Read Tian’s invariant of the Grassmann manifold text version

The Journal of Geometric Analysis Volume 16, Number 3, 2006

Tian's Invariant of the Grassmann Manifold

By Julien Grivaux

ABSTRACT. We prove that Tian's invariant on the complex Grassmann manifold Gp,q (C) is equal to 1/(p + q). The method introduced here uses a Lie group of holomorphic isometries which operates transitively on the considered manifolds and a natural imbedding of (~1 (C))P in G p,q (C).

R~sumg. On prouve que l'invariant de Tian sur la grassmannienne G p,q (C) est 1/(p + q). La m~thode prgsentge dans cet article utilise un groupe de Lie d'isom~tries holomorphes qui opbre transitivement sur les vari~t6s consid6r~es ainsi qu'un plongement naturel de (IP1(C))P dans Gp,q (C).

1. Introduction

On a complex manifold, an hermitian metric h is characterized by the 1-1 s y m p l e c t i c form o9 defined by o9 = i g ~ d z ~ A d ~ ~, where g ~ = h ~ / 2 . The metric is a K~ihler m e t r i c if o9 is closed, i.e., do9 = O; then M is a K i i h l e r m a n i f o l d . On a Kahler manifold, we can define the R i c c i f o r m by R = i RAg d z ~ A d i ~, where Rzff = -0xff log Igl. A K~ihler manifold is E i n s t e i n with f a c t o r k if R = kog. For instance, choosing a local coordinate system Z = (zl . . . . . Zm), the projective space ~m (C) with the Fubini-Study metric 09 = ia01og(1 + IIZII 2) is Einstein with factor m + 1. On a K~ihler manifold M , t h e f i r s t C h e r n class C 1 ( M ) is the cohomology class of the Ricci tensor, that is the set of the forms R + iO0~o, where ~0 is C ~ on M. If there is a form in C I ( M ) which is positive (resp. negative, zero), then C 1(M) is p o s i t i v e (resp. negative, zero). If a Kahler manifold is Einstein, then C 1(M) and k are both positive (resp. negative, zero). In the negative case, it was proved by Aubin [1], see also [4], that there exists a unique Einstein-Kahler metric (E. K. metric) on M. It is so for the zero case too [1, 15]. The question for the positive case is still open: Some manifolds, such as the complex projective space blown up at one point, do not admit an E. K. metric (for obstructions, see [11] and [9]). Aubin [2] and Tian [14] have shown that for suitable values of holomorphic invariants of the metric, there exists an E. K. metric on M.

Math Subject Classifications. 53C55, 32M10. Key Words and Phrases. Kahler manifold, Einstein-K~ler metric, first Chern class, admissible functions, Tian's

invariant, Grassmannian.

9 2006 The Journal of Geometric Analysis ISSN 1050-6926

524

Julien Grivaux

For co/27r in C 1(M), Tian's invariant ot(M) is the supremum of the set of the real numbers

satisfying the following: There exists a constant C such that the inequality L . e-a~ < C holds

d/F/

for all the C ~ functions ~0 with co + iO0~o > 0 and sup~o > 0, where co = i gx~dzXA d~ ~ is the metric form. Such functions ~0 are said co-admissible. In [14], Tian established that if a ( M ) > m/(m + 1), m being the dimension of M, there exists an E. K. metric on M. This condition is not necessary: It does not hold on the projective space, where Tian's invariant is 1/(m + 1). In the same article, Tian introduces a more restrictive invariant ot6 (M), considering only the admissible functions ~0 invariant by the action of a compact group G of holomorphic isometries. The sufficient condition for the existence of an E. K. metric on M remains aG (M) > m/(m + 1); it is more easily satisfied if the group G is rich enough. In many cases, the group G is a nondiscrete Lie group. The invariant OtG(M) can be computed using subharmonic functions methods and the maximum principle (for effective examples, see [5, 6, 7, 8, 13]). In this article, we prove the following theorem.

Theorem 1.1.

Tian's invariant on G p,q (C) is given by ol (Gp,q (C.)) = 1/(p + q).

