through regular values of a(z) for each t . Therefore V(r,z) -* V(z) and Ch(E,z,r,t) -> Ch(E,z,t) converge uniformly in N(p) as r -> 0 through regular values of a(z) for each t ; where i V(r,z) is the V(r,z)-derivation of E | ^ ^ ^ induced from o~^(x,z) , and Ch(E,z,r,t) i s the Chern character of V(r,z) . Moreover <)>t(r,z) -> <f>t(z) converges uniformly i n N (p) as r -> 0 through regular values of a(z) , where (6^_(r,z) and <f ) t^ z) a r e t h e geometric endomorphisms corresponding to a^(r,z) and a t ( z ) respectively. Now i f L(r,z) is the transpose of the V(r,z)-derivation of * i - I t T X , . induced from (da (r,z)) , then L(r,z) = L(z) , since 'N(p) s L(r,z) depends only the derivatives of a^(r,z) = a^(z) - r^ . Hence Td(z,r,t) = Td(z,t) , where Td(r,z,t) i s the Todd class of L(r,z) . Since a l l the zeroes P ^ ' - ' ' P M °^ a(r,z) are simple isolated, then we have for t e W - 0 ; (i.e. sufficiently small values of t =)= 0) 1 56 (2) m I Res j=l trace ( ^ ( r j Z j d z ^ . . . .dz^ z -0 (r,z),..,z -0 (r,z) I t n t m . n I Res j=l Ch(E,z,r,t).Td(z,r,t)dz dz n a (r,z), ,a (r,z) n For each f i x t e W - 0 , let r -* 0 through regular values of a(z) . Then we see that, by the continuity of Grothendieck residue, the right hand-side of (2) tends to the right-hand side of (1) and similarily the l e f t hand-side of (2) tends to the left-hand side of (1) (a (r,z) 0 (z) as r 0 , since V(r,z) -> V(z) as r -> 0) . t t Hence we have the claim. Q.E.D. Remark: Theorem 1.4.1 generalizes the theorem of N.R. O'Brian p -.^ t I D * ID * [34], where he actually proves this theorem for (A do^ ) : A^T X ->• A^T X We would like to thank L. Roberts for simplifying our complicated looking Todd class for us. We wouldn't have had the following result without his simplification. We keep our notation as before. Proposition 1.4.1. Let A,,..,A be the characteristic roots I n of L(p) = v(p) . 9a. 9z. (P) , and l e t u,,..,u. be the characteristic roots of 1 q 57 We set X.(t) = 1 — — i f X. f 0 and X. (t) =0 otherwise, tX. i i and 1-e tp. Y_. (t) = e 3 for 1 < i < n and i < j < q . Then for t e W - 0 , we have vpU>t,E) = — P(Y 1(t),...,Y g(t),X 1(t),..,X n(t),t) where P i s a polynomial i n n + q + 1 variables, in particularily v (<b ,E) is a meromorphic function of t i n W . p t Proof. By theorem 1.4.1 we have for t e W - 0 v (<f> ,E) P — Res t n p Ch(E,z,t)Td(z,t)dz 1 dz^ a n (z) , ,a (z) 1 n but then by the algorithm .(i) v (<j> ,E) = l / t n . {the coefficient of p u m i m n z ....z in the power series expansion of Ch(z,t)Td(z,t)det(B. .(z)} i>3 for some holomorphic functions B. .(z) and for some HL > 1 k = l,..,n. 1,3 K -Therefore i t i s sufficient to prove for each multi-index (i , . . , i n ) i n+..+i , 1 n ^ 1 „ n 9z, ...3z 1 n (Ch(E,z,t)Td(z,t)) z=0 is a polynomial in t, X_^(t) and Y_. (t) , 1 < i < n , 1 < j < q . But then by the product rule - of 58 differentiation i t i s enough to show N 1 n ~ 1 ~ n dz, .. . 9z 1 n (Ch(E,z,t) ) and z=0 i,+..+i , 1 n r i i dz, . . . dz 1 n (Td(z,t)) are i n this form. z=0 For i , +..+i , 1 n i 1 ^ n dz, ...dz 1 n (Ch(E,z,t)) , by the chain rule i t i s enough to t=0 . 1 n show q tv.( Z) T — (ChCE,z,t) = I e 1 ) i=l 9v,1...9v n 1 n is in the desired z=0 form, which is obvious. Now for l , +..+i , 1 n 1 1 1 3z, . . .9z n 1 n (Td(z,t)) , i t i s again by the chain rule, t=0 enough to obtain the desired form for > 1 n 11 Xn 9w, ...9w 1 n tw. (z) (Td(z,t) = n -i=1 tw. (z) 1-e , hence z=0 we only need to consider one dimensional case where the result may be checked by direct differentiation, considering the case A = 0, A =f 0 separately. Q.E.D. 59 Remark: It is actually obvious by Theorem 1.4.1 and by the algorithm (i) that v^((j>t,E) i s a meromorphic function of t in W with a pole at t = 0 . v^ (<(>t,E) has a unique analytic continuation as a meromorphic function on the whole complex plane. n Now let us look at the fixed-point formula. L(<j> ,E) = £(-1) k=0 k trace H^fa ,<}> ) is a holomorphic function of t in the complex plane. On the other hand we have the meromorphic functions v^($^,E) on the complex plane such that L(<j>t,E) equals to E Vp(cJ)t,E) for sufficiently peZ small [t| , by the fixed-point theorem, where Z i s the zero set of V , which i s f i n i t e . Hence by the uniqueness of analytic continuation the fixed point formula (1) L(<|>t,E) = I V (<f>t/E) pez then holds for 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 for t = 0 <|> = i d : a E -* E , hence the l e f t member of (1) reduces to -n L (<j) ,E) = 1 ( - l ) k dim Hk(X,0 (E)): = X (0 (E) ) 0 X X By theorem 1.4.1 we have for t e W - 0 v (d^jE) = —• Res p y t ' t n p Ch(E ,z,t)Td(z,t)dz ..dz n a l ' 60 Since Res (E,t) = Res < P P Ch(E,z,t)Td(z,t)dz^....dz^ > i s a V n holomorphic function of t c W (by the algorithm (i)) . Then the coefficient of t n \ in the power series expansion of Res^(E,t) w i l l be the constant term of V (E,t) . But the coefficient of t n in the power series P expansion of Res^(E,t) equals Res Ch(E,z)Td(X,z)dz ...,dz a i ' - - " a n t where Ch(E,z)Td(X,z) = the coefficient of t i n the power series expansion of Ch(E,z,t)Td(z,t) (by the linearity property of the Grothendieck residue). Hence we get (2) X(0 (E)) = I Res peZ Ch(E,z)Td(X,z)dz1 .. dz 1 n V Now R.H.S. of (2) can be viewed as the value of the global residue on a section of 0 . This operator, Res: T(Z,0 ) -> C , i s Z Z defined as follows: Let w be a holomorphic function i n a neighbourhood N(p) of p e Z representing the function s e T(Z,0 ) at p . Then Z Res(s) = £ Res peZ w(z)dz„....dz 1 n ci^ / > • • • f 3. 61 i s a well defined linear map. Moreover by [9] we have a commutative diagram r ( z , o ) R e s ) C Hn(X,ftn) X If we apply (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 Ch(E)Td(X) Remark: We believe our concepts are justified by this formula. R. Bott has obtained (3) i n [5] for the t r i v i a l line bundle. It i s this beautiful work of R. Bott, who let us study this subject. Historically, during the course of this work, the need for Theorem 1.4.1 came as follows; We wanted to prove Riemann-Roch and Hirzebruch formula for line bundles by using Bott's method for the t r i v i a l line bundle. We started with a projective variety X and a line bundle L on i t . We computed trace <|>t(p): L^ by using N Blanchard equivariant imbedding theorem X -* (P , because we already N knew how to compute this trace i n case X = P (see Example 1.3.1 ( i i i ) and Lemma 1.3.2). To our suprise, we found exactly the expression which is 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 particularly useful in analyzing the cohomology of a compact Kaehler manifold which has a vector f i e l d with isolated zeros. These were introduced in [11] and studied further in [13]. We w i l l begin by recalling these spectral sequences and some of their interesting consequences. Let V be a holomorphic vector f i e l d on a compact complex manifold X of dimension n . Then we have the anti-commutative t diagram of sheaves. Q + ftr 4-n,0 0 -y Or 0 -y tt v P-1 cp,0 9 -> £ y v p-1,0 9 , i + v 0 + 0 X 0 + 0 . 1 \ v c0,0 Is -y t where £ ° ' q = £°' q / i„(£ 1 , q) i s the sheaf of £°'0-module associated Z v -to the presheaf £°' q / i v(£ 1 , q) for each q = 0 ,. .. ,n 63 Fi r s t spectral sequence of V . Take the q-th cohomology of the p-th row £r + t in the diagram. By the Dolbeault lemma, we have H q(X,fi P) = ker{3: £ P' q(X) -> E P' q + 1(X)} / 3(& P' q _ 1(X)) Then the above diagram gives the f i r s t spectral sequence of V , I E 1 P , q = H q(X,fi P) => H q P(K,D) where K i s the total complex associated n — v r to the diagram with the derivation D = 3 + i , namely K = I K , k=-n K r = I K P' q and K P' q = £" P' q(X) . p+q=r Second spectral sequence of V. We now take the q-th cohomology of the p-th column on the global sections, namely " E ^ ' - * = ker{i v : £ q' P(X) + e^'^X)} / i v(£ q + 1' P(X) Then the diagram gives the second spectral sequence of V , 2\ P'- q=> HP-q(K,D) By [13] H q = ker{i : ftq -> fiq-1} / i v ( f i q + 1 ) i s supported on the set of zeros Z of V aftd 1 1 ^ ''^ = HP(X,Hq) for each q = 0,1,...,n . We note that tf° = 0 =0 / i (fi1) i s the structure sheaf of Z . Hence -z "x v we have two spectral sequences. 