Cohomology of Quotients in Symplectic and Algebraic Geometry. (MN-31), Volume 31 9780691214566

Citation preview

Cohomology of Quotients in Symplectic and Algebraic Geometry

by

Frances Clare Kirwan

Mathematical Notes 31

Princeton University Press Princeton, New Jersey 1984

Copyright (c) 1984 by Princeton University Press All Rights Reserved

Printed in the United States of America by Princeton University Press, 41 William Street, Princeton, New Jersey 08540

ISBN 0-691-08370-3

The Princeton Mathematical Notes are edited by William Browder, Robert Langlands, John Milnor, and Elias M. Stein

Library of Congress Cataloging in Publication Data will be found on the last printed page of this book

Contents

§1.

Part I.

Introduction,

1

The symplectic approach.

§2.

The moment map,

21

§3.

Critical points for the square of the moment map,

35

§4,

The square of the moment map as a Morse function.

44

§5.

Cohomological formulae.

62

§6.

Complex group actions on Kahler manifolds.

79

§7.

Quotients of Kahler manifolds.

95

§8.

The relationship with geometric invariant theory.

102

§9.

Some remarks on non-compact manifolds.

115

§10.

Appendix.

121

Morse theory extended to minimally

degenerate functions.

Part I I . The algebraic approach.

§11.

The basic idea.

138

§12.

Stratifications over arbitrary algebraically closed fields.

143

§13.

The strata of a nonsingular variety.

159

§14.

Hodge numbers.

167

§15.

Calculating cohomology by counting points.

176

§16.

Examples.

189

References.

208

§1.

Introduction The aim of these notes is to develop a general procedure for computing

the rational cohomology of quotients of group actions in algebraic geometry. The main results were announced in [KiJ. We

shall

nonsingular

consider

linear

actions

complex projective

of

complex

varieties.

reductive

groups

on

To any such action there is

associated a projective "quotient" variety defined by Mumford in [ M ] .

This

quotient variety does not coincide with the ordinary topological quotient of the action. For example, consider the action of space n,

Pn

where

Pn

on complex projective

is identified with the space of binary forms of degree

or equivalently of unordered sets of

The orbit where all

SL(2)

n

n points on the projective line

P-j.

points coincide is contained in the closure of every

other orbit and hence the topological quotient cannot possibly be given the structure of a projective variety.

To obtain a quotient which is a variety

such "bad" orbits have to be left out. The quotient variety can be described as follows.

Suppose that

projective variety embedded in some complex projective space G

is a complex reductive group acting on

GL(n+1).

If

A(X)

invariant subring

X

via a homomorphism

denotes the graded coordinate ring of

A(X)G

is a finitely-generated graded ring:

associated projective variety.

The inclusion of

Pn

A(X)G

G-invariant surjective morphism ty from an open subset

in X

X

is a

and that

t :

G •*

X,

then the

let

M

A(X) of

be its

induces a X to

M.

- 2 (The points of subset M1 of

X

are called semistable for the action).

M which is an orbit space for the action of

There is an open G

image under \p, in the sense that each fibre is a single orbit of So we have two "quotients" on

X.

M

and

on its inverse G.

M1 associated to the action of

G

Our main purpose here is to find a procedure for calculating the

cohomology, or at least the Betti numbers, of these in the good cases when they coincide. quotient

This happens precisely when

XSS/G.

stabiliser in

M

is topologically the ordinary

In fact we make the slightly stronger requirement that the

G

of every semistable point of

X

should be finite; this is

equivalent to requiring that every semistable point should be stable.

Under these conditions an explicit formula is obtained for the Betti

numbers of the quotient cohomology of

X

classifying spaces of

M

(see theorem 8.12.)

n.

This formula involves the

and certain linear sections of

X,

together with the

G and certain reductive subgroups of

For example, consider again the action of degree

(properly)

Then good cases occur when

non-zero Betti numbers of the quotient dim H 2 j (M;Q) =

n

SL(2)

G.

on binary forms of

is odd, and one finds that the

M are given by

1 + ^min(j,n-3-j)

for 0 p = i/2ir

with respect to local coordinates

^

dxj Adxj

(x-j,...,x n ) + (1 :x^ : . . . : x n )

these cooordinates the vector field induced by

a

on

Pn

near p.

But in

takes the values

- 26 Also

d((2iri||x1 2 f 1 x # t ax#) = (2irif 1 ^

at p. This corresponds to the vector (a

,...,a

(aojdx. • a]Q dx.)

) under the duality defined

by to at p. Hence 2.3 holds and p is a moment map.

2.6. Remark. An alternative proof runs as follows. It is known that there is * a natural homogeneous symplectic structure on any orbit in u(n+1) of the co-adjoint action of

U(n+1), and that the corresponding moment map is the

inclusion of the orbit in u(n+1)*. (This is true for any compact group K by [Ar] p. 322).

We can identify

u(n+1)#

with

u(n+1)

using the standard

invariant inner product on u(n+1). Then the map from Pn to u(n+1) given by

is a U(n+1)-invariant symplectic isomorphism from skew-hermitian matrix rank 1 with

x#

(2iri)

Pn

to the orbit of the

diag (1,0,...,0). For x* x" #

as an eigenvector with eigenvalue

follows from this because the inner product of x*3T*

|x # |

is hermitian of .

Lemma 2.5

with any a e u(n+1)

- 27 — #t

.

is x

•

ax .

To sum up: variety

XC P

by 2.4 and 2.5, given a nonsingular complex projective and a compact group

K acting on

: K -*- U(n+1), a moment map y: X + k 2 7

X

by a homomorphism

is defined by

y(x) . a = (2Tri||x*||2)"1 7 # t 4>#(a) x*

- -

for each a e k and x e X. This moment map is functorial in X and

2.8.

Consider the example 2.2

sphere R^,

Pt

and

acted on by P4

of configurations of points on the complex

SU(2).

The Lie algebra of

y:

(Pt)

+ su(2)

in such a way n

(up to a scalar factor of

Henceforth we shall assume that a moment map K on

R^

is isomorphic to

sends a configuration of

on the sphere to its centre of gravity in R^

of

SU(2)

can be identified with the unit sphere in

that the moment map

K.

y

points n).

exists for the action

X.

Fix an inner product on the Lie algebra adjoint action of

k

K and denote the product of

product to identify For example if

k with its dual K C U(n+1)

which is invariant under the a and b

by

a.b.

Use this

k*.

we can take the restriction to

k of the

standard inner product given by a.b = -tr(ab) on

u(n+1).

Then 2.6 implies that for each x e X the element

y(x) of

k

- 28 is identified with the orthogonal projection of the skew-hermitian matrix (2 1 ri||x # || 2 )" 1 x # 7

# t

onto

k.

Also choose a K-invariant Riemannian metric on particular if

X

If

X

is Kahler (in

is a projective variety) then the natural choice is the real

part of the Kahler metric on

2.9.

X.

Definition. Let f:

X.

X •+• R be the function given by

f(x) = Hu(x)||2 for

x e X,

where

||

||

is the norm on

k

induced by the fixed inner

product.

We want to consider the function f: For any

x e X

let

{ x t | t >_0}

X •*• R as a Morse function on

be the trajectory of -grad f

x 0 = x, i.e. the path of steepest descent of w(x) = { y e X| every neighbourhood of points x t

f starting from x.

X.

such that

Let

y in X contains

for t arbitrarily large}

be the set of limit points of the trajectory as

t •+ °°.

Then

u)(x)

is closed

and nonempty (since X is compact) and is connected. For suppose there are disjoint open sets y t UW V Xt t Wy such that

U and V

there is some for

t >_ty •

t >J

in X such that

t v >^0

But

w(x)C UV/V.

and a neighbourhood

X - (UVy V)

implies x t e U V V .

Wv

Then for every of

y

such that

is compact so there is some

Since the set

{x t 11 >^T}

T > 0

is connected

- 29 it is contained either in

U

or in

either

U or V .

We conclude that

2.10.

For every

x e X

of

o)(x) is critical for

If

f

[A & B] §1)

V,

the limit set

and thus

u)(x)

is connected.

then the set of critical points for

implies that for every

is contained in

C.

f

on

{C e C}

x e X

S~

Definition.

stratification 1 of

X

S

corresponding to any u)(x)

contained in C.

A finite collection X

if

X

3 e B.

{S

3

| 3 e B}

of subsets of

is the disjoint union of the strata

(For

C

and

in the following sense.

> on the indexing set

sfi c U every

Given such a

there is a unique C e C such

The Morse stratum

and there is a strict partial order

for

would be a finite

retract onto the corresponding critical submanifolds

form a smooth stratification of

2.11.

X

of X.

C e C is then defined to consist of those x e X with The strata

Also every point

were a nondegenerate Morse function in the sense of Bott (see

function, 2.10 u)(x)

is also contained in

f.

disjoint union of connected submanifolds

that

w(x)

the

Morse

{S

X

p

forma

|3 e B},

B such that

s

stratification

nondegenerate Morse function the partial order is given by

associated

to

a

- 30 C> C where for C e C, f(C)

if

f(C)

> f(C')

is the value taken by f on C).

The stratification is smooth if every stratum

So

is a locally-closed

P

submanifold of

X (possibly disconnected).

In fact the set of critical points for the function singularities in general so that in the sense of Bott.

f

has

cannot be a nondegenerate Morse function

Nevertheless we shall see that the critical set of

a finite disjoint union of closed subsets takes a constant value. there is a unique

f = Jp|

{C3 | 6 e B}

on each of which

Because of 2.10 it follows that for every

3 e B such that

f [s_

co(x) is contained in C o .

f

x e X

So X is the

P

disjoint union of subsets set

co(x)

{S3 | 3 e B}

where

of its path of steepest descent for

x e X lies in S f

is contained in C

shall find that for a suitable Riemannian metric the subsets form a smooth K-invariant stratification of

2.12.

Example.

action of

SU(2)

sphere in R

n points in

|| ||

P

\i

.

We

{S | 3 e B} 3

associated to the

identified with the unit

is given by

( x r . . . / x n ) + ||x1 + x 2 + . . . + x j where

X.

The norm-square of the moment map on sequences of

if the limit

is the usual norm on

takes its minimum value on centre of gravity at the origin.

y

(0)

, R

.

As is always the case ||y||

which consists of all sequences with

Note that if n is even \x (0) is singular near

- 31 configurations containing two sets of n/2 coincident points. One can check that the critical configurations not contained in some number

r > n/2

and the other

n-r

of the

n

u

(0)

are those in which

points coincide somewhere on the sphere

coincide at the antipodal point.

The connected

components of the set of non-minimal critical points are thus submanifolds and are indexed by subsets of The union of the Morse strata

{1,...,n}

of cardinality greater than

corresponding to subsets of fixed cardinality

consists of all sequences such that precisely somewhere on P

can build up the cohomology of

3 e B

r

X

{S | M 3

B}

of the manifold

X

one

This is done by using the Thom-Gysin sequences which for each relate the cohomology groups of the stratum So

and of the two open

P

^J Y . x x p —

On the other hand suppose

k e K is such that

steepest descent from kx for the function u

3

by

definition

u

p (kx) = ii(kx).3 P

takes

P

the value

kx e Y . . Then the path of P

has a limit point in 1

||fl|f

on

Z p

.

p

, and

Thus as

2 we have p(kx),3 > ||3|| • But

||u(kx)||2 = |,i(kx)| 2 = | | e | | 2 . Together these imply that follows that

ii(kx) = 3, and since y(kx) = Adku(x) = Adk3

k e Stab3.

Now suppose a e k is such that

a x

e T Y. For t e R we have x

M((exp ta)x) = 3 + tdu(x)(a x ) + e(t) where e ( t ) = O ( t 2 ) as t ^ 0; and du(x)(a x ) = [a,u(x)J = [3,3] because y is K-equivariant.

As [ a , 3 ] . 3 = a . [ 3 , 3 ] = 0, it follows that

u ( ( e x p t a ) x ) = ||3|| 2 + 3 . e ( t ) . P

But also ||y(exp ta)x||

2

= ||ji(x)||

2

2 = ||3|| for all t e R , so that

| | 3 | | 2 = ||3 + t [ 3 / 3 ] + e ( t ) | | 2

it

- 50 O(t ) as t > 0.

Thus 23.e(t) = -t2||[a,3]||

+

O(t3)

as t -> 0

and hence y 3 ( ( e x p t a ) x ) = | | 3 | | 2 - - | ^ U t a ^ l l ^ + OCt3) But by assumption

a

e T Y

eigenspaces of the Hessian

as t - 0 .

which is the sum of the nonnegative

H (u ) x p

of

nondegenerate in the sense of Bott.

u

at

3

x

because

y

3

is

The last equation shows that this is

impossible unless [a,3] = 0, i.e. unless a e stab3. This completes the proof.

4 . 1 1 . Corollary.

The subset

KYD p

of —

X

is a smooth submanifold when

restricted to some K-invariant neighbourhood of

]i

C . = K ( Z . f\ p

p

(3))

in

—

X.

