Invariant Forms on Grassmann Manifolds. (AM-89), Volume 89 9781400881888
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Table of contents :
CONTENTS
PREFACE
GERMAN LETTERS
INTRODUCTION
1. FLAG SPACES
2. SCHUBERT VARIETIES
3. CHERN FORMS
4. THE THEOREM OF BOTT AND CHERN
5. THE POINCARÉ DUAL OF A SCHUBERT VARIETY
6. MATSUSHIMA’S THEOREM
7. THE THEOREMS OF PIERI AND GIAMBELLI
APPENDIX
REFERENCES
INDEX
Citation preview
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