64 \ ™ = H q(X,fi P) => Hq-P(K,D) I ] : E P ' - q = HP,(X,Hq) => H P - q(K,D) If X i s Kaehler, then a theorem of J.B. Carrell and D. Lieberman [11] says that the f i r s t spectral sequence degenerates at E 1 , namely the linear operator i ^ : H q(X,fi P) -»• H q(X,fi P 1) i s zero. In particular, i f the dimension of X is zero, then H P(X,fi q) = 0 for p =|= q (by comparing with the second spectral sequence) . Therefore we get a graded ring isomorphism. (1) 9 """E P ' q = ® H q(X,fi P) = 9 H P(X,fi P) = H*(X ,C) = gr(H°(K,D)) oo I 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 ) 2 ' Z Z Hence © " E E ' " * = XIE°J° = r(Z,0 ) = gr (H°(K,D> . Z 0 • 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 r j j ( H (K,D)) = H (K,D) , hence the edge morphism defines a ring isomorphism 0 • e : T(Z,0 ) = H (K,D) which is given as follows; Let s e T(Z,0 ) , and let p ,...,p denote the zeros of V . Choose _L JC z smooth functions f on X , whose germ at p^ represents s(p^) in 65 k 0 . Then g = £ f. e £ ' (X) represents the image of s i n ~Z i = l 1 " e°'°(X) / i v ( £ 1 , 0 ( X ) ) . But then 3g Q = 0 i n £ 0 , 1 ( X ) / i v ( £ L f l ( X ) ) . Since the bottom row i n the diagram i s a f i n e r e s o l u t i o n of 0 by [13] Z and H q(X,0 ) = 0 for q > 0 , we get an exact sequence (2) 0 •+ r(Z,0 ) + £°'*(X) / i ( ^ ^ ( X ) ) . z Therefore there e x i s t s g., e £ 1 , 1 ( X ) such that i (g_) = 3g^ . But then 1 - v 1 0 i 3g n = - 3i g n = 0 , hence 3g_ e ker i : £ 1 , 2 ( X ) + £°' 2(X) . V 1 V 1 1 v - -Since 0 -+ H 1 + ker i I . i (£ 2'*) v 1 i * / v " i ' i s a f i n e r e s o l u t i o n of H 1 and H^CXjh'1) = 0 for every q > 0 , 2 2 -then we can f i n d g^ e £_ ' (X) such that 3g^ = i^Q-^ ' Proceeding i n t h i s manner, we are able to construct a sequence of forms g. e f 1'1(X) such that i -(3) 3g. = i (g.,,) f o r i = l , . . , n - 1 1 V 1 + 1 n i 0 n But (3) says that the form G = ^ ( - l ) 1 g. e K = \ £ P # P ( X ) i s a 0 1 p=0 cocycle f o r the t o t a l d i f f e r e n t i a l D = 3 + i . I t i s easy to check 66 that the cohomology class G i s independent of the particular choices 0 • made. The edge morphism e : T(Z,0 ) ->- H (K,D) i s precisely the map ~Z e(s) = G , i t ' s inverse i s defined as follows; given an arbitrary S o -cocycle G = l g. e K , then 8g Q = - xv^9j^ > hence 3g Q i s zero i=0 £°' 1(X)/i v(£ 1 , 1(X)) . By the exactness of (2) there exists a unique s e F(Z,0 ) whose image is g Q . Then G -+ s provides the inverse. Z Now 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 0 • -n+1 0 • -1 0 • 0 O f H (K,D) = H (K,D) D H (K,D) D ... OH (K,D) 5 H (K,D) 3 0 0 • By the edge isomorphism e : r(Z,0 ) = H (K,D) , we get a f i l t r a t i o n T(Z,0) = F n O F n + 1 o ... O F 1 ^ F° ? 0 such that 0 • n 0 • -p 0 • -p+1 gr (H (K,D) ) = © H (K,D) * / H (K,D) ^ 1 p=0 n - - + 1 = gr(T(Z,0 ) = © F P / F P Z p=0 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 in gr(T(Z,0 ) ? This was the our original question which lead ~Z 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 • -p 0 • Let us f i r s t look at the f i l t r a t i o n H (K,D) ^  of H (K,D) which arises from the f i r s t spectral sequence. For each p > 0 we have a f i l t r a t i o n F i=0 F i „ - P , o = e ° ' ° ( x ) + ^ ^ ( x ) +...+ a P ' P (x) c K ° = I t L , ± W i=0 n-1 I W - P . I = £ 0 , 1 ( x ) + £ 1 , 2 ( x ) + ..+ £ p , p + 1 ( x ) c K 1 = I e'i+1(x) F " " " i=o •" Then from the general theory of spectral sequences, we get H°(K,D) P = the image of { Ker{D| _ n= IF P'° + IF Prl} / D(XF P p P' in H°(K,D) = ker{D: K ° + K1} / D(K 1) , and VP'P = HP(X,ftP) = H°(K,D)- P / H°(K,D)- P + 1 Let G = q„ + g n +..+ g e P'° be the cocycle for 0 1 p D| „ then 3g = 0 , hence g defines a cohomology class [g 1 'l F-p,0 yp ^p ^ ^P J i n HP(X,ftP) . Now i f g^ e £. P , P(X) i s a smooth form representing the cohomology class [g ] i n HP(X,f2P) then 9g^ = 0 . But then 68 3i (g ) = - i 9g = 0 (since 3g = 0) , hence i (g ) e fcP 1 , P ( X ) v p v p p v p -represents the cohomology cla s s [ i v ( g p ) ] £ H P ( X , f i P "*") = 0 . Therefore there e x i s t s g , e £ P 1 , P 1 ( X ) such that i (g ) = - 3(g ,) . Consider p-1 - v p P _ l i (g n) e 8 P" 2' P _ 1(X) , since 3i (g .) = - i 3g = i i (g ) = 0 , v p-1 - v p-1 v p-1 v v p then i-(<3^ j) represents the cohomology cla s s [ i ^ ( g ^ ^)] e H P ^ ( X , n P 2 ) hence there e x i s t s g . e £ P 2 ' P 2(X) such that 3(g „) = - i (g .) . p-2 - P _2 v p-1 Proceeding i n t h i s manner we are able to construct a sequence of forms g^ e £ 1 , : L(X) such that 3g^ = - i v ( 9 ^ + 1 ) f o r i = 0 >•••' P - l • But G= Tg. e I F P ' ° i s a cocycle for DI „ , hence the l i n e a r map S 1 Vp'° H ° ( K , D ) ~ P ^ H P ( X , f i P ) [G] [g p] i s s u r j e c t i v e P Moreover i f G = Y g. e F P ' i s a cocycle for DI 0 1 VP'° and g = 3(f) for some f e £ P ' P - 1 ( X ) . ( i . e . [G] —>- 0 e H P ( X , f i P ) ) , P Then G^ = g Q + ...+ gp_2. + g ' ^ e Vp+1'° , where g ' ^ = gp_1 - ±v(f) i s a cocycle f o r DI n • . Since G ~ Gn mod D (G - G, = 'lp-p+1,0 1 1 g + i (f) = Df = (3+i ) f = 3f + i (f) ) , and [ G j e H ° ( K , D ) ~ P + 1 we p V V • • V 1 get [G] e H ° ( K , D ) ~ P + 1 . I t i s c l e a r that H ° ( K , D ) ~ P + 1 Q ker \\i , hence ker(i);) = H°(K,D) P + 1 . Therefore induces an isomorphism (5) lp: H°(K,D) P / H°(K,D) P + 1 = HP(X,ftP) I t i s easy to see that \\i i s well defined, and that" the edge isomorphism 69 IE P'P = HP(X,ftP) = H°(K,D) P / H°(K,D) P + 1 i s precisely <ft . OO Let/ d>: g£ (C) x...x g£. (C) -»- C be a k^linear invariant form r • r (i.e. <f> (g A g 1, . g^g 1) = <j> (A^,.. ,A^) for any g e GL^_(C)) , and let E ->- X be a V-equi variant holomorphic vector bundle of rank r - Then by theorem 1.3.1, there exists a hermitian metric h on E such that i ( 9 ) = 9 ( D for some L e £(X, Hom(E,E)) , where 0 is the canonical curvature matrix associated to h . k k—i k — i Consider <fr^_^ = ( J <j)(L,L,.,L, 0 ^ 0 ^ . , 0 ) e £ ' (X) for 'i ' ' k-i' i = 0,...,k (0: = 1) . Since 9 0 = 0 , we get 9<|> = 9 (<j> ( 0 , . . , 0 ) ) = 0 . By the derivation property of i and the symmetry property of <j> , we have v k-i v (.) (f)(L,.L, 0 , . . 0 ) = ( J (k-i) <j)(L L 1^ 1L,i v ( 0 ) , 0 , . . , 0 ) , and k-i-1 = 9 ( ± + 1) -ftCLj^L, 0 , . . 0 ) i+1 •' (i+D ^(L^^L^L,©,..©) , Since 9(L) = i v ( 0 ) / we get i (<(>. ) = 3(4 ,..,,.) for each i = 0 ,. .. ,k - 1 . v k-1 k-(i+l) i I -k 0 Therefore G = £ .(-1) (JK e F ' i s a cocycle for D l j _ k 0 * B u t i=0 7 0 then iK[G]) = (-l ) k [ < t > v ] = (-l ) k [ 4 > ( 9 , • . 0 ) . ] e H k(X , f t k ) . Now consider the 0 * edge isomorphism e: T(z , 0 ) = H (K,D) . If V i s the V-derivation of Z 0 (E) associated to L (i.e., L = V - i D) then LI = VI (i i s X zero operator on Z , i.e. L i s the smooth extension of v| ) . Hence we get e (((> (V,. . ,V)) = G eH (K) ^  . Therefore the cohomology class k k k (-1) ( f ) ( 0 , . . 0 ) e H (X,ft ) can be represented by the element <|>(V,..,V) e F(Z , 0 ) in the isomorphism ~Z * n ( 6 ) gr( (Z , 0 ) = H (X , C ) = © H P(X , f i P ) . Z p=0 In particular i f : g£^ ( C ) x . . xg£^ _ ( C ) -> C i s the polarized invariant j-linear form associated to invariant homogeneous polynomial r c . : ql -> C defined by det(I+XA) = Y c . (A) Xs (i.e., c . (A) = $ . (A,.. .A)) , D r 1 o 3 ' 3 3 1 1 j i i then the j-th Chern class c. (E) = (—- ) [ $ . ( 0 , . . 0 ) ] e HJ(X,ftJ) of E j 1 j can be represented by the element (-1) (——r-) $.