Proof. by

Since

Y

3

a(k,x) = kx

is invariant under induces a map a:

Stab3, the map a: K x

P

neighbourhood of

{y e Y o | v i . ( x ) < l|3|f p 3 — in Y

Z 3

in the subset

that if

-• X given

+ e}

of

e > 0 is sufficiently YD 3

is a compact

. Moreover its complement in Y D 3

is contained

3

{y e X | |y(y)|| >_||3|| + ||3||

K-invariant and does not meiet

3

, Y o •* X whose image is KY. . btaDp p p

It is easily checked from the definition of Yo small the subset

K x Y

Z o C\ P

\i

e} (3).

of

X, which is closed, From this one can deduce

- 51 easily that if point in

x e Z

K x

the image

,

KY

Y of

P

K

k x T Y . x

c «.

X

UQ Y o

Dtaop

\i

(3)

then

maps each neighbourhood of the

(1,x)

onto a neighbourhood of

a

at any point of the form (1,x) sends a vector (a,£) a

+ £ e T X.

The tangent space of

XX

at a point represented by (1,x) is the quotient of

p

x in

a.

to the tangent vector P

a

represented by

The derivative of in

f\

P

the subspace consisting of all

(a,£)

such that

a e Stab

It x T Y o

by

x p

and

£ = -a

P

Thus 4.11 shows that the derivative of x e 1

point represented by (1,x) with neighbourhood image

5(V)

V

of

of this point. V

under

a

a:

P

K x r\

v

. (3),

Y

-• X is injective at a

and hence also in some

x in KY

it follows from the inverse function theorem that the image

We have thus shown that that

KYO

P

.

The preceding paragraph shows that the

is a neighbourhood of

smooth in some neighbourhood of

x

.

P

Therefore

KYO

of

P

a

is

x.

KYO

P

is smooth in some

is smooth near 1OC\ P

K-invariant

p

(3).

It follows

neighbourhood of

Co

P

=

K(Z Q f\ ]x" ( 3 ) ) , as required. P

We are aiming to show that the intersection sufficiently small neighbourhood of Co . M

Co

P

lo

p

KYrt

with a

P

is a minimismg manifold for f

The last corollary shows that the condition that

closed submanifold of

of

X can be satisfied.

Z

3

along

be a locally-

For the other conditions we need

two technical lemmas.

4.12.

Lemma. ————

Z

p

is an almost-complex submanifold of

X.

Moreover

- 52 T Yo x p

is a complex subspace of

Proof,

Suppose

x e Zo . P

acts on the tangent space

T X x

for every

x e Z

.

o

P

Then t h e compact torus

To P

g e n e r a t e d by

3

T X, which decomposes into the sum

V n @ V ^ © . . . @V 0 1 p of complex subspaces where

\/

is fixed by T o P

0 Z

while for each

P

j > 1 —

nontrivial c h a r a c t e r . scalar iX. w i t h have u

P

A

Thus

= 0

g r a d y o ( y ) = i3 at

T Z

x

3 acts on

= V

and

T

and

acts on

p 3

V. j

as scalar multiplication by some

acts on each X.

V.

as m u l t i p l i c a t i o n by some

real and nonzero f o r j > J . Also by 4 . 3 w e

f o r all

y e Y.

T h e r e f o r e t h e Hessian

y V . as m u l t i p l i c a t i o n by J

T Yg

and is the tangent space to

of

x p ( c f . [ A 2 ] lemma 2 . 2 ) . Thus

X. J

is t h e sum of those

H (u )

V.

such t h a t

X. >^0, so both

a r e complex subspaces of T X X . The result f o l l o w s .

4.13.

Lemma.

Suppose

x e C

t h e symplectic f o r m u to -——-——-----~—-—----—---—-—— x —

Moreover

since

x e ZoHu"1(3)/

o>

is

and then

p

T (KY O ) X

p

KYO P

X).

x

is orthogonal to

Suppose that

H

M~ ( 3 ) ) . Then t h e restriction of

under

T (KYJ

denotes the metric then x p

0 = = < £,£ >

by 3.19, by the assumption on a.

Hence g = 0. But then as k

CT(KYJ x— x 3 0 = u> (a ,b ) = dy(x)(a ).b X

X

X

X

for every b e k (see 2.3), so

because ji(x) = 3. Thus a e stab3 and hence a e T Y. by 4.10. But by x x p assumption a is orthogonal to T Yo so a = 0 . X

X

p

X

This completes the proof.

4.14.

Remark. Lemma 4.13 implies that there is an open neighbourhood Zo P

of the critical subset

Co

P

in

KYO

P

such that the restriction of the

symplectic form co to the tangent bundle TZ that

co

P

is nondegenerate. It follows

and the metric together induce a K-invariant almost-complex (cf. 4.1). It also follows that the normal bundle I

structure on I P

can be identified with the co-orthogonal complement restriction of

TX

to

Z

3

.

Since

TZ

l

P

OJ is nondegenerate on

complex structure to this normal bundle as well.

to

TZ l

TZ P

P

p

in X in the

it gives a

- 54 At last we are in a position to prove

4.15,

Proposition.

in KY

Proof.

There is a K-invariant open neighbourhood

Z

of

C

along C ft .

which is a minimising manifold for f

It follows from 4.11 that there is a K-invariant neighbourhood

Z

of P

Co 3

in KYO 3

which is smooth.

To show that

Z

satisfies the definition of a minimising manifold, we

3

must check that the restriction of C Q* 3

But if

x e Y 1

to a point of

f

, and by definition

the value taken by

f

on

Z

x e KY

x e C

of

f

U

P

at

T Z x 3

and as the metric, points x e 1

P

C\ y

in Z

3

T X x and

(3)).

and hence for

3

x

x e C

3

13II

Since

x e Z

converges on Z

.

3

.

, which is f

is

K-invariant

Moreover equality

then the restriction to T Z . of x 3

H (f)

is negative definite. f

>J|3||

is positive semi-definite.

show that the restriction of the Hessian subspace to

u 3

3

C o = K(Z_ O

. 3 It follows immediately that if H (f)

takes the value

and so f(x) = ||y(x)||

P

the same is true for

the Hessian

takes its minimum value on

3

y(x).3 = y (x) >_||3||

holds if and only if

Z

then its path of steepest descent under

3

p

Hence

to

So it remains to

to some complementary As C o = K f Z . f l p " 3 3

(3))

are all K-invariant it is enough to consider

(3).

By lemma 4.13 the restriction of the symplectic form o> at x to T Z x 3 is nondegenerate. Therefore the orthogonal complement T Z L to T Z in x 3 x 3

- 55 T X with respect to m

is a complementary subspace to T I

£ e T X and a e k then

By the definition of a moment map, if

So if

£

then

is w-orthogonal to the image

dy(x)(£).a = 0

invariant under

for a"

K we have

a e k k

x

.

k

in

T X

of the Lie algebra

and so dy(x)(£) = 0 .

G T I , so a fortiori if x 3

Since

U T I > x 3

I

k is

3

then

dy(x)U)=0. Let Then if

Exp:

TX -• X

U T X

be the exponential map associated to the metric.

and t e R, y(Exp t £ ) = 3 + tdy(x)(£) + e (t)

where e (t) = O(t ) as t -• 0 . Therefore if

5 e T IQL

s>

then

x 3 y(Exp t £ ) = 3 + e ( t ) ,

so that 2 2 3 f(Exp t £ ) = ||3 + e (t)|| = ||B|| + 2 3 . e (t) + O(t ) as t -• 0 . On the other hand y (Exp t 5 ) = y(Exp t £ ) . B = | 3 | | 2 + 3.e ( t ) . P s. It follows that the Hessians factor of KYO , 3

H (f) x

and H ( y o ) x 3

2 on the subspace T I L of T X. But 2 is an open subset of x 3 x 3

so

T

x

L1- C T Y.1. 3 ~~ x 3

stratum of the function

yn 3

Moreover by definition

negative definite.

Yo 3

is the Morse

associated to the critical submanifold

which implies that the restriction of the Hessian

definite.

agree up to a scalar

Thus the restriction of H (f) x

H (y ) x 3 to T IOL x p

Zo , p

T Y L is x 3 is also negative to

- 56 -

Therefore

Z

is a minimising manifold for f along C

We have thus shown that

the function

degenerate along each critical subset C o . P

f = |^i||

is minimally

By theorem 10.2 of the appendix

this implies the existence of Morse inequalities for Morse inequalities.

as required.

f,

and also of equivariant

Indeed, by theorem 10.4 and lemma 10.5 we have the

following result.

4.16.

Theorem.

Let

compact Lie group action.

3

I 3eB},

stratification SQ

be a compact symplectic manifold acted on by a

K, and suppose

y:

X •* k*

Fix an invariant inner product on k.

for the function {C

X

f = ||y||

on each of which {S

|3eB}

of

is a moment map for this

Then the set of critical points

is a finite disjoint union of closed subsets f

takes a constant value. There is a smooth •

X such that a point

:

x e X lies in the stratum

if and only if the limit set of the path of steepest descent for f = ||y||

from

x

(with respect to a suitable K-invariant metric) is contained in C

•

v

For each

3 e B

the inclusion of

C

m

S

is an equivalence of (Cech)

cohomology and also K-equiva riant cohomology.

Theorem 10.4 shows in addition that

4.17.

if

B e B

then the stratum

S

coincides in a neighbourhood of

C

- 57 with the minimising manifold

t

(which is an open subset of KY. P

Yo

is defined as at 4 . 6 ) . In particular if

P

where

P

T S

x 6 2

x e Z A y P

( 3 ) / then

TXZB .

From this together with remark 4.14 we deduce that

4.18.

both the tangent bundle and the normal bundle to each stratum Sn P

have K-invariant complex structures in some neighbourhood of the critical set

Theorem 4.16 implies immediately the existence of equivariant Morse inequalities for the function f = |p|

. We shall not state these explicitly

until the next section, where it will be shown that they are in fact equalities.

We shall conclude this section with some remarks about the codimensions of the components of the strata

So p

and the equivariant cohomology of the

critical sets Q . P

Recall that when stating the Morse inequalities induced by a smooth stratification of

X

in

§2, we made the simplifying assumption that every

stratum was connected and hence had a well-defined codimension in X. fact the stratification disconnected strata.

{S I 3 e B} 3

In

defined in theorem 4.16 may contain

Therefore it is necessary to refine it so that the

components of any stratum all have the same codimension.

- 58 For M B ,

where

Z. 3

the critical subset C

is the union of certain components of the critical set of the

nondegenerate Morse function H (u ) at any critical point x

was defined at 3.14 by

3

p

t h e t a n g e n t space

T X x

u

P

x for

.

Recall that the index of the Hessian y

is the dimension of any subspace of

p

to which the

r e s t r i c t i o n of

definite and which is maximal with this property. codimension of a maximal subspace of semi-definite.

Since

Bott the index of of

y . P

y

P

T X x

H (yo) x p

is n e g a t i v e

This is the same as the

on which

H (y ) x

p

is positive

is a nondegenerate Morse function in the sense of

H (y ) is constant along any component of the critical set x 3

Its value is called the index of

y

P

along this component.

So we

can make the following definition.

4.19.

Definition. For any integer m > 0, let — —

connected components of

Then each

Z

p,m

Zo g

Z

p,m

be the union of those

along which the index of

is a symplectic submanifold of

disjoint union of the closed subsets

X,

yrt g

and

is m. __

C

Let —

is the

P

{C

p/m

| 0 _||3|j > 0. o

= S _ 0,T

x does not lie in the minimum stratum

for the torus then there exists some nonzero 3 e B such that

xeSOTCTYfl.

||y||

T

Since 6.14 holds trivially for tori, proposition 6.18 is valid for the

Proof.

x!T

X

Thus

is the same whether the group is K or T;

by corollary

6 . 1 1 if

yeGx

then

Since the path of steepest descent for the function •

from x is contained in Gx, we deduce that x cannot lie in X

The proof of theorem 6.18 for any group, torus or not, is now complete.

6.20. for

Remark. f

By theorem 4.16 the inclusion of the minimum set

in the minimum stratum

cohomology.

X

\i

(0)

is an equivalence of equivariant

So 5.10 and 5.16 may be interpreted as formulae for the

equivariant Poincare series

F^X™1").

These formulae can also be derived

directly from theorem 6.18. If X

/G

G

acts freely on the open subset

Xmm

of

X then the quotient

is a complex manifold, and it would be natural to hope that the

- 93 rational cohomology of this is isomorphic to

H*(X

proved by showing that the quotient map

•* X

X

;Q). /G

This could be

is a locally trivial

fibration.

However this is unnecessary because in the next section we shall

see that

Xmm/G

is homeomorphic to the symplectic quotient

y

(0)/K.

This reduces the problem to the action of a compact group.

Let us conclude this section by considering how the stratification is affected if we alter the choice of moment map or of the invariant inner product on

k.

(From the algebraic point of view, changing the moment map

on a complex projective variety embedding of

X

corresponds to changing the projective

X).

