Invariant Forms on Grassmann Manifolds. (AM-89), Volume 89 9781400881888

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Annals of Mathematics Studies Number 89

INVARIANT FORMS ON GRASSMANN MANIFOLDS BY

WILHELM STOLL

P R IN C E T O N

U N IV E R S IT Y

PRESS

AND U N IV E R S IT Y

OF

P R IN C E T O N ,

TOKYO

NEW

1977

PRESS

JERSEY

Copyright © 1977 by Princeton University Press A L L RIGHTS RESERVED

Published in Japan exclusively by University of Tokyo Press; In other parts of the world by Princeton University Press

Printed in the United States of America by Princeton University Press, Princeton, New Jersey

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

CONTENTS

PREFACE

vii

GERMAN LE T T E R S

ix

IN T R O D UC T IO N

3

1. F L A G SPACES

11

2. SC H U B E R T V A R IE TIES

27

3. C H ER N FORMS

35

4. TH E THEOREM OF B O T T AND C H ER N

43

5. TH E PO IN C AR E D U A L OF A SC H U B E R T V A R IE T Y

57

6. MATSUSHIMA’S THEOREM

64

7. THE THEOREMS OF P IE R I AND G IA M B E L LI

82

A P P E N D IX

103

REFERENCES

110

INDEX

113

V

PREFACE

Schubert v a r ie t ie s d e s c rib e the cohom ology of G rassm ann m anifolds. T h e P o in c a re d u a ls of the Schubert v a rie tie s gen erate the vector s p a c e of invariant form s, w h ich as an exterior a lg e b ra is isom orphic to the cohom ology ring of the G rassm ann m anifold. In fac t, G ia m b e lli’s theorem a s s e rts that this a lg e b ra is generated by the b a s ic Chern forms.

Thus the

theory of in variant forms on G rassm ann m anifolds is important for the study of vector b u n d les and ch a ra c te ristic c la s s e s . I becam e in terested in this s u b je c t matter b e c a u s e of its ap p lic atio n s to v a lu e d istribu tio n theory.

Bott and Chern stu died the eq u idistribu tio n

of the z e ro e s of holom orphic s e c tio n s in holom orphic vector b u n d les.

R e­

cen tly, C ow en co n sid ered Schubert z e ro e s of holom orphic vector bu n dles on p se u d o c o n c a v e s p a c e s .

In order to extend this theory to p se u d oc o n vex

s p a c e s , a “ d e fic it term ,” w hich requ ires the c a lc u la tio n of a certain in­ variant form on a G rassm ann m anifold, has to be id en tified .

On this matter

I co n su lted Y . M atsushim a who obtained a very u s e fu l theorem. is lon g, d iffic u lt and u s e s d eep re su lts about L i e a lg e b ra s .

H is proof

L a te r I found

a sim p ler proof w hich u s e s only fib e r integration and elem entary c o n s id e ra ­ tions.

I em ploy the sam e method to obtain a number of known re su lts such

as the d uality theorem, the theorem of P ie r i and the representation theorem of Bott and Chern.

P i e r i ’s theorem e a s ily im p lies G ia m b e lli’s theorem.

T h e proof of M atsu sh im a’s theorem given here is about as d iffic u lt as the proof of the d uality theorem. sectio n diagram .

It u s e s fib e r integration in the d ou ble inter­

T h e theorem of P ie r i is much more d iffic u lt to obtain

and requ ires fib e r in tegration in a triple in tersection diagram .

F ib e r inte­

gration a lo n g smooth fib e rs is w e ll known, but d oe s not s u ffic e here.

The

ex ten sio n to fib e rs with s in g u la ritie s is not triv ia l, but has been e s ta b ­ lish ed in Ch. T u n g ’s th e s is .

vii

viii

PREFACE

A fte r this monograph w a s w ritten, but befo re it w a s sent to the pub­ lish er, M atsushim a re ceiv e d a latter from J. Damon in w hich Damon p roves M atsu sh im a’s theorem b a s e d on Damon [9 ] and [1 0 ].

T h is proof

u s e s the G y s in homomorphism computed by a resid u e c a lc u lu s . M ost of the re su lts in this monograph are known.

T h e method of proof

is new , e s p e c ia lly in the c a s e of M atsu sh im a’s theorem. e a s ily a c c e s s ib le .

T h e topic is not

T h erefo re an introduction has been w ritten here in

order to provide a c le a r, coherent, in tra n s ic a lly formulated account w hich w ill b e u s e fu l for a p p lic a tio n s to va lu e distribution theory, and w h ich may have a w id er ap p e a l as w e ll. I thank the N a tio n a l S c ien c e Foun dation for p a rtia lly supportin g this rese arch under Grant M PS 75-07086.

WILHELM S T O L L

GERMAN LETTERS A

SI

a

o

B

93

b

b

C

©

c

c

D

$

d

b

E

©

e

e

F

S

f

f

G

©

g

g

H

§

h i )

I

8

i

i

J

S

j

i

K

Si

k

I

L

8

I I

M

1

m m

N

91

n

n

0

S3

o

o

P

%

P

9

Q

£l

q

q

R

SR

t

r

S

‘ " > a p ) < n -p . If

Q = a 0 +***+ap.

v = ( v Q,-* *, v p)

with

w here

a^ < b^

for

T h e fla g m anifold

vq f G g + q (V )

for

q = 0 , l > , “ »P

such that E ( v 0) C E ( v x) C ••• C E ( v p) .

T hen

F (C t )

d(a),

and

is a com pact, connected, com plex m anifold of d im ension U

acts tra n s itiv e ly on

Schubert v ariety E ( v q ) > q+1

S (v , a )

for a ll

s u b s e t of d im ension

S(q)

c o n s is t s of a ll

q = 0, -- -, p . a

F(a) .

of

Then

G p (V ) .

and

x e G p (V ) S (v ,C t)

veF(ct). su ch that

The dim E ( x ) fl

is an irred u c ib le an aly tic

T h e Schubert fam ily

= ! (x , v ) f G p ( V )

is an irred u c ib le, an a ly tic s u b s e t of n ' S ( a ) -> G p (V )

Take

X

F ( a)|

X

f S (v, a ) !

G p (V ) x F ( a ) • T h e p ro je ctio n s

a:S(Ct)^F(a)

are lo c a lly triv ia l and s u rje c tiv e

3

INVARIANT FORMS ON GRASSMANN MANIFOLDS

4

(C o w e n [8 ] and Lem m a 2.1). and

a*

Qa >0

H ence the fib e r integration operators

rr^

are d efin ed (T u n g [3 3 ]).

T h ere is one and only one volum e form

on

is invariant under the action o f

F(q)

such that

Qa

U

and

F( Ct) T hen

c( a ) =

77* c t * (Q q

a = d (p ,n ) - a .

) > 0

T h e form

is the P o in c a re d ual o f

is a form of b id e g re e

c(a)

S(v,a)

gen erate the cohom ology of Take

a e @ (p ,n ) .

A ssu m e that

G p (V )

with

is invariant under the action o f U

and

for each

VfF(a).

A

Let

r be an in teger w ith

s = d (a )-r > 0

be a form of c la s s

on

T h e forms

c(a)

G p (V ) .

.

C °°

invariant under the action of

o < r < d (p ,n ) - a .

D efin e

A ( a , r ) = \ b € @ (p , n ) |b > a Let

(a ,a )

and d egre e

U .

D e fin e

and 2s

b = a + r! . on

F(a) .

A =

.

Theorem [2 1 ] a s s e rts the e x iste n c e o f constants

A =

^

A ssu m e

A

is

M atsushim a’s

such that

yj, c ( b )

b e A ( a ,r) (T heorem 6.12). is that yj, = 0

C le a rly , o u tside

A

is a lin ear com bination o f

A(a,r) .

c(b) .

T he point

T h e proof given here u s e s the w ed ge

product form ula o f fib er integration [2 2 ] and the d ou ble in tersection diagram . T h e ta u to lo g ic a l bun dle Sp ( V ) = { ( x , b ) f G p ( V ) x V| b e E (X )!

is a su b b u n d le of the triv ia l bundle quotient bundle.

G p (V ) x V .

T h e herm itian metric

metric alo n g the fib e rs of

G p (V ) x V

I on and

V

Let

Q p (V )

be the

d e fin e s a hermitian

S p (V ) .

T h e bun dle

S p (V )^

INTRODUCTION

orthogonal to

S p (V )

is isom orphic to

fined a lo n g the fib e rs of c( M

" - ’ cn -p M

c [p ] = 0

if

of

q n -p .

T h e Theorem of G ia m belli [1 3 ] a s s e rt s

that c ( a ) = d e t c n_ p _ a . _ i + j [p ]

(T heorem 7 .5 ), and is b a s e d on the Theorem of P i e r i [2 1 ], w hich com­ putes

c( a )

a

c ^ fp ]

V e s e n tin i [3 4 ]).

in terms

c(b)

T h e proof given here (Theorem 7.4 ) u s e s fib e r integration

in the trip le in tersection diagram . c la rify

(C h ern [5 ], H o dge [1 8 ], and

S p e c ific care is taken (Lem m a 6 .7 ) to

the gen eral p o sitio n mentioned by Chern [5 ].

Let

p, q and

p

b e in tegers with

o < q < p < ^ t < n .

D e fin e

F p q = ! ( x , y ) < G q( V ) x G p ( V ) | E ( x ) C E (y )j .

Let

t

: F p q -*

G p (V )

and

n : F p ^ -> G q (V )

be the p ro je ction s. T h e B ott-

Chern R ep re se n tatio n Theorem [3 ] sta te s that

D (q - l , p - l )c ^ _ p[p] =

r^ 77*(c^i_ q [q ]

a C l [q ](P “ q ) q ) .

T h e proof announced in [2 8 ] w ill be fin a lly given here. T h is monograph is alm ost s e lf-c o n ta in e d .

Some elem entary fa c ts on

G rassm ann m anifolds and fla g m anifolds are not proved here. that the open Schubert c e ll s u b d iv is io n of s e e M iln o r-S tash e ff [2 3 ]. the open Schubert c e l l

T h e proof that *

S

G p (V ) S (v, a )

d e fin e s a

F o r a proof C W -com plex

is irred u c ib le, and that

is bih olo m o rp h ically eq u ivalen t to

C

a

fo llo w s Chern [7 ] and is given in the ap pendix for the co n ven ien ce of the reader.

A pp lica tio n to va lu e distribution T h is paper w a s written with a p p lic a tio n s to v a lu e d istribu tio n in mind. Such an ap p lic a tio n s h a ll be outlined here. Let

F o r d e ta ils s e e [3 2 ].

W be a holom orphic vector bun dle of fib e r dim ension

k over an

INVARIANT FORMS ON GRASSMANN MANIFOLDS

6

irred u c ib le com pact com plex s p a c e

N .

T h e vector s p a c e

holom orphic se c tio n s has fin ite d im ension > 0 .

T h e e v alu a tio n map

A ssu m e that d efin e

e

e : N x V -> W

is s u rje c tiv e .

e x : V -> Wx

by

v = (Vq , •• •, Vp) e F ( a ) .

T h en

n+1 .

V

A ssu m e that

is d efin ed by

Take

p = n -k

e (x ,s ) = s (x ) .

W is s a id to be am ple.

ex ( s ) = s ( x ) .

o f g lo b a l

a =

If

x e N ,

ap ) e @ (p , n )

and

T h e Schubert ze ro s e t P

s w (v > ° ) =

n

lx e N |dim e x ( E ( v q ) ) < aq !

q= o is an aly tic in Let

N .

M be an irred u c ib le com plex s p a c e o f dim ension is given.

A ssu m e a

holom orphic map

f : M -» N

many

v e F(a)?

T h e fam ily

[33 ].

T h erefo re a so lu tio n can be given a lo n g the fo llo w in g lin es.

T a k e a herm itian metric herm itian metric

I

S(C t)

I

on

f(M ) ft S ^ (v , a ) =(= 0

q < 0

or if

q > k .

b id e g re e

(s ,s )

with

Take

for

is a d m is s ib le in the s e n s e of T u n g

V .

By

e:NxV->W ,

is d efin ed alo n g the fib e rs of

be the a s s o c ia t e d Chern forms of if

When is

m .

W for

W .

q = (),•••, k .

a quotient

Let

Cq(W, I ) > 0

D e fin e

Cq(W, I ) = 0

a 6 @ (p ,n ) .D e fin e a form of c la s s

s = d (p ,n )-a

C °° and

by

cw( a ) = d e t c k_ a . _ i + j( W, I)

on

N .

T hen

c^(ct)> 0

A ssu m e that

is c lo se d .

M c a rrie s a p seu d ocon vex exhaustion

that a no n -n egative function ddcr > 0

r of c la s s

C °°

r .

is given on

T h is m eans,

M such that

and such that M [r] = {x 6 M |r (x ) < r!

is com pact for each

r> 0 .

A ls o defin e

M (r) = lx £ M |r (x ) < r! D e fin e that

v = d d cr .

v > 0

A ssu m e that

M< r> = Sx e M |r (x ) = r! . q = m+ a -d (p ,n ) > 0 .

on som e non-empty open s u b s e t o f

M .

If

D efin e

q > 0 ,assum e

INTRODUCTION

A f(r, a ) =

f*(cw( a ) ) A « q > 0

J* m

T f (r, a ) =

7

[ r]

f

A f(t, a ) d t > 0 .

J 0

H ere

Tj:(r, a )

is c a lle d the c h a ra c te ristic o f

that there e x is t s at le a s t one one point

x e M ,

T f( r , a ) -> °o for

su ch that r -> oo .

v e F( Ct )

f

for

a e @ (p , n ) . A ssu m e

for w hich there e x is t s at le a s t

f - 1 (S w (v , a ) )

has d im ension

q

at

x . T hen

In [3 2 ], the fo llo w in g theorem is proved:

If

A f( r , b ) for

r

T f( r , a )

for eac h ]= 0

b e @ (p , n )

for alm ost a ll

with

b > a

b = a + 1 , then

and with

f(M ) fl Sw (v , a )

v e F(Q) .