This generalizes the known result on ~ m ( c ) [14], see also [3]. Let us also mention that Tian's invariant has been computed on ~>m(C) blown up at one point and on certain Fermat hypersurfaces using H~rmander L 2 estimates for the J-equation [14]. We first compute the volume element of the metric ~p,q ; then we will establish some general preliminary results concerning Tian's invafiant as well as imbeddings of {71 (C)}P in Gp,q (C) which allow us to deduce a (Gp,q (C)) from cr (]?1(C)).

2. Basic properties of the Grassmann manifold

We propose here a short survey of the properties of the Grassmann manifold (for more details, see [ 10]). We denote by Gp,q (C) the set of the subspaces of dimension p in C P+q; in particular, G l,m (C) is the complex projective space of dimension m. It is known (see [3]) that on IPm(C), the Fubini-Study metric is Einstein with factor m + 1 and that Tian's invariant is 1/(m + 1). Now, let M* (p + q, p) be the set of the matrices of rank p in Mp+q,p (C). The group Glp (C) acts by multiplication on the right on M*(p + q, p). More precisely, (M*(p + q, p), Jr, Gp,q(C)) is a principal fiber bundle with group Glp (C). The group Glp+q (C) acts by multiplication on the left on M*(p + q, p) and induces an action on Gp,q(C); so does the unitary group U(p + q). These groups act transitively on Gp,q (C), which shows that Gp,q (C) is compact. We denote by Z the set of all increasing-ordered subsets of p elements in {1 . . . . . p + q}. Let P be an element of M*(p + q, p), P = (Pij)l<i<p+q. By Cauchy-Binet formula we get:

l<j<p

det(tpfi) = ~ l e i I d e t m , ( P ) l 2, where ml(P) is the matrix (Pij). i d 9 The form w, where

l<j<_p

co = i 0~ log det( tp ~ ), is invariant by the action of Glp (C) on M* (p q- q, p), and so it projects onto a form Gp,q. The metric Gp,q is a Kahler metric form on Gp,q (C.). For p = 1, this metric on Gl,m (C) is the Fubini-Study metric on the complex projective space. The action of the unitary group U(p +q) on Gp,q (C) preserves the metric ~p,q so that U(p +q) is a group of holomorphic isometries which operates transitively on Gp,q (C).

Tian's Invariant of the Grassmann Manifold

525

For I in Z, let UI be the set o f the matrices P in M* ( p + q , p ) such that det(m I (P)) is nonzero. Then 7r(Ul) is a coordinate open set on Gp,q ( C ) , the matrix ZI in Mq,p(C ) is the coordinate, the inverse o f the chart ~Ol sends M*(p + q, p) onto zr(Ul) and we have ml (~oI 1(ZI)) = I (p) where I (p) is the p x p identity matrix, and m l c ( ~ o l l ( Z l ) ) L e m m a 2.1.

= Zl.

For I in Z, let )~i be the map from Jr ( U1) to ~+ defined by k l ( Z l ) = [det ( I d + t Z l Z l ) [ - ( P + q ) .

Then ()~1) I ~z are the components of a maximal differential form ~1on G p,q ( C ), namely: r1 = ~.1(i/2) pq ( d Z A d Z ) , . Proof.

It suffices to show that the following transformation rule holds: for every I, I" in Z, ~.I is equal to )~7"x Let P1 be the matrix r det O Z T 2 O ZI on 7 f ( U I ) CI ~ ( U T ) .

Then PI{m"f(P1)} -1 = P-f, so Z 7 = m ic(Pl) {m?(Pl)} -1. The

differential o f the map which sends Z1 on PI is the map which sends H on/-), where m 1c (I~-I) -m- H and ml(lYl) ----0. The change o f charts sending Z1 on Z 7, we obtain

D ZT(H) = m p ( / 4 ) { m ' f ( P l ) } -1 - m ic(Pl){m~'(P,)l-lm'f(IYt){m)'(P/)} -1 = (m Zc(IY-I)- ym'i'(IYI))a -1 ,

where ot = m'f(Pl), fl = mic(Pl) and y = flot -1 .