First consider the inner product.

Clearly if the group is a torus then any

inner product is invariant, and different choices give different stratifications. For an example take (C # )

acting on

P

via the map

: (C*)

+ GL(2)

given by

0 where

a.:

(C*)

•*• C*

o^h)

is the projection onto the (i+1)th factor. Then the

stratum to which an element

( x n : X -i)

closest point to 0 in the convex hull of

e

P k

K preserves the Kahler form co on

GO by its average over

exists for the symplectic action of

Any torus in

G

K), and that a moment map

K on

X.

will always have fixed points in

to give the topological quotient

X/G

X

so we cannot hope

the structure of a Kahler manifold.

However in good cases there is a compact Kahler manifold which it is natural to regard as the "Kahler quotient"

of the action of

complex projective variety on which

G

G on X.

When X is a

acts linearly, this quotient coincides

with the projective quotient defined by Mumford using geometric invariant theory.

The good cases occur when the stabiliser in K of every

is finite.

x e \i

(0)

Recall that this is the condition needed for there to be a symplectic

quotient associated to the action. As before let X

m

be the subset of

X consisting of points whose paths

of steepest descent under the function p

(0).

when M

By 6.18

Xmm

\i

can be identified with

X

/G

The symplectic form induced on

p

(0)/K

V

(0)/K

have limit points in

is a G-invariant open subset of

K acts with finite stabilisers on

(0)/K

f = |p||

(0)

X.

We shall see that

then the symplectic quotient

and thus has a complex structure. is then holomorphic and makes

into a compact K3hler manifold except for the singularities caused

by finite isotropy groups.

(Manifolds with such singularities have been well

- 96 studied; they are sometimes called V-manifolds). This is the natural Kahler quotient of X by G. The rational cohomology of this quotient can be calculated by using 5.10 or 5.17.

Recall from 5.5 that the condition that ]i

(0)

implies that

y

(0)

K acts with finite stabilisers on

is smooth. The inclusion of

y

(0) in X

induces a natural continuous map u"1(0)/K-Xmin/G. In order to show that this map is a homeomorphism we need some lemmas. The first is

7.1. Lemma. G = K expik. Proof.

The left coset space G/K

is a complete Riemannian manifold (see

[He]), so that the associated exponential map the tangent space at the coset

K onto

G/K.

Exp: T (G/K) •*• G/K maps K Moreover

and by [He] p. 169(4) we have Exp(a+k) = (expa)K for any a e g. Since g = k + ik the result follows. Next we need

7.2. Lemma. If x e y~1(0) then Gx Hy" 1 (0) = Kx.

T (G/K) = g/k

- 97 Proof.

Suppose

there exists

g e G

k e K

is such that

such that

gx e y

gx = kx.

(0).

Since

y

We wish to show that (0)

is

K-invariant, by

7.1 it suffices to consider the case g = expia where a e k. Let at

0

h: R •*• R

be defined by

h(t) = y ((expiat)x).a.

and 1 because x and (expia)x

is some t e (0,1)

Then h vanishes

both lie in y~ ( 0 ) . Therefore there

such that

0 = h'(t) = dy(y)(ia ).a = u> (ia ,a ) = < a ,a > y y y y y y where

y = (expiat)x

structure. then

Thus

a

and =0,

< , >

denotes the metric induced by the Ka"hler

so that

expiaR

fixes

y

and hence also x.

But

(expia)x = x e Kx, so the proof is complete.

It is necessary to strengthen this result.

7.3.

Lemma.

Suppose

x

and y

lie in

exist disjoint G-invariant neighbourhoods of

Proof.

Since

K

neighbourhood

V

is compact and of

x

in

y

by 7 . 1 , it suffices to show that

x i Ky

(0)

y

(0)

x and y in

Then there

X.

there is a compact K-invariant

not containing y.

(expik)V

x £ Ky.

and

Since G = (expik)K

is a neighbourhood of

x

in

X

and that y £ (expik)V. To see that a: k x y

(0) + X

(expik)V

is a neighbourhood of

which sends

(a,w)

x

in X consider the map

to (expia)w.

This is a smooth map of

smooth manifolds, so it is enough to show that its derivative at (1,x) is surjective.

If

not, there

exists some nonzero

£ e T X

such that

- 98 < £,£ > = 0 for all

for all a c k ,

c in the image of

da(1,x).

In particular

But then

for all a e k.

Thus £ e ker dp(x) = T (p

and hence

£ lies in the image of

Therefore

da(1,x), which is a contradiction.

if W = exp {ia | a e k, ||a|| < J } V

then W is a compact neighbourhood of e = inf{< a

w

If

w e W

then

w

x in X.

Let

,a > I w e W, a e k, llall = 1} . w

'

' ii ii

lies in the G-orbit of some

z e p

easily from the proof of 7.2 that the stabiliser of Therefore a

#0

w

Now suppose the function 7.2, if

t e R

h'(t) >^0 Since

h:

whenever 0 / a e

then

given by

and

We deduce that if ||a|| >J\. p

(0).

Hence as V Since

>

h'(t) >_e

where

then

is compact

yep

(0)

is finite.

||a|| = 1 . Consider As in the proof of Therefore

by the choice of h(t) >_e

e.

when t > J .

when t >_1.

|p( expiaz)|| >^e

(expik)V

and

G

w = (expita)z.

when t e [0,1]

||p(expita)z|| >_e

z e V

in

h(t) = p((expita)z).a.

the mean value theorem implies that

As ||a|| = 1 it follows that

and it follows

k, and so e > 0 . '

h'(t) = < a ,a

for all t e R,

h(0) = 0

w

z lies in V and a e k is such that R -• R

(0)

whenever

a e k and

is closed in a neighbourhood of

y ft (expik)V by 7.2,

it follows that

- 99 y t (expik)V.

Now we can prove the result we were aiming for.

7.4.

Theorem.

Let

X

be a Kahler manifold acted on by a group G

which

is the complexification of a maximal compact subgroup K that preserves the Kahler structure on this action of

X.

u: X + k

Suppose that a moment map

exists for x e \i

K and suppose that the stabiliser in K of every

is finite. Then X m l n = Gy~ 1 (0)

(0)

and the natural map y~ 1 (0)/K + X m i n / G is_

a homeomorphism.

Proof. M

Gy

(0).

(0)CX

because

Conversely if

x e X

X

is G-invariant by 6.18 and contains

then there is some

the closure of the path of steepest descent for path is contained in the orbit either

y e Gx

in

is finite, and this implies that

K

or

Gx, so that

We conclude that

surjective. /G

y e Gx, so that

X m m = Gu"1(0),

Thus

X

dim Gy < dim Gx.

||y||

y e Gx.

yep from x.

(0)

lying in

By 6.7 this

Then G y C G x ,

so that

But by assumption the stabiliser of dim Gy = dim G >^dim Gx

xeCp

y

(see 7.2).

(0).

so the natural map

y~ 1 (0)/K •* X m m / G

is

Lemma 7.2 implies that it is injective, while lemma 7.3 shows that

is a Hausdorff space.

Thus the map is a continuous bijection from a

compact space to a Hausdorff space, and therefore it is a homeomorphism.

- 100 It follows easily from the proof of 7.2 that if then

G

Xmm

acts freely on the open subset

structure on

X

X

(0)/K.

/G = y

K acts freely on y

of

X,

so that the complex

induces a complex structure on the topoiogical quotient The symplectic form on

y

(0)/K

holomorphic with respect to this complex structure because on

X,

and indeed is a Kahler form because

quotient

X

/G = y

(0)/K

(0)/K

u>

induced by

y

(0)

u)

is

u> is holomorphic

is Kahler.

is a compact Kahler manifold.

when the stabiliser of every point in = y

(0)

Hence the

More generally

is finite the quotient

X

/G

can be thought of as a Kahler manifold with singularities caused

by the finite isotropy groups.

7.5.

Remark.

The proof of lemma 7.2 is independent of the assumption that

the stabiliser of every point in

y

(0)

is finite, and it is also possible to

prove lemma 7.3 without using this assumption. a e k

One uses the fact that if

then the function

Morse function on

X.

y defined by y (x) = y(x).a is a nondegenerate a a This implies that given any point y e p (0) and any a

neighbourhood U of y in X, there is a smaller neighborhood V e > 0 grad y

such that the intersection with a

which passes through a point of

of this when

y

is not critical for

y

a

y [-e,e] a V

the facts that

is contained in U .

G = C*,

y

and

of any trajectory of (The proof

is easy: see the proof of 7 . 3 ) .

this the argument of 7.3 gives the result when also follows without difficulty.

of

Using

and the torus case

The general case can then be deduced from

G = KT K and that

K is compact.

- 101 From this it follows without the assumption of finite stabilisers that any x e X lies in G|T (0) closed in X m m ;

if and only if

x lies in X m m

and also that the natural map \i

and its orbit Gx is

(0)/K + Gy

(0)/G

is a

homeomorphism. In particular when X is a projective variety on which G acts linearly one finds that

y

(0)/K

is naturally homeomorphic to the

geometric invariant theory quotient of X by G.

- 102 -

§8. The relationship with geometric invariant theory From now on we shall assume that our Kahler manifold nonsingular complex projective variety and that complex group acting linearly on

X

M

where

A(X)

general

M

(see [ M ] ) .

In fact

M

is in fact a

is a connected reductive

as in example 2 . 1 .

invariant theory associates to the action of variety

G

X

Then geometric

G on X a projective "quotient"

is the projective variety

Proj A(X)

is the invariant subring of the coordinate ring of has bad singularities even though

X

is nonsingular.

X.

In

However in

good cases M coincides with the quotient in the usual sense of an open subset X

of

X

implies that

by

G

and the stabiliser in G of every

x e X

is finite. This

M behaves like a manifold for rational cohomology.

It turns out that the geometric invariant theory quotient with the symplectic quotient

y

(0)/K,

M

coincides

and that the good cases occur

precisely when the stabiliser in K of every

x e \i

(0)

is finite.

So the

work of the preceding sections can be used to obtain formulae for the Betti numbers of

M

in these cases.

The formulae involve the cohomology of

and various subvarieties, together with that of the classifying space of

X G

and certain reductive subgroups.

8.1.

Remark.

acts on

X

generality.

The example of

PGL(n+1)

via a homomorphism

shows that the assumption that

: G + GL(n+1)

However the finite cover SL(n+1)

of

G

involves some loss of

PGL(n+1)

has the same

- 103 Lie algebra, moment map and orbits on

X

as

PGL(n+1).

Moreover if

G is

a connected reductive linear algebraic group acting algebraically on a smooth projective variety

Xcp

: G •+ PGL(n+1),

provided we assume that

hyperplane.

then the action is given by a homomorphism

G on the Picard variety

enough to show that every Borel subgroup theorem 10.4

B

Applying this with bundle on

Pic(X)

that the image of Thus as

B

corollary 1.6).

X

replaced by

fixed by B. B

Pic(X)

of

B of

G

Pic(X)

For it is

acts trivially.

But by

By the theorem of [G & H] p. 326 it follows

in the group of automorphisms of

Now let

action of any

X

given by an element of

Then

g*L = L

PGL(n+1).

inverse image of this in

for all

X£P

, which has

g e G,

so that the

So we

get

a well-defined

by its image in

SL(n+1)

L

and hence is

This element is unique because X is not

which induces the action of G

is discrete.

(Alternatively see [M]

is covered by an automorphism of

in a hyperplane.

We may now replace

Pic(X)

L be the hyperplane bundle on

GL(n+1).

: G -• PGL(n+1)

X is trivial.

we see that there is an ample

is connected it must act trivially.

g on

First we note that the

has a fixed point on each component of Pic(X).

automorphism group

contained

is not contained in any

The argument for this runs as follows.

induced action of

[B]

X

homomorphism

G on X. PGL(n+1)

and then by the

to obtain a linear action on

essentially the same properties as the original action.

X

with

- 104 The inclusion of morphism

\j>: XSS -• M

We shall see that Xmm

A(X)

in

A(X)

induces a surjective XSS

from an open subset

X

of

X

on

f =||u|f

X.

i|> contains more than one orbit of

However there is an open subset

M

Xs

X

is a single G-orbit (see

is an open subset

M.

Therefore §5 and §6

give us formulae for the equivariant Betti numbers of

meets

to the quotient

always coincides with the minimum Morse stratum

associated to the function

that a fibre of

G-invariant

M1 of

8 . 2 . Definitions (see

of

X

G,

X

•

It may happen

so that

M 5* X

/G.

such that every fibre which

[M] theorem 1.10).

The image of

Xs

in

M and M1 = X / G .

[M] definitions 1.7 and 1.8, noting that Mumford calls

stable points "properly stable": this seems to be no longer the accepted terminology).

A point

constant polynomial action of

G

on

is semistable if there is a homogeneous non-

F e C[X Q ,...,X ]

C[X

there is an invariant

x e X

U

...,X ] n

F

with

which is invariant under the natural

and is such that F(x) 5* 0

F(x)^0.

x

such that all orbits of

is stable G

if

in the

affine set X p = {y e X | F ( y ) * 0 } are closed in X_ and in addition the stabiliser of X

is the set of semistable points of

x in G X

is finite.