T h e proof is o f in terest here, s in c e it in v o lv e s in variant forms on G rassm ann m anifolds and the T heorem s of G ia m b e lli and M atsushim a. H en ce a short outline of the proof s h a ll be given . Take

x e N .

T h e kernel

lin ear s u b s p a c e of

V .

Sx

of

e x : V -> Wx

One and only one

E ( 0 ( x ) ) = Sx .

T h e map

0 * (c (a ))

Sw ( v , a ) = 0 _ 1 ( S ( v , a ) ) .

used.

is a (p + l)-d im e n s io n a l

0 on

and

cf> : N -> G p (V )

F o r d e ta ils s e e [3 2 ].

on

F (a )-M

F(a)-{v|

S(v,Q),

e x is t s for each

and such that

d efin e

Ay =

S(v, a ) .

D e fin e

cw ( a ) .

Take

g * (A g ^ ) = Ay

0 .

A w y = ^>_1 ( A y ) > 0

v e F(a)

veF(a)

T h en on

and assum e that

such that

for a ll

ddc Ay = 0 a

g e U.

dd c A y = c ( a )

N - S w(v,a)

On on

with

z e A,

a m ultiplicity

0 \ (z ) > 0

A = f - 1 (Sw (v , a ) )

is a s s ig n e d .

G p( V ) ~

ddcA w y =

empty or pure q -d im en sio n al, w hich is the c a s e for alm ost a ll F o r each

G p( V ) -

is either v e F( Ct ) .

T h e counting

8

INVARIANT FORMS ON GRASSMANN MANIFOLDS

function, the v a le n c e function and the d e ficit o f for a ll

r > 0 by

J

nf(r.v) =

f

for

(v, a )

are defin ed

uq > 0

A (1 M [ i ]

N f(r ,v ) =

r

nf (t ,v )d t > 0

J o

J

D f(r ,v ) =

f * ( A ff>v) a uq+1 > 0 .

M [r]

F o r alm ost a ll

r > 0,

the com pensation function is d efin ed by

mf (r ,v ) =

f* (A W v)

J*

a

dc r

a

uq >

0

.

M< r >

The F ir s t Main Theorem T f (r, a ) = N f (r ,v ) + m£(r,v) - D f (r ,v )

holds and extends the d efin itio n of

m£(r,v)

to a ll

r> 0 .

T h e in tegral a v erage

f

A (x ) =

Av ( x ) f l a ( v ) > 0

F(a)

e x is ts and is a form on A = 7r;+,cr*(A) > 0 b id egree

(s ,s )

F(a)

invariant under the action of

is invariant under the action o f with

s = d (p ,n ) - a .

M atsu sh im a’s Theorem constants

D e fin e

U

on

U .

G p (V )

A(a) = A(a,l).

y a jj > 0 e x is t such that

T h e form and has By

INTRODUCTION

A =

2

9

ya b c( a ) •

beA(a) D e fin e

A w = (£ * (A ) .

mf( r , a ) =

An ex ch an ge of integration sh o w s that

mf ( r , v ) f l Q =

J '

f* (A ff)

F(q)

D f(r, a ) =

f* (A w )

F(a)

2

=

dc r

a

vq

M< r >

D f ( r , v ) Q fl =

J

a

a

yq+1

M [r]

y a b A f(r' b > -

BfA (a) S to k e s ’ Theorem im p lies that

D f ( r , a ) = mf ( r , a) .

T f( r , a ) =

j

T h erefo re

N f (r ,v )Q 0 .

F(a) The set

B = l v e F ( a ) | f(M ) fl Sw (v , a ) =(= 0 !

O < b f( a ) = J

is

m easu rable.

Then

n tt < 1 .

B

B ecause

N f (r ,v ) = 0

eq u a lity

N f (r ,v ) < T ^ r , a ) + D f:(r,v)

O
0 .

For

INVARIANT FORMS ON GRASSMANN MANIFOLDS

16

T h e flag s p a c e o f s y m b o l 21 is d efin ed by F (2 l)=

T hen

F (2 I )

G™-(V)

l ( v 0 , - , v p ) f G g j ( V )| E ( v 0) C - C

E (v p )l .

is a connected, com pact, smooth, com plex subm an ifold of

with dim ension

P d (2 t) = dim F ( 2 I ) =

( n - a q) ( a q - a ^ )

.

q= o If

v € G g j(V ) , then the co o rd in ates o f

V

‘ , vp

such that

for each

g e G L (V ) .

tran sitiv ely on

F (2 I)

e e F (2 I) .

c lo se d subgroup of

acts on

G L (V )

Ggj-(V)

with

G L (V ) .

G L (V )/ P e

bun dle

77 : G L ( V ) -> F (2 I )

Let

t : G L ( V ) / P e -> F ( 9 f )

77 ( g )

e x is t s such that

d epends on the b a s e point T h e isotropy group Gs -a (v ) p p-1

such that

=

T hen

e G n( V ) .

is a

be the left c o s e t o f

be the left c o s e t s p a c e .

p = 77 .

V

F (3 I) . is a g e

A holom orphic fib e r

= g (e ) .

l ° p = 77 .

One and only one H ere

F ( 9 f ) = G L ( V ) / P e can be id en tified such that

identity and such that

If

P Q = { g e G L ( V ) | g (e ) = e!

p (g ) = g P e

is d efin ed by

g (F (2 t )) =

acts holom orphically and

acts tran sitiv ely on

T h e isotropy group

and let

H en ce

T hen

1 I(V )

G L (V )

map

G L (V )

as a group of biholom orphic maps.

hermitian vector s p a c e , Take

are a lw a y s denoted by

v = ( v 0 , - , v p) .

T h e gen eral lin ear group F (2 t )

v

1

is b ije c tiv e .

becom es the

1

T h is id en tificatio n is not in trinsic, but

e .

P A can be computed e x p lic itly . e

-c

a--*ac

for

P ic k

p = (),•••, p + l

c

e

with

e F

^

V = E ( c 0) © E ( Cl ) © • • • © E ( Cp+ 1 ) .

Let

j

fi

: E (c ) p

tion for each L ( V c jx) '

V

be the in clu sio n and let

/n = 0, - " > P + 1 • F o r

T hen

77

p

: V -> E (c ) p

g ( G L ( V ) , d efin e

g

be the p ro je c=

° g 0)v t

1. FLA G SPACES

17

p+l £ =

>

° TT

1 ° £

IL,V=0 Write

g = matrix ( g ^ ) • T hen

Pe = { g e G L ( V ) |

Let

I

= 0 V n > v\ .

b e a p o s itiv e d efin ite herm itian form on

U(V,I)

is d efin ed .

P ic k

e rankx g = n + q - dimx g_ 1 (y ) .

H en ce

dim x g_1 (y ) > q .

and contains T h erefo re

g_ 1 (y ) ,

M = B

L E M M A 1.3. Let

N

B ecause

w e have

f _ 1 (y )

is irre d u c ib le , q-d im en sio n al

g_ 1 (y ) = f —1(y ) .

H en ce

z e g- 1 (y ) C B .

is irred u c ib le and (n + q )-d im e n sio n a l; q.e.d .

M

Let

m .

be an irreducible com plex sp a c e of dim ension

be a con nected, sim ply con nected, com plex manifold of dim ension

n with

m -n = q > 0 .

holom orphic map.

Let

Then

f : M -» N

f - 1 (y )

be a s u rje c tiv e , loca lly trivial,

is irreducible and

q-d im en sion a l for all

y L

D e fin e

h : L -> N

be the branch of

d efin ed by

g- 1 (h- 1 ( U ) )

g

g (x ) = ( f (x ), g0( x ) )

is continuous.

is open.

h (y ,z ) = y .

f ~ 1( f ( x ) )

Introduce the quotient top o lo gy on nected and

by

F o r each

co ntaining

is s u rje c tiv e with L .

T h en

If

U

is open in

T h erefo re

h

is continuous.

L

N ,

x .

x eM ,

T h e map

h° g = f .

is a rc w is e con­ then Let

f""1^ ) = V

be open in

L .

INVARIANT FORMS ON GRASSMANN MANIFOLDS

22

T h en set h

U = g_ 1 ( V )

is open in

h (V ) = h (g (U )) = f ( U )

M with

is open in

g (U ) = V . N .

Because

H en ce

h

f

is open, the

is open.

T r iv ia lly ,

is su rje c tiv e . Take

b e N .

e x is t s such that F = f _ 1 (b )

T h en

L (b ) 4 0

is at most co u n table.

L ( b ) = }F ^ |v e Z [ l , p ] }

with

is the d is jo in t union o f the

F

with

T h erefo re for

v e Z [l,p ] .

f - 1 (U ) ( x ,b ) and

and

77:

for a ll V

F x U -> U

x e F .

such that

is the projection .

T hen

V

U

of

a = f , w here

A ls o w e can assum e

= a -1 ( F x U )

is the d is jo in t union o f the

77 °

Then

B ecau se

is lo c a lly triv ia l, there e x is t s an open, connected neighborhood and a biholom orphic map a : V -* F x U

p < oo

fi =|= v .

Take

b

V = a (x ) =

is open and connected in

for v e Z [ l , p ] .

f

V,

y eU .

Then v „ n f - 1 (y) =

is

a branch o f f _ 1 (y )

with

F ^ (y )

F ^ y ) = a ~ 1(F v x i y ! )

4F

(y )

for

v 4= fi • H ence

L (y ) =

T h e s et

.

W, = g f y j = !(y , F ^ (y )) | y U

continuous, and is therefore a homeomorphism. T h erefo re , provided that

L

is b ije c t iv e , O b s e rv e that

is a H a u sd o rff s p a c e ,

open, and h- 1 ( U ) = W .

h : L -> W

is a c o v erin g

sp a c e .

Take c^,

and c2 in L with Cj 4 c2

neighborh oods h_ 1 ( A

)

M

of

h (c ^ )

• ^

e x is t such that

is an open neighborhood o f

c

fl

in

K ci)

fl A 2 = 0 . L

with

h (c 1) = h (c 2) = b , the previou s construction h olds. W

with ^11

c ^

e W vn

e x is t s for

fi = 1,2 .

^ M c2) > open

B ecau se

Then

B . fl B 0 = 0 . l

z

B^ = If

One and only one h|W ^u

is in jec tiv e ,

1. FLAG SPACES

v . =1= v 0 . and

T h erefo re

h : L -> N

s in c e

L

W

fl W

= 0 . H en ce

is a c o v e rin g s p a c e .

is sim ply connected,

# L ( y ) = 1 for a ll

y e N .

tiv e ly Take

MQ = M - S

and

b e M .

D e fin e

neighborhood 77°a = f ,

of

Let

w here

if

F .

a-1 (T x U )

and

N

are connected and

be the s e t of s in g u la r points of

F o r each

yeN

F^Cy)

T h en

let

L (y )

T x U

M.

(r e s p e c ­

(re s p e c t iv e ly

f ~ 1(y ) ).

L ( b ) = !F ^ |v e Z [l, p ]S

and a biholom orphic map a : V -> F x U

V = f '^ U )

T h en

and

with

v 4= ii • T h ere e x is t s an open, connected

and

w e can assum e that a ( x ) = ( x ,b )

T h en

S

f Q = f |MQ .

F = f"”1^ ) .

b

is a H a u sd o rff s p a c e

T h e lemma is proved in this s p e c ia l c a s e .

F^ =(= F^

U

L

L

is a homeomorphism. T h erefo re

L Q(y ) ) b e the set of bran ch es o f

p < oo and w ith

s e t of

Since

h : L -* N

C o n sid e r the g en era l c a s e . D e fin e

23

n : F x U -> U for a ll

F H S = T .

D e fin e

L ° ( b ) = lF ^ | v e Z [ l , p ] i

where

is the projection . M oreover

x e F .

is the s in g u la r s et of

such that

Let

T

FxU .

be the sin g u la r H en ce

V fl S =

F ° = Fy ~ T = F ^ - S = Fv n MQ =j= 0 . F® =|=

f ° r v 4= #i ■ T h erefo re the

fo llo w in g three co n s e q u e n c e s have been e s ta b lis h e d : 1)

If

y 6N ,

2)

T h e holom orphic map

3)

F o r each

C o n se q u en tly , T h erefo re

then

# L ( y ) = # L Q(y ) .

y e N ,

# L Q(y ) = 1

f - 1 (y )

f Q : MQ ^ N

the bran ch es of for a ll

y e N .

is s u rje c tiv e and lo c a lly triv ial. f ^ 1( y ) H en ce

are d isjo in t. # L (y ) = 1

is irre d u c ib le and q -d im en sio n al for each

T h e s e Lem m ata w ill be quite u se fu l.

for a ll

y e N ; q .e .d .

T h e first alread y help s in prov­

ing the fo llo w in g resu lt:

L E M M A 1.4.

Take

p^ e Z [0 ,n ]

for

N k = | ( x , v l f - , v k) £ P ( V ) x ' is an irreducible ana lytic s u b s e t of

Then

I ]

Gp (V)

#x= 1 k Y k = P (V ) x H G_ (V ) . #i=l p#i #i=l

y e N .

^

24

INVARIANT FORMS ON GRASSMANN MANIFOLDS

P ro o f.

If

k = 1 ,

then

is bih olo m o rp h ically eq u ivalen t to

F_ P1

1 under the map G_ (V ) . Pi

(y ,x ) .

A s a fla g s p a c e ,

irred u c ib le . G p (V )

(x ,y )

N 1 is an an a ly tic s u b s e t o f P ( V ) x

H en ce

Fn Px

is a connected m anifold.