L e t u s define a m a p u from M ~ p ( C ) to Mq,p ( C ) by u(H) = m ic(IYt) - y m'f(IYI). We can choose ..... q+p}and/={1 ..... r}tA{q+l+r ..... q+p},whereO<r<inf(p,q). (kxS) by (E(kxt)~I).U = (~ik (~jtz" We have We define the k x l matrix Ei, j x i,j I={q+l

mT (~(qxp)]I = E!P.xp) i,j z,j

and

if

i -- r, and 0 if < i > r,

i > r ' i _< r. Hence, if i < r, and 0 elsewhere.

micx(/~(qxp)]i,j = E(qxP)i-r,j if i

and 0 if

(F mT [ ~ (i,j x p)]'~ ~. q liar = Yai m _I[\ ~(q x p)h 8J3 = Fai 6j3 i,j ]ij

Now the map which sends H to Y m~'(/~) can be restricted if 1 < j < p to the span Bj of the (Ei,j)l<_i<q. The r first columns of its matrix are those of y, the others are 0. The map which sends H to F m ic ( H ) maps also Bj into itself. The right upper block of its matrix is I (q-r), the other elements are 0. This allows us to compute the matrix of the restriction of u to B j, whose de-

terminant is ( - 1 ) r·

det(Fij )q-r +l<_i<_q. S o d e t u = ( - 1 ) pxr·

l<j<_r be the span of the (Ei,j)l<j<_p. Each

[ det(Fi j )q-r +l<_i<_q] p.

l<j<r a

For 1 < i < q, let Ci Ci is stable by the map from Mq,p(C ) to Mq,p(C ) which sends H to H ot-1 . The matrix o f the restriction is ot -1, so the determinant of the map is (det a)-q. Hence, I det DZ~'(H)12 =

det(yi,j)q_r+l<_i<_ q 2p x [ d e t a [ - 2 q .

l<j<_r

Let A be the right r x r upper block of or. The left (p - r) x (p - r) lower block of or is I (p-r) and the right ( p - r) · r lower block is 0, so detot = ( - 1 ) r(p-r) d e t A . The left r x ( p - r) lower block of/3 is 0, the right r · r block is I (r) so that the left r x r lower block of V is A -1 .

526 From this we deduce ~.?=

Julien Grivaux

]det DZT(H)[ 2 = I detot [-2(p+q).

Since PI et-1 = PT, we have det OzIOZ'~-21.

det(tpTp~ ") -~P+q)=ldetotl2(p+q)~,i=

[]

L e m m a 2.2.

The unitary group U (p + q) preserves rl.

Proof.

Wecall I the set {q + 1 . . . . . q+p}. We define Pl in zr(U1) by Pt = {071(Zl). L e t U

be an element in U(p + q) such that mI(UPI) is invertible. Let/5/ = UPt {mI(UPI)} -1 and

ZI = mlc(Pl). We have Z1 = mlc(U) PI {mI(U)PI} -1. So

D2I( H~ = mIc(U) [I:I{m,(U)ej }-1 _ PI {mI(U)PI }-lmI(U) 1:1{mI(U)PI }-1].

Thus, DZI(H) = S/-)8 -1, where 8 = mI(U)PI and X -- mlc(U)[I (p+q) - PI 8-1 mi(U)]. Let X1 be the q x q matrix of the q first columns of X. Then, X/-) = X1H and we get DZI(H) = X1 H8 -1. The determinant of the map from Mq,p(C ) to Mq,p(C ) which sends H to H8 -1 is (det 8) -q. The determinant of the map from Mq,p(C ) to Mq,p(C ) which sends H to X1H is (det X1) p, so det DZI = (det X1) p (det 6) -q. We divide U into four blocks:

U:(

Up,qUq Uq,Pup ) , Uq E Mq(C), Up E Mp(C), Up,q E Mpq(C),,

Uq,p E Mq,p(C).