(V,...,V) e T ( z , 0 ) in •27T1 : - z the isomorphism ( 6 ) . Hence we have the following theorem which i s also proved in [12] independently. Theorem 1.5.1. Let V be a holomorphic vector f i e l d on a compact Kaehler manifold X with isolated zeros, and let E--»- X be a V-equivariant holomorphic vector bundle with V-derivation V . Then i j (——) c . (V) e T(Z,0 ) represents.:the j-th Chern class c . (E) of E J Z J in the isomorphism 71 gr(T(z,0 ) = H (X,C) where c . (V) i s defined by the Z 3 equation det(I+XV) = Y cAV)X3 . j 3 Remark: This Lemma c l a r i f i e s our notation in section 4. Before giving examples, l e t us mention the following result in [13] F = ker{Res: T(Z,0 ) -> c} and (1) „ , Res r (z-,o ) y c H (x,n i s commutative -0 • where 3 i s the natural map induced from e': T(Z,0 = H (K,p) ~Z Example 1.5.1. Let V be the holomorphic vector f i e l d on P induced from 0 --x a t = e X. + ^ • for i + j l ' 3 1 Let V be the V-derivation of 0(U, .,) = 0(-l) given in 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 1 the function s n e T(Z,0 ) , s n(e.) = ^- (-X.) = -1 - -1 j 2TT j 2TTI Z j = 0,...,n 72 represents the Chern class c (u, n ) . Similarily s e T(Z,0 ) , 1 l,n+l Jc -s, (e.) = kA . j = 0,..,n represents the Chern class of 0(k) for k 3 2TT J each k =)= 0 e Z . i . Let us compute the cohomology algebra of p n . We have seen that the function s e T(Z,0 ) , s(e.) = A. j = 0,...,n represents "z 3 2 7 7 3 * 1 n 1 -1 the (U .,) e H ( P ,ft ) , hence s e F c T(Z,0 ) . Therefore 1 l,n+l - z n -n i n s e F = r(Z,0 ) , and s (e.) = (—) A., j=0,...,n . Let us compute Z 3 3 n v Res (s ) = I Res i = 0 s (z)dz,....dz 1 n v a (z),...,a^(z) By the algorithm (i) in section 4, we get Res s (z)dz,...dz 1 n e . 1 3 S n(e.) 3_ 3a. • — (e.)) 3 z v 3 det( i n j (—) V2TT n A . - A . i 3 since v l = I ( A . - A .)z.3/3z. ' n, ^ 1 3 1 1 ( IP ) 1 x . 3 Hence n n A n Res (s = (—) y — J — v2TT ; . L n n (A.-; 1-0 . ; , - i - x j > 73 Let JXL be the n x n matrix obtained by deleting the i+l-th row and n+l-th column of the matrix M = 1 0 0 1 X . X T 1 1 1 X X n n for i = 0,1,.. ,n We note that det(M) = II X . - X . and (- l ) n det(M) = Y (-1)1 X n det(M.) • • 3 i • n i i 3>i J 1=0 moreover det(M.) l It i s easy to see that II (X.-X.) det(M.) = (-1) det(M) . But then • 1 • 3 1 ' 1 we have > _ n ( X . - X . ) 1 = 0 3 1 n X n(-l) 1det(M.) . l Ji=0 i=0 det(M) I (-1) X" det(M ) det(M) (-1) det(M) det(M) = ( - D n, 1 n Therefore Res(s n) = (-^-) ( - l ) n = (-T—r) =(= 0 . Hence s P represents a non-zero element in F P/F for each p = l,..,n . But the dimension of F P/F P + 1 can't be bigger than 1, 74 since dimr(Z,0 ) = n+1, and we have n + 1 subspaces in the f i l t r a t i o n ~Z of T(Z,0 ) , we conclude that F~ P/F~ P + 1 = C. sP(s°=l) . Therefore gr(T(Z,0 ) = © Z p=0 F P/F P + 1 = C[s]/(s'n+^") which i s the cohomology algebra of P . Remark: the argument of the non-vanishing of s in F~ P/F P + 1 i s similar to the argument that =f= 0 in HP(X,ftP) p < dim X for the Kaehler class w , since w11 is a volume element. In fact from the above computation and diagram 1, we get . n 1 n , I , ResCs ) = (——) = (^ -) ( c l ( U l , n + l ) ) ' h S n C e 1 l,n+l P P i f we take n = 1, then c^(U^ ^) = - 1 , namely in the cohomology sequence -* H 1( Pn,0) H 1( Pn,0*) H 2( P n,z) 6(0(1)) = 1 where 0 (U., , .,) l,n+l 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 = e , where M = \ ° 0 ' A e gX (C) A. =(= A . for i 4= j n l ' j 1 75 Then U. x -> n > e Gr, : det k,n nxk *1 x. -> + P is the standard open affine covering of Gr^ n f o - """l """2<""<ik - n ' where x^ •> denotes the vector in C . Let z = Z l , l ' - " Z l , k Zp,kj x. -> 3i JP J X . + 1 3 P J -1 be the holomorphic local coordinates on U. , p = n - k . If A t ] _ , . . , t k denotes the k x k matrix which i s obtained by taking the t ^ . - j t ^ - t h rows of the n x k matrix A . Then d tM tM Vtz) = ((e t MX) (e^X) ) ar 31'"" p where X = t=0 X l * x + n nxk hence tx. tx. l e x.. e . ="l • •tx. J X p e x. -> e •' D' P x.~ + x. -> V -z V(z) = lim t->0 76 lim t->0 t(X. -X. ) 3 i x i 1/1 t(X. -X. ) 31 \ Jl,k t(X. -X. ) t(X. -X. ) 3P " l DP Xk e z • • • s e z _ p,l p,k - z But then (1) V(z)= (X . -X. )z. . . =>1 Xl 1 ' 1 (X . -X, ) z . Dt ^ t , l (X . -X. )z Dp i x P,l " A i ) Z1 k D l Xk 1 , k . (X . -X. )z^ 3 t Xk fc'k .(X . -X. )z D i , P,k p •- k namely V(z ) = (X. -X. )z r,s i i r,s r s 0 e l " Therefore e. . = < 1 l ' - " 1 k Sk + > . (e„ = (0...1..0) in the i ^ - t h r o w and zero elsewhere) i s the only zero of V on U. . , since 1 l ' - - ' \ (V(Z )) i s equal to the maximal ideal (Z ) r,s r=l,..,p ^ r,s r=l,..,p s=l,..,k s=l,..,k at e. . the multiplicity of the zero e. . i s one, namely i 1 , . . , i k i 1 # . . , i k e. vector f i e l d on Gr. is a simple isolated zero of V . Hence V is a nondegenerate with the zeros Z = {e. . 1 < i . < i <..<i, < k,n l ^ , . . , i k ' - 1 2 k -77 Then we get T(Z , 0 ) = © Ce. . • C e. Z K i <i <..<i<n l k 1 k = c Now let us compute the Chern classes of tl in T(Z,0 k,n If a : U, -> U, i s the natural extension of a, , then t k,n k,n t cr (e. . ) : (U. ) t ,. . k,n e. 1 k i 1 , - . , i k -»• (U, ) k,n e-. is given the matrix f(t) e GL (C) such that k But e t M e. . = e. . f(t) V - " 1 * 1 i ' - - ' 1 k tM e e. e. 1' ' k 1' ' k tA. tA. A. 0 Hence a (e. . ) t i ± l . . , i k = e -t V(e. ) " " " l ' " ' k = e by section 3. Therefore V(e. 1 l ' - - ' 1 k -A. 0 0 -A. If c.(V) s T(Z,0 ) i s the function given by 1 ~Z 78 c (V)(e. . ) = (-1) cr. (A ,A. ) , where a. is the j-th elementary symmetric function in k-variables. Then by Theorem 1.5.1. i j i 1 j (—) (-1) a. = (——) a. e'.T(Z,0 ) represents the j-th Chern class of 2TT J 2TTI J -k ,n If V i s the V-derivation of U, associated to k,n a : u -*.U, (cr = (a ) ) , then V (e. . ) = - V(e. . ) t t,n k,n s s I , . . , 1 ^ l ^ , . . , ! ^ A. 0 11. 0 '"-A. * class of U, k ,n i 3 Hence (—) a. e T(Z,0 ) represents the j-th Chern 2* 1 - z 1 3 By the similar reasoning we get easily s = (-r^r) cr. e T(Z,0 ) , 1 7 1 D Z where cr. (e ) = jth elementary symmetric function in A ,..,A : t 1 , . . , t k r x r p , 1 < r <...<r < n and {r ..r t,..t,} = {l,..n} represents the j-th - 1 p - l p l k Chern class of the universal quotient bundle Q K / n Remark: The same computations are done in [12] by a different technique. Let us f i n a l l y compute b (Gr ) the 2p-th Betti number of Grassmann manifold Gr, . If V is a nondegenerate holomorphic k,n vector f i e l d on a compact Kahler manifold X with zeros, then by the vanishing theorem of J.B. Carrell and D. Lieberman [11], we have b 2 k + 1 ( X ) = 0 , and by the theorem of C. Kosniowski [27], we have 79 t b (X) = {the number of zeros x e Z of V such that the 2p Real part of 0 (x) > 0 for exactly p indices i 1 < j < n = dim X , where 0 . (x) i s the J - - 3 9a. n eigenvalues of ( j ^ — (x)) ' VIN(X) = I a i %/dzA j 1 Now, let V be the holomorphic vector f i e l d on Gr induced K. f n n oi 2. 0 '-n from e . Then we have Z = {e. . : 1 < i < i_<..<i, < n} , V . . , l k - 2 k -and by (1) {i - i } , ' are the eigenvalues of V at the zero t s s=l,. . ,k t=l,-. ,P e. . Hence we get r r . . , r k b (Gr ) = the number of e. . 1 < i <..<i. < n such that 2p k,n- i 1 - . i k - 1 k -there are exactly p positive integers in the set t s s_'1/ • • /k t=i,..,p Therefore b„ (Gr, ) = the number of 1 < i . < i<..<L < n such that 2p :.k,n - 1 2 k -there are exactly p positive integers i n the set {j - i } _-, , t=l,..p where { j j , i n ,. . i , } = {1,2, ..n} and p = n - k . 1 p 1 k Let us count this number: for 1 < i < i <..<i < n , there exist n.- i elements in the set {l,2,..n} which are greater than i ^ . Hence there exists exactly n - i ^ - (k-1) = n - i ^ + k + l elements in the set {1,2,..n} \ {i^,..which are greater than i ^ . 80 Similarly there exist exactly n - elements in the set {l,2,..n} which are greater than , hence n - i ^ - (k-2) = n - i 2 ~ k + 2 elements in {l,2,..n}\ -fi- 3, - - - r4-^ } which are greater than i.