(Lem m a 1.2 a ls o could have been ap p lie d , b e c a u s e

is a s u rje c tiv e , p -fib e rin g map with irred u c ible fib e rs

N o w , under the induction h y p oth esis for proved for tt:

k .

Fpk -* P ( V )

of

H en ce

T hen

N jc_ 1 is irred u cible.

be the projection.

( p >77) ( s e e T u n g [3 3 ] § 8 .1 ). N k = ! ( z , w ) f N k_ j x Fp

Let

k- 1 , Let

the lemma w ill be

Then

|p (z ) = 77(w)i

X

(G p k( V ) x P ( V )

xf n E(yi ^=i

v = ( v 1 ,* •*, v jc_ 1) • H ere

A =

T h en

A

V

morphic map

j : A -» Y k

Y k_ j x G k( V ) x P ( V )

and

Fp^

(1 -4 )

N k = j ( N k)

x e P (V )! .

A b ih o lo ­

is d efin ed by

is an aly tic in

be the p ro je ction s.

D e fin e

w ith N k C A .

j ( ( x ! V J > '"> Vk_ 1), (v k , x )) = (X j V ^ -'-.V J j)

T hen

)

is an an a ly tic su bset.

' ‘ ,vk- l ) ’ ( v k >x))|vft e G p ( V )

is an aly tic in

and

be the re la tiv e product

T a k e the standard model.

( ( X , v ) , ( v k , x )) f Y k l

w here

r : Fp

E (x ) .)

p : N j

is

A ssu m e

q > 0 .

B y Remmert, the s e t L q = | z f N k |dim z f - 1 ( f ( z ) ) > q - l ! = Iz f N k | rankz f < dim N k - q + 1}

is an aly tic (s e e [1 ] §1).

T h erefo re

( v l » ' * ' » v k)

v € ^ (k q )

( x ,v ) e L q

6^ k * T hen w ith

dim^x

f(L ^ )

is an aly tic.

if an^ only if

Take

v =

x e P ( V ) e x is t s such that

f _ 1 (v ) > q - 1 , w h ich is the c a s e if and only if k

(1 .5 )

dim x

f|

E ( V/i) > q - 1 •

H =1 S ince H en ce

E ^ )

fl — fl E (v ^ ) = E

v e f(L q )

if and only if

is a p ro je ctiv e p lan e, dim E > q - 1 ,

f|

E (v p > q .

H =1 T h erefo re

M

= f(L

)

is an a ly tic in

X^ ;

is irred u cible.

w hich h o ld s if and only if

k

dim

E

q .e .d .

INVARIANT FORMS ON GRASSMANN MANIFOLDS

26

A subset if

M -U

and if

U

o f a com plex s p a c e

is an aly tic.

U ^ 0

M

is s a id to be Z a risk i open in

A Z a r is k i open s u b s e t is open.

is Z a r is k i open in

M ,

then

U

If

M

M is irred u c ible

is d e n se in

M .

2.

a)

S C H U B E R T V A R IE T IE S

Schubert fam ilies Let

V

p e Z [0 ,n ]

be a com plex vector s p a c e of dim ension and

the s e t of a ll

(2 .1 )

a G p( V )

T hen

T h e re presen tatio n s of

G p (V ) x

G L (V )

a s a group of biholom orphic maps.

tran sitiv e in gen eral.

(2 .2 )

such that

is an an aly tic s u b s e t of

(x , e 0 ,***,ep) e S ( a ) . H en ce

g(S(a))=S(a)

m orphically on

e

P

T h e gen eral lin ea r group with

is

#x = 0 , l r #,,P •

p = (),•••,p .

A- - - At t b ) 0

F(q)

S(Ct)

)

a

D e fin e

X = P (n b

for

for

Take

of sym bol

v = (v 0," * ,v

dim E ( x ) n E (v ^ ) > p + 1

N o w Lem m a 1.5 im p lies e a s ily that

n+ 1 > 0 .

S(a)

g°

77

G p (V )

=

77 °

and

< j : S ( a ) -> F ( a )

g

and

F(a)

go /x+ 1 ; q .e .d . j*

S (v ,0 ,•••,()) = Svp ! .

dim E ( x ) fl E (v Re

dim E (v ) > n -p +

r

dim E (v ) = u + 1 for jr = 0 ,1 ," - . p . f* )>M+1

f ° r /x=0,l,-*-,P

x e S (v, a ) ,

then

dim E ( x ) = p + 1 and

If

if and only if

x e G D( V ) , r x = v

*

then

; q .e .d .

E ( x ) c E (v p) .

dim E ( x ) fl E (v p ) > p + 1 imply that

E (x ) C

E (v p ) , q .e .d . Ex

a m p l e

3.

Take

r r + 1

w ith E ( v r) C E ( x ) C dim E ( x )

E (x ) C

imply that

E ( v p) . If

a^ = 1

E (v p ) . A ls o dim E (v f ) = r + 1 E ( x ) DE (v r) . T a k e

0 < /x < r , then

fl E (v ) = dim E (v ) = /x+ 1 . fx f*

If

.

E (v ^ ) C E (x )

t < ^ < p ,

x e Gp(V ) .

then

dim E (x ) fl E (v ^ ) > dim E (x ) + dim E (v ^ ) - dim E (v p) = ( p + l ) + (/x+1) - (0 + 2 ) = fi + 1 .

H ence

29

2. SCHUBERT VARIETIES

H en ce

XfS(v,a) ;

q .e .d .

E x a m p l e 4. S ( v , l , “ -,1) = i x r G p(V)| E ( x ) c E (v p)l . C o n sid e r the short fla g m anifold - G p +1( V )

and

tt :

F p+1 p

F p+1 p -» G p( V ) .

w ith p ro je ction s

’■; F p + i p

Then’ S ( v , l , - , 1 ) = ^ ( V p ) ) •

E x a m p l e 5. If 0 < q < p , then S (v ,a Q, - •• ,a q , n - p , ’ ** ,n -p ) = { x f G p ( V ) | d i m E (x ) f l E ( v ^ ) > M + 1 V /x = 0,*•• ,q! . Take

P ro o f. Take

x

G p(V )

^ e Z [q + l,p ] .

with

dim E ( x ) fl E (v ^ ) > /* + 1 for a ll

fi e Z [0 ,q ] .

T hen

dim E ( x ) fl E (v ^ ) > p + l + n - p + j K + 1 - (n + 1 ) = /z + 1 .

H en ce

X f S (v ,a 0 , - , a

REM ARK. 4> : F ( Q )

,n -p ,-,n -p ) ;

In the c a s e o f E xam p le 5, d efin e F(b )

be the p rojection

a 0 = ^ oor ; S ( a ) -> F ( b ) . a ll

g e G L (V ) .

E x a m p l e 6.

T h en

B y Lem m a 1.1

E ( v 0) C P ( V )

b e the com plete fla g s p a c e ( n + 1 be the s e t of a ll

dim E ( x ) fl E (v g ) > ju+ 1 for that

b = ( a 0 ,--* ,a q) e @ (q ,n ) .

0 ( v Q, --- ,v p) = ( v 0 ,-**,V q) .

C o n sid e r the short f l a g m anifold

b)

q .e .d .

S

^ = 0, l , - - - , p .

is an a n a ly tic s u b s e t o f

z e ro e s ).

(x ,v ) e G p ( V ) x F**

T ak e

such that

Lem m a 1.5 irhplies e a s ily

G (V ) x F

.

C o n sid e r the commuta­

tive diagram (2 .5 ), w here a ll maps are the natural projections^

30

INVARIANT FORMS ON GRASSMANN MANIFOLDS O'

S ------------------------------------

F

„

D i a g r a m (2 . 5 )

For

v € F n d efin e

(2.6)

a = ( a 0 ,--- ,a p ) = S < dj a > .

d e fin e

If

a j _ 1 = aj , d efin e

dj S < v, a > = S < v, dj Q >

a j _ 1 = aj .

(2.8)

if

a j_ 1 < a j ,

and

(9jS=0.

If

( 9 j S < v , Ct> = 0

vf F if

O b s e rv e

d jS < v , a > c S < v , a >

d-S c S

.

D e fin e * S

(2.9)

=

S

-

u P

a ,s < v ,o >

j= 0

* S

(2.10)

= S -

P

[J

djS

.

j=o

H ere

S < v,a>

sym bol S

a .

is c a lle d the open Schubert c e ll for the fla g v and the * ^ O b s e rv e that S < v , Q > and S are Z a r is k i open in

re sp e c tiv e ly

S .

The reader should note that d j S ( a ) ,

31

2. SCHUBERT VARIETIES

S ( a ) , d : S (v , a ) w = = S (w , a ) and S < v , G > is Z a r is k i

then

but

L E M M A 2.2.

S (v,a)

S *< v , a >

Take

is not in trin sica lly

v e F n and

P

*

S < v,Q> =

Pj

d efined by w

x e S *< v , a > ,

A b b re v ia te

D e fin e

m^ = dim E (x ) fl E (v ^ )

proved.

= a

(i

(i

= a + ju for (1 and

0 < m - n < 1 . fi fi

(i = 0 .

Now

= 0 .

and

jz =

N o w , c o n sid er the c a s e

m^ > f i + 1 for

nQ = 0, w hich

fi + 1 .

(),•••,p .T a k e

m = fi + 1 and fi

fi> 1 imply

im p lies

x eS .

n = fi fi

Then

have to be

fi .

A t firs t c o n s id e r the c a s e a Q = 0 .

1 = dim E (v k ) > m0> 1 • T h e re fo re nQ

= fi\ .

n^ = dim E (x ) fl E (v ^ _ x) .

T h e p ro o f p ro ceed s by induction for

A ss u m e

alone.

^

P ro o f.

m > /Lth- 1 and fj.

b

and a

{ x ^ S C v, a > |dim E (x ) fl E (v g

dim E (x ) fl E (v g ) =

then

open in

d e ® ( p , n ) . Then

(£= 0 If

and

mQ = 1 .

A ls o

T h en b Q = 0

and

b Q- 1 = - 1 , hence

a Q = b Q> 0 . A ssu m e

nQ > 0

. T h is

X £ S < v , d 0 Ct> w hich is wrong. H en ce

mQ < 1 .

T h erefo re

mQ = 1 .

T h e c a s e fi = 0

is

proved. fi > 1 and assu m e that the c a s e s

A ss u m e

first co n sid er the c a s e m i = fi . fi i

A ls o

a _1< a . fl L fl

if

and

n = fi fl

and

n = fiand fi

m

for a ll

A ssu m e

n fl

v e Z [0 , p ] . H e n c e m < n + l fl fl fi

f i = 0 , ' ' ’ ,P -

T h erefo re

> fi .

0 /z+ 1 . fl

m if

n = fi

N o w co n sid er A ls o

m > v + 1 1/

w hich is wrong.

= fi+ 1 . fL ^

x e S < v, a > .

and d efin e n fi

and m fi

as before.

for a ll

fi ~ 0, - - - , p .

A ssu m e

m^ = f i + l

for a ll

fi = 0, - - - , p .

A ssu m e

fi

H en ce

B y induction

fi + 1 and fi

b -1 = fi

= / z + l . T h erefo re

= fi+ 1 for a ll

C o n v e rs e ly , take T hen

- = a . Then fi i fi

m < n +1 = fi + 1 . fi fi

the c a s e v =(= fi

a

0, 1, -•*, fi-1

n = /z fi

INVARIANT FORMS ON GRASSMANN MANIFOLDS

32

x e ( 9j S< v , a > H en ce

L E M M A 2.3.

s*< v,

P ro of.

fpr som e

x e S * ;

g> n

V f F n and a

Take

s*< v,

A number

j e Z [0 ,p ] .

T h en

j - nj > j + 1 w hich is wrong.

q .e .d ,

b

and

in @ (p ,n )

with a

^b .

Then

b> = 0 .

q e Z [0 , p ]

e x is t s such that

a

= b for 0 < /li< q and M M ^ a q < b q - 1 . A ssu m e x e S < v , t t > fl

/v

3q ^ bq*

W .l.o .g .

a q < ^q • Then

S * < v ,b >

e x is ts .

T h en

q + 1 < dim E (x ) fl E (v ~ ) < dim E (x ) fl E (v t

q

= q

q 1

w hich is im p o s sib le ; q .e .d .

T h eo rem

2.4.

Take

v f F n and

Q f g(p,n) .

S*< v, Q >

S

is an

G p (V ) • The open Schubert

&-d im en sion a l irreducible^ analytic su b se t of c e ll

Then

C °

is biholom orphically eq u iva len t to

and is d en se in

S< v,G> .

T h e proof is lon g and com p licated and has therefore been put into the appendix.

It fo llo w s Chern [7 ] §8.

Theorem 2.4 has important co n seq u en ­

c e s for Schubert v a rie tie s and th ese w ill b e d is c u s s e d later.

T H E O R E M 2.5.

Take

v eFn .

Then

G p (V )

can b e rep resen ted as a d is ­

jo in t union of open Schubert c e lls Gp(V ) =

|J

s*

.

C U 0 (p ,n )

P ro o f. b

^

Take

e Z [0 ,n ]

x e G p (V ) .

such that

dim E (x ) fl E ( v l )

and

p

B y defin ition o f

For

p e Z [0 ,p ]

dim E (x ) fl E fv ^ ) > /z+ 1 .

[L

n

= dim E (x ) fl ECv^

^

b

M

there e x is t s a s m a lle s t integer

w e have

*) . fl

n < p . M

A ls o d efin e

H en ce

T hen

m

^

=

0 < m -n M

< 1.