Then 8 ---- Up,q Z 1 3t- Up, so X1 = Uq - (Uq Zl -~- Uq,p) (Up,q Z, -]- Up) -1 Uq,p. Let Z in Mp+q,p+q(C) be the matrix with blocks Zq = I (q), Zp,q : O, Zq,p -- Z1, Zp : I (p), the notations being the same as above. Writing det U = det(UZ) and using the column transformation

C1 ~ C1 - C2 (Up,q ZI + Up) -1 Up,q where C1 is made of the first q columns and C2 of the

remaining ones, we get

det U : det[Uq -Hence, Idet

(Uq Z1 + Uq,p) (Up,q ZI + Up) -1 Up.q] · det(Up,qZi W Up). el = API8 -1, so

DZII 2 =

[det t~[-2(p+q). We have

~.~"= det(t/51 -~l) -(p+q) = det(tpl PI) -<p+q) x Idet 812Cp+q) = ;~1 Idet DZl1-2 , which proves the result.

[]

Proposition 2.3.

(1)

dV (~p,q) :- rI.

I f I ~ Z, ]Gp,qlI = {det(l (p) +tZIZ1)} -~p+q). (3) 7~ (~p,q) : (p + q)~p,q.

(2)

Proof

(1) Let I in Z. It is easy to compute ~p,q at the point Z I : 0: ~p,q(H, K) = Tr(HK). Then dV(Gp,q)lz,= o = (i/2) pq (dZ/x dZ-)l = rl[z/=0. Since dV(Gp,q) and/7 are invariant by the transitive action of U (p + q), we have dV (~p,q) : rl. (2) Since dV (Gp,q) -- IGp,qII (i /2)Pq (dZ /x dZ-)l, property (1) gives the result. (3) Remark that Gp,q = i 0-0 log{det(I (p~ +tZtZt)}. Since T~ (Gp,q) = - i OO log IGp,qIi, we obtain 7~ (~p,q) ~ (p ~ q)~p,q, which expresses that ~p,q is Einstein, with factor p + q. []

Tian'slnvariantof the GrassmannManifold

3. Some general results about Tian's invariant 3.1. Tian's invariant with a normalization on a finite set

527

If X is a manifold, we will denote b y / z x a measure on X compatible with the manifold structure.

Theorem 3.1. Let M be a compact Kiihlermanifoldand G a compactLiegroup ofholomorphic isometries. Let An = {P1 . . . . . Pn} be a finite subset of M. Let ot(w) (resp. Ot~n(W)) be the supremum of the set of the nonnegative real numbers ot satisfying the condition: There exists a

i*

constant C such that the h~equality ],. e -~r <_ C holds for all the og-admissible functions go with

sup go > 0 (resp. with go(Pi ) >- 0 for 1 < i < n). Suppose in addition that the orbit of each Pi

under the action of G has positive measure. Then a(w) = a~n (w).

We first establish a few lemmas which will be useful for the proof.

L e m m a 3.2. Let (gon)n>0 be a sequence of admissible functions with nonnegative maxima. Then there exists a subsetT2 of M, with izM(f2) = I~M(M), and a subsequence gonk of gon, such that for every p in f2, the sequence (gonk(P))k>0 has a finite lower bound (depending on p).

Proof.

It is sufficient to assume that gon has null maxima. Let Qn be a point such that {o (Qn) n vanishes. Green's formula runs as follows:

gon(Qn) = --~

gon +

G(Qn, R) Agon(R)dV(R) ,

with G(Q, R) >_ 0 and fM G(Q, R) d V ( R ) = C, where C is a positive constant (see [4]). Since gon is admissible, Agon is less than m, m being the dimension of M. Thus, fM Igonl < C m V.

Furthermore, fMAgon =O, SO fMlAgonl = 2 f{ A

Vgon(Q)

~v.>0}

Agon < 2mV. Forevery Q i n M , wehave

] V Q G ( Q , R)Agon(R)dr(R), so that JM

fM IVgonI < f M [ f M IVQG(Q' R)Idv(Q)]IAgon(R)Idv(R) < 2 m C V '

f since ].. [VQG(Q, R)I d r ( Q ) is a continuous, hence a bounded function on M. Thus, (gon)n>_O is bounded in the Sobolev space H 1'1 (M). By Kondrakov's theorem, we can extract from (gon)n>__o a subsequence which converges in L 1(M), and after another extraction we can suppose that this sequence converges almost everywhere to a function go of L 1(M). Since q9 is finite almost everywhere, we get the result. []

Lemma 3.3. Let (gon)n>0 be a sequence of admissible functions with nonnegative maxima and suppose that there exTsts a compact group G of holomorphic isometries of M such that the orbit of each Pi has positive measure. Let qb : G ~ IR U {-cx~} be the map defined by ~ ( g ) = inf inf (gok o g). Then there exists g in G such that ~ ( g ) is finite.