X

and

is the set of stable

G

of every semistable point in

points.

8.3. Remark.

Suppose that the stabiliser in

- 105 X is finite. G-invariant

Then if

x eX

polynomial

X = {y e X | F(y) 5* 0}

there exists some homogeneous non-constant

F

such that

F(x) 5* 0.

is semistable, so every

Every point in

G-orbit in X

dimension as G. This implies that every orbit is closed in X

has the same and thus that

x is stable. Hence X = X .

We shall use the following facts which follow from [M] theorem 2.1 and proposition 2.2.

8.4.

A point x e X is semistable for the action of G on X if and only if

it is semistable for the action of every 1-PS (one-parameter subgroup) X: C* > G of G on X. 8.5.

If

X: C*^GL(n+1)

is given by z - diag(z r °,...,z r n)

with r ,...,r

e Z, then a point

x = (x :...:x ) e P

is semistable for the

action of C* via X if and only if min{r. | x. ?* 0} #(k)

Let

aek

is the subspace of

u(n+1)

be a basis element of norm 1 .

x = (x : . . . : x ) e X then

r |x | 2 ) ( ^

M(x) = ( ^ 0.||3|| 2 } ^ ( Y e k| C.Y >^6}.

U

y

K.

P

p

t.

for

S_ = G Y m m and this is the same as P P p is invariant under the parabolic subgroup PD and

x e So.

G = KP . Therefore Now

does not lie in

Xmm Q Xmm

by

my

is a lattice point of

compatible with T.

for a suitable integer t

m > 0

we may assume that

and hence corresponds to a complex

Since kx e Y

, by 6.11 we have P

1-PS

of T

- 109 £ U In particular

e k|

y ( y ( C )kx), which is the projection along y of

does not contain with

e M(X)|C.B >_||3|| 2 }C U

0.

Let

K such that

X = Ad(k)y-

0 £u

(X(C*)x)

Then

X is a 1-PS of

and hence

x

™

n

cxmm,

Any 1-PS

G compatible

x i X1™" .

A

Q

U ( Y ( C )kx),

Therefore

A

and the proof is complete.

has a conjugate Ad(g)X = g X g " 1 : C * + G which

X: C * -• G

is compatible with

K.

Therefore from 8.4, 8.6, 8.8 and the fact that

X

is G-invariant we can deduce the following

8.10. let

Theorem. Let X C P

G

be

a

complex

homomorphism subgroup points of function and ||

K

such that

2

reductive

: G •*• GL(n+1).

X |y|

be a nonsingular complex projective variety and algebraic

Suppose that

(K) c U(n+1).

on X, where y: X •*• k

Suppose now that the stabiliser in G Then by remark 8.3 we have \|K X

•*• M

each fibre which meets theorem 1.10).

G

acting

from Xs

Therefore

X = X X

on X

via

a

has a maximal compact

Then the set

XSS

coincides with the minimum Morse stratum

|| is the norm associated to any

morphism

group

of semistable X

of the

is the moment map defined at 2.7 K-invariant inner product on

k.

of every semistable point is finite. .

But we know that there is a

to the projective quotient

is a single orbit under the action of

M such that G (see [M]

J \> induces a continuous bijection $: X / G + M.

- 110 We saw in §7 that

X /G

is a compact Hausdorff space, and so is the

project!ve variety M. Hence $ is a homeomorphism. Thus we obtain formulae for the rational cohomology of the quotient variety

M.

Before stating these formulae in a theorem, let us review the

definitions of the terms involved and interpret them in the case of a linear reductive action on a projective variety.

First recall from 3.5 that the moment map

vu. for the action of the

compact maximal torus T on X is given by

•,. . j where

a .,...,a

c_ J

l*,|2«,

are the weights of the action.

Choose an inner product which is invariant under the Weyl group action on the Lie algebra

t

of

T

and use it to identify

t*

with

minimal combination of weights is by definition the closest point to convex hull of some nonempty subset of

{a , . . . , a } .

t.

Then a 0

of the

The indexing set

B

consists of all minimal weight combinations lying in the positive Weyl chamber

Note that if we assume the inner product to be rational (i.e. to take rational values on lattice points) then each 3 e B is a rational point of Thus each subgroup expRfl of T is closed and hence the subtorus T

of T P

generated by

3 is one-dimensional.

t.

-111 We saw in 3.11 that for each

3 e B the submanifoid Z.

P

of X is the

intersection of X with the linear subspace {xePJx. =0 of

P . n

Recall that

Z.

p

2 unless a..B = ||B|1 }

was defined as the set of points in Z_

paths of steepest descent for the function points in Z o f \ y

(3).

P

whose

P

2 |y-B|

on

Z

have limit

Let StabB be the stabiliser of 3 under the adjoint

action of G and let

Stab.,3 be its intersection with K. By 4.9 y - 3 is K a moment map for the action of Stab 3 on Z . K

8.11.

p

In order to interpret the inductive formula of 5.10 we want to define

a subset

Z

of

P

coincide with Z

P

Z.

somehow in terms of semistability so that

P

1

P

will

. There are at least two alternative ways to do this. One

way is to let G

be the complexification of the connected closed subgroup 3 of Stab,.3 whose Lie algebra is the orthogonal complement to 3, and to let Z*S be the set of points of ZQ which are semistable for the linear action of P

G

P

on Z

=Z by theorem 3 3 8.10 because the projection onto the Lie algebra of KAG O of y restricted P

defined by the homomorphism (j>. Then Z

P

P

to 1

P

is y - 3. Another way is to note that since 3 is a rational point of

the centre of stabB there is a character x- StabB + C a positive integer multiple rB of semistable points of

Zo

P

whose derivative is

3. One can define ZSS to be the set of

under the action of

P

StabB, where the action is

linearised with respect to the rth tensor power of the hyperplane bundle by the product of

with the inverse of the character x» The corresponding

- 112 moment map Zo

P

is then

r\i - r3

so that again

Z o = Z_ P

P

.

However the

details are unimportant.

can be reinterpreted as the union of those components of 1

Z

which are contained in components of 3-sequences 3 = ( 3 . . , . . . , 3 ) — I q Z

of

Yo

P

of real codimension

m.

Finally Zo p

and the corresponding linear sections

and

X and subgroups Stabj^ can be defined as in §5.

The theorem for which we have been aiming all along can now be stated.

8.12.

Theorem.

linearly

Let

X£P

be a complex projective variety acted on

by a connected complex

equi variant Poincare series for

X

reductive

algebraic

group

G.

The

is given by the inductive formula

f

t

e;

3,m where the sum is over nonzero M B and integers 0

if and only if the closures in

But by remark 7.8

x e X

= X

steepest descent for the function U

(0),

X SS ,

is closed in

||p|f

Gy

(0)

so the map

It follows that

h

of

Gx

consists of those y

(0)/K

+

M

is

then the closure of the path of from

x

contains a point of

and by 6.7 this path is contained in the orbit

surjective.

h:

X

Gx.

Thus

h

is

is a bijection from a compact space to a

Hausdorff space, and hence is a home omorp his m.

- 115 -

§9.

Some remarks on non-compact manifolds So far

we have considered, only compact symplectic manifolds

projective varieties. a compact group

and

Now suppose X is any symplectic manifold acted on by

K such that a moment map

u: X •* k

exists.

Then one

can obtain almost the same results as for compact manifolds subject only to the condition that

9.1.

for some metric on

function f = ||y||

X,

every path of steepest descent under the

is contained in some compact subset of

X.

One simply checks that all the arguments used in §§3,4,5 appendix are still valid with trivial modifications. is theorem 5 . 8 . total space

X

and the

The only result which fails

This says that the rational equi variant cohomology of the is the tensor product of its ordinary rational cohomology with

that of the classifying space of the group K; i.e. that

Pf(X) = Pt(X)Pt(BK). Thus in the formulae obtained for the equi variant rational cohomology of y

(0)

(see 5.10 and 5.16) one must now always use the equivariant Poincare

series I*/(X)

rather than the product

P (X)P (BK).

Otherwise the formulae

are correct and in good cases give the Betti numbers of the symplectic quotient

y~ ( 0 ) / K .

- 116 9.2.

Example: cotangent bundles.

The examples which motivated the

definition of symplectic manifolds and moment maps were phase spaces and conserved quantities such as angular momentum. The cotangent bundle T*M of any manifold M has a natural symplectic structure given by

a) = ^_

where

(q 1/ ..., c l )

are

dp.Adq.

local coordinates on

M

and

(p ,...,p )

are the

induced coordinates on the cotangent space at

(q ,...,q ). Any action of a

compact group K on M induces an action of

K on T M which preserves

this symplectic structure.

Moreover it is not hard to check that there is a

moment map y: T*M •* k* for this action defined as follows. If m e M and £ e T M then m 9.3 m for all

a e k,

between k# T*M

where

and k

and T M.

.

on the left hand side denotes the natural pairing

and on the right denotes the natural pairing between

So a general moment map is of the form y + c where c

lies in the centre of k* (see §2). The condition 9.1 holds for each of the moment maps on T*M provided that M is compact. To see this one fixes a metric on induce a Riemannian metric on T*M. steepest descent for the function

M and uses it to

It can then be shown that the path of

f =||u + c|f

from any point £ e T*M

-117consists of

cotangent

depending only on The function

vectors

whose norm is bounded by some number

£. f = \\i\

where

\i

is given by 9.3 is not an interesting

Morse function because the only critical points are the points in u reason for this is that by lemma 3.1 if vector field induced by y(£)

=0,

so if we put

However if centre of

y(£)

on

£ e T* M

T M

is critical for

vanishes at

a = y(£)

£.

f then the

Thus in particular

in 9.3 we obtain

||y(S)||

K is not semisimple then it is often possible to choose k

( 0 ) . The

c

=0. in the

such that the norm-square of the moment map p + c has non-

minimal critical points. For example, consider the action of the circle S 2 about some axis.

rotation of the sphere in the Lie algebra of

for any

m e S2

Let

S1 on

T*S 2

induced by

c be an element of norm 1

S^ and let f = ||y + c|| • Then from 9.3 we have

and

5 e T* S 2 ,

So f ( £ ) = 0

which means that the minimum set for

f

if and only if

£.c

=-1,

is homeomorphic to a line bundle

over the sphere less two points and hence is homotopically equivalent to

S1.

Since the circle action on this is free the equivariant cohomology of the minimum set is trivial. By lemma 3.1 the other critical points These are the two points of

S2

£ for

f

are those fixed by S^.

fixed by the rotation.

Hessian at each of these is 2 . Thus we obtain

The index of the

- 118 P^ (S2) = P* (T*S2) = 1 + 2t 2 (1-t 2 )" 1 = (1+t 2 )(1-t 2 )" 1 = Pt(S2)Pt(BS1) S2

as one expects from proposition 5.8 since

has a symplectic structure

preserved by the action of S^. As a second example, consider the linear action of the torus

ei

L° on the unit sphere S 3 9 C .

By 9.3 if m e S 3 and £ e T * S 3

y(£) = (a where a =

'

and

b =

n

m

. £ ) a + (b

m

then

.£)b

• • Consider the function

f = «li • a • b||2 on T * S 3 .

If

a

Any

= 0

mm

£ e T*S3

or

satisfies f ( £ ) = 0 if

y(£) = -a - b, i.e. if

a .5 = - 1 = b .5 . m m = 0 these equations for

b

%

have no solution, and

otherwise they define an affine line in T*S 3 . So the minimum set f m acted on freely by

T

(0) is

and its equivariant cohomology is isomorphic to the

cohomology of the quotient by T of

S3

with two circles removed. This

quotient is an open interval, so its cohomology is trivial. From lemma 3.1 we see that if % e T*S 3 is a non-minimal critical point m ^ for f then either £ is fixed by a and u(£) + a + b is a scalar multiple of a, or

£ is fixed by b and y(£) + a + b is a scalar multiple of b. In the

first case

£.b = -1

and

£ e T*S1

and the second case is similar.

where

S1

is the circle fixed by a,

So the non-minimal critical points form two

-119circles in

T*S3,

each of which is fixed by one copy of

S1

in the torus

and is acted on freely by the other. The index of the function

f

T

along each

of these circles is 2 . Thus we obtain pj(S3) = 1 + 2 t 2 ( 1 - t 2 ) " 1 . Note that this is not equal to

P (S 3 )P (BT);

this does not contradict

proposition 5.8 because S 3 is not a symplectic manifold.

9.4. Example: quasi-projective varieties. compact

symplectic

manifolds

are

Other obvious examples of non-

nonsingular

quasi-projective

complex

varieties. Suppose G K,

is a complex reductive group with maximal compact subgroup

acting linearly on a nonsingular locally closed subvariety

complex projective space

P .

semistable point is finite.