M

m < n + 1 < p + 1 . T h ereM M

2. SCHUBERT VARIETIES

33

fore

m = ( i + 1 and n = /z for /x = 0 ,l,* **,p . A ssu m e that b < b . M M r r* for /x > 1 . T hen E (v ^ ) C E (v ^ ) . T h e re fo re /x + 1 < m < m x = fx , VM -l ^ w hich is im p o s sib le . T h ere fo re 0 < b Q < b x < •*'• < b < n . D e fin e a^ = b

.

m

T h en

>/x+l

0 < a 0 < a^^ < ••* < Up < n - p

for

=

0

, — ,p ,

w e have

"

and

Q e @ (p , n ) .

x e S < v ,Q > ,

Since

and b e c a u s e

n

*

for fx = (),••*,p ,

w e have

X fS

= fi

M

< v ,Q >

by Lem m a 2.3; q .e .d .

O f co u rse this s u b d iv is io n o f the G rassm ann m anifold into open Schubert c e lls is w e ll known and in fact is a s e e M iln o r-S tash e ff [2 3 ] Theorem 6.4. G p (V )

M oreover

H m(G p (V ), Z )

H ^m C^pOO, C )

and in fact

S < v, a >

(o r

S < v, G > )

b e taken as a b a s e o f m is odd and d uality If

( 2 .1

1

w ith

is b ije c t iv e w ith

H 2 m(G p( V ) , C ) . (2 .1 2 )

m is odd.

veF

n

fix e d ) can

H m(G p( V ), C ) =

# @ (p ,n ,m ) . # (g (p ,n ,q )

0

if

B y P o in c a re

with

q = d (p ,n )-m .

Then

: 0 (p ,n ,m ) -> @ (p , n, d ( p , n ) - m) = cfi ,

statem ent on B etti numbers. the vector s p a c e

h a s dim ension

< £ (C t )= Q * .

0

0 _1

if

T h ere is

In fac t the c e lls

(and with

B y duality

h as a ls o dim ension

d efin e

)

H m(G p (V ), Z ) = 0

Q e ® (p , n , m )

H 2 m( G p ( V ) , C )

t t f0 (p ,n ) ,

is a c y c le .

is d efin ed a s in § l . b .

H 2m(G ( V ) , C ) .

H 2 m(G p (V ), C )

S < v ,a >

is a com plex vector s p a c e o f dim ension

# @ (p ,n ,m ) , w h ere @ (p ,n ,m ) *

F o r a proof

T h is s u b d iv is io n im p lies that

is sim ply connected and that eac h

no torsion in

CW -Com plex.

w hich is in agreem ent with the p reviou s Sin ce

Inv 2 m( G p (V )

G p (V )

is a com plex sym m etric s p a c e ,

of in variant forms is isom orphic to

H en ce dim Inv 2 m(G p ( V ) ) = # © ( p ,n,m ) .

N o w , w e s h a ll return to Schubert v a rie tie s and the c o n seq u en ces of Theorem 2.4.

2. SCHUBERT VARIETIES

34

T H E O R E M 2.6. 1)

sio n

S(v, a )

v e F (a ).

Then:

is an irreducible analytic s e t of

o'. S ( G ) -> F ( a )

has irreducible fibers of dimen-

G . The Schubert family

d (a ) + 4)

S (a )

is irreducible and has dim ension

a . (F o r the definition of d ( a ) s e e § l b ). Th e project ion

n : S ( 0 ) -» G p (V )

has irreducible fibers of dimen­

d ( a ) + q - d (p ,n ) = d ( a ) - a * .

5) d (a ) +

P roof.

and

Q .

T he projectio n

3)

s io n

a < f @ ( p ,n )

The Schubert variety

dim ension 2)

Take

If

x € G (V ) , f

then

S (a ) x

is irreducible and has dimension

a - d (p ,n ) .

1)

S< w ,a >

A fla g

W f F n e x is t s such that V-

ab b rev iate

77

= n „ 0n . V

Then

bp =

5. THE POINCARE D UAL OF A SCHUBERT VARIETY

59

J F(6)

J

P * ” * r * ( c n_ s [ s ] s * ) %

=

1

; q .e .d .

F(6) Because

1 2 (a )

every point of

b)

is in variant under the unitary group,

-> G ( V )

p

6

Z [ 0 ,n]

and

form of s y m b o l

a

a

a e @ (p ,n ) .

be the projection .

Let

cr:S(a)->F(a)

b id e g re e

(a

h as fib e r dim ension ,a

)

w ith

a

0

mute with this action.

bidegree

(a, a)

d ( a ) + a - d (p ,n ) ,

= d (p ,n ) - a .

O b v io u s ly

Take on

C le a rly

with

Proof.

c(a)

(ft )

c(a)

c(a)

h as

is of c la s s

U , because

i/r be a form of c la ss

di// = 0 .

C °°

tt, t com­

C1

and

Then

Gp(V)

is the P o in c a re dual of

T h e function

the form

dc(a) = 0 .

v e F ( a ) . Let

G (V)

s (v, a )

Therefore,

:S(a)

.

and in variant under the action o f the unitary group

T H E O R E M 5.2.

77-

C o w e n [ 8 ] d e fin e s the non-n egative Chern

e ( a ) = rr^a*^la ) >

n

and

by

(5 .2 )

B ecau se

at

F(a) .

The Chern form of s ym b ol T ake

1 2 (a ) > 0

S (v, a )

(C o w e n [ 8 ]) .

e x is t s and is of c la s s

C1

on

F(a)

w ith

INVARIANT FORMS ON GRASSMANN MANIFOLDS

60

dg^tt ((A) = o^tt (d^r) = 0 . N = a-

1

(v )

H en ce

is a constant y .

S (v, a ) = tt(N ) .

and o b s e rv e that

y

((A)

a n

=

u ^ t t' ( f t ) ( v )

=J ';

r*(ft)

D e fin e

H en ce

=

N

ft

J ' S(v,Q)

F ib e r in tegration (T u n g [3 3 ]) im p lies that

J *

c(a)Aft

Gp(V)

J*

=

n^cr*(Q,a )

a ft

J*

=

Gp(V)

a * (Q a )

a

it*(ft)

S(Q)

F(a)

q.e.d .

L E M M A 5.3 (C ro fto n ’s form ula). {[/

be a form of cla s s

C1

q e Z [0 ,n ]

T ake

and bidegree

(q ,q )

on

and P (V )

v e G ^ (V ) . L e t with

dy = 0 .

Then

/

"

n~ , A * -

P (V )

Proof. T hen

In Theorem 5.2 take S(v, a ) = E (v ) .

unitary group, and e x is t s such that

* ■

E (v)

p = 0

B ecau se

and

c (q )

a>n-c* gen erates

c (q ) = ycon~ ^ .

a = (q ) . T a k e v e F ( a ) = G ( V ) .

is invariant under the action o f the H 2 n~ 2 cl ( P ( V ) , C ) ,

a constant

Theorem 5.2 im p lies that

JY=y

E(v)

f

P (V )

y

5. THE POINCARE DUAL OF A SCHUBERT VARIETY

H ere

y

is independent o f y .

T h e c h o ic e

y = CO** s h o w s

O f co u rse there are many proofs o f Lem m a 5.3.

61

y = 1 ; q .e .d .

F o r in stan ce the un­

integrated F ir s t M ain Theorem [ 2 5 ] T heorem s 4.5 and 4.6 im ply Lem m a 5.3. A direct proof is b a s e d on cohom ology and S to k es’ Theorem . gen erates that

H 2c* ( P (V ) , C ) ,

y = a co* + d P ( V ) and r : F -> G ( V ) be the p ro je ction s. B y P P P S (v, a ) = r77_ 1 ( v 0) . O b s e rv e that a = ( n - p ) ( p + l ) - h . L e t i

0

be a form o f c l a s s

T hen

n^ r

^

d 77 r ( 0 ) =

(iJ/) 77.

j* E (v o}

v = ( v Q, ***, v )

— >

C

and b id e g re e

is a form of c la s s r (d 0 ) = 0 .

7J,* r * ( lA ) =

C

1

(a ,a )

on

G p (V )

and b id e g re e

(s ,s )

with on

d0 = 0 .

r._

P (V )

H en ce

^

A

P (v>

(0 ) =

J F

G (V) pv

77*7

J

77*(0JP + l1 ) A 7

p

G (V) pv

'

(0 )

with

INVARIANT FORMS ON GRASSMANN MANIFOLDS

62

Let map

X

and

f : X -» Y

set

A

of

Y

(Y - A)

be com plex s p a c e s .

A proper, su rje c tiv e holom orphic

is c a lle d a modification, if there e x is t s a thin an a ly tic s u b ­ such that

B = f - 1 (A)

is thin in

X

and such that

f:(X-B)

is biholom orphic.

Because sio n

Y

s

E ( v Q)

is a com pact, connected, com plex m anifold o f dim en­

and b e c a u s e the fib e rs o f

dim ension

(n -p )p ,

are connected com plex m anifold s o f

77

the an aly tic s e t

N = r T 1^

) )

= { (x ,y ) f G p ( V ) x E (v 0) |y e E (x )}

is irred u c ib le and has dim ension

(n -p )p + s = a

by Lem m a 1.2.

E xam p le 6

o f §2 im p lies that r ( N ) = 1 x 6 G p ( V ) |E (x) fl E ( v Q) 4 0 I = S (v, a ) .

T h e restriction ? = r : N -» S (v, Q) fib e r ? - 1 ( x ) = E ( x ) fl E (v Q) and

S (v, a )

ficatio n . and

r,

is holom orphic and s u rje c tiv e .

is a connected m anifold.

h av e the sam e dim ension

a

r,

n,

n

H ence, b e c a u s e

and are irred u c ib le,

In the commutative diagram 5.3, the maps

E ach

j, j, j

r

N

is a modi­

are in c lu s io n s

are p ro jection s.

P (V )

G n( V )

j

j

77

T

N

S(v, o )

E (v 0>

D ia g ra m 5.3

H ere

tt :

-» P ( V )

N

E ( v Q)

is the pull b a c k o f the holom orphic fiber bun dle

under the in clu sio n

j .

T h erefo re

tt :

Fp

5. THE POINCARE DUAL OF A SCHUBERT VARIETY

j * 7 7 * 7 =

Since

r

77* j

=

63

77*7*j*(lA)

is a m odification , this im p lies that

=J i*7r*r*(|/f)= J

J

E ( v 0)

E ( v q)

7] /

j

OA)

E ( v q)

= J ? * j*(i/r) = N

t/r

.

S(v,a)

T h erefore

^

Ch M A l/> =

G p(V)

for a ll forms

t/r o f c la s s

C1

•A S(v, 0 )

and b id e g re e

(a , a )

with

dt/r = 0 .

Be­

c a u s e the cohom ology c l a s s o f the P o in c a r e d ual is unique, and b e c a u s e c(a)

and

c ^ [p ]

are invariant under

U (V ) ,

w e have

c ( a ) = ch M

>

6. M ATSUSHIM A’S TH EOREM

a)

D o u b le inters ec tion Let

V

be a com plex vector s p a c e o f dim ension

a p o s itiv e d e fin ite herm itian form on b

in

®>(p,n) .

V .

Take

and

is an an aly tic s u b s e t o f

G p (V ) x F ( q ) x F ( b ) .

intersec tion diagram

are p ro jection s.

6 .1

T h e diagram commutes.

(6 .2 )

If

p e Z [0 ,n ] .

Let

I

be

T ake

a

and

T hen

S ( a , b ) = {(x , v ,w )| (x , v ) £ F ( Q )

( 77^ , 77a ) .

n + 1 > 1.

(S (c t , B ),

(v,w )eF (a)xF (b),

^_

1

)

(x,w)(rF(b)S A ll maps in the double

is the re la tiv e product of

then

(v ,w ) = ( S ( v , a ) n S ( w , a ) ) x i ( v , w ) i

64

.

6. MATSUSHIMA’S THEOREM

65

H en ce the re striction

(6. 3)

p : f

1

(v , w ) -> S (v , a ) (~l S (w , a )

is biholom orphic.

LEM M A 6.1.

The analytic s e t

S(a,b)

is irreducible with

d ( a , b ) = dim S( a , b ) = d ( a ) + d ( B ) + a + b -

/3a

A lso

and

d (p ,n ) .

are s u r j e c t i v e and lo c ally trivial, and have irreducible

fibers.

Proof.

B ecau se

the maps /3 a 77 ^

(S ( a , b ),

and

/3^

(re s p e c t iv e ly

is the re la tiv e product o f

are lo c a lly triv ia l and s u rje c tiv e . are isom orphic to the fib e rs o f

) . H e n c e the fib e rs o f

S(b)

, /3^ )

are irre d u c ib le ; a ls o

/3a

and

S(d,b)

T h e fib e rs of

77^

are irred u c ib le.

( 77^ , 77^ ) ,

(re s p e c t iv e ly

S ince

S(a)

and

is irre d u c ib le by Lem m a 1.2 with

dim S ( a , b ) = dim S ( a ) + fib. dim ^ = dim S ( a ) + fib. dim

77 ^

= d ( a ) 4- a 4- d ( b ) + b - d (p ,n ) ; q .e .d .

O b s e rv e that (6 .4 )

dim F ( c t ) x F ( b ) = d ( a ) + d ( b ) .

dim S ( a , b ) - dim F ( a ) x F ( b )

P R O P O S I T I O N 6.2.

Proo f.

H en ce

= a + b - d (p ,n ) .

c ( a ) a c ( b ) = p * £ * (a * (C lQ) n a j ( f t 6 )) .

Theorem 4.1 im p lies that c(a) a c(b)

=

TTaik a * ( Q , a ) a ^

= P*£*(aa (na> A ab(nb^ ;

q .e .d .

66

INVARIANT FORMS ON GRASSMANN MANIFOLDS

L E M M A 6.3.

If £

P roof.

C= b

D efin e

e 0 '” ’ , e n

.

3 ^ase

Then

V-

vq = P ( e 0 A - A

for

q =(),•••,p .