An k>O

Proof.

Suppose that 9 = -cx~. For i = 1 . . . . . n, let Ai be the set of the g in G such that

528

k>0

Julien Grivaux

inf (~0k o g)(Pi) = --oo. The sets Ai are measurable and t.Jn_l Ai = G, so there exists i such that

Ai has positive measure. From Lemma 3.2, Ai .Pi is a subset of ~2c. Since f2 and M have the same measure, the measure of Ai.Pi vanishes. Let ui be the map from G to M which sends g to g(Pi). Then ui has constant rank on G. Indeed, ui o L(g) = Crg o ui, where L(g) is the left translation by g and crg the map from M to M which sends x to g.x. Since G.Pi has positive measure, ui is a submersion on G, so that ui (Ai) has positive measure. This is a contradiction

since Ui ( A i ) = A i .Pi. We can now prove Theorem 3.1.

[]

Proof.

Itis clear that or(co) < otA,(co). Conversely, let e > 0. There exists a sequence (~On)n>_ o goes to infinity as k goes

of admissible functions with positive maxima such that ] . . e -(u(~

to infinity. Replacing q9 by q9 -- sup g)n, we can take sup ~0n = 0. First we apply Lemma 3.2. n n For the sake of simplicity, we take q)nk = q)k. From Lemma 3.3, there exists an element g in G such that qb(g) is finite; we define qJk by qJk = ~Ok o g -- ~ ( g ) . Since g is an isometry, qJk is co-admissible, and from the very definition of ~, qJ~(Pi) is nonnegative. Furthermore,

fMe-(~(o~)+~)*~= e('~(o))+~*(g)fMe-(~(o~+e)~ok.

ThisprovesthatfMe-('~(~

infinity as k goes to infinity. Then, otz~"(co) _< or(co) + e. This inequality holds for every positive e, and so ~A, (co) _< or(co). []

3.2. Tian's invariant on a product

For a K ~ l e r form w on a compact K ~ l e r manifold M, or(co) is defined as in Theorem 3.1.

Proposition 3.4. L e t (Mi)l<i< n be compact K ~ l e r manifolds with metric f o r m s (coi)l<i< n . We endow the product M1 x ... x Mn with the metriC col ~ . . . @con. Then ot(Wl ~ . . . ~ Wn) = inf ot (coi).

l<_i<n

Proof.

It suffices to make the proof when n = 2, the general result will follow by induction.

(1) Suppose that ot(coa) < ot(w2), and let e > 0. There exists a sequence (~On)n>_Oof coladmissible functions on M1 with positive maxima such that [

dM 1

e -(~(~~

P

goes to infinity

when n goes to infinity. We define gin on M1 x M2 by aPn(ml, m2) = qgn(ml). Thus, aPn is (o91 @ wz)-admissible on M! · M2, with positive maximum, and ] V(M2) ] e -(~(~ , so that ]

J M 1xM2 e--(et(COl)+e)IPn

JM lXM2

e -(a(~~

goes to infinity when n goes to infinity.

JM 1

We have therefore Or(CO1 ~) O92) ~ Or(COl) -~- 6. This yields or(w1 @ w2) < ~(col). (2) Let us now prove the opposite inequality. Let ot be a real number such that ot < inf(ot(Wl), ct(co2)) and tp an (COl ~ co2)-admissible function on M1 x ME. If m2 is in M2, the function which sends ml to tp(ml, mE) is wl-admissible. The same holds for M1. Let (u, v) in M1 x M2 be such that ~0(u, v) __ 0. Then

fMtxM2e-C~~

dVldV2=fMle-U~(rnl'v)(fM2e-C~[~(m1'm2)-~('nl'V)ldV2)dV1 < C2 [

dM 1

e -a~~

dV1 < C1 C2

Q

Tian's Invariantof the GrassmannManifold

Thus, cr _< or(o) 1 ~ o92) and we get inf(a(wl), a(O)2)) < Ot(W1 ~ 092).