^ . By proceeding in this manner, there exist exactly n - i g - k + s elements in the set {l,2,..n} \ {i , i ,0»-.i, } which are greater than i , for s = 1,..,k . But the eigenvalues of V at e. . are { A . - A . } , { A . - A . } , , {A . - A . } 3 t 1 l - n : t X2 ^ n 3t \ ^ . t=l,..p - t=l..p t=l..p n-i -k+1 positives n-i -k+2 positives n _ 1 v positives Therefore we get b„ (Gr, ) = {# 1 < i <- -<i, < n such that Y (n-i -k+s) = p} 2p k,n - l k - , s * s=l But k k k k r 2 v v . , , > k(k+1) v • ) n - i - k + s = kn - k + ) s - ) i = k(n-k) + > l i i i s 2 i s Hence k k (k+1) r (2) b 0 (Gr ) = {# 1 < i < . -<i < n: k (n-k) + \ i = p} 2p K / I l X K Z ^ S In particular for p = 1 , b (Gr ) = 1 ; since i = n - k rc / n X i = n - k + 2,...i = n i s the only one which satisfies (2) . 2 k 81 CHAPTER II APPLICATIONS OF G -ACTIONS TO LINEAR ALGEBRAIC GROUPS m 1. THEOREM OF BIALYNICKI-BIRULA A structure theory for complete varieties with G -actions was G m initiated by Bialynicki-Birula in [3]. If G^ acts on X and x e X is a fixed point then one can find coordinates for 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 n v ) , where dA (x): T (X) + T (X) t i n 1 n t x x is the differential of <j>, : X'-»- X at x , and <J> : G xX -> X is the t m given G -action. Let T (X) be the invariant G submodule spanned 3 m x m by those v e T (X) such that d<J>, (x)v = t m v for some m < 0 . Let x t U and W be the complete varieties with G -actions. Define a G -m m equivariant morphism f: U -> W to be a G^-fibration i f there exists a representation a: G -> GL(V) and an open cover {w.} of W such m l that f "''(W^) = x V with the action given by a on the second factor. In this case the dimension of V i s said to be the rank of the G -fibration f: U W . m Let X be a complete variety defined over an algebraically closed f i e l d k . For a given G -action cb: G x X X , let m m G r X = KJ Z. be the decomposition of the fixed point scheme into i=0 1 G connected components. For p e X , we define d(p) = dim Tp(X) G By [25] d i s a locally constant function on X . Hence d(p) = d(q) for any p , q e Z^  , since Z^ i s connected. We denote this common 82 value by d(Z.O . 1 Theorem 2.1.1 (Bialynicki-Birula [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 , let X m = (j Z. be the decomposition of m m . _ I i=0 the fixed point scheme into connected components. Then there exists a unique locally closed G^-invariant decomposition of X X = [J U. (called the (-) decomposition) i=0 1 and morphisms v.: U. -> Z. for i = 0,...,r so that i i I G (i) (UJ m = Z ± (ii) y. : U. -»• Z. i s a G -fibration of rank d(Z.) - ' i i i m l ( i i i ) for any 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. B-B's decomposition. i=0 1 F i r s t of a l l , we would like to discuss the relation between B-B's decomposition on X and p n under the G^-equivariant closed immersion ij;: X -*• . For this we need to know the existence of an everywhere defined map $: P 1 x X ->- X such that the diagram 1 (b P x X > X G x X m 83 1^ is commutative, where P = G u{0} u {«>} . We w i l l f i r s t show this m result on IPn , following [22] . Let eft: G x p n p n be a given G -action on p n . Then <f> m m defines a linear algebraic group homomorphism <j>: G Aut( (Pn) = PGL n + 1 t <ft : P n + p n . Hence <ft (G ) lie s i n a maximal torus of PGL T t m r since G i s a torus. Therefore up to the change of bases of k* m m m <ft: G m x p n -> p n i s given by cb (t, [x Q, . . . ,x^]) = [t x Q, .. ., t x^] for some e 2 . Now for a given x e p n , consider the orbit map cbX: G -»- G .x ci X(t) = cb(t,x) e p" . Define cbX: G u {0} + P H , m m 1 m X- Tl X I X cb : G u {°°} -> P where cb. =,<(> for i = 1, 2 and 2- m 1 Q m m -k m -k cj>X(0) = [t x Q,...,t n x n ] | t = Q "evaluation at t = 0" m -£ m -l cbX(°°) = [t x , . . . , t n x n] | ^  "evaluation at t = ~" n+1 n+1 where k = min: {m.: x.4=0, i=0,...,n}, I = max{m.: x.fO i=0,...,n} and x = [x , ...,xn] . It is easy to see that the <JK are well defined X I X for i = 1, 2 and that they patch in the intersection (<j>. |„ = § 1 o m for i = 1, 2) to give a morphism <j>X: P 1 -> P R such that $ x | = cj)X . m Hence {<j)X} defines the required map eft: p"*" x p n -+ p n xe P Let cft:G xx->X b e a G -action on a projective variety X m m Then i t is 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 • y X m G x p > p m <p commutes, where d) is a G -action on - m For x e X we have G <J> , G ...x CU X m > m .. V m . 4*(x) m > G .i> (x) c P N Since ^ ( X ) ( P 1) = ^ ( X ) (Gm) C ^ ( x ) (G ) = (cbX (G ) ) c i|) (x ) = * (x ) , - - m >- m m (x) f a c t o r s through X , say <j)X: P 1 -> X , such that ip o <$*• = (jS^5^ ~x ~ 1 Hence {<b } defines the required map <j>: P x x-> X . Moreover (j) : P x x y X N N (j) : P x p • p is commutative. 85 Proposition 2.1.1. Let <J>: G x x + X b e a G -action on a ^ m m complete variety X . Then there exists an everywhere defined map eft: P 1 x x •> X such that the diagram 1 <ft P x x —-—> X G x x m is commutative. Proof. By the equivariant Chow Lemma, there exists a surjective birational morphism TT: Y •+ X , and a G^action eft on Y such that the diagram G x y m G x x m -> Y -* X commutes where Y i s a projective variety. For a given x e X , there exists y e Y such that TT (y) = x , since TT i s surjective. Then we ~y 1 y have tft : IP -> Y which extends <ft : G -> G .y c Y . Hence - - m m X *" V 1 X (ft = TT o eft : P + X i s the extension of eft : G G .x C X . Hence y T m m ~x 1 {eft '}" define the required map eft: P x x -*• X . Moreover the diagram xeX 86 is commutative. Q.E.D. Now, let us find B-B's decomposition of the G -action m n n m0 m d>: G x P IP , d>(t, [x-,...,x]) = [t x . . . . , t nx ] , m. :e Z . m U n : u n 1 Without los of generality, we may assume m^  > m^  > m^  > ... > . Let s„, s,,...,s be the partition of n such that m„ = m. = ... m 0 1 r 0 1 s ^0 m s n = m s +1 = ••• = m s + s - 1 m l = m£+l = ••• = m n 0 0 1 0 ' ^ . s r Consider the following f i l t r a t i o n (D.Lieberman1s filtration) on P n. lP n = X„ oX,3 — OX , where XN = V(x ,.. . ,x . ) , . . . , 0 1 r 1 o s o i-2 Xi = X i-1 A V ( X £ '•••'Kl +s. n - l } w h e r e A-2 = ^ s for i > 2 1 1 1 i-2 1-2 i - l 1 2 k=0 K and V(f (x ,...,x n),...,f (XQ,...,x )) i s the set of common zeros of the homogeneous polynomials f ,...,f in (Pn . . Then the X^ are •t G -invariant, irreducible, closed subschemes of IP . Let m <f> I = d>. : G x x. -> X. for i = 0, l , . . , r . It i s easy to see that y |X. y i m I I 1 l i-1 h = V ( V % + j f ° r I \ - j < I \ 5 i = 0,...,r are the connected components of the fixed point scheme G r G Z = ( IPn) m = [J Z , and (X ^ X i + ) m = z i i = 0,...,r. One can check easily that the morphism y. = (<b.) : X. v. X. ,. ->• Z. , l I °° l l+l i 87 l y. (x) = A. (°° ,x) defines a G -fibration of rank d ( Z . ) = n + l - Y s, 1 1 m l , L„ k k=0 and T ( X . - X . ,) = T (Z.) ffi T ( X ) ~ for p £ Z. where d>. i s the map p l l+l p i p l I p 1 x X . -> X . associated to d>. : G x X . X . . Then by the uniqueness i i • I m I I r of the B-B decomposition we conclude that IP = I ) ( X . s X . ,) i s the • ~ i i+l 1=0 B-B decomposition. One notes that X ^ - X _ ^ + 1 i s the maximal open subset of X . such that (d>.) i s a morphism. 1 1 °° Now, i f we take m > m, > m„ > ...> m . Then X , = V(x ) , 0 1 2 n 1 0 X „ = V(x .x ) , . . . , X . = v(x n,...,x. , ) , . . . , X = e = [0,...,1] . In this 2 0 1 l 0 i i n n case D. Lieberman's 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 n and B-B's decomposition i s the natural cellular decomposition of p n (Bruhat decomposition of p n) . Let I|J : X -> pn be a G^-equivariant closed immersion, and l e t r s pn = Uu., x = U x. be the B-B decomposition of P and X 0 1 i=0 1 r respectively. If Z = \^_J Z. i s the decomposition of the fixed point i=0 1 G scheme ( P ) = Z into connected components, and y^ = cb^ : -> Z^  i s the G fibration on U. as given above, -<t>: G x (pn -s- p n . Then m l m r _ n we have a locally closed G invariant decomposition of X = I j ij> (U.J i T 0 Moreover the morphism tft: ij> "*"(IL) IL factors through IJJ ~^(Z^) • Z. l Namely we get a morphism X^: ±(V^) -> ift ^( Zj_) such that the diagram 88 ip """(U. ) — ^ — y U. 1 1 4> is commutative, since (|) (ip (x)) £ ip ^  (Z) for any x e X . Now let ip L(Z. ) = W u ... u W be the decomposition of ip "*"(Z.) into connected 1 1 K. . 