A ls o

n- b

r

4

T hen

= c

a > b

is s u r j e c t i v e , then

- bp_q

f ° rq = 0 , “ * , p . L e t

D efin e e Sq )

Wq

= P ( c n A ••• A en_ b q )

v = ( v 0, - - , v

+ q = c

4

Cq = n - p

) e F(a)

and

w = (w 0>- - , w ) e F

with

4

E ( vq) = C e 0 + •" + C e aq

E (w p -q ) = C e n +

B ecau se

is s u rje c tiv e ,

£

x e G p (V )

+ C c cq '

e x is t s such that

(x, v, w ) € S ( a , b ) .

H ence dim E ( x ) fl E (V q ) > q+ 1

In p articu lar

E ( x ) C E (V p )

dim E ( x ) fl E (W p _ q ) > p-q + 1 .

E ( x ) C E (W p ) .

and

T h erefo re

dim E (x ) fl E (V q ) fl E (W p _ q) > q+1 + p - q + 1 - p -1 =

H en ce

dim E ( v q ) D E ( w p_ q ) > 1 and this im p lies that

aq

q = 0, - * * , p .

for

T h erefo re

P R O P O S I T I O N 6.4. c(b) ^ 0 .

Proof.

Then

a >

b*

A s su m e that

an aly tic s u b s e t o f Let

Take

a

t:S(a,b)-*F(a)xF(b)

Because

a * (Q q

) a

A a b (nb

a jj(flj> ) ) = 0

b

b >

in

@ (p ,n )

d (p ,n )

is not su rje c tiv e .

F (fl)x F ( b )

or

and

£

. A s s u m e that c( a )

cq
a ) A a k ( n 6 ) ) = 0 .

B y P ro p o s itio n 6.2

c(a)

a

c

(6 )

= 0

is s u rje c tiv e , and co n seq u en tly

contrary to the assum ption. T h erefo re

^

a > b

by Lem m a 6.3.

H en ce

^^

a > f)

= d (p ,n ) - h ; q .e .d . T h e c a s e of an in je c tiv e map f

b)

needs more preparations.

Genera l po sit io n Take

p q + 2

is an aly tic.

H (a )= S (a )-L . com plete fla g

H en ce

w eFn

is the p rojection . H en ce

H(a)

L^

of a ll

H en ce H(a)

n

(x ,v ) e S ( a )

is Z a r is k i open. v = a (w )

S (v, a ) = S < w , a > . 7r- 1

77- 1

(v )

(v ) .

+ 0

T hen

dim E ( x )

is a n a ly tic with

Take

v eF (a).

where

A

c£a : F n -> F ( a )

B y Lem m a 2.2,

S C

is Z a r is k i open and d e n se in the irre­

In p articu lar

d e n se in the irred u c ib le an a ly tic s e t ( x ,v ) e S ( a ) .

such that

L = L Q U ••• U L p

e x is t s such that

T h en

d u c ib le an aly tic s e t

T ake

Then

is a non-em pty Zarisk i open s u b s e t of

is d e n se in

d e n se in

H(a) .

a e @ (p , n ) .

dim E ( x ) fl E (V q ) = q + 1 for a ll

b e the s e t o f a ll

L E M M A 6.5.

Proof.

and

H ( a ) 4 0 • H en ce

H(a)

is

S(Ct) ; q .e .d .

(£ , b )

is c a lle d a representation o f

(x ,v )

if the fo llo w in g con d itio n s are s a tis fie d : (1 )

We h av e

dent v ecto rs in (2 )

We h ave

£ = (£ V

w ith t) =

d e p e n d e n t v e c t o r s in

0

r * £ p) > where x = P

a •••

(£ 0

^

V , w ith

v = ( v o »’ ” >v p ) •

‘ “ i Sp

t)n , • • • , ! ) -

= P ( t ) n a •• • a

v

are lin e a rly in depen­

£p) .

( t)n, ••• , t> - ) , w here u p "

M oreover

a

are

lin ea r by in-

P b - ) for q

q =

0

," ' f p .

INVARIANT FORMS ON GRASSMANN MANIFOLDS

68

(3 )

We have

£

= bg

4 L E M M A 6.5.

for

q = (),•••, p .

q (x ,v ) e S ( a ) .

Take

P roo f,

a)

A s s u m e that

( x ,v )

Lq

of

V

of dim ension

aq - aq

for

q = 0, l , * * * , p .

(x ,v )

of

is in general posit ion if ex is ts.

is in general position.

Because

-1

A lin ea r s u b s p a c e

e x is t s su ch that

E (v q) = E C v ^ j ) © L q

dim E (x ) fl E ( v q l ) = q

dim E ( x ) fl E (v ) = q+1 , w e have dependent v ectors

( x ,v )

Then

(£,b)

and only if a representation

b 0 ,--*, b g

dim E (x ) fl L

and b e c a u s e

= 1 .

can be taken such that

H en ce lin e a rly inb 0, ‘ **» b £

p E (v

)

and

bg

" ) and span for

and

E (x ) .

q = (),•••, p .

tion of

E (x ) fl E (v ) q

£q = b g H en ce

.

x = P

D e fin e

Then (£ 0

for each

q =( ) , • ••, p .

£0 , - " , £p are lin e a rly

A *“ A £p ) • A ls o

£ = ( £ 0 ,-**, £ p) .

Then

D e fin e

independent

vq = P ( b Q

(r,b)

b =

a

•••

a

bg )

is a represen ta­

(x ,v ) .

b) If

sp a n s q

span q

(£,b)

A s s u m e that

q = p ,

q < p .

then

Let

Mq

aq + 2 , “ *,ap .

q+l,***,p

E ( x ) C E (V p )

with

and

dim E (x ) fl E (V p ) = p + l .

be the lin ea r s u b s p a c e spanned by

Then

(x ,v ) .

is a representation of

y E ( v q)

for /z = (),•••, q

Mq fl E ( v q ) = 0 .

b^

and

Take

q € Z [0 ,p ].

A ssu m e that

for p = a q + 1,

£^ 6

Mq

for

p=

H en ce

E ( x ) = ( E ( x ) n E (v q ) ) © ( E ( x ) n Mq) .

Because

dim E ( x ) fl E (v ) > q + 1 and 4

dim E (x ) = p + 1 ,

w e obtain

dim E ( x ) (1 M

4

dim E (x ) fl E ( v q ) = q + 1 .

> p -q

w h ile

T h erefo re

(x ,v )

is in gen eral p ositio n ; q .e .d . Take

a

and

b

in

@ (p , n ) .

Then

(x ,v ,w ) e S ( a , b )

is s a id to be in

general posit ion if the fo llo w in g conditions are s a tis fie d :

1

(G l)

(x ,v ) € S ( a )

(G 2 )

T h e vector s p a c e

for

q =

0

,'“ , p .

and

(x ,w ) f S ( q ) L

HI

are in gen eral position.

= E (x ) fl E (v ) fl E (w 4

r

4

)

has dim ension

6. MATSUSHIMA'S THEOREM

(G 3 )

If

< m < q < p , then

0

69

L m fl L q = 0 .

O b v io u s ly (G 3 ) is eq u iv a le n t to (G 3 ' ) Let

E (x ) = L

H(a,b) Take

tion o f

q

© •■• © L p .

b e the s e t o f a ll

(x ,v , w ) e S ( d , b )

( x ,v ,w ) e S ( a , b ) . T h en

(x ,v ,w )

(£,b,to)

( £ , b ) is a representation o f

(R 2 )

For

q=0,---,p

d efin e

6 .6

.

Lq

a).

A s s u m e that

(x ,v , w )

D e fin e

£ 0' “ ’ ' ^ p

Th en

that

q . If

b 0, * " , b g

1

.

Take

fl E (W p _ q) C L q .

L q = E (x )

n

E ( Vq 1)

w hich is im p o s sib le . e x is t for

p = a

bo,-*-,bg

4

span

n

dim L q = 1 ,

- +l,-*-,a E (V q ) .

ex ists.

T h e vecto r s p a c e

(£ 0

B y (G 3 ')> the A ••• A £ p) •

E (v ^ )

4

£ e 4

such that

with

b0, - * - , b g t)g

= £^

£ Q e E ( v 0) • A ssu m e

q < p .

A ssu m e that £ q e

w e h ave

0 =(= Eq t E ( x ) fl E ( v q_ 1)

w e have

E ( Wp_ q ) C E ( x )

T h erefo re

1

x = P

this is triv ia l s in c e

J q e E ( x ) fl E (W p _ q)

S ince

is in general p o s i ­

0 =(= £ q e L q .

span

are constructed w ith

E (vq _ l) • B ecau se

•

N o w , lin ea rly independent vecto rs

q = 0 ,

) .

t) -

(x ,v , w )

of

are lin ea rly independent w ith

£ = ( £ 0 , - ,#, £p ) .

p =

(x ,v , w )

(£,b,to)

s h a ll be constructed such that for

£ = ( r Q, ••*, £

is in gene ral position.

d efin ed in (G 2 ) h as dim ension

v ecto rs

w ith

(x ,w ) .

( x ,v ,w ) e S ( a , b ) .

Take

tion if and only if a representation

Proof,

(x ,v )

b q = £ p_ q - D e fin e

is a represen tation o f

LEM M A

is s a id to b e a rep res en ta­

if the fo llo w in g con d ition s are s a tis fie d :

(R l)

T h en

in gen eral positio n .

n

E (V

l )

E (v ) - E (v 4

£

4

= b~

dq

4

fl E ( w ,) .

^

^ q+1) = L

V e c to rs b

q_ 1

r

and such that

B y induction lin ea rly independent v ecto rs

4 o for

•• •, b ~ ap

are constructed su ch that

q = (),•••, p .

tion o f

(x ,v ) .

D e fin e

£_ = b 4

b = ( b 0,-**,bg ) .

T h en

and

v _ = E ( b n a ••• A b 4 u •c*q

(£,b)

)

is a re p resen ta­

INVARIANT FORMS ON GRASSMANN MANIFOLDS

70

D e fin e

>)q = S p _ q for

x = P ( t ) 0 a •■•a t)q)

q=0,--,p.

D e fin e

t) = (t ) 0 , " M ) p) • T h en

and 0 + t)q r E ( x ) n E (w q ) n E ( Vp _ q) .

H en ce the sam e re aso n in g as a b o v e produ ces lin ea rly independent vectors such that tn = representation o f b) D efin e

Let

(b,to)

fore

(x ,v )

for

Then

(£,b,to)

*)q = £ p_ q

and

(x ,w ) .

tog )

for

(£,b,to)

q=0, — ,p.

D e fin e

/x = 0, ** *, q

(x ,w ) with

(x ,v )

is a

(x ,v , w ) . T h en

(x , v , w ) . %-

t) = ( b 0 r - - , b p ) • T h en

and

(x ,w )

are in gen eral p ositio n .

re sp e c tiv e ly .

We have

dim E ( x ) fl E (v ^ ) = q + 1 .

are lin ea rly independent.

(t),to)

is a representation o f

be a representation of

are represen tatio n s o f and

and such that

(S,b) T h e re ­

e E ( x ) fl E (v ^ )

T h e vecto rs

Sq,***, £ q

T h erefo re

E ( x ) n E ( v q ) = C S0 + ••• + C Eq .

B y symmetry

E ( x ) PI E (w q ) = C t)Q + ••• + C l}q .

E ( x ) fl E ( w q) = C s p + -

T h erefo re

+ Csq

L q = E ( x ) n E ( v q ) n E (W p _ q) = c Eq .

H en ce

dim

T h erefo re

= 1 for (x ,v ,w )

LE M M A 6.7. Zariski open

q=0,---,p

and

L m fl

= 0

if

0 < m < q < p .

is in gen eral position.

Take

a and

s u b s e t of

b

in @ ( p , n ) .

Then

S ( a , b ) . In particular

H ( a , b ) is anon-empty

H(a,b)

is de n s e in

S(a, b) . Proof. H(b)

C o n sid e r diagram 6.1.

Since

are thin an aly tic s u b s e ts o f

No =

Na = S (a )-H (a )

S(a)

and

S(b)

and

N^=S(b)-

re sp e c tiv e ly

71

6. MATSUSHIMA’S THEOREM

is a thin an a ly tic s u b s e t o f

(x,v,w)f S ( Q , b )

S(d,b) .

D e fin e

H Q = S(Ct, b ) - N Q .

For

d efin e L q (x ,v , w ) = E ( x ) n E ( v q) n E (W p _ q ) .

B y Lem m a 1.5, the s e ts

p

=U

N 1

i ( x , v , w ) I dim L q(x ,v , w ) >

2

}

q= o

U

=

n 2

i(x ,v ,w )| dim L m(x ,v , w ) n L q (x ,v ,w ) >

1

}

m^q

are an aly tic s u b s e t s of T hen

H (a ,b )= S (a ,b )-N .

Take

(x ,v , w ) e S ( Q , b ) .

representation o f order tions o f (

,

a ll

S(a, b) .

( x ,v ) tip)

and

with

j

(x ,w )

O nly

N = NQ U Nj U N2

H(a,b)4=0

T h en

(£,t>)

if and only if

and

for

q =

0

is c a lle d a double

and

are rep resen ta­

£ =

, l , •••, ] ' .

is an a ly tic.

rem ains to be shown. (t),tt))

(£,b)

re s p e c t iv e ly , and if

£ q = t)p _ q

( x ,v ,w ) e S ( a , b )

H en ce

£ p)

and

t) =

HO') be the s e t o f

Let

for w hich a d ou ble represen tation o f order

j

e x is t s .

C le a rly H0 = H ( - 1 ) 2 H ( 0 ) 2 H ( 1 ) D - O H ( p ) = H ( a , b )

H ere

0

HO) 4 (t) , to )

H ( - l ) = H Q =j= 0 . s h a ll b e proved.

Take

0 < j < p

A ls o

b =

tog

for

and

t)p) )

and

with

hq

eC

W .l.o.g.

P r o o f of Cla im 1.