X

of some

Suppose also that the stabiliser of every

If condition 9.1

is satisfied then we obtain

formulae for the Betti numbers of the symplectic quotient

\i

(0)/K

which is

homeomorphic to the quotient variety produced by invariant theory (cf. §8). There is also a more algebraic condition for these formulae to exist which is an alternative to 9 . 1 . It is described as follows. When

X

is a closed subvariety of

stratification of the stratification

P

acted on linearly by G then the

X induced by the action is just the intersection with {S

|3 e B }

induced on P .

can still define a stratification of

X

If

of

X is quasi-projective we

with strata

{X f\ S

13 e B } . 3

Moreover by 6.18 and 8.10 we have

X

- 120 -

for each

$ eB

where \

is a nonsingular locally-closed subvariety of P

and

P_

P

is a parabolic subgroup of

G.

Since X is invariant under G

P

n

this

implies that XOS

Z

P

of

P

(XOYSeS).

x

Y * s -• Z

There is also a retraction p : 3 of a linear subvariety

= C

B

of

Y

P

3

onto the semistable points

under the action of a subgroup of

G.

Provided that

9.5.

p (x) e X

xeX0Y*S

whenever

P

for each

3 e B,

P

one can check that each

p_

induces a retraction of

XHYO

P

and that all the results of

One

can

§8 hold for

use quasi-projective

onto

P

XHZO p

X.

varieties

satisfying

this condition

to

rederive Atiyah and Bott's formulae for the cohomology of moduli spaces of vector bundles over Riemann surfaces (see [Ki3]).

For this one considers

spaces of holomorphic maps from Riemann surfaces to Grassmannians. can

be

embedded

as

quasi-projective

subvarieties

of

These

products

of

Grassmannians. The results of Part I also apply to reductive group actions on singular varieties satisfying appropriate conditions Goresky on C * actions [C & G ] ) .

(see the work of Carrell

and

- 121 -

§10. Appendix: Morse theory extended to rninimally degenerate functions Given any nondegenerate Morse function with isolated critical points on a compact manifold, one has the well-known Morse inequalities which relate the Betti numbers of the manifold to the numbers of critical points of each index. Bott has shown that this classical Morse theory extends to a more general class of Morse functions (see [Bo]). The functions which are nondegenerate in the sense of Bott are those whose critical sets are disjoint unions of submanifolds along each of which the Hessian is nondegenerate in normal directions. The associated Morse inequalities relate the Betti numbers of the manifold to the Betti numbers and indices of the critical submanifolds. The purpose of this appendix is to show that Morse theory can be extended to cover an even larger class of functions.

10.1.

Definition. A smooth function f:

X + R on a compact manifold X

is called minimally degenerate if the following conditions hold. (a) The set of critical points for f on X is a finite union of disjoint closed subsets

{C e C}

subsets

C

on each of which f takes a constant value f(C).

are called critical subsets of

f.

If the critical set of

reasonably well-behaved we can take the subsets

The f is

{C e C}

to be its

For every C e C there is a locally closed submanifold Z

containing

connected components. (b)

C and with orientable normal bundle in X such that

- 122 (i)

C is the subset of

I

on which f takes its minimum value,

and (ii)

at every point

x e C

the tangent space T Z^

subspaces of T X on which the Hessian H (f)

is maximal among all

is positive semi-definite.

A submanifoid satisfying these properties is called a minimising manifold for f along C.

Thus minimal degeneracy means that critical sets can be as degenerate as a minimum but no worse. The purpose behind this definition is to find a condition on

f

more

general than nondegeneracy which ensures that for some choice of metric induces a Morse stratification whose strata are all smooth. shows that minimal degeneracy is such a condition.

f

This appendix f

is any

the

strata

themselves are minimising manifolds provided that the Hessian at

every

function

which

induces

a

smooth

Morse

Conversely if

stratification

then

critical point is definite in directions normal to the stratum which contains it. We do not demand that the minimising manifolds be connected.

However,

this extra condition is always satisfied if we replace each critical subset by its intersections with the connected components of assume that the index of the Hessian of along any

C e C,

the submanifoid

Any function degenerate.

f

Z .

C

Hence we can

takes a constant value

X(C)

because by 10.1 (ii) it coincides with the codimension of

2 . C,

We shall call

X(C)

the index of ———

f

along C.

f which is nondegenerate in the sense of Bott is minimally

For by definition the set of critical points of

f

is the disjoint

- 123 union of connected submanifolds of subsets of

f.

X

If we fix a metric on

and these can be taken as the critical X

then the Hessian of

f

self-adjoint endomorphism of the normal bundle

N

submanifold C.

splits as a sum

Because f

is nondegenerate

where the Hessian is positive definite on N * It is easy to check that locally the image of

Nr

induces a

along each critical N* © N I

and negative definite on N*

N~.

under the exponential map

induced by the metric is a minimising manifold for f

along C.

We wish to show that any minimally degenerate Morse function on

X

induces Morse inequalities in cohomology, and also in equivariant cohomology if

X

is acted on by a compact group

K

which preserves the function.

These inequalities are most easily expressed using Poincare polynomials

Pt(X) = Y. x* dim J>0

and equivariant Poincare polynomials

jX) Our aim is to prove the following theorem.

10.2.

Theorem.

Let

with critical subsets numbers of

f: X -• R {C e C}

be a minimally degenerate Morse function

on a compact manifold

X.

Then the Betti

X satisfy Morse inequalities which can be expressed in the form

- 124 -

^_

t X ( C ) P t (C) - P t (X) = (1 + t ) R ( t )

CeC where

X(C)

is the index of

f along G and R(t) >_0

in the sense that all its coefficients are non-negative. acts on

If a compact group K

X preserving f and the minimising manifolds/ then X also satisfies

equivariant Morse inequalities of the same form.

When f

is nondegenerate one method of obtaining the Morse inequalities

is to use a metric to define a smooth stratification

(S

|C e C}

of

X.

This

is perhaps not the easiest approach, but we shall follow it here because the stratification of the particular function relevant to us is interesting in its own right.

A point of

gradient field subset C.

X

-grad f

lies in a stratum

S

if its trajectory under the

converges to a point of the corresponding critical

For a general function f such a trajectory may not converge to a

single point.

However the limit set of the trajectory is always a connected

nonempty set of critical points for

f

(see 2 . 1 0 ) .

Therefore if

f

is

minimally degenerate then any such limit set is contained in a unique critical subset.

So we make the following definition.

10.3.

Definition.

Suppose

function with critical subsets Riemannian metric.

f: X -• R {C e C}

Then for each

is a

minimally degenerate Morse

and suppose

CeC

let —_

S_ Q

X

is given a fixed

be the subset of

X

- 125 consisting of all points of

-grad f

X

x e X such that the limit set u>(x) of the trajectory

from x is contained in C.

is the disjoint union of the subsets

{S

|Ce C).

We shall see that if

the metric is chosen appropriately they form a smooth stratification of such that each stratum _c.

S^

coincides near

C

X

with the minimising manifold

The condition which the metric must satisfy is that the gradient field

grad f

should be tangential to each minimising manifold

2 .

We shall show

that such a metric exists, and then prove the following theorem.

10,4.

Theorem.

Let

with critical subsets

f: X •• R

{C e C}

that the gradient flow of {_c(CeC}.

on a compact Riemannian manifold.

C e C the stratum S r C.

V

equivalence of Cech cohomology. X

such that the function

invariant under

{Sc|CeC}

defined at 10.3 form a smooth

X, called the Morse stratification of the function

in some neighbourhood of ———_——____—__^__

Suppose

is tangential to each of the minimising manifolds

Then the subsets

stratification of For each

f

be a minimally degenerate Morse function

f

on

X.

coincides with the minimising manifold 2 Moreover each inclusion _____________________________________

C + Sr ^

If there is a compact group

is an ______

K acting on

f, the minimising manifolds and the metric are

K then these inclusions are also equivalences of equi variant

cohomology. In order to be able to apply this result to any minimally degenerate function we need the following lemma.

- 126 10,5.

Lemma,

Let

f: X + R be a minimally degenerate function on X.

Then there exists a metric on X such that near each C e C the gradient flow of

f

is tangential to the minimising manifold

Z r.

If

f

and the

minimising manifolds are invariant under the action of a compact group K then the metric may be taken to be K-invariant.

Proof.

A standard argument using partitions of unity shows that it is enough

to find such metrics locally. The only point to note is that one should work with dual metrics because gradpf is linear in p* but not in p. Suppose x is any point of a critical subset

C.

Condition (ii) of 10.1

implies that there is a complement to T Z r in T X on which the Hessian K * x C x H (f)

is negative definite.

It follows from the Morse lemma (lemma 2.2 of

[Mi]) that there exist local coordinates minimising manifold I

(x ,...,x ) around x such that the

is given locally by 0=X

d+1

= X

d+2 = -

=

V

and such that 2 f ( x r . . * / X n ) = < : (x r ... / x d ) - (x rf+1 )

2 - . . . - (x n ) .

(To prove this regard x ,...,x . as parameters and apply the Morse lemma to x

,...,x ).

metric on R Finally a over K.

Then the gradient flow of

f

with respect to the standard

is tangential to Z~. K-invariant metric is obtained by averaging the dual metric

- 127 Because of this lemma, theorem 10,2 can be deduced from theorem 10.4 by the standard argument using Thom-Gysin sequences (cf. §2).

The rest of

this appendix is devoted to the proof of theorem 10,4, The most difficult part of the proof of this theorem will be to show that for each

C e C

the stratum

in some neighbourhood of C. will follow easily that

coincides with the given submanifold

Sr

Once we know that

S

S^

is smooth near

C

2^ it

is smooth everywhere, and the cohomology

equivalences are not hard to prove.

First we shall show that the subsets of

X

in the sense of 2 . 1 1 .

{S

|C e C}

form a stratification

It suffices to prove the following lemma, which

depends on the assumption 10.1 (a) but not on the existence of minimising manifolds.

10 7

' *

Lemma. For each C e C, f(C')>f(C)

Proof.

If a point

x

lies in S

of steepest descent for

c

for some C e C

then by definition its path

f has a limit point in C, and hence f(x)M(C)

since f decreases along this path.

Moreover

f(x)>f(C) unless x e C. If

x

lies in the closure S~ of

of steepest descent.

Sr

then so does every point of its path

Hence the closure of this path is contained in

Sc»

It

- 128 follows that

x e Sr,

is not critical for Since {U If

the

|C e C} x

for some C

So if

x e L

and x

f then f (x) > f (C). subsets

{C e C}

are

compact

x e 3U H~S

3Ur

are

open

sets

for each C e C .

of some U^ then x is not critical for

then f(x) > f ( C ) .

it follows that there is some

there

UC3C

whose closures are disjoint such that

lies in the boundary

Hence if

with f(C') >_f(C).

6>0

Since each 9U H ^

such that if

C e C

and

f.

is compact x e 3U

fl Sc

then f(x) M ( C ) + 6. Now suppose that x e SpO Sr. x = x.

Let

C

and

{x | t >_0}

T ^0

are distinct and that there is some

be the path of steepest descent for

Then the limit points of

So there exists

C

as t + »

{x 11 ^ 0 }

such that

x

e U^,

implies that there is a neighbourhood

V

f

are contained in

and f(x ) < f ( C ) + 6. of

x

in

with

X

such that

C.

But this y

e U^,

and f(y ) < f(C') + 6 whenever y e V. Since points as

xeS t •*• »

UpHUp, = 0 implies that

there is some of

{y 11 >_0}

y e VOS

the path { y J t ^ O }

Then y

are contained in

there must exist some f(y ) >_f(C) + 6

.

t>T

C.

such that

by the choice of

e U , but the limit Since by assumption y

f(y ) > f(y ) > f(C) + 6, so that f(C)>f(C). This shows that if

SflnSf

is nonempty then f(C) < f ( C ) .

the disjoint union of the subsets {S

|C e C}

the result follows.

Since

X is

- 129 Now we shall begin the proof that each stratum with the corresponding minimising manifold

10.8 z

Lemma. w

r

'tn

a

For each

C e C

S~

the intersection of the minimising manifold

sufficiently small neighbourhood of

C

is contained in the Morse

Sr.

Proof.

As in the proof of 10.7 choose open subsets

whose closures are disjoint and such that Z

If

C e C

subset of then

U DC

Z

Zc C\ U~

{ U f | C e C}

for each

is a submanifold of some neighbourhood of

small enough then

C, if

U^

X

Since

is taken

is closed for each C e C.

on which f takes its minimum value. and so as

of

C e C.

then by the definition of a minimising manifold

f(x) > f(C),

C

2 .

stratum

each

coincides near

Z 0 3UC

Hence if

C

is the

x e Z f\ 3 U C

is compact there exists y > 0 such

that f(x)M(C) + y whenever C e C and x e

Z C\ 3 U C . Then for every C e C the subset

V_ = U

f l { x e X | f ( x ) < f ( C ) + y}

is an open neighbourhood of C in X. Suppose Then as {x | t ^ 0 }

x

grad f

lies in the intersection of this neighbourhood V^ is tangential to

of steepest descent for

Z

and f

from

remains in U^.