1 1 components and let V. = X 1(W.) for j = 1, — ,k.. Then i t follows D 0 0 3 1 from Lemma 4.2 in [3] that V. = X. for some 0 < i . < s . Hence 3 1 j " 1 ~ ip "''(U.) = LJ X^  . In particular ip "'"(U.) = <J> i f and only i f 1 j j tp "'"(Z^ ) = cp . This observation essentially shows how to prove B-B's theorem from the decomposition of IP11 for projective varieties. In fact the morphisms y. : X. W. i s the restriction of X : ip "'"(U.)-Mp "*"(Z 1 . 1 . "J o o r l T 3 3 • on xT [30]. j Let d>: G XX->X b e a G -action on a complete variety with m m r isolated fixed points {pQ,...,p^} . If X = IL is 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 the Betti numbers of X . Let d>: G x X X b e a G -action on a compact Kaehler manifold r m m X with isolated zeros {p ,...,p } . Then for each p. we can find o r l coordinates for the tangent space T (X)' so that the induced action of P i 89 m i m n G on T (X) i s of the form dd> (p. ) (v, ,... ,v ) = (t v, , . . . , t v ) m p . t i l n 1 n 1 m. e 2 since G i s diagonizable and dd> (p.): G + GL(T (X)) i s a l m t i m p. I representation of G . But then the induced X-action on T (X) via m p. l exp: C G i s of the form m m^ z m z (*) (z,(v.,... ,v )) = (e v ,...,e v ) 1 n 1 n Now consider the C-action a: C x X -* X induced from <b via exp: C + i.e.; a(z,x) = <b(eZ,x) . Then the representation C + GL(T (X)) , P i d z z z -* da (p.) i s given by (*) , since — (e ) = e . Therefore we get z ^ i J dz 9z. I m i o m2 0 *'m n -1 n 9 A for some A e GL^ where v | N = I a± p i i=l V i s the vector f i e l d induced from a . + . m Hence dim T (X) = dimlveT (X): deb (p.)v = t v for some p. p. t I m > 0} = the number of positive m. , i = l,...,n . Since the eigenvalues of the matrix M and B "*"MB are the .same, we get dim T (X) = the number of eigenvalues of (- (p.)) which have p. 9z. I i 3 positive real part = the number of positive rru i = l,..,n. But then by the theorem of Kosniowski (see Chapter 1 section 5) the 2m-th Betti number of X is given by b (X) = Card{ie{0,...,r} such that there 9 ak exists exactly m eigenvalues of (. (p^)) with positive real 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 b^ (X) = b' = vCard{ie{0,. . . ,r} such that l 2m 2m dim T (X) + = m} . By [3] we have b' = b„ . Hence b„ i s the p. 2m 2m 2m I 2m-th Betti 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 for 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 in 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 i s greater than or equal to one, then T fixes at least two points of X . Proof. If X(T) = Hom(T,Gm) i s the character group of T and Y(T) = Horn(G^,T) , then the composite of X e Y(T) with x e X ( T ) yields a morphism of algebraic groups G m G m ' i-e., an element of X(G ) = Z . This allows one to define a natural pairing X(T) x Y(T) -* m denoted <x,A>•, under which X(T) and Y(T) becomes dual Z-modules. Therefore there exists X e Y(T) such that <X-/^> + KX • f o r 1 ^ J k 1 3 where {x-} i s a basis for the free abelian group X(T) . It i s 1 i=0 easy to see that the fixed points of T and the fixed points of G = X(G ) coincide. Hence we can assume T = G . Now i f the m m m G dimension of X > 1 , there is nothing to prove. Otherwise we can G ~ find x e X - X m , since dim X > 1 . But then <t>(°°,x) =j= <|>(0,x) and 91 they are the fixed points of G . Hence we have the claim. m Q.E.D, 92 2. BRUHAT DECOMPOSITION Let G be a connected reductive linear algebraic group defined over an algebraically closed f i e l d k , and let B be a Borel subgroup of G , T a maximal torus contained in B . Then B i s a semidirect product T n B where B i s the unipotent radical of B . u u Let 0 be the set of roots of G with respect to T and let W = N (T)/T G be the corresponding Weyl group We shall denote by the same symbol an element of W and a representative in N (T) when this can be done without ambiguity. G Let A be the set of simple roots of $ for the ordering associated to B , and let $ + be the set of positive roots. Then we have g = t © 1L g b = t f f i b ,b = © g , dim g = 1 , where . a u u +^ a ae$ ae$ t, g, b, b^ are the Lie algebras of T, G, B 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 1 = B } and G u u u N (b ) = {geG: Adg(b ) = b } . G u u u Proof. We have obviously B c N (B ) c N (b ) because B • G u G u u is a normal subgroup of B . Now N^(b^)/B i s a closed subvariety of G/B and T acts on i t via l e f t multiplciation. Since T i s connected, 93 i t stabilizes each irreducible component of N (b )/B . Let Z be any G u irreducible component of NG(b^.)/B . Then there exist fixed points of T on Z and they are of the form aB , for some a £ W . Now, i f cr B e Z i s a fixed point of T , then Ad (a) (b ) = b , because .. u u cr e N (b ) . But this implies a a e $ + for a l l a e $ + , hence a i s G u the identity. Therefore {[B]} is the only fixed point of T in Z , hence Z = {[B]} by Lemma 2 . 1 . 1 . Since Z i s an arbitrary irreducible component of N (b )/B , we get N (b )/B = {[B]} , which means N,_ (b ) = B . G u G u G u Since B c N„(B ) c N_(b ) we have the claim. G u G u Q.E.D. This Lemma gives an elementary proof of the well known theorem (C. Chevalley) N (B) = B as follows. G Corollary 2 . 2 . 1 . Let G be a connected linear algebraic group defined over an algebraically closed f i e l d k and let B be a Borel subgroup with the unipotent radical B^ . Then B = N G ( B U ) = NQ^ U^ where b is the Lie algebra of B . In particular B = N_(B) . u u G Proof. Let R (G) be the unipotent radical of G , then u G/R (G) i s a reductive connected linear algebraic group. We have the u natural epimorphism of algebraic groups cb: G G/R^(G) ~. Hence cb (B) i s a Borel subgroup of G/R^G) with the corresponding unipotent, radical <j)(B) [ 2 2 , p. 1 3 6 ] . For g e N (b ) we have <j> (g) E N _ . / t j r r M (L(d>(B ))) U G U Cv K 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) . Since R (G) c B i t follows that g £ B . This, together with the fact u that B c N„(B ) c N (b ) implies B = N (B ) = N (b ) . Since G u G u G u G u B c N (B) c N (B ) , we have in particular N (B) = B G G u G Q.E.D. 94 Let G be a connected reductive linear algebraic, group defined over an algebraically closed f i e l d k , and let $ = {a.} g = kv 1 i - i a. a. x ^ x 1 m g = kv i = l , . . , d and t = © k w. . Then by Lemma 2.2.1 the -a. -a. • , 3 x x 3 = 1 morphism Ad: G ->- GL(g) induces a one to one morphism Ad: G/B -> GL(g)/P where P = St(b ) = {-LeGL (g) such that L(b ) = b } . It is easy to see u u u that Ad: G/B GL(g)/P i s a closed immersion. By the Plucker imbedding d d of GL(g)/P = Gr -* P ( A g ) , the morphism <j>: G/B ->- P ( A g ) , "d,2d+m tf) (gB) = [Adg v A....*Adg v ] e P ( A g ) i s a closed imersion. The a l a d 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_^ is given on the basis {v_ ,w_.} by x. X u u u . . -a. - 3 i=l I +a  Ad(t)w. = w. Ad(t)v = + a.(t) v . Hence this induces naturally 3 3 - x J J +a. +a. x x an action of T on A^g which descends to P(A^g) . It i s easy to see that d> is T-equi variant, since 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(Ag) i s G -equivariant. Therefore the B-B decomposition of G/B , associated to a G -action on G/B induced by T can be computed from the B-B m decomposition on P(Ag) . Now we w i l l give a special 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 disjoint union G/B = u B^.OXQ (xQ=BeG/B) where is the unipotent radical of the opposite Borel subgroup B of G 95 and B . ax„ i s the orbit of ax„ under the l e f t action of B on G/B . U 0 0 u To find the right G -action, we need to compute d(Z.) . m l If a e W has a coset representative n e N (T) , then the following G formulas yield an action of W on X(T) = Horn (T,G ) and Y(T) = Horn(G ,T), m m independent of the choice of n . (OX) (t) = x(n tn) t e T , x e X(T) (aX) (a) = nX (a) n -1 a e G , X e Y(T) m Moreover <a\,oX> = <X/^ > a e W , x e X'(T) , X e Y(T) (see section 1 for the definition of < >: X(T) x Y(.T) -> Z) . Let b e B , and consider i , : G •+ G i , (g) = bgb^1 and l^-.G/B -> G/B b b D l^(gB) = bgB . Then the commutative diagram G G/B -> G/B where TT (g) = gB induces the commutative diagram on the tangent spaces Ad(b) g —.