If

j = 0 ,

£q = i)q_ p

£p

e x i s t s u c h th at

hq = 0

j - 1 of

to = (tD0>--*, tog )

q = 0, 1, " - , p • B e c a u s e

numbers

C L A I M 1.

t) =

fo r

and

H ( j - 1)

(x ,v , w ) e HO - 1) • L e t

be a d ou b le represen tatio n o f order

U 0 , - - * , £ p)

E (x ) ,

A s su m e that

and

1

0

and

T hen

, l , - - - , j-

£q = t a n d h 0»*’ ‘ ^ p

£ = 1

. t)q =

eac h span

£j = hQ t)0 + ••• + hp t)p .

q = p - j + I ,- * - , p

claim

q =

4 0 • Then

(£,t))

( x ,v ,w ) .

for

with

.

is vacu o us.

c a n b e a s s u m ed . A ssu m e that

j > 0 .

72

INVARIANT FORMS ON GRASSMANN MANIFOLDS

Define

£ q = Sq for q =(= j

and

j - 1

5j = Ej “ 2 q=

0

j - 1

hp-qt,p-q

q=

Then JQ a ••• a a ••• a %'• . Because £n = b~ = q

J

vq =

P(Uq

a

dq

C laim 1 is proved and W .l.o .g .

Proof of Claim 2. q4

hp-q sq '

. Define b^ = b^ if n =(= aj and bg = ^or p = (t,o A " ‘ A W

j

+

0

6. MATSUSHIMA’S THEOREM

and

x = P ( t ) Q a •••

M = b p_ j •

If

a

t)p ) .

D e fin e tt^ = ffl^

if

n 4= b p _ j

and VD^

=

5

j

if

q > P - j , then

t t ' a --- a t t e = tt„ a - - - At t r 0 bq bq

T h erefo re

73

w = P (t o g

a

•••

a

to g

) for a ll

h

•+ 0

.

P-J

q =( ) , • • • , p .

D e fin e t)' =

q ( t ) o , ’ *’ , ^ p ) of j

(x ,w ) . of

anc* A ls o

(x ,v ,w ) .

= ( to o , *“ ' to g ) • P ( £ , b ) and ( t ) ', t o ')

T h erefo re

B ecause

S ( Cl , B )

d e n se in

S ( a , b ) ; q .e .d .

0

H (j) 4

.

T h en

( t ) ', t o ')

is a representation

is a d o u ble representation o f order B y induction

0

H ( a , b ) = H (p ) 4

is irred u c ib le the Z a r is k i open s u b s e t

H( a , b ) 4

0

. is

T h e concept o f a m o dification w a s e x p la in ed in the proof o f P r o p o s i­ tion 5.4.

T h e con cept of a rank o f a holom orphic map and the properties

of the rank are giv en in [1 ]. set

T ( a , b ) = £ (S ( a , b ) )

THEOREM

6 .8

B ecau se

is irre d u c ib le with

. In diagram 6.1 a ssu m e that

b * > a and f : S ( a , b ) -> T ( a , b )

P ro of.

S(a,b)

R e c a ll

is irred u c ib le, the an aly tic dim T ( a , b ) = rank £ .

rank

= dim S ( a , b ) .

Then

is a modification.

d ( a , b ) = dim S ( a , b ) . T h e an a ly tic s e t E Q = { z e S ( Q , b ) | rankz f < d ( a , b )!

is thin in S(a, b)

S(a,b) .

Let

re sp e c tiv e ly

E2 = Eq U E 1 U

and

T(a,b).

S(a,b)

T(a,b)~ D . m ension map.

T hen

d (a , b )

w ith N

and

and

M M

= S( a , b ) - H (a , b ) . S(a,b) .

T(a,b).

E D E2 .

£ :N

b e the s e ts o f s in g u la r p oints o f

D e fin e

is thin a n a ly tic in

is a thin a n a ly tic s u b s e t of an a ly tic in

X1

D e fin e

T h en

H en ce E = £-

T hen

D =

.

1

A ssu m e

Then

if

that

0 4 A. e C . •••

£ '0 a

£ p) .

T hen

4= 0 if

H en ce

|A| < r .

If

(x " (A ),v ,w ) e £ -

1

(v ,w ) .

H en ce

B ecau se

x '(A ) = x

Let

1

|A|

< r , d e fin e

x '(A ) = P (

-» x for A

0 .

for a ll

H en ce

E ( x '( A ) ) D E (w ^ )

for

£ "0 a

(v ,w ) e M .

1 = dim E (v = aq + q+

H en ce

4

If

) fl E (w

T h e map

P r o o f of Claim 2. b e lo n g to f _

1

r

4

w hich im p lies

w hich is wrong.

x '(A ) C laim 1

4

(v ,w ) .

1

- n-

w hich means

f : N -> M

T ake

then

) > dim E (v ) + dim E (w

+ b p _ q + p—q +

1

n - p - b p_ q > a q

C L A I M 2.

0 < q < p ,

4

) - dim V

= a q - ( n - p - b p_ q) +

1

b

r

1

.

> a .

is biholom orphic.

(v ,w ) e M .

A ssu m e

(x ,v , w )

and

a

/x =( ) , • • • , p .

is proved. Take

•••

( x '( A ) , v )

is zero dim en sion al,

|A| < r ,

, b,tu)

A ls o

for f i = 0 , • • • , ? •

(v ,w )

5

r > 0 e x is t s such

( x '(A ),v , w ) e S ( a , 6 ) ,

M

is in je c tiv e , hence b ih o lo -

C laim 2 is proved.

B y claim 2,

T H E O R E M 6.9.

£ :S (a,b)^T (a,b)

a and b

Take

rank £ = dim S ( d , b )

is a m odification , q .e .d .

@ (p ,n )

in

b

if and only if

= a

with

a + b = d (p ,n ) .

(,Diagram 6.1).

b

If

Then = a ,

T(a,b) = F (a )x F(b) .

then

B y Lem m a 6.1

Proof. sio n o f

F ( a ) x F ( b ) . H e n c e if

an aly tic s u b s e t o f Since

dim S ( d , b ) = d ( a , b ) = d ( a ) + d ( b ) rank f = d ( a , b ) ,

F (a )x F (b )

F (a )x F (b )

with the dim ension of

is irre d u c ib le , w e have

a > b

Lem m a 6.3 im p lies that v- * b - a .

then

.

Theorem

is the dimen­ T(a,b)

is an

F (a )x F (b ).

T(a,b) = F (a )x F (b ).

6 .8

im p lies that

> a . H en ce

b

j|s

A s su m e that

b

= a .

for

ap

and

[i =

Vq

=

P ( t >0

Let

e 0 , - - - , e n be a b a s e o f

= e n_ ^

a —

a t >s

for

wq

)

V .

D e fin e

q =( ),••■,p .

D e fin e

5

^ = e-

£ 0’" ‘' £ q

s Pan

T hen for

P (t o 0

=

A —

a ftg

E ( x ) fl E ( v q ) .

b q + q = n - p - a p_ q + q = n - S p _ q J p> S p _ i > '“ >£p _q S(b)

and

s Pan

and

x = P(f

H ence and

E ( x ) n E ( w q)

z = (x ,v ,w ) £ S ( d , b ) .

)

q

v = ( v 0,---, v p) e F ( a ) q=0,---,p

=

^ = 0, - *•, b p . D e fin e

Q

for

b

and 0 a

w = ( w 0 , - " , w p ) e F (f> ). a

(x,v)eS(Q).

efor

r p)

q=0,-",p.

an

bq =

T h erefo re

H en ce

en, " ’, e z u

T h e v ecto rs

O b s e rv e

= e a p_ q = S p - q -

T h e vecto rs

.

span

( x ,w ) e E (v _ ) 4

INVARIANT FORMS ON GRASSMANN MANIFOLDS

76

and the vectors

c n > e n_

E (W p _ q ) = C e g

= C

E ( v a ) > q-f 1 and

i .

ea

Take

s Pan

(x ',v ,w )

T h erefo re

Then

E (V q ) fl

q = 0,-,p

,

w hich im p lies that

h ave the sam e dim ension,

H en ce

(v ,w ) = ! z\

rank

H en ce

E (x ) C E ( x ' ) . S ince

E(x)=E(x')

is zero d im en sion al.

and

x = x '

C on se qu en tly ,

> rankz £ = dim S ( a , b ) - dim z

-

(v ,w )

1

= dim S ( a , b ) > rank £ . H en ce

rank f = dim S ( a , b ) ; q .e .d .

T h e b a s ic geom etric re su lts h avin g been obtain ed , w e can apply the c a lc u lu s o f exterior forms.

c)

Th e D u ality Theorem

T H E O R E M 6.10 (E hresm an n [1 1 ]).

,

— >

a + b = d (p ,n ) . If c(a)

a

Take

^

a =(= b

,

then

b

and

c ( a ) fl c ( b ) = 0

in

@ (p , n )

5j(

a = b

if

,

with then

c ( b ) = c n_ p [p ] P + 1 • In particular

r

(0

I c( a )

A

c (b ) =

G (V )

T h en

2 (d ( a ) + d (b ) ) .

then

If

a 4 b* ,

dim S ( a , b ) = d ( a ) + d ( b )

P ro p o sitio n 6.2 im p lies

c (a)

a

if

a+b*

if

Q= b* .

) (l

P r o o f .C o n sid e r diagram 6.1.

where

a

Q = a a ( ^ Q) A a g ( ^ b ^ rank

< dim S ( a , b )

by Lem m a 6.1.

c(b) =

H ence

(fl)= 0 .

^ as d egree

by Theorem 6.9, (0 ) = 0 .

A ssu m e that

a = b .

6 . MATSUSHIMA’S THEOREM

T hen

rank f = dim S( a , b )

and

f :S (Q ,b )^ F (Q )x F(B)

tion, by Theorem 6.9 and Theorem

J'

c(a)

a

c(B) =

G p( V )

77

6 .8

J*

.

is a m o d ific a­

T h erefo re

pj*(& ) =

J'

G p(V )

£*(ft)

S (Q ,B )

a * (Q a ) A a b(flb )

F(a)xF(6)

=

Because

J

J

F (q>

F(B)

c n_ p [ p l ^ + 1 > 0 > a function

y

fiB =

e x is t s su ch that

c ( a ) a C( B ) = y c n_ p [p ] p + 1

Because y

c(d),

is constant.

4.4

c(B)

and

Integration ;ration ove over

.

are in variant under the unitary group,

G p (V )

s h o w s that y = 1 w here Theorem

w a s u sed ; q .e .d .

T H E O R E M 6.11. d (p ,n ) - m .

Take

Define

i c ( a ) 5a e @( p n

p c Z [0 ,n ]

P roo f.

m e Z [0 , d ( p , n ) ] . D e f in e

*s a ba se of the v ect o r s p a c e

B y (2 .1 1 ) and (2 .1 2 )

s u ffic e s to sh o w that the d e @ (p ,n ,m )

and

@ (p ,n ,m ) = j d e @ ( p , n ) | a = mi .

fore a co hom olo gy base o v er

each

c n_ p [p ]

•

1

C

for dimension

Then the family

In v 2 s (G p (V ) ) 2

are lin ea r independent.

there e x is t s a constant

ya

6

and there­

s .

dim In v 2 s ( G p ( V ) ) = # @ (p ,n ,m ) .

c(d)

s =

C

H en ce it

A ssu m e that for

such that

78

INVARIANT FORMS ON GRASSMANN MANIFOLDS

€

ya c( fl) = 0 •

ct 0(p ,n ,m ) Take

e @ (p ,n ,m ) .

6

Then

J '

yjj =

ya c ( a ) a c(6*) = 0 ;

^

G (v ) a e @( P « n>m)

q.e.d.

P

Take

a e @ (p ,n )

and

V fF (a) .

T hen Theorem 5.2 and Theorem 6.10

im ply that

/

if

0

H a *

c ( b ) = a and

T H E O R E M 6.12

(M atsushim a [2 1 ]).

Take

Let

r be an integer with

0 < r < d (p ,n ) - a

Let

Abe a form of cl a s s

C° °

and de gree

and with

invariant under the action of the unitary group F(a) 77-* 0 .

on F ( a )

such that

U (V )

F(a) .

yc e C

such that

A =

and

on

b e the projections.

Then there ex is t s one and only one

D e fin e

yc c( O •

Let

D e f in e

for each

A

A =

ce

is

6 . MATSUSHIMA’S THEOREM

M oreover, if A each

yc

is real, anc/ if A > 0 ,

f/ien y c > 0

for

C e A ( a , r) .

P r o o f . D e fin e T h e form g

is rea l, each

79

77^ 0

A

m= a + r .

has d e g re e

(A ) = 77-^0 (g

S ince

n h as fib e r dim ension

2 (d (p ,n )-m ) .

A) =

Take

g e U (V ) .

(A ) = A . U n iq u e numbers

77^ 0

d ( Q ) + Cl - d (p ,n ). T h en yc e C

g * (A ) = e x is t

such that (6 .6 )

A =

^

yc c ( c ) .

c € @ (p ,n ,m )

Take and

c e @ (p ,n ,m ) o = oa .

D e fin e

.

'oC.=

c

*

.

C o n sid e r diagram 6.1 with

Theorem 6.10 and Theorem 4.1 im ply that

=

/

A

a

c( c )

G p (V )

=

J O p (V )

V V >

op T ( a , b )

cT (Q ,b)-»F (Q )xF (b)

5|(

dim S ( a , b ) ,

b e the in clu sio n .

5|(

T h e d egre e o f a a (A )

a

is irred u c ib le with

(0 ^ )

is

2dimS(a,b).

be the restriction

Then

£ = t °£ 0 •

H en ce if

rank £


0 ,

If y c ^ o , then rank 0 ; q .e .d .

and

is a modifica­

Therefore

(6.8)

J

yc =

a*(A ) Aa J t f l j ) .