Hence if the path leaves

x e 3Ur0 2r.

This implies that

Uf

Z

is closed in x

stays in

there exists

Z

Ur

with

Z .

the path

as long as it

t >0

such that

- 130 f(x)>f(x

)>

which contradicts the assumption that for all time.

x e V .

Since the only critical points for

So the path remains in

f in

Ur

U^

are contained in

C

it follows that the limit points of the path lie in C and so x e S~.

10.9.

Remark.

neighborhood

U

such that if

x e V

Note that

the same argument shows that

of

X

C

in

f\ S r

from x is contained in

given any

there exists a smaller neighbourhood

V_

then the entire path of steepest descent for

f

U_.

In order to prove the converse to the last lemma we need to investigate the differential equation which defines the gradient flow of coordinates near any critical point

x.

f

in local

We shall rely on the standard local

results to be found in [ H ] . Recall that if of

f at

x e X

is a critical point for

f

then the Hessian

x is a symmetric bilinear form on the tangent space

T X

local coordinates by the matrix of second partial derivatives of Riemannian metric provides an inner product on

T X

identified with a self-adjoint linear endomorphism of eigenvalues of

H (f)

eigenspaces of

H (f).

are real and

T X

xeC

the subspace

T 2_

T X.

H

given in f.

The

can be

Then all the

splits as the direct sum of the

The assumption that the gradient field of that for each

so that

H (f)

of

f

is tangential to

T X

£r

implies

is invariant under H (f)

- 131 regarded as a self-adjoint endomorphism of complement

T Z *-.

eigenvalues of of

By the

H (f) restricted to L

restricted to T I

H (f)

Now fix

definition

C e C

T Z

T X. of

Hence so is its orthogonal a minimising manifold

the

are all non-negative, while those

are all strictly negative.

and a point

Then we can find local coordinates

x e C.

Let

(x , . . . , x )

d be the dimension of in a neighbourhood

W

Z . of

x such that

10.10. given by

(i)

x

is the origin in these coordinates and the submanifold

xrf+1 =

X(j+2

Z^

is

= . . . = X R = 0;

(ii) the Riemannian metric at

x is the standard inner product on R ;

and (iii) the Hessian H (f)

is represented by a diagonal matrix

where x

V' # # / X d-°

and

Let

P

be the diagonal matrix

diagonal matrix diag (-X .

,...,-X

diag (-X , . . . , - X ,) ).

Then

and let

Q

be the

- 132 -

-Hx(f) in these coordinates. z

= (x

P 0]

For ( x ^ . ^ x ) e R

.,...,x ); Then the trajectories of n d+1

write

y = (x ..,.•.,x ,)

and

-grad f in these coordinates are -g

the solution curves to the differential equation

10.11.

y = Py + z = Qz + F 2 (y # z)

where

F

and

F

are

C

and their Jacobian matrices

3F

and 3F2

vanish at the origin (cf. [H] Chapter IX,§4). By reducing the neighbourhood W

of x if necessary we may assume that

over

R

F^ and

F2

extend smoothly

in such a way that there exist complete solution curves to 10.11

through every point (yo^ 0 ) e R , given by t * (y t ,z t ) say, for t e R (see [H] IX 3 and 4). Then we have

10.12.

y t = e P t y 0 + Y(t,y o ,z o ) \

= e Q t z 0 + Z(t,y 0/ z 0 )

for all t e R, where Y,Z and their partial Jacobian matrices. 3

v

3 Z vanish at the origin. Yo/Zo

o/zo

Y and

- 133 We want to show that if a point

x does not lie in

steepest descent stays well away from then it has a well-defined distance show that near

C.

If

d(x,Z )

Z

then its path of

x is sufficiently close to Z from

Z .

It is sufficient to

C this distance function is bounded away from zero along all

paths of steepest descent not contained in

Z~.

We can do this by working in

local coordinates near each x e C. The submanifold by

z = 0.

Z

is defined in the local coordinates

Therefore in the standard metric on

is given by

|zj.

in W

the distance from Z

Moreover the coordinates were chosen so that the given

Riemannian metric at

x

It follows that given any

10.13.

R

(y,z)

(1

+ £

coincides with the standard inner product on e >0

we may reduce W

R .

so that

) " 1 ||z| ;£d((y,z),Z c ) 1 , which depends only on the critical

over

R

of

xeC.

If the

x is taken sufficiently small and the extensions of F

are chosen appropriately, then for every

we have

Ikll > blkll

(yo,zo)

e

R

- 134 where zx = e^z 0 + Z(1,y o ,z o )

as at 10,12,

Proof,

f

The gradient field of

F 2 (y,0) = 0 extension of

whenever F

(y,0)

to R

is tangential to the submanifold lies in

W

(see 10.11).

can be chosen so that

lc

so

Therefore the

F 2 (y,0) = 0 for all y e R

.

This implies that Z(t,y 0 ,0) = 0 for all y 0 e R

and t e R

(see 10.12). Now for each x e C let c

be the minimum eigenvalue of

e .

Recall

that Q = diag ( ~ x d + 1 / " * / - * ) where T Z c

x

X .-/...,X d+1 n ,

are the eigenvalues of the Hessian

H (f) x

restricted to

and that each of these eigenvalues is strictly negative.

Hence

> 1 . Let c = inf{c

since c > 1. b>1

C

is compact and

So we can choose

c

|x e C};

depends continuously on

6 >0

x

it follows that

such that c - 6 > 1 . Set b = c - 6; then

and b depends only on C. By 10.12 the partial Jacobian

3 V

t e R.

Z

vanishes at the origin for all

0>ZQ

Hence by reducing the neighbourhood

extensions of

F

and F

and choosing the

appropriately we may assume that

||a Z i Z(1,y o ,z Q )|| £ 6 (cf. [H] IX §4).

W

It follows that

for all (y o ,z o ) e Rn

- 135 for all (y o ,z o ) e Rn .

||Z(1,y o ,Zo)||£e||z o ||

Since every eigenvalue of e ^ is at least c, for any (y 0 / z 0 ) e R we have || Zl || = ||e Qz «

+

Z(1,y 0/ z 0 )||

>_ c ll z oll " 9| z oll = b||zo||. The result follows.

10.15.

Corollary.

The intersection of the Morse stratum

sufficiently small neighbourhood of

C

S^

with a

in X is contained in the minimising

manifold Z .

Proof.

It follows from 10.13 and 10.14 that given

neighbourhood

W

descent with x e W

of

C such that if when 0 1

is independent of

e.

there is a

is any path of steepest

then

d ( x x , l r ) > b(1 + e) where

e>0

If

-2

, d(x 0/ Z )

e

is chosen sufficiently small we

have b(1 + e ) " 2 > 1 . By remark 10.9 there is a neighbourhood V^ x0 e V flSp

its entire path of steepest descent

of C in X such that if {x I t ^ 0 }

is contained in

W . Then for each n > 1 d(x

°' z c ) *

But we may assume without loss of generality that

d(x,Zc)

is bounded on

- 136 W .

Hence we must have

d(x o ,Z

) = 0,

i.e. x 0 e

Zr.

From 10.8 and 10.15 we deduce that each stratum Zc

in a neighbourhood

But any point of

S

Uc

of

C,

and hence that

is mapped into S f f t U

This shows that

Sr

coincides with

Srf\Uc

is smooth.

by the diffeomorphism

x •* x

of

S p induced by flowing for some large time

f.

So we have the following

10.16.

Lemma.

For each

with the minimising manifold

CeC Z

t

the stratum

along the gradient field of

Sr

is smooth.

It coincides

in some neighbourhood of C.

We have now proved that the subsets

{S | C e C}

form a smooth

stratification of X, and it remains only to prove one more result.

10.17.

Lemma.

For each

for Cech cohomology. on

X

CeC

the inclusion

C + Sc

is an equivalence

More generally if a compact connected group K acts

in such a way that the function

f

and the Riemannian metric on X

are preserved by K, then each stratum S^ is K-invariant and the inclusions C •> S

are equivalences of equivariant cohomology.

Proof.

We need only consider the second statement.

definition that the Morse strata

It is clear from the

{ S r |C e C} are K-invariant.

- 137 For each sufficiently small

6 ^0,

N { = { x e S c | f ( x ) 0

v

M

N

6

=

r

C

'

So the continuity of Cech cohomology implies that the inclusion C • S^ is an V

equivalence of equivariant Cech cohomology (see [D] V I I I 6.18). problem is that EK to

X X..EK

is not compact.

The only

This can be overcome by regarding

as the union of compact manifolds which are cohomologically equivalent EK up to arbitrarily large dimensions.

10.18.

Rema rk.

When f is nondegenerate in the sense of Bott each path of

steepest descent under

f converges to a unique critical point in X. Thus the

strata retract onto the critical sets along the paths of steepest descent. fails in general for minimally degenerate functions:

This

there exist minimally

degenerate functions with trajectories which "spiral in" towards a critical V

subset without converging to a unique limit. used above.

This is why Cech cohomology is

However it is unlikely that the square of a moment map has such

bad behaviour.

- 138 -

Part I I . The algebraic approach. §11. The basic idea. In Part I a formula was obtained in good cases for the Betti numbers of the projective quotient variety associated in geometric invariant theory to a linear action of a complex reductive group projective variety

X.

G

on a nonsingular complex

The good cases occur when the stabiliser in

every semistable point of

X

topologically the quotient

XSS/G

is finite.

G

of

The quotient variety is then

of the set of semistable points by the

group. The formula was obtained by employing the ideas of Morse theory and of symplectic geometry.

We shall now approach the same problem using

algebraic methods. The basic idea common to both approaches is to associate to the group action a canonical stratification of the variety X. The unique open stratum of this stratification coincides with the set (provided X

X

of semistable points of X

is nonempty) and the other strata are all G-invariant locally-

closed nonsingular subvarieties of

X.

There then exist equivariant Morse-

type inequalities relating the G-equivariant Betti numbers of X to those of the strata. It turns out that these inequalities are in fact equalities, i.e. that the stratification is equivariantly perfect over the rationals.

From this an

inductive formula can be derived for the equivariant Betti numbers of the semistable stratum X

which in good cases coincide with the ordinary Betti

numbers of the quotient variety.

- 139 The difference

between the two approaches lies in the way the

stratification of X

is defined.

In Part I symplectic geometry was used to

define a function f

(the norm-square of the moment map) which induced a

Morse stratification of X. In Part II the stratification will be defined purely algebraically.

The main advantage of this method is that it applies to

varieties defined over any algebraically closed field.

On the other hand the

approach of Part I generalises to Kcihier and symplectic manifolds. The algebraic definition of the stratification is based on work of Kempf. It also has close links with the paper [Ne] by Ness.

Suppose that we are

given a linear action of a reductive group G on any projective variety singular or nonsingular, defined over any algebraically closed field. shows that for each unstable point

x eX

X,

Kempf

there is a conjugacy class of

virtual one-parameter groups of a certain parabolic subgroup of

G

which

are "most responsible" for the instability of x. (The terminology "canonical destabilising flags" is also used).

The stratum to which

x

belongs is

determined by the conjugacy class of these virtual one-parameter subgroups in

G.

Over the complex field the stratification is the same as the one

already defined in Part I. Just as in Part I the indexing set

B

of the stratification may be

described in terms of the weights of the representation of G which defines the action. An element 3 e B may be thought of as the closest point to the origin of the convex hull of some nonempty set of weights, when the weights are regarded as elements of an appropriate normed space (see 12.8).

- 140 In §13 it is shown that if

X

is nonsingular then the strata

nonsingular and have the same structure as in the complex case. each Y

8

of P

in the indexing set

B

S. are also 3 That is, for

there is a smooth locally-closed subvariety

X, acted on by a parabolic subgroup Po of p

There is also a nonsingular closed subvariety

G, such that

Z.

of

P

X

and a locally

trivial fibration

whose fibres are all affine spaces* of

Z

P

Here

Zo

P

is the set of semistable points

under the action of a reductive subgroup of

P . P

These results were precisely what was needed in Part I to show that the stratification

{S

p

I 3 e B}

is equivariantly perfect and hence to derive an

inductive formula for the equivariant Betti numbers of

X

.

Thus the reader

who is interested solely in complex algebraic varieties can avoid the detailed analytic arguments needed for symplectic and KaTiler manifolds by using definitions and results from these two sections.

It will be found that at times

the algebraic method is neater while at others it is more elegant to argue analytically. In §14 we shall see how the formulae for the Betti numbers of the quotient variety

M

can be refined to give the Hodge numbers as well.

We

- 141 use Deligne's extension of Hodge theory to complex varieties which are not necessarily compact and nonsingular. In §15 an alternative method of obtaining the formulae is described, though without detailed proofs.

This method was suggested by work of

Harder and Narasimhan [H & N ] .

It uses the Weil conjectures which were

established by Deligne. These enable one to calculate the Betti numbers of a nonsingular complex projective variety by counting the points of associated varieties defined over finite fields.