529

[]

3.3. Tian's invariant on Gp,q (C)

Since there is a natural duality isomorphism between Gp,q (C) and Gq,p (C), we can assume that p < q without loss of generality.

Imbedding of {~I(c)}P into Gp,q(C) when p 5 q For w in C p(q-1), W = (Wi,j) l<i<q, we define the map 15w from {C 2 \ (0, 0)} p to Mp+q,p(C) by l<j<p i#j

Pw ( ()~i,tZi)l<_i<_p ) =

{

~.i ~ij Wi-p,j;~j

/z i

if if if

i < p i > p and i # j + p i > p and i = j + p

Wemake, f o r p + l

< i < p + q , the following row transformations: Li +-- L i -

We get a matrix

(ciJ)l<_i<_p+ with r q

~ Wi-p,j Lj. l<j<_p iCj+p = ~ij ~'i if 1 < i < p and Cij = ~i--p,j ~j if p + 1 <

l<j<_p i _< p + q, which has rank p. tSw induces a map from {I?1(C)} p into Gp,q(C ) as shown on the following diagram, where y is the projection of the principal fiber bundle {C 2 \ (0, 0)} p onto {1?I(c)}P. Remark that 15w sends [0, 1] · ... x [0, 1] onto zr(a), where m{p+l ..... 2p}(a) = I (p) and m{p+l ..... 2p}C(A) = 0 (qxp).

{C2\(0,0)} p

Pw > M * ( p + q , p )

{171(c)}P

pw

We have

>

Gp,q(C)

1

(7~ o Pw)* (~p,q) = i O-a log(det(t~w~w))

= i O-Olog ( p det (~w~w)

) + ~ - ~ i O 0 1 o g ( l ~ . k l 2 + l # k [ 2)

\l-I(l

k=l

= i O0 log ~ +

l + I. l

y*(FS1 ~ " " ~ F S 1 ) ,

where FS1 is the Fubini-Study metric on 171(C). @ is invariant by the action of the structural group C* x . . . x C*, so it induces a map qb from {171(c)}p into C. Note that qb([O, 1] · · [0, 1]) = 1. Then (Jr o Pw)*(~p,q) = zr*(i O0 log 9 + FS1 @... @ FS1), so that p*(~p,q) = i Oa log qb +

FS1 ~ ... ~ FS1.

530

Lower bound of ~(~p,q)

.Iulien Grivaux

For I in Z, we define

P1 by ml(Pl) :- I (p) and m l c ( P l ) = 0 (q· I f n = (P+q), we set An = {PI}I~I. Since U(p + q)isatransitive group of holomorphic isometries of Gp,q (C), we know from Proposition 3.1 that ot (~p,q) = aA, (Gp,q). We set I = { p + 1 . . . . . 2p}. Let gobe an admissible function on Gp,q (C), nonnegative on An. The last equality of the precedent section shows that the function goopw + l o g q~ is (FS1 @... ~ FS1)admissible for every w in cp(q-1). Furthermore, (go o Pw + log ~ ) sends [0, 1] x ..- x [0, 1] to the nonnegative number go(P~). It is known that a ( F S 1 ) = 1 (see [3]). Proposition 3.4 yields

ot(FS1 ~ . . . ~ FS1) = 1.

Let a be a real number such that ot < 1. There exists a constant C, independent of go, such that

f{~ ~(c)}pe-Ot~oop~ ~-ot < C. We define the map Fz from 7"t'(Ul) to N+ by F I ( Z I ) =

det (Id+tZlZ1). On { p I ( c ) } P , we work with the coordinates/zl . . . . . # p in the chart ),1 = . . . . 3.p = 1. Thus,

r

-- p

k=l

o Pw(,)

,

so that

I-I (1 + [/zkl 2)

q We have the inequality 1 + E

l+l/zkl 2 <

f Ju~CP e-~176176

dV.(CP)

I-I (1 + ]/zk 12)2-u (F,o

k=l

P

~C .