—y g dTT (e) dir (e) T (G/B) T (G/B) x o x o where T (G) = g , 7r(e) = x„. Hence we have e 0 96 Ad(b): g/b g/b (1) d £ : T (G/B) b x Q diT T (G/B) V since ker (dTT ) = b and IT i s separable, e For any regular 1-parameter subgroup X of T (i.e., X e Y(T) such that <a,X> =f 0 for any a e $ ; such X. • exist) , via l e f t mu lt i p l i c a t i o n , we get a G -action X: G x G/B -> G/B •. X. (gB) = X(t)gB m m t on G/B such that (G/B) = (G/B) = {cfxQ: aew} . Now let us compute d(ax ) = dim{veT(G/B): dX (ax,Jv = t m v for some m<0} . By the diagram 0 aXQ (1) we have d(x Q) = dim{weg/b: AdX(t)w = t w for some k < 0} Since g/b = b g , and AdX"(t)x = a(X(t))x = t < a , X > x for any t e G and x e g , we get 1 m a a d(x Q) = the number of a e $ such that <-a,X> = - <a,X> < 0 Therefore (2) d(x Q) = Card{ae$+: <a,X> > 0} Let n e N (T) be the representative for a. e W , then we have G 97 the commutative diagram G/B -> G/B A G/B G/B for each t e G , £ (xB) = nxB , a H (xB) = n 1A(t)nxB, A (xB) = m n t t A(t)XB . Hence we get a commutative diagram on the tangent spaces d£ (x ) T (G/B) — - — y T (G/B) x„ crx„ 0 i 0 d a - \ ( x 0 ) dA t(ax Q) d£ (x ) T (G/B) — T (G/B) x o CTXo ~ 1 rn Therefore d(ax„) = dim{veT (G/B): da A (x„)(v)=t v for some m<0} , 0 x Q t O .since d£ (x^) , dA (ox) and da "*"A (x^) are isomorphisms of vector n 0 t o t o spaces. Hence we have (by taking a X\ instead of A in (2)) (3) d(ax Q) = Card{ae$+: <a, a 1A > > 0} = Card{ae$+: <aa, A>>0} since <a ^aa, a 1A> = <aa,A> By [22, p. 166] we can pick a regular 1-parameter subgroup A e Y(T) such that <a,A> < 0 i f and only i f a s $ + . In this case the formula (3) gives us 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 Betti number of G/B , b„ (G/B) = the number of a e W such that £(a) = p 2p which i s a classical theorem of Borel. We have the similar situation for G/p where P i s a parabolic subgroup of G . ' To do this we need the following important lemma. Lemma 2.2.2. Let H be a connected solvable linear algebraic group acting on the complete varieties X and Y . If the morphism / H H TT: X -> Y i s surjective and H-equivariant, then TT ( X ) = Y Proof. Let x e X . Since TT i s H-equi variant we have h.Tr(x) = Tr(h.x) = IT. (x) for a l l h e H , hence TT(X) e Y . Now let H —1 i y e Y ; then IT (y) f <j> i s a H-invariant closed subvariety of X . By the Borel Fixed Point Theorem we have (TT ^ ( y ) ) H =j= cb , which means H H ' ' Y c TT(X ) , and hence we have the claim. ' Q . E . D . Let P o B be a parabolic subgroup of G . Then the morphism IT: G/B + G / P , Tr(gB) = gP is surjective and T-equivariant (T acts via l e f t multiplication on G/B, G / P ) . Hence by Lemma 2.2.2 T T ( ( G / B ) T ) = ( G / P ) T = {crP: aeW(P)} where W(P) c W i s such that T + + (G/p) = {aP: aeW(P)} contains distinct points. Let $ (P) C $ be such that p = b © g , where p i s the Lie algebra of P . ae$+(P) " a For any regular 1-parameter subgroup A of T , the morphism 99 TT: G/B -> G/P is G -equivariant. Hence we have (G/p) m = {aP: aeW(P)} m i where X: G x G/P -> G/P i s given by X (gP) = X(t)gP . By a similar m t argument above, i t i s easy to see that d(aP) = Card{ae$+(P): <a,a 1X> > 0} = Card{ae$+(P): <oa,X> > 0} for a e W(P) . If 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 (G/P) i s given by P^ (4) b„ (G/P) = Card{aeW(P): £ (a) = p } 2 P $+(P) Example 2.2.1. Let B = < * • * * V 1 > C GL n , T = * 0 0 * C B then. $ = {a. .: Ki,j<n, i=)=j} where a. i , J i f . *1 ° 0 t = t . t : 1 r}+ = {a. .:j>i} , $ = {a. .: i>j} . The map 1 0 0 1 ,2,....,n o•• ,a ,. .. ,a 1 2 n a • -> -.1 y 5 -* • " n , < = a e N(T) , where a j _ = e a = ( 0 , ... . ,1 , . . , i f i defines an isomorphism between the symmetric group and the Weyl group W . It i s easy to see that the action of W on $ is given by a.a. . = a . Now, i f we take A: G -> T by i , 3 a i ' C T j m A(t) = t n 0 . n-1 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 , aj = Card{(i,j): n>j>i>l, a^>o^} • Therefore b„ (GL /B) = Card{aeS : there exists exactly p Ki<j<n such 2p n n - -that < a_.}. Similarily- i t can be shown by formula (4) that b„ (GL /P) = Card{Ki <...<i <n: k(n-k) + -±= '- - \ i = p} , where P = A * 0 * AeGL, ^  c GL , Kk<n-1 k n - -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 algebraically closed f i e l d k , and P be a parabolic subgroup of G . Then N G ^ P U ^ = p • w n e r e p u 1 S the set of unipotent elements of P and N (P ) i s the normalizer of P in G . In G u u particular N (P) = P . G Proof. Let B be a Borel subgroup of G contained in P , and B be the unipotent radical of B . Now B acts via l e f t u u multiplication on G/B and on G/B so that TT: G/B ->- G/p , Tr(gB) = gP B is B -equivariant. By Corollary 2.2.1 we have (G/B) U = {[B]} . But U B then Lemma 2.2.2 gives (G/P) U = { [P]} . Since B^ C p u • w e have P B P (G/P) U c (G/P) U , hence (G/P) Q = {[p ]} which proves N (P ) = P . G U Since P C N (P) c N (P ) we have in particular N (P) = P . G G u G Q.E.D. Now we can start 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 of T such that <X,ct^ > = iru > 0 for each i = l,...,d where $ + = {ct^: i=l,...,d}. We have seen that the G -action X: G x G / B G/B , X(t,gB) = X(t)gB l i f t s to a G -m m m d d d action X: G x p(Ag) + P(Ag) so that cb: G/B +P(Ag) m cb(gB) = [Adgv A . . . A A d g v ] e P(y\q) becomes G -equivariant. Take ot _ ot -i in 1 d i d a basis L y v : k=0,...,N} of A g the exterior products of d elements in {v , w.: i - l , . . . , d ; j=l,...,m} where g = kv , g = kv , — j a. a. -a. -a. +ct. 1 1 1 1 I m X, t = ® kw. . Then the action of X on Y i s given by X(t,Y ) = t Y j = 1 D k k y N 102 <a.,X> m. for some I, e Z , since X(t,v ) = a.(X(t))v = t v = t v , k a. i a. a. a. 1 i i i <-a.,X> -m. X(t,w.) = w. X(t,v ) = - a. (X(t))v = t 1 v = t . Let i i -a. i -a. -a. -a. I I I i {Y ,Y , ..,Y } be the ordered bases of A g for which SL > I > I >...>J \J -L. J.N \J -L & d d (£. = 7 m. > Jt > £„ > > I . > I = - Y m. , Y„ = v A . . . A V 0 . L, l 1 - 2 - - N-1 N . ^ , I 0 a, a, 1=1 i=l 1 d N Y = v A . . . . A V ) . Let x. be the homogeneous coordinate of P N -a. -a l J- d corresponding to the basis vector Y^  in the natural isomorphism , d , „ N (P ( Ag) = P . N . r. N Let P = i_J U. be the B-B decomposition of P induced i=0 1 G by X , and (Uj m = Z^ . Then Z Q = {e Q •= [1,0,...,0]} i s the unique G connected component of ( P ) such that d(e^) = N , and moreover Q U Q = ( P N ) x = {[1,*, , * ] } ^ P N . Now (<J)"1(U0)) m = {xQ = [B] e G/B} . Therefore <j) ^ (UQ) i-s a member of the B-B decomposition (by section 1). -1 d n Let us compute <j> (UQ) = {gBeG/B: A(Adg).Y Q = £ a±Yi a o ^ ° ^ " W e c l a i m i=0 that <J> "^(UQ) i s invariant under the l e f t action of B^ on G/B where B i s the unipotent radical of the opposite Borel subgroup B of G . U - - d - N For u e B , i f A(Adu ).Y = y a. , (u.) Y. , then by [18, p. 9] we u k ^ i,k l have a. , (u ) =0 i f z(i) > z(k) or i f z(i) = z(k) , i =t k a n d a , (u ) =1 where z(s) = Card{ae$ -.such that v A Y = 0} . Now i f k ,k a s -1 d N gB e <j) (UQ) , then A(Adg).Y Q = £ a j _ Y i a o ^ ° " T n e r e f o r e f o r a nY 1 0 3 - - d - • ? d -u £ B u , A(Adu g).Y Q = I a. A <Adu ).Y. = a ^ + a ^ +a'Y 2 +. . ,+a'^ since z ( 0 ) = d > z (i) for any i = 0 N . Hence ugB e § "*"( UQ) namely d> ^(U„) invariant under B . Since the orbit of a unipotent 0 u group on an affine variety i s closed [ 2 2 , p. 1 1 5 ] we get b U * X Q = $ "^ ) * Because they have the same dimension, d , and ^ ^ U 0 ^ ^ S a n a ^ ^ i n e v a r i e t y (as a member of the B-B decomposition or, since d) i s a closed immersion N and U Q i s open affine subvariety of IP ) . We w i l l proceed in the same manner to obtain the other members of the B-B decomposition. Let us put the connected components of ( IP ) in an order, so that N = d(Z Q) > d f Z ^ > d(Z2)>...