In our case it is possible to count points

by decomposing these varieties into strata and using 11.1 and 11.2.

However

the Weil conjectures only apply when the quotient variety is nonsingular. Finally in §16 some examples of stratifications and of calculating the Betti numbers of quotients are considered in detail. given by the action of

SL(2)

on the space

P

The first example is

of binary forms of degree n,

which can be identified with the space of unordered sets of points on the complex projective line sets of points on

P .

P .

(P )

of ordered

These have been used as examples throughout Part I.

The good cases occur when quotient varieties

We also consider the space

n

is odd, and then the Hodge numbers of the

M are given by h p ' p = [1 + 1 / 2 min (p,n-3-p)]

for the case of unordered points, and hp'p = i

+ (n-i) + v

'

(n;1) v 2 '

+

for ordered points. The Hodge numbers we generalise

(P1)n

...

+ (

h p '^

v

. ; n " \ j min(p,n-3-p)'

with

p^q

all vanish. Then

to an arbitrary product of Grassmannians.

That is, we

- 142 consider for any Grassmannians

m

the diagonal action of

G(£.,m)

il-dimensional subspaces of

to

£_

&..

where C

•

G(£,m)

SL(m)

on a product of

denotes the Grassmannian of

The good cases occur when

m is coprime

The associated stratification is described in Proposition 16.9 and

i it is shown how in good cases this provides an inductive formula for the equivariant

Betti

numbers

of

the

semistable

stratum

in terms

of

the

equivariant Betti numbers of the semistable strata of products of the same form but with smaller values of products of

m.

Explicit calculations are made for some

P2«

One reason for studying products of Grassmannians in depth is that it is possible to rederive the formulae obtained in [H & N] and [A & B] for the Betti numbers of moduli spaces of vector bundles over Riemann surfaces by applying

the

results

of

Grassmannians (see [Ki3]]L

these

notes

to

subvarieties

of

products

of

- 143 -

§12.

Stratifications over arbitrary algebraically closed fields Let

k be an algebraically closed field.

k-variety acted on linearly by a reductive shall define a stratification of Part I for the case when

X

X

Suppose that

X

k-group

In this section we

G.

is a projective

which generalises the definition given in

is nonsingular and

k

is the field of complex

numbers. The set

X

of semistable points of

stratum of the stratification.

X

under the action will form one

To define the others we shall use work of

Kempf as expounded in a paper by Hesselink (see [K] and [Hes]). associates to each unstable point

x

of

X

Kempf

a conjugacy class of virtual one-

parameter subgroups in a parabolic subgroup of

G.

responsible" for the instability of the point x.

These are the ones "most The stratum to which

x

belongs will be determined by the conjugacy class of these virtual oneparameter subgroups in

G.

We shall find that each stratum

So

can be

P

described in the form S B

where

Y.

P

-

GY $

e

is a locally-closed subvariety of

X,

itself defined in terms of

the semistable points of a smaller variety under the action of a subgroup of G.

From this it will be obvious that the stratification coincides with the one

defined in Part I in the complex nonsingular case. First we shall review briefly Hesselink's definitions and results and relate them to what we have already done in the complex case: this is completed in

- 144 lemma 12.13.

Note that in [Hes] arbitrary ground fields are considered. We

shall

ourselves

restrict

to

algebraically

closed

fields

for

the

sake

of

simplicity.

Remark. k

The definition of the stratification given at 12.14 makes sense when

is any field.

There is also a stratification of the variety

over the algebraic closure

K

of

k.

When

k

5.6).

X.

defined

is perfect it follows from

[Hes] that this last stratification is defined over first stratification on

X x, K

k

and coincides with the

However this fails in general (see [Hes] example

In §15 where finite fields occur it will be necessary to avoid certain

characteristics where things go wrong.

In [Hes]

Hesselink studies reductive group actions on affine pointed

varieties.

We shall apply his results to the action of

X £k

on X.

X:

k •* G

of

For each nonzero x G

every

x and

12.1.

]f

X:

m(x;A).

x

in

X

r ...,r

k* + GL(n+1)

then

x

is given by

m(x ; X ) .

and hence can

The following two facts determine

X.

e Z

and one-parameter subgroup

determined by

z + diag ( z r ° , . . . , z r n ) with

on the affine cone

Hesselink defines a 'measure of instability'

This depends only on the point also be written as

in X

G

m(x;X)

for

- 145 m(x;X) = min{r.| x. ^ 0} if this is non-negative and m(x;X) = 0 otherwise. Also m(x;gXg

12.2.

Definition.

xeX

) = m(gx;X)

for any g e G.

is unstable for the action of

some one-parameter subgroup X of

G if m ( x ; X ) > 0

for

G.

In [M] theorem 2 . 1 , Mumford proves that 12.3.

x e X

is semistable if and only if

parameter subgroup X of

12#4#

Definitions

G; that is, if and only if

([Hes] §1).

k* + G

Let

Y(G)

x is not unstable.

be the set of one-parameter

X:

of

with the natural numbers by the equivalence relation -

(X,£)-v(M,m)

if

G, and let M(G)

for every one-

subgroups Y(G)

of

m(x;X) q ( 3 ) .

then either

of Conv{a.|x. ? 0}

0

T

is optimal for

If

of

Conv{a.|x. 5^0}.

3

3 ' = 3 or q ( 3 ' ) > q ( 3 ) .

0

T is optimal for

Therefore x

or there is some

and

3

3' > 3

is the closest such that

x

Sol. P

Hence

S C GW £ y P

3

3»>3

SR, 3

so the proof is complete.

This lemma shows that the subsets

| 3 e B} 3 X in the sense of definition 2 . 1 1 , and in particular

is open in its closure So

P

{S

form a stratification of

for each 3 e B.

We next want to describe the stratum

S 3

in such a way that it is clear

that when k = C this stratification coincides with the one defined in Part I.

- 153 12.18.

Definition. Let = {(x o :...:x n ) e X | x. = 0 if

ct..3 5*q(3)}

and let Y B = { ( x Q : . . . : x ) e X | x. = 0

x. * 0

if a . . 3 < q ( 3 ) ,

for some j with a..3 = q ( 3 ) K Zo

is a closed subvariety of

X

and Y o

P

is a locally-closed subvariety.

P

Define p ^

where

Yg + Zg

x! = x. if J J

as a map Y lies in X.

p

by

a..3 = q ( 3 ) and x! = 0 otherwise. This is well defined J J

•* Z

since if

ye Y

P

then

p o ( y ) e Gy

P

Let Stab3

be the stabiliser of 3 under the adjoint action of G

on M ( G ) . Then Stab3 The definitions of

and in particular

P

is a reductive subgroup of G which acts on Z . 3 Zo/ P

Yo

P

and

po

P

depend only on

3.

They are

independent of the choice of coordinates, and indeed of the maximal torus chosen except that

3

T

must lie in M(T). Moreover by 6.5 when k = C and

X is nonsingular they coincide with the definitions made in Part I.

12.19.

Lemma. If

x e Z

Proof.

If

then

x e Zo

then StabB 3

fixes

is optimal for x.

x so 3 e M(P(x))

by 12.13 (iii).

Also

P

A^(x) £ M(P(x)) Li

p e P(x)

by 12.13 (iv)

X e A_(x)

there is some

LI

such that

pXp

pXp"1 e M(Stab3)OA^(x) Li

required.

so that if

and

3

commute.

by 12.13 (ii) so

Stab3

But this implies that is optimal for

x

as

- 154 Note that if

x e I

then by definition

P

m(x;3) = min{a..3 | x. 5* 0} = q ( 3 ) . Thus in particular when is an open subset of

M O Zo

P

no point in Z

P

is semistable.

However there

whose elements are unstable "only insofar as

3

makes them unstable". The neatest definition of this subset is the following.

12.20. x e ln

3

Definition. such that

Since

Stab3

Let

Zo

be the subset of

Z.

consisting of those

3/q(3) e A~(x). * is optimal for

x

the condition that

3 / q ( 3 ) e A~(x)

is

equivalent to the condition that m(x;X) . («i) y e S & . (Hi) y e Y * s . (iv)

x e Z*S. P

(v) x e S^.

Proof,

(iii) and (iv) are equivalent by definition, while (i) implies (ii) by

definition 12.14 and the converse follows from 12.17 since 12.17 again if Then

y t S.

then y e S o l

P

x e S , since 3

P

Yo Q W o . p

for some 3' satisfying q(3') > q(3).

x e Gy, and by lemma 12.16 this implies that

Therefore (v) implies ( i i ) .

By

P

xi S . 3

It follows straight from the definitions that (iv)

implies ( v ) . Finally suppose that

x i 1

.

Since

T

is a maximal torus of

Stab3

3 there is some s e Stab3

such that

T is optimal for sx. By 12.6 3

is not

the closest point to 0 of Conv{a. | (sx). 5*0}. Moreover (sx). 5* 0 if and only if both (sy).^O and a..3 = q ( 3 ) because p o (sy) = sx (see i i 3 definition 12.18).

So it follows from the geometry of convex sets that

not the closest point to drawing a picture).

0

of

Conv{ot. | (sy). 7^0}.

3 is

(This is best seen by

Thus by 12.6 and 12.7 A (sy) 5* ( 3 / q ( 3 ) }

and hence by

- 157 12.17 sy e S o , for some 3' > 3. P

So y i S o . Thus (ii) implies (iv) and the P

proof is complete.

12.25. Corollary. A

V 20JJ

If

—

3 £ 0 then y e Y ! S if and only if T is optimal for p

T (y) = {3/q(3)>,

or equivalently if and only if

3/q(3) e A G (x).

Thus So = GY* S for any 3 e B. p1

3

Proof. It is obvious that GY*S = XSS = S_. If 3 10 — — — — — u u A (y) = {3/q(3)> then by 12.6 and 12.7 3 is the closest point to Conv{a.|y. ^ 0 } . i

i

Thus y e Y

and 0 of

so the result follows straight from lemma P

12.24.

We have now proved the following theorem.

12.26.

Theorem.

Let

X G. P be a projective variety over n

be a reductive k-group. parameter subgroups of

k and let G

Fix a norm q on the space M(G) of virtual oneG. Then to any linear action of

associated a stratification

{S. | 3 e B} 3

G on X there is

of X by G-invariant locally-closed —

s bvarieties described as follows. If T is a maximal torus of G the indices 3 e B are minimal combinations of weights in a fixed Weyl chamber of M(T) and while if 3 £ 0 S. = CY?S

- 158 where Y* S = { x e X | B / q ( 3 ) e A G ( x ) } . When

k = C

and

X

is nonsingular the strata

coincide with those defined in Part I.

S o and the subvarieties p

Yo p

- 159-

§13. The strata of a nonsingular variety Now suppose that

X

is a nonsingular project!ve variety over

section we shall see that the strata

{S | 3 e B}

k.

In this

of the stratification

P

associated in §12 to the action of a reductive group subvarieties of

X.

G

are all nonsingular

To prove this we shall show firstly that the subvarieties

Z

and Y defined at 12.20 are all nonsingular and secondly that each 3 3 stratum S. is isomorphic to G x n Yrt . In addition we shall see that each P ^ P morphism p : Y o + ln is an algebraic locally trivial fibration such that P p P every fibre is an affine space.

The following facts about linear actions of the multiplicative group on nonsingular projective varieties such as X will be needed. to Bialynicki-Birula (see [B-B] especially theorem 4 . 3 ) . to certain one-parameter subgroups of

G.

13.1.

X.

Suppose

k*

acts linearly on

These are due

We shall apply them

Then the set of fixed points is a

finite disjoint union of closed connected nonsingular subvarieties of be one of these.

For every

x e X

extends uniquely to a morphism

0

tx.

Let

the morphism

k* -* X

k •> X; the image of

Y consist of all x e X such that

X; let Z

given by

t + tx

0

will be denoted by

- tx

lies in Z . Then Y

is a connected locally-closed nonsingular subvariety of p: Y + 1 defined by

k

X

and the map

- 160 -

P(x) = J™ tx is an algebraic locally trivial fibration with fibre some affine space over

13*2.

Corollary*

For each 3 e B the subvarieties

12*18 are nonsingular*

Y

and Z

k.

defined at

The morphism p : Y

-• Z

is an algebraic locally trivial fibration whose fibre at any point is an affine space* The same is therefore true of its restriction v ss Y

to the open subset Y* S

of —

P

Proof*

Fix

$ e B

_ss

B *ZB

V Yo. p

and let

r>0

be an integer such that

r3 e M(T)

corresponds to a one-parameter subgroup of T. This one-parameter subgroup acts on X as t where

a

...,a

The definition of

Zo p

and

.

tx e Z r t , p

Yo

shows that

P

fixed point set of

n

tran-e}

are the weights of the representation of

the

t"Mi

diag(tra«'B

•

Zo

T

on

k

is a union of components of

P

this action and that

x e Y

if

3

in which case this limit coincides with p (x)* p

and only

if

So the result is

an immediate consequence of 1 3 . 1 .