T( a, B) with

b=

C

.

T h erefo re

yc

can

be e x p re s s e d d ire ctly by

Aand b .

M atsushim a [21] ga v e a to tally differen t proof which u s e s L i e a lg e b ra and d iffic u lt re su lts o f Kostant.

A fte r this paper w a s written in its fin al

form, but b efo re it w a s sent to the p u blish er, M atsushim a re ceiv e d a letter from Jam es Damon in which Damon p roves Theorem 6.12 u sin g Damon [9 ] and [1 0 ].

H is method em ploys the G y sin homomorphism computed by a

re sid u e c a lc u lu s .

T h u s Damon u s e s deep re su lts in cohom ology theory.

T h e proof provided here rests on fib e r integration for fib e rs with s in g u la ri­ tie s. [3 3 ]).

T h is operator is e a s y to understand but d iffic u lt to construct (T u n g O nce this operator is accepted and once the elem entary but tedious

6 . MATSUSHIMA'S THEOREM

81

geom etric c o n sid eratio n s Lem m a 6 . 3 - Theorem 6.9 are se ttle d , the proof of M atsu sh im a’s Theorem beco m es a triv ia lity and the sam e remark a p p lie s to the D u ality Theorem and to P i e r i ’s Theorem , w here the latter requ ires som e ad d ition al geom etric in q u iries.

7. T H E THEOREMS O F P IE R I A N D G IA M B E L L I

T h e theorem o f G ia m b e lli can b e reduced to the theorem o f P ie r i by elem entary c o n sid eratio n s.

Both theorems h ave been proved on s e v e r a l

o c c a s io n s ; s e e G ia m b e lli [1 3 ], H o dge [1 8 ], Chern [5 ] and V e se n tin i [34 ]. H ere fib er in tegration and the triple in tersection diagram w ill g iv e a proof o f P i e r i ’s Theorem .

a)

Triple in tersec ti on Take

p e Z [0 ,n ]

s e t o f a ll

(x ,v , w , z )

(x,w)eS(b) su b set.

and

and f

a , f>, C in

G p (V )

x

F(a)

(x,z)fS(c).

x

@ (p ,n ) . F(B)

D e fin e

F(c)

x

O b v io u sly ,

S (a , B , c )

su ch that

S(a,b,C)

C o n sid e r the triple intersec tion diagram 7.1.

as the

(x,v)«S(a),

is an an aly tic T h e s p a c e s are

enum erated by

0 = Gp(V ) 1 = S(o,B,c )

8 = F(o)x F(B)x F(c)

2 = S(b,c)

9 = F(B)x F (c )

3 = S(c)

10 = F ( c )

4 = S( a , c )

11 = F ( a ) x F ( c )

5 = S(a)

12 = F (

6

14 = F ( b )

7 = S(b)

[y ,x ] .

)

13 = F ( a ) x F ( B )

= S( a , B )

A map from a s p a c e numbered

q

x

into a s p a c e numbered

A ll the maps in the diagram are p ro je ctio n s.

82

y

is denoted by

T h e diagram commutes.

7. THE THEOREMS OF PIERI AND GIAMBELLI 0

Triple Intersection Diagram 7.1

83

INVARIANT FORMS ON GRASSMANN MANIFOLDS

84

A ls o d efin e

C

=

[ 8,1] : S ( a , b , c ) - F ( a ) x

r,

=

[ 0 ,1 ] : S ( a , b ,

da

=

[12,8] : F ( a ) x F ( b ) x F ( c )

C)

- Gp(V )

= [14,8] : F ( a ) x F ( b ) x F ( c )

ec

LEMMA 7.1. of

S( a , b , C )

F (b)x F(c)

-

F(B)

[10,8] : F ( a ) x F ( B ) x F ( c )

=

S(a,b,C)

is irreducible.

Let

F(a)

-

F(c)

be the dim ension

d (a ,B ,C )

and de fine fi =

(7 .2 )

0a

(fia ) a

0 6

(1 2 b ) a ^ c ( Q c )

Then (7 .3 )

d (a , b ,c )

(7 .4 )

=

d ( a ) + d ( b ) + d ( c )

c(a)

P roof.

a

c(b)

a

C( c )

+

=

-

a + b + c

r)^C

(P)

2 d (p ,n )

•

C o n sid e r the subdiagram S(a,b)

S ( a , b , c ) ----------

[6 . 1]

[5 ,6 ]

[4 ,1 ] [5 .4 ]

S( a )

S(a,c)

S ( a , b , C ) = j (x , v , w , z ) j (x ,v , w ) ( S ( a , b )

is the re la tiv e product o f the maps -» S ( a , c )

and

[5 ,4 ] and [5 .6 ].

is a holom orphic fib e r bun dle with

(x , v , z ) e S ( Q , C ) ]

H en ce [4 ,1 ] : S ( 0 , b, C )

7. THE THEOREMS OF PIERI AND GIAMBELLI

dim S ( a , 6 , C ) = =

85

dim S ( a , C ) + fib e r dim [4 ,1 ] dim S ( a , C ) + fib e r dim [5 ,6 ]

= d(a, c ) + d ( a , b ) - d(a) - a =

d ( a ) + d ( B ) + d ( c ) + a + B + c - 2d(p,n)

b y L em m a 6.1 and T h e o re m 2 .6. ir r e d u c ib le .

H e n c e th e fib e r s o f [4 ,1 ] are ir r e d u c ib le .

ir r e d u c ib le (L e m m a 6 .1 ), ( 7 .4)

B y L em m a 6 .1 , th e fib e r s

S(C t, 6 , C.)

S in c e

o f [5 ,6 ] are S (a , C )

is ir r e d u c ib le b y L e m m a 1.2.

is

O n ly

rem ain s to b e p ro v e d .

T h e d o u b le in t e r s e c t io n s u b d iagram fo r

S(a,B)

y ie ld s

c ( a ) a c ( 6 ) = [ 0 f6 ] J 1 3 , 6 ] * ( [ 1 2 , 1 3 ] * ( O a ) a [ 1 4 , 1 3 ] * ( a & ) )

M o re o v e r

c ( c ) = [0 , 3 TJjC[1 0 , 3 ] * ( f i„c ) .

C o n s id e r th e d iag ram 7 .5

S ( a , & , c ) -------------------------- S ( c ) ---------------------------- F ( c )

D i a g r a m 7. 5

O b v io u s ly [0 ,3 ].

(S(a,b,C),

[3 ,1 ], [ 6 , 1 ] )

is th e r e la t iv e p ro d u ct o f

[0 ,6 ]

and

T h e r e fo r e T h e o r e m 4.1 im p lie s th at c( a )

a

c( B)

= i?j|t( [ 6 , l ] * [ 1 3 , 6 ] * ( [ 1 2 ) 1 3 ]* (Q a )

a

a

c( c ) =

[1 4 ,1 3 ]* (O j ) )

a

[3 ,1 ]* [1 0 ,3 ]* (Q C ) )

= 7y+( ( [ 1 2 , 1 3 ] [ 1 3 , 6 ] [ 6 , l ] ) * ( n a ) a ([1 4 ,1 3 ] [1 3 ,6 ] [ 6 , l ] ) * ( p B ) a

= r j^ ([1 2 ,8 ][8 ,l])* (1 2 a ) =

a

( [ 1 0 , 3 ] [ 3 , 1 ] ) * ( Q C) )

([1 4 ,8 ][8 ,l])* (n b )

v ^ * ( 6 * c a a) A 0 j ( f i b ) a 9 * ( 8 c )) =

a

([1 0 ,8 ][8 ,l])* (n c » q .e .d .

INVARIANT FORMS ON GRASSMANN MANIFOLDS

86

b)

Th e Theorem of P i e r i

L E M M A 7.2.

Take

a and

B in

@ (p , n ) .

h e Z [0 , n - p ] .

Take

Assum e

y

a + b = d (p ,n ) + h . D e f in e

that

the project ion

£ = [8 ,1 ]

a j ^ < bj < aj

for

[8 ,1 ]

,

£

Then

j = 0, l , - - - , p . A s before

are s u rje c tiv e .

[1 3 ,6 ] 0

is surjectiv e.

is a modification and

a _ x = 0 and

C o n sid e r the trip le in tersection diagram 7.1.

Proof.

j =

c = (n - p - h , n -p ,--* , n - p ) . A s s u m e that

is su rje c tiv e .

H en ce

[1 3 ,6 ]° [6 ,1 ]

Lem m a 6.3 im p lies

T h e maps [1 3 ,8 ] and

is su rje c tiv e . b

bj = n - p - b p _ j .

< a .

C on se qu en tly

H en ce

b j < aj

for

,---, p .

1

—>

Since

C = d ( p , n ) - h , w e have

dim S ( a ,

6

, c ) = d ( Q , B , c ) = dim F ( a ) x F ( b ) x F ( c )

R e c a ll the d efin itio n o f is thin an a ly tic in S(a,b,C)

and

H(a,b)

S(ct,b) ,

A q = £ ( A Q)

a n aly tic s u b s e t

A^

of

in

a ls o

6

.b ).

Sin ce

A Q = [6 .1 ]-

1

is thin an aly tic in

F (a)x F (b )

.

A '0 = S ( a , b ) - H ( a , b )

(Aq) F(

is thin an aly tic in

q

)

x

F ( b ) x F ( c ) . An

is d efin ed by

P A 'l

U

=

i ( v , w ) f F ( a ) X F ( b ) | dim E (v q ) n E (W p _ q) > a q - b * + 2 !

.

q= 0

Let G g q (V ) F(a)

,---,en

2 0

and

and

wq = P ( en

a

•••

E ( v q ) n E (W p _ q) r

4

)

a

•••

a

V .

D e fin e

e n_ £ ^ )

in

vq = P ( e Q

Gg (V ) .

a

T hen

•••

a

eg )

in

v = (v Q,---, vp )

w q = P ( e n A ‘ " A e n - b q> •

0.

ze F (c

7. THE THEOREMS OF PIERI AND GIAMBELLI

T h en

v = ( v 0 ,---, v p) e F ( c t )

is span n ed by

and

w = ( w 0 ,---, w p ) e F ( b ) .

for ^ = b q ,---, n .

Mm( v ,w ) H M q (v ,w )

is contained in

spanned by

/x =

for

fj. = n>' ” >b *

with

+1

0

T a k e any

E (W p _ m+1)

Sm = am+ m < b *

E ( v m) n E ( W p _ m+1) = 0 F ( a ) x F ( B ) - A '3 .

and

+1

thin in

F (a )x F (b )

N - A3

is open, connected and d e n se in

Take

re s p e c tiv e ly in

( v , w , z ') e N - A

3

.

T hen

M

M q = MQ + ••• + M q

dim Mq < h-h ph- 1 .

A ls o

-1

= E (v

F ( C)

such that

) fl E (w

M = MQ + ••• + Mp

+ M q + 1 + ••• + Mp

q =

Take

( v ,w , z ) e N - A 3

(x ,v , w , z ) x

Then

and for £p

and x '.

L q ( x ,v ,w )

C M

E ( x ') C M .

and

lin ea r s u b s p a c e T h erefo re e C q

are

T h erefo re

)

h as p o s itiv e

4

is a direct sum with

for

with

1

.

q = (),•••, p .

Then

dim E (z 'Q) = n - h - p + 1 .

and such that

dim M q fl E ( z Q) = 0

( x ', v , w , z ) Take

is a b a s e o f

B ecause

\

6

, l,---,p .

0

hold for

(v ,w )

z ' , there e x is t s a point z = ( z 0 , - * - , Zp) in

dim M H E ( z q) = 1 for

for

A 3 = A '3 x F ( c )

4

z ' = (z q , * ■•, z p )

T h erefo re arbitrary c lo s e to

is

. H en ce

+1

We have

dim M = a + B - d (p ,n ) + p + l = h + p +

D e fin e

E ( v m)

F (a )x F (B )x F (c ).

4

a q - b q + 1 and

b*

F (a )x F (B )x F (C ).

*

dim ension

and

T h en

w here

+ m = b *+1 - l
G q ( V * )

n-p=q-fl

p+l = n-q.

The

is d efin ed and the p u ll b ack

S * : Inv 2 m(G q ( V * ) ) -

is an isom orphism .

and

Inv 2 m(G p ( V ) )

B e c a u s e o f (7 .2 9 ) and (7 .3 0 ) the b a s e o f

Inv 2 m(G q ( V * ) )

as given by Theorem 7.6 and (7 .2 1 ), (7 .2 2 ) and (7 .2 3 ) p u lls b ack to a b a s e of

Inv 2 m( G p (V ) ) If

n

oo ,

as d e s c rib e d in Theorem 7.9, q .e .d .

the restriction

j j + •••+ j ^

< n-p

can be lifted and the

cohom ology ring o f the in finite d im en sion al G rassm ann-m anifold o f order p is isom orphic to the free exterior a lg e b ra gen erated by

l , s x[ p ] , * •• ,s ^ [ p ]

INVARIANT FORMS ON GRASSMANN MANIFOLDS

102

over

C .

From this the cohom ology ring o f the

G p (V )

< oo is obtained by truncating the ring by the rule

with

dim V = n +

j x + ••• + jp + 1 < n - P •

O f co u rse a ll this is w e ll known, for in stan ce s e e Chern [4 ] or MilnorS tash eff [2 3 ], w here the c o e ffic ie n t ring is

Z .

1

A P P E N D IX

T h e Schubert variety is an irred u c ib le an aly tic s e t and the open Schubert c e ll is bih olo m o rp h ically eq u iv a le n t to an E u c lid e a n s p a c e . T h is is w e ll known.

A proof can b e found in C hern [7 ].

F o r the co n ven ien ce

of the reader, a proof is given here fo llo w in g the lin e s of C h e rn ’s proof.