>d(Zr) = 0 . We w i l l 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 - 1 - 1 Gm - 1 d> (U ) , i t i s easy to see that (d> (U ) ) = d> hence <j» (U ) = d> - 1 Gm Let us assume that (6. (n ) ) = {a,x-, . . . ,cr x.} •. Then U„ i s in the 2 1 0 k 0 2 form { [ 0 , , 0 , 1 , * , ,*]} u { [ 0 , . . . _ . , 0 , 1 , * , ...,*]} W S o + 1 s +k U { [ 0 , , 0 , 1 , * , ,*]} = O U . • Let us assume that w „ j = 1 2,j s o + s i + k - 1 4>(a.xJ e n . for i = 1 , ,k ; 1 < j < s + k . Note that T l 0 2,3^ - l - x s +k U = K^J U i s a disjoint union, and d> (U . ) = X for each 2 , 2 , 3 ' - 1 i i 0 1 = l , ,k . Now we w i l l show that <j> 1(U . ) i s B -invariant for '3 2_ 104 each i = l,....,k ; in particular cb "^ (U^ ) w i l l be B^-invariant. For example, let us show cb 1(U„ . ) i s B -invariant. In this case 2,:k Then vv 1 ,-1... . r d. f (0 2 ) = {gBeG/B: A(Adg) .Y = \ a. Y ± , a g + - =f= 0} . i = s 0 + j k _ ! 0 Jr If gB i s an element of <j> ±(U^ . ) , then A(Adg) .Y„ = a Y + . .+a y ' 2 ' \ . 0 0 s 0 +j k _! - y N ' ~ — d "~ a^/0 and Y =v A - . - A V • F o r u e B , / \ (Adu g) . Y = ° W 1 a k a l ° k a d U ° b„Y„ + b Y...+b Y +...+b Y + a^Y +...+a'Y , since 0 0 1 1 y y s 0 + J . k _ 2 s 0 + j k _ 2 0 SQ+J.^ ! Y N z(s +-i -1) > z(s) for any s = 0,....,N , the coefficient of Y 0 k " . s o j k - i d i n the expansion of /\(Adu g),Y stays as a Q which i s not zero by assumption. Hence u gBgfcb' (U . ) i f and only i f there exists 0 < y < s„ +-! -2 such that b =1= 0 . We w i l l show now that a l l - - o Jk y b =0 for 0 < y < s„ + \- _ o . F i r s t of a l l b = 0 , otherwise y • - ~ 0 Jk 0 u gB e cb "*"(UQ) '> hence (u ) X u gB = gB e cb "*"(U ) since cb ^ (UQ) i s B -invariant, which contradicts the choice of gB e cb ^  (U . ) . Hence u . 2 , 3 k b = 0 . Now i f b ^ 0 for some 1 < y < s then u gB e cb 1 (U ) 0 M — 0 I which can not happen since cb "*"(U^ ) = cb . Similarily b^ = 0 for s < y < s ^ + i - 2 > otherwise u gB e cb 1 (U ) which can't 0- - 0 Jk 2 happen, since cb ^ ( ^ J does not contain such an element, (otherwise [0,....,0,1,0,....,0,1,0, ,0] i s a fixed point of A on f K. 105 G/B , but there i s no such fixed point, since the fixed points of G/B in p N correspond to e. = [0,.. .-,0,1,0,... ,0] for some 0 < i < N) i+1 Therefore .b =0 for a l l 0 < y < s_ + -1, - -2- • Hence u gB e <j) ^  (U u - - 0 •Jk namely d> ^ (U„ . ) i s invariant under the l e f t action of B . But then B .a x„ i s a closed subvariety of the affine variety u k 0 -i d(akV (j) (u . ) = A , since the orbit of a unipotent group on an affine variety i s closed. But the dimension of B .a, x_ is equal to * u k 0 Card{ae$+: such that a, a > 0} = d(a x ) . Therefore k k 0 X = d>-1(U„ • ) = B-.cf x„ . Similarily we have X = cf> 1 (U„ . ) a kx Q 2, D k u k 0 0.xQ 2,D. = B .a.x„ . u i 0 Proceeding in this manner, we see that each <j) ^ (U. , ) is 1 ,K B~ invariant where (|)~1(U.) = L J <!)~1(U. . ) , U. , ) = X u I i,k i,k T X Q Then by comparing the dimensions of the closed subvariety B ^ . T X ^ of the affine variety X , we get X = B . T X . Hence the B-B T X Q T X Q U 0 decomposition of G/B induced by X i s the Bruhat decomposition of G/B , G/B = L J B .crx0 . u 0 aeW Remarks: (i) The only place we have used B-B's theorem in the above proof i s to compute the dimension of d> ''"(U. ) = X . Now we w i l l 1 ,K 0 X Q 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 {ax > — y z. o i It induces on the tangent spaces a commutative diagram T (cb ?"(u. ,)) — ^ — y T . . (u.) <Jx„ i,k <H0xQ) I 0 dy. + T c b ( c r x 0 ) (V Since dy. :T, . , (U. ) = T . (Z. ) © T . , ( PN) + T, f r l ,(Z.) I $(axQ) i <KaxQ) I <f>(ax0) <b(Ox 0) I i s the projection to the f i r s t factor (see section 1) we get d c i ( T (cj, ^  (U. ,))) C. T . ( P N)' OxQ i,k <('( c rx 0) Hence T„ (cp 1(U. )) C T n (G/B)' Ox„ i,k c r X o Therefore dim(B . CTx ) < dim <b 1(U. , ) < dim T„ (cb 1(u. , )) < u 0 - T i,k - cr x Q i/k dim T (G/B)~ = d(Ox^) . Since d(cbc) = dim(B~.Ox ) , we get <JXQ 0 0 u 0 dim <b_1(u. , ) = dim(B~.axJ = d(c x ) . i,k u 0 o With this argument,, the above proof should be considered as an elementary proof of the Bruhat decomposition. 107 ( i i ) The p r o o f o f the B r u h a t d e c o m p o s i t i o n g i v e s more i n f o r m a t i o n about the c o e f f i c i e n t s a ,.... ,a i n the e x p r e s s i o n /?(Adu ) . Y = I a Y i=0 ( i i i ) L e t P be a p a r a b o l i c subgroup G c o n t a i n i n g B . I t i s c l e a r t h r o u g h G - e q u i v a r i a n t morphism IT: G/B -> G/P , Tr(gB) = gP , B-B's d e c o m p o s i t i o n o f G/P , a s s o c i a t e d t o G - a c t i o n A: G x G/P -* G/P m m i s the B r u h a t d e c o m p o s i t i o n o f G/P , i . e . G/P = B .ax„ where aeW(P) u A: G T i s a r e g u l a r one parameter subgroup o f T such t h a t m < A, a > > 0 f o r a l l a e $ + . L e t us g i v e a s i m p l e example. Example 2.2.2. Take n = 2 i n the example 2.2.1 and f 0 1 0 o" f 1 0 f \ 0 0 v a = E 1 2 = 0 0 , V = E = ' -a 21 1 0 ' w l = 0 0 0 1 A: G -> T , A(t) = m t 0 0 1 E>+ = {a} . Then <|>: GL^/B + I P ( A g ! l 2 ) = P i s g i v e n by cj> (gB) = [ a Q , a , a 2 ,a 3] where Adg E 1 2 = a ^ E ^ 2 + a . ^ + a ^ + a 3 E 2 1 * L 6 t g X l X 2 X 3 X 4 e G L 2 , then Adg E 1 2 = g K 1 2 g _ 1 = U\ E 1 2 - x 1 x 3 w i + X i x 3 w 2 - x ^ } 2 2 Hence <j> (gB) = [ x ^ - x 1 x 3 ' x i X 3 ' ~ X 3 ^ • C o n s i d e r the n a t u r a l isomorphism GL 2/B = P , X l X 2 X 3 X 4 B -> [x ,x ] . V i a t h i s n a t u r a l isomorphism 108 1 3 2 2 cb: P -> P i s given by ^([x^x.^]) = [x^, - x 1 x 3 ' x i X 3 ' ~ X3^ "Twisted Veronese mapping". The corresponding G^-action on fl?"*" = GL^/B 3 1 is simply (t, [x ,x ]) = [ t x ^ x ^ , and on P = P( Ag^) (t, [x^x^x^x.^] -1 ' 3 [tx ,x ,x2, t x^] . Then D-L's f i l t r a t i o n on P is given by X l = V ( X 0 ) ° X2 = V ( X0' X1' X2 ) ° * ' d> ^ ~ (X ) = V(y ) c PX i f . y^ are the homogeneous coordinates of P 1 , <b_1(X ) = e = [0,1] . P 1 = ( P 1) u (e } = G au{°°) i s the 1 1 Y0 Bruhat decomposition of GL^/B = P"*" . In fact for any G -action on p n , i f Z = I j Z. i s the 0 1 irreducible decomposition of the fixed point scheme, then the B-B. decomposition i s of the form r P = [J B .z. , where B i s the unipotent radical of some u i u i=0 Borel subgroup B of GL^+^ (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 i s of m r s the form G/B = i I - „ , „ ~ > • — V T . \-Jrs B .Z. (resp. G/P = VJ B .W.) . i=0 u l i=0 u l I am very grateful indeed to D. Lieberman for the several helpful discussions I had with.him while this work was in progress at IHES. I am also grateful to B. Iversen for mentioning this problem to'me. 109 Let us fi n a l l y give one more application 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 linear algebraic group defined over an algebraically 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 identity e e G . In u e u u particular P = N (P) = {geG: Adg(P) = P} where P = L(P) i s the Lie G i algebra of P . Proof. We w i l l f i r s t prove Lemma when G i s reductive. In this case, l e t B be a Borel subgroup of G contained in P , and let $ 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 via l e f t multiplication. By Lemma G u 2.2.2 the fixed points of T on N (P )/p are of the form oP for G U some a e W . Now i f oP e N (P )/P is 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 having Lie algebra g' [22, p. 161] for each a e $ and n e N (T) i s a a G representative for a e W , then nU n 1 = U for each a e $ . Let a aa I(J = {a a^} and consider the morphism of varieties d>: U x . . . . xrj -> P : <j> (x, ,.. . ,x, ) = x,'. . x, . It is an injective morphism. T a , a, u 1 k l k 1 k Since they have the same dimension, the constructible set i> (U