Now X

D rg

Y S

! p

we want where

to show that

PD p

each stratum

is the parabolic subgroup of

So

P

is isomorphic

G defined in 12.11.

to For

- 161 simplicity we shall assume that the homomorphism defines the action of

G

on

X

is faithful.

immediately from this except that

Po

: G + GL(n+1)

which

The general result follows

must be replaced by

| ((PO)),

P

P

which is also a parabolic subgroup of G.

13.3.

Definition

and for each

([B] 3.3).

3 e B

let

Let

p

P

g

be the Lie algebra of the

k-group

G

be the Lie algebra of the parabolic subgroup

V As a k-vector space g is just the tangent space to the group G at the origin. The action of G on X induces a k-linear map

from

g to the Zariski tangent space T X for each x e X.

13.4.

Lemma. Suppose G is a subgroup of GL(n+1).

If x e YQ

then

{ g e C | g x e Y* S } and

Proof P

(compare the proof of lemma 6.15).

so P

£ {g e G | gx e Y

By 12.24

x e Y* S

A_(x) = ( 3 / q ( 3 ) } . •

P

} and p

£ {£ e g | £

if and only if

Suppose that

x

By 12.23 Y* s is invariant under 3

and

T gx

eT Y

}.

is optimal for both lie in

Yo P

x

and

for some

- 162 g e G.

Then

B/q(3) e A_(gx)

so that

lie in A ~ ( x ) . Therefore g e P o VJ

of

By

both

by 12.13 (iii).

P

It remains to show that 13.2 if

and ' Ad(g" x )B/q(B)

3/q(3)

{£ e g | £ e T Y ( S p , x

x p

r is any positive integer such that

As in the proof of

p

r3 is a one-parameter subgroup

T then r3 acts on X as

12.11 the

subgroup

(r3(t))g(r3(t)) an element

P P

consists

of

all

g e G

such

that

tends to some limit in G as t e k* tends to 0 . Hence

g of

G

lies in P

P

if and only if it is of the form

g = (g..) IJ

with g.. = 0 when a..3 < a . . 3 . |J i J

Let

g = t + £_

g

respect to the Lie algebra 13.18). all

If

n e t

£eg

be the root space decomposition of

t

of the maximal torus

T

g

with

(see [B] theorem

has a nonzero ij-component then as [ n , 5 ] = 0 . —

and £ e T Y

then

a.3 « X V M' = M O p

(So)

A

and hence is an open subset

P

We have M1 = ( ( g P g / y ) | g ~

which is isomorphic to

G xp

Y

lemma 13.4 the restriction of

y e Y " }

and hence is nonsingular.

pv

to

M'

is a bijection onto

Moreover by So.

A

since

G/P

is complete

M1 + So

pv: A

G/P

X

x X.

To show that

pv: A

is a closed map, so that

P

is a homeomorphism

P

P

G / P o x X -*- X

p :

P

Indeed

P

because

M1 + So

P

M'

is locally

closed in

is an isomorphism it therefore

suffices by [Ha] ex I 3.3 and lemma II 7.4 to check that the induced maps of Zariski tangent spaces

(p,.) • T M' + T , .S o A m Px/Cro) "

are all injective.

A

It is only necessary to consider the case when y e Y^ S . 3

Then an element of

a + p o e g/Po/ P

P

T M1 m

S e T X

0 = ( p x ) # ( a + Pg,S) = K

then

is of the form

and

V

a

m = (Pg/y)

e T Y

-a ,

V

+^eT

(a + p o / £ ) P

V

Y? S . P

for some where So

if

and hence by lemma 13.4

- 166 a epn

P

(p ) A

so that

(a + p n , £ ) 3

is the zero element of M1

is injective everywhere in

isomorphism.

and

We conclude that for each

nonsingular and isomorphic to G x D

T3

Y.

P

T M1. m

hence that p v : A

3 e B

It follows that M1 •> S

the stratum

is an

p

S

.

Thanks to corollary 13.2 the proof of the theorem is now complete.

P

is

- 167 -

§14. Hodge numbers Suppose now that

X£P

is a nonsingular complex projective variety

acted on linearly by a connected complex reductive group that the stabiliser in

G

of every semistable point of

obtained a formula for the Betti

X

G.

is finite.

numbers of the quotient

associated in invariant theory to the action of

G

on

X.

Suppose also We have

variety

M

In this section we

shall see that this formula can be refined to give a formula for the Hodge numbers of

M.

We shall use Deligne's

extension of Hodge theory which applies to

algebraic varieties which are not necessarily compact and nonsingular (see [D1] and [ D 2 ] ) .

If

Y is a variety which is not nonsingular and projective it

may not be possible to decompose H P / (Y)

H (Y;C)

as the direct sum of subspaces

in a way which generalises the classical Hodge decomposition.

However Deligne shows that there are two canonical filiations of

H (Y;C),

the weight filtration

which is defined over Q, and the Hodge filtration -^ c

"•"} c

p-1-

"*} IT

p~

T5

p+1

giving what Deligne calls a mixed Hodge structure on define the Hodge numbers

hp'q(Hn(Y))

of

H n (Y)

H (Y).

to be the dimension of

appropriate quotients associated to these filtrations ([D1] Hodge numbers satisfy

One can then

II 2 . 3 . 7 ) .

The

- 168 -

If

h P ' q (H n (Y)) ?0

min(n,dim Y),

and

dim Hn(Y;C) = $ [

h P ' q (H n (Y)).

then

lie between

p

p + q _n

Y

the

Y

is

is nonsingular and

p +q = n

f: Y

if

and

+ Y

are the same as is a morphism of

induced

homomorphism

is strictly compatible with both the Hodge filtration and

the weight filtration (see [D1] II 3 . 2 . 1 1 . 1 ) . Suppose now that

Y

is acted on by a group

G.

Recall that its

equi variant cohomology is defined to be H # G ( Y ; Z ) = H*(Y x G EG;Z) where

EG * BG

is the universal classifying bundle for G .

Although

BG is

not a finite dimensional manifold there is a natural Hodge structure on its cohomology (see [D1] I I I 9 ) . finite dimensional varieties BG

Indeed

BG

may be regarded as the union of

M n such that for any n the inclusion of

induces isomorphisms of cohomology in dimensions less than

preserve the Hodge structure.

In the same way

Y x_ EG

n

Mn

in

which

is the union of

Li

finite dimensional varieties whose Hodge structures induce a natural Hodge structure on the cohomology of

Y XQ EG.

Hodge numbers (Y)

= hp'q

(H

Thus we can define equi variant

- 169 for

Y. In particular there are equivariant Hodge numbers for each stratum

SD P

of the stratification associated in §12 to the action of variety

X.

G

on the projective

These strata may be disconnected so it is convenient to refine

the stratification as follows.

For each integer

union of those components of

S

m>0

let

SQ p,m

be the

whose complex codimension in

P

X

is

) where

d(3,m) = m - dim G + dim Stab3

(cf. §§4 and 8). In §8 we saw that 14.1

*—

VJ

for

each

p,m

where

the sum is over

The argument

| 3 e B, 0 < m < dim X} —

indexing set some

u

3,m

n ^0,

0 hl P ' q ' n)V * d(6 ' m) (Sfi )

h ^ W - h ^ V V Li

C

Li

—

LJ

3,m

where the sum is over all nonzero

By theorem 13.7 for each

P,m

3 e B and integers 0 < m H " ( X S S ; Q )

is surjective, since it is the

Mr>(T.;Q) + HU(T. VJ

14.9

), and hence also of

is finite.

could of course also be deduced directly from [D1] that

which are the

h p ' q (M) when M is smooth.

classical Hodge numbers

true by induction of

Thus we obtain a formula for calculating

VJ

I ""I

*Q)

for

1J\

the number of points of

elements is

}•_ ( a . ) " - ^_ ( 6 . ) " . i

)

a ,...,ot , Y

- 178 We may assume that each

a.

form

q ^'

the

a. ^ 3.

is of the form 2

where

for every

q

n(i)

and the absolute value of each and

n(j)

(2k)th il-adic Betti number of

absolute value 3.'s

q ,

and its

with absolute value q

i and j . Then the absolute value of

k+V2

are non-negative integers.

Y

(2k+1)st

3.

is of the Moreover

is equal to the number of ct.'s with Jl-adic Betti number is the number of

.

We shall use the Weil conjectures in a slightly different but equivalent form.

15.3.

Definition.

For

r > 1 —

let

which are defined over the field of

N (Y) r q

be the number of points of

elements.

If

n

Y

^

is the dimension of

Y let

R r (Y) = q " m N r ( Y ) .

15.4.

It follows easily from Poincare duality and the Weil conjectures as

stated above that we can write the series

Nr(Y)t7r) in the form Q 1 (t)Q 3 (t)...Q 2n-1 (t)/Q o (t)Q 2 (t)...Q 2n (t) where

- 179 -

Q.(t)= I T i y..

for complex numbers

and where deg Q.

satisfying

is the

ith Betti number of

Y.

We shall use 15.4 to calculate the rational Poincare polynomial of the quotient variety

M

associated to the action of

G

on

X.

(It seems to be

natural to use this dual form of the Weil conjectures here.

This is what

Atiyah and Bott do when comparing their methods with those of

[H & N ] .

Using the ordinary form corresponds to using cohomology with

compact

supports, and it is difficult to make sense of this for the infinite-dimensional manifolds in [A & B]).

For simplicity suppose that that

G acts freely on X

G

is a subgroup of

GL(n+1).

We assume

. The argument we shall use runs as follows.

We may assume throughout that the action of

G

on

X

is defined over

R and that all the (finitely many) quasi-projective nonsingular subvarieties of X and subgroups of G which we shall need to consider are also defined over R and have nonsingular reduction modulo

IT. We may also assume that their

dimensions are unaltered by reduction modulo conjectures still hold if

q

IT.

is replaced by some power

Moreover the Weil q .

Hence we may

- 180 assume that

all

subvarieties

of

X

and subgroups of

consideration are defined over the field

F

SS

( X-) £ R(XSS)) = N N(X) £_

15.5

r

X

induced by the action of

TT of the stratification of

of G, and hence using the results of

r

—

§13

d 3

3 e B

gives us an inductive formula for

N (X

formula for the Poincare series

P. (X

explicit formula can be derived for

X induced by the action

m

ss

R ( z R ss m )R(G/P ft ) R(z r p ,m

each projective variety for

P (M)

r

p

and integers 0 f

G

on

X

to obtain stratifications of

X

G

and 7T

on X.

X

and to that

It is necessary to

- 181 investigate the relationship between these stratifications. check that they can be indexed by the same set set for the stratification of

X

M(T) SB Y(T) ® Q where Y(T)

B.

Recall that the indexing

is a finite subset of the is the free

First we must

Q-vector space

Z-module consisting of all one-

parameter subgroups of the maximal torus T. Since T

has the same rank

as T there is a natural identification of M(T) with M(T ). The WeyI group actions coincide under these identifications, and so do the weights a ,...,ot of the representations of T and T X .

which define their actions on X and

We may assume that the norms chosen on

M(T)

and

M(T )

IT

also

IT

coincide.

Hence the stratifications of

X

and X

may be indexed by the

same set B (see 12.8). Let ( S j B e B} 3

be the stratification of X and let {S

13 e B} be 3 /ir

the stratification of X .

Under the assumptions already made the following

lemma follows without difficulty from the definitions of §12. 15.6. Lemma. The stratification {S |3 e B} is defined over R and 3 (S ) Moreover

(Yf S ) , (zf S ) p

IF

p

= S.

for each

and (P-) IF

M B ,

coincide with the subvarieties of X

p IT

and parabolic subgroup of

G

defined in the corresponding way for the

on X . Finally the quotient variety M = XSS/G satisfies

action of G 11

for every r > 1 .

IT

IT

Rf(M) = Nr(XSS) N r (G)" 1

- 182 In order to apply 15.4 we need to calculate last lemma suggests that we should investigate each

N (M) N (X

for each r >^1. The ).

It also tells us for

N (S ) is the number of points in the stratum r 3 which are defined over the field of q elements, and so

X

3 e B that

N (XSS) = N (X) - Y

15.7

r

r

^--~ 35*0

S

of

P,TT

N (S o ). r

p

Moreover

by the lemma together with theorem 2.26, and so

15.8

for each

$ e B.

open subsets

P/m

§4

we can decompose Y

{Y | 0 < m < dim X} p,m — —

real codimension GY

As in

m

in

X.

Then

P

into a disjoint union of

such that each component of So

which have complex codimension

Yo has P/iTi

is the disjoint union of open subsets l

/ 2 d(3,m) = V2m - dim(G/P_) p

in

X. There is also a locally trivial fibration

P

6=

,.,ss (Y

M

such that each fibre is an affine space that

(see 13.2),

from which it follows

- 183 -

for each r >J\. So by 15.7 and 15.8 we have

Rr(XSS) = Nr(X) -

15.9

for each

r >^ 1, where the sum is over nonzero

$ e B

and integers

0