THEOREM . p e Z [ 0 ,n] S < v, a >

Let

and is an

V

a e @ (p ,n ) .

v eFn

Let

Proof.

If

p = 0 ,

a = 0 ,

then

Then

G _ ( V ) . The P -» • ft is biholomorphically equivalent to C and

* S

then

a = a Q = a e Z [0 , n ]

S*

is d e n se in

E ( v Q) - E ( v &_ 1)

=

Ca .

p- 1 .

If

Ct = 0 ,

theorem is triv ia lly true. has b een proved for a ll A b b re v ia te

a = a

then

A ssu m e

and

b

fi

= a

v = ( v 0 , - * s v n) e F n

(A l)

w ith

p.

= a

If

D e fin e

aQ = 0 ,

p >

1

and is

p.

p=0.

under the assum ption that and the

b < Q . + p

for

p = 0,***, p .

O b s e rv e

A b b re v ia te

EOx) = E (v ^ )

Q = S < v, a > - .

then

S

and assu m e the theorem a lrea d y

is given .

EGz) = E (v ^ )

p = 0 , -'-, n .

(9Q S < v , a > =

S < v, a > = {v ^ i = S*< v, a > a > 0

e @ (p ,n )

6

then

T h e theorem h o lds in the c a s e

N o w , the theorem s h a ll be proved for

that the f la g

S < v,Ct> = E ( v a ) . If

a > 0 ,

b ih olo m o rp h ica lly e q u iv alen t to

it h olds for

and

E ( v Q) = { v QS = S *< v, a > . If

and

S .

be a com plete flag.

Take

S .

is de nse in

for

n+1.

Cl -dimensio nal irreducible analytic s u b s e t of

open Schubert c e l l

E ( v q_ 1 )

be a complex v e c t o r s p a c e of dimension

(9Q S < v , a > = 0

then

103

and

Then

Q

Q =S.

is open in If

a Q> 0 ,

104

INVARIANT FORMS ON GRASSMANN MANIFOLDS

(A 2 )

Q = !x e S < v, a > | dim E ( x ) fl E (b Q- l ) = Ol

C L A I M 1.

Q

is d en se in

S .

O b v io u sly ,

P r o o f of Claim 1.

a Q > 0 can be assum ed.

dim E (x ) fl E (b - 1 ) = (i fi and

fi =

r 0 ,* " , r n

^

dim E ( x ) fl E (b ) = fi + 1 fi

dim E ( x ) fl E (b Q- l ) = 1 .

anc*

E (b 0 -

1

) = C r0 + - + C r b

0

E (x ) = C r b

x

+•••+ C r b

E ( x ) fl E ( b Q- l )

for

,+Ccj + Ctb

+

i- C r b p

2

° + e C r b 0 + i + 4= A e C ,

n = 0,

= C r y ,

o

0

T h erefo re a b a s is

e x is t su ch that

E( b ) = C r

If

+ C r b1 ’

then

? /V A = ( rb U Q - l1 + A r b u Q ) A q A rb u 2

D e fin e

x .

for

Take

x^ = P ( r ^ )

6

E ( x x ) n E (b A

G p (V ) .

Then

) 3 C(rb _ fl

u0

°0

dim E (x ^ ) fl E (b ^ ) > fi+ 1 .

H ence

dim E (x ^ ) fl

T h erefo re d en se in

E ( b Q) > 1 .

H ence

x^fQ

S < v , ^ Qa > .

Take a base

e 0r " > e n

rb - i + ^rb

T h erefo re N ow ,

of

A -> 0

(9Q S < v , a > C Q . V

1 < fi < p, then

+ - + C r b

such that

€ ^ ( XA^ ^ A ls o

im plies that

C laim

.

fl

x^ e S < v , a > .

B y induction assum ption

H en ce

If

2

A ls o

by (A 2 ).

S *< v, (9Q a > C Q .

for A -* 0 .

+Arb ) + C q + C rb

1

1

H en ce

E ( b Q- l ) = 0 .

x^ -> x

+ ° -

Up

' E (x ^ ) fl x e Q .

S * < v , 1

v^ = P ( e Q A

is

is proved. A e^)

for

APPENDIX

fi =

0

,---,n.

D e fin e

(A 3 )

for

w

[1

= P (e

fi = a 0 ,--*, n -

=v

0

for n = 0, I , - ’ *, a n - 1 and U

[A

A -A

. D e fin e

1

105

e a (r l A c a o+ 1 A - A

W = E ( w n_ 1)

and

e^+ 1 )

L = C

ea

.T h en V =

W ® L . D e fin e (A 4 )

A = E ( a 0) -

(A 5 )

B = {y . G ^ C W ) ! E (y ) n E ( a Q- l ) = Oi .

T hen

B

is open in

G

1

(V )

and

b ih olo m o rp h ically eq u iv a le n t to A holom orphic map T -

is an aly tic.

f :Q

Take

E ( a Q- l ) = 0 .

H en ce

G ao

C

AxB

x e Q .

T hen

H en ce

g

T h erefo re

O b s e rv e

is d efin ed , w h o se graph

x

6

Q .

T h en

g (x ) = P ( f l )

T h erefo re one and only one

dim E ( x ) fl

E (x ) fl E ( a Q) =

T fl ( Q x E ( a Q) )

T h erefo re

is an alytic.

with

0 + g

6

E (x ) n ( E ( a Q) - E ( a 0 - 1 ) ) .

g \ W .

h (x ) e G p _ 1 (W )

H en ce

dim W fl E (x )

e x is t s such that

E (h (x )) = E ( x ) 0 W .

B ecause

E (h (x )) fl E ( a Q- l ) = E ( x ) fl W fl E ( a Q- l ) = 0 ,

h : Q -> B

Take

is d efin ed .

xQ e Q .

T h en and

We w ill show that

A n open neighborhood

phic v ector function

E ( a Q- l )

n : T -» S < v , a >

g (x ) \ E ( a Q- l ) .

W fl E ( a Q) = E ( a 0 ~ l ) , w e co n clu d e that

U .

is

is holom orphic.

(A 6 )

6

the p rojection

dim E ( x ) fl E ( a Q) = 1 .

g : Q -» A

A map

H ere A

dim E ( x ) fl E ( a Q) > 1 and

a map

x

is an aly tic.

.

S ,

c o n s is t s o f e x a c tly one point.

S ince

(V )-B

s h a ll be defin ed. B y Lem m a 1.4

{g (x )S

Take

1

| ( x , y ) ( S < v , a > x E ( a 0) | y f E ( x ) i

B y the d efin itio n o f

is s u rje c tiv e .

= p .

E (a 0 - 1 )

g : U -> V

U

of

e x is t s such that

h

w e have

h (x ) e B .

is holom orphic.

x Q in

Q

and a holom or­

g (x ) = P ( g ( x ) )

for a ll

g ( x ) e E ( x ) fl ( E ( a Q) - E ( a Q- l ) .

H olom orphic maps

gQ : U

g ( x ) = g Q( x ) + g Q( x ) e

C-{0!

e x is t such that

g Q : U ->

INVARIANT FORMS ON GRASSMANN MANIFOLDS

106

for a ll

x e U .

Since the ta u to lo g ic a l bun dle is lo c a lly triv ial,

U

taken so sm all that there e x is t holom orphic vector functions for

such that

fi =

over

U ;

i.e.

e„ on ^ 0 T hen

,

*5 p

g ( x ) , ^ ( x ) , - •*, t )p (x )

H olom orphic maps y

0

: U -» W

: U -> C

U .

B ecause

E (a n- 1 ) C W , u

is holom orphic.

F o r each

x e U ,

b a s e of E ( x ) x e

U and

W

D

=

E (h (x )) .

h|U is holom orphic.

h (x )

=

T h erefo re

E (x )

5 1

(x ),--* , a

3

••• a

+

t )^ =

p (x ) 3

S p (V )

x^U.

g n(x ) e W if u

P ( a x( x ) h

for each

e x is t such that

w e h ave

the vecto rs

H en ce

: U -* V

is a holom orphic frame o f

is a b a s e o f

and

can be

x e U .

d efin e a

(x ) )

is holom orphic on

if Q .

A

holom orphic map (A 7 )

f = (g ,h ) : Q -» A x B

is defin ed. C l a im

2.

T h e map

is in jec tive .

Take

P r o o f of Cla im 2. g(x) = P (g)

f

with

x

g J- W .

and

x

A ls o

is a total fla g for

W .

fl = 0, I , - - - , p - 1 .

Then

the Schubert variety

n-

1

.

T hen

T hen g (x ) =

T h erefo re

the seco n d claim is proved.

R e c a ll the d efin itio n o f

im p lies that

h (x ) = h(5T) .

f(x)=f(x).

g = ( E ( X ) n W) © C g = E ( x ) .

= E (h ( X ) ) © C x = x ;

Q with

( E(x) nW) © C g = E (h (x )) © C g =

E (x ) =

H en ce

in

G p_

1

w^

D e fin e

in ( A 3 ). c

fl

= a

fl~rI

Then w = (w 0,-*-, w n - 1 ) e F - and

d = c n + fi = c for fl fl_ fl

C = ( c 0,-*-, c p l ) _ 1 ( V ) .

n_1

G p _ j(W )

is d efin ed.

D e fin e

E 'Q i ) = E (w ^ )

is a sym bol and H ere

WC V

for fi = 0, I , - * - ,

107

APPENDIX

(A 8 )

E(/0 n w

(A 9 )

EGx)

D e fin e

n

= E '(/ x - 1)

if

a0 < M < n

W = EGt) = E'(/x)

if

C = B (lS < w ,C > .

C L A IM 3.

If

x e Q ,

P r o o f of Cla im 3. im p lies that

then

Take

W fl E (b

h (x ) e C .

^ € Z [0 ,p -1 ] .

T h en

b^ + 1 > a Q .

x) = E '(b ^ + 1 - 1 ) = E '(d ^ ) .

dim E ( h ( x )) n E '(d ) = dim E ( x ) n W R E ( b (J.

> dim E ( x ) fi E (b

H en ce

0 < M < aQ .

h (x ) e S < w , C > .

C L A IM 4.

T h e map

P r o o f of C la im 4. w ith

0 =|= t)

that

b t W .

such that

6

A ls o

f : Q -> A x C

T ake

E ( a Q) - E ( a Q- l ) .

If

1 < /x < p ,

I1

/x

x e S < v , Cl > .

E ( z ) fl E ( b 0 - 1 ) = 0 . d efin ed w ith

[i

= E (h (x )) .

We h ave

C laim 3 is proved.

E (z ) C W

T h e map

e x is t s

t) e E ( x ) fl E ( b Q) . H en ce dim E ( x ) N ow (A

in

x e Q

w e have

8)

im p lies that E (b

fl ) fl

z e B , by (A 2 ).

.) > n . /xx

dim E ( x ) fl E (b ^ ) > M + 1 •

w e h ave N ow

E ( x ) 'f l E ( a Q) = lg(x)S

( y ,z ) = (g (x ), h (x )) = f (x ) .

f

y = P(t})

x e G p (V )

E (x ) fl E (bQ —1) =

f (x ) = (g (x ), h (x )) and with

T h e map

C laim 4 is proved. C L A IM 5.

and

W f l E ( a 0) = E ( a Q- l ) , this im p lies

) = dim E ( z ) fl E '(d

Because

H en ce

y = P(t))

j ) + dim W - dim V > fi + 1 .

T h en

b^ > a Q .

t) e ( E ( x ) - E ( z ) ) fl E (b ^ ) ,

T h erefo re

)

T h erefo re

1

dim E ( z ) fl E (b

B ecause

[A~r 1

One and only one

T h en

then

W = E '(b - 1 ) = E '(d _ - ) .

.

S ince

t) ^ E ( z ) .

E ( x ) = C i) + E ( z ) .

T h erefo re

is b ije c t iv e .

(y ,z )f A x C

T h erefo re

E ( b Q) > 1 .

h (x ) e B .

H en ce ( A 8 )

r e s t r ic t s to a b ih o lo m o r p h ic map

is

E (z ) = E (x )f lW f

is s u rje c tiv e .

INVARIANT FORMS ON GRASSMANN MANIFOLDS

108

(A 1 0 ) and

f : S*

S* < v , a >

is bih olo m o rp h ically eq u iv a le n t to

Take

P r o o f of Claim 5. p lie s that

y e S *< w, C > .

W n E (a 1 ) = E '( a r

0 = dim E (y ) H E '(d Q- l )

T h erefo re

y f

D e fin e and

AxS*

l )= E '( d

l).

E (y ) < W .

N o w ( A 8 ) im­

H en ce

= dim E ( y ) n E ( a j ) > dim E (y ) n E ( a Q- l )

B flS < w , C> = C .

M = A x S *< w , C > .

b j - l > a 0 .

0-

Then

C Q .

S *< w, C > C C .

H en ce

Take

B y ( A 8 ) w e have

.

x f S

flQ.

T hen

1< j < p

W n E ( b j - l ) = E '( b j - 2 ) = E '(d

-

1

) .

T h erefo re dim E (h (x )) n E 'C d j ^ - l ) = dim E ( x ) D W f l E ( b j - l ) > dim E ( x ) fl E ( b ~ l ) + dim W - dim V > ( j - l ) + 1.

Take

(i c Z [ 0 , p - 1 ]

w ith

4

n

E (b ^ + 1 ) n W = E X b ^ - l ) = E '(d ^ ) .

j • T hen

b^ + 1 > a 0

ant* 0^8 ) imp lie s that

T h erefo re

dim E (h (x )) n E '(d ) = dim E ( x ) D W n E ( b ^ + 1 ) > /x+ 1 .

H en ce

h (x ) e S < w, (?j_j C > .

Take

x e Q

w ith

( E ( a Q) - E ( a Q- l )

e x is t s such that

g if E ( h ( x )) fl E (b — 1) h av e

h (x ) eS < w , .

g (x ) = P ( g ) .

g