X . . . X T J ) • 01, OL 1 k contains a dense open subset of P , say V . But then u ' -1 i (V=P ) c i (V) c <t(U X . . . X T J ) = p where i (g) = gng . Therefore •n u • - v " - a a f c u n 110 i (P ) = P , namely n e N_(P ) . By Corollary 2.2.2 we get n e P . n u u G u Hence {[P]} i s the only fixed point of T on N CP )/P . By Lemma x G u 2.1.1 we get N (P ) = P . G u Now, let G be any connected linear algebraic group, and let R U ( Q ) be the unipotent radical of G . We have natural morphism of algebraic groups TT: G -»• G' , G' = G/R^ CG) . Hence <J> (P) i s a parabolic subgroup of G' , and cb (P ) = (<b(P)) i s the set of a l l unipotent elements of u ' u <b(P) . For geN (P ) we have cf) (g) e N (T ((cb(P)) ) where G u G e u dir (e) (T (P )) = T ((<b(P)) ) . Since G' i s reductive, we get cb (g) e cb (P) . e u e u This means g = pr for some p e P and r E R (G) . Since R (G) lie s u u in every Borel subgroup of G , in particular R U(G) C P , i t follows that q e P . This, together with the fact that P c N (P ) implies P = N (P ) 3 G u G u Since P c N (P) C N (P ) we have i n particular P = N CP). G G U G Q.E.D. Remark: N CP) = P i s a well known result. In fact one can G 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) is the unipotent radical of P. I l l 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 is greater than or equal to one. In this section we w i l l show the existence of a G -action on G/P with only one fixed point i f the ground a f i e l d k i s nice. This w i l l be suprisingly an application of Lemma 2.1.1. I am very grateful indeed to 0. Loos for several helpful discussions I had with him while this work was in progress. Let G be a connected reductive linear algebraic group defined over an algebraically closed f i e l d k , and let B be a Borel subgroup of G , T a maximal torus contained i n B . Then B i s a semidirect product T x> B^ where B^ i s the unipotent radical of B . Let $ be the set of roots of G with respect to T and let W = N (T)/T G be. the corresponding Weyl group. We shall 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 B , and let § + be the set of positive roots. Then we have g = t ffi JJL g , b = © g , dim g =1 where t , g , b • a u .+ a a u aes> ae$ are the Lie algebras of T , G , and respectively. Now we have the following important lemma. Lemma 2.3.1 (E.Y. Akyildiz). For any n e b , X /B is 'a u n closed subvariety of G/B , where X = {geG: Adg(n) e b } . n u Proof. If n = 0 , there i s nothing to prove. Assume n =(= 0 . 112 Let d = dim.b = dim b , m = dim(t) , where hx = © g - Then u u u .+ • -a ae$ g = t © b © b . Consider Ad: G -> GL(g) C g&(g) and define u u . • P = (TegA(g): T(b ) c b } , Y = {Teg£(g): T(ri)eb } . Let P = P A GL(g) u " " u n u and Y = Y f\ GL(g) , we have clearly X = Ad 1(Y ) and B = Ad~1(P) n n n n because of Lemma 2.2.1. We claim Y .P = { T O S : TeY , SeP} i s closed in g£(g) . To n n see this extend n 4= 0 to an ordered bases {Ofn=vn ,. ,v,,v. ,v~,.. ,v n,t n ,. 1 a 1 2. d 1 of g such that {v,,..,v,} ,•{v.,..,v.J , {t.,..,t } are bases of I d I d 1 m b , b , and t respectively. With respect to this ordered basis P u u is the set of matrices of the form d+m * * 0 * £ qSL 2d+m and Y i s the set of matrices of the form n d+m 2 d+m Let Z be the set of matrices of the form d+m { e gZ , with rank (C) < d 3 2d+m 113 j Then Z = Y .P . n For i f A = A. A. •e Y , B = n B B 1 2 0 B, 4, e P , then A.B = * * A B, * 3 1 Because the f i r s t column of A^ i s zero, and rank ( A 3 B 1) < rank (A3) < d . Hence A.B e Z . Conversely, given e Z via column operations on C^ we can find an invertible d x d matrix Q such that, C3Q Now take B = Q 0 0 Id A .B = e P , and A = -1 e Y n 2d+m Therefore (Y .P) A GL (g) = (Y A GL(g)).(P (\ GL (g)) = Y .P is closed n n n in closed i n GL(g) . The morphism Ad: G + GL(g) induces a morphism Ad: G/B -y GL(g)/P which, as easy to see, i s a closed immersion. Since GL(g)/P has the quotient topology and Y P i s closed in GL(g) , i t ~ -1 follows that Y /P i s closed i n GL(g)/P . But then X /B = Ad (Y /P) n n n 114 i s closed i n G/B , which completes the proof. Q.E.D. Let G cGL^(k) be a connected linear algebraic group, and let n c g c. gl^ be any nilpotent element, where g is the Lie algebra of G . Then, i f the characteristic of k i s zero, there i s a well defined algebraic group morphism e: G •+ G such that de(l) = n , where de: k -»- g is the cl differential 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 linear algebraic group defined over an algebraically closed f i e l d k of characteristic zero, and l e t B be a Borel subgroup of G . Then there exists a G -action on G/B , induced by 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 g for each aeS . Now we have an algebraic group homomorphism e: G -> G such that ds(1) = n . e Si induces, via l e f t multiplication, a G -action on G/B . We claim that cl {[B]} i s the only fixed point of this action. To see this, consider X /B in G/B which i s a closed sub-n variety of G/B by Lemma 2.3.1. The torus T acts on X /B via n l e f t multiplication, hence i t acts on each' irreducible component of X /B . Let Z be any of them. Fixed points of T on Z are of the n form aB , for some a e W . Now, i f aB e Z i s a fixed point of T , then Ada(n) = Ada( Y x ) = Y x e b because a e X . But then „ « r. act u n aeS aeS we must have aae$ + for a l l aeS . Since S contains the simple roots 115 A , a must be the identity. Hence {[B]} is the only fixed point of T i n Z . By Lemma 2.1.1, we get Z = {[B]} . Since Z i s an arbitrary irreducible 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 fixed point of G , then cL g 1 e (x)g e B for a l l x e G . Hence Adg 1(de(l)) = Adg 1(n)eb This implies g 1 e X n = B , therefore 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 in g is conjugate to an element in b^ , i t i s clear from the proof of this theorem that any G -action, induced by G , on G/B with only one fixed point, i s a obtained.in this way. Corollary 2.3.1. Let G be a connected linear algebraic group defined over an algebraically closed f i e l d k of characteristic zero, and let B be a Borel subgroup of G . Then there exists a G -action on G/B , induced by G , having only one fixed point, a Proof. Let R (G) be the unipotent radical of G . Then u G' = G/R^ CG) i s a ,reductive group, and B' = <j> (B) i s a Borel subgroup of G' . The natural epimorphism of algebraic groups <j>: G G' induces an isomorphism <J>: G/B ^  G'/B' , since R^(G) c B , d> i s bijective and separable. Let n 1 = £ x e g ' where S i s any aeS subset of $ + containing A , and 0 =]= X e g ' for each aeS . Then 116 there exists, a nilpotent element n of g such that dtp (n) = n' , since R (G) lies in B for a l l Borel subgroups B of G and u u dtp: g -* g' i s surjective. Hence we have an algebraic group homo-morphism e: G G 'such that de(l) = n . Let £' = cb o E : G ->• G' a a Then e(resp. e 1) induces via l e f t multiplication, a G -action on 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'/B1 induced by e' has only one a fixed point. Therefore the G -action on G/B induced by e has only 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 linear algebraic group defined over an algebraically closed f i e l d k of characteristic zero, and let P be a parabolic subgroup of G . Then there exists a G -action on G/P , induced by G , having only one fixed point, a Proof. Let B be a Borel subgroup of G contained in P , and le t e: G * G/B -> G/B be the given action in 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 for reductive groups would go through for characteristic p > 0 also, i f we knew the existence of an algebraic group morphism e: G -> G with a de(y) = £ X for some y E k .and for some s D A . But, generally aeS such E does not exist. A. Borel.and T.A. Springer give a sufficient condition for the existence of e i n [4, p. 495]. 117 BIBLIOGRAPHY M.F. Atiyah and R. 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