Pw(tZ)) ~

p

lz,Jl 2 --- F,(z,).

l<j<p ir

l<i<q

In particular, for every k in {1 . . . . . p},

i=1 j = l

Fiopw(lZ),andFiopw(IZ) > 1 + E

IWiJ[ 2" Thus, f o r x > 0 a n d w 6 C p ( q - l ) ,

P 1-I (1 +

k=l

Iz~kl2)2-~

< 1 <

1

1

( F 1 o pw(ls

tc+p+q-~ - (F, o pw(l~)) tc-p+q+~

-

1+ Z Iw/Jl 2

l<_i<q l<j<p i#j

(l+llwll=)

We have, according to Proposition 2.3,

e_C~~ (UI) F~[

e-et ~oopu,(lz )

(X, OPw(l~)) r+p+q

e-a~~176(it) Pu'

NV#(CP)MVw(CP(q-1))

= f~c~-l' f~c~ (fi(1

k=l P

+ I/zkl2) 2-~ (FI

o

Pw(tZ)) a

)

1 7 ( 1 + I/zkl2) 2-~

x

(F1 o Pw (['1"))tc+p+q-~

k=l

dVl.t (C P) d V w ( C p(q--l))

= fU, ECP(q--1)

(

Tian'sInvariantof the GrassmannManifold

e-Ct~oopw (tx) ECp

531

f~ P

1-I(l+llzl~12)2-a(fiopw(p,))a

k=l

dVl~(CP )

X

dVw(Cp(q_l))

(I+

Ilwl15

<C

--

fw

cC p(q-1)

dvm(cpCq- < C' )) (1 + IIw[15

- -

if

x >p(q-1).

Thus, we obtain that for all I in 77, f~

(u~) F[ -

e - ' ~ < C, where C is independent of ~o.

Since Gp,q (C) is compact, there exists a family (V/)te:r of open sets of Gp,q (C) such that VI is relatively compact in Jr(U/) for every I e 77, and U / e I v/ = Gp,q(C). There exists M > 0 such that F/ < M on VI for every I ~ 77. Thus,

fG p,q(C) e - ~ ~ -We deduce that ~(~p,q) > 1.

I~Z

--_<M~CZf fvie-C~~ F[ IeZ

rr(UI)

--<ce-~~ M~:(P; q). F~

-

U p p e r bound o f t~(~p,q) We use here a method which can be found in [13] for the complex projective space. Let I in 77. We define/r from M*(p + q, p) to ]?1 (R) by the relation K (M) = [ I det m ~( H ) 12, det tMM ]. K is invariant by the action of the structural group Gp (C), so it induces a C ~ map K from Gp,q ( C ) to ~l (]~). Remark that ~t = log K is a K~hler potential on UI for the metric ~p,q.

There exists a decreasing sequence (~on)n>_oof admissible functions with positive maxima which converges pointwise to -ap on zr ( Ui ).

L e m m a 3.5.

Proof.

We construct a decreasing sequence (fn)n> 0 of C ~ convex functions on R+ satisfying

the conditions 1 + f~ > 0, fn(x) = - ( 1 - 1/n)x f o r x in [0, n] and fn(x) = - n f o r x > 2n. Let y be an element of re(U1) C and ~"~n the set of the elements x in re(Ul) such that ~ ( x ) > 2n. Since FI(y) = [0, 1], there exists a neighborhood V of y such that the inequality z > e 2n holds for every point [1, z] in FI(V). Thus, V N re(Ul) is included in f2n. We have proved that Wn = ~2ntOre(UI) c, so that Wn is an open neighborhood ofzr (UI) c. We define ~n by ~0n = fn o 7t on re(Ul) and ~0n = - n on Wn. Thus, ~0n is well defined and ~0n(0) = 0. It remains to show that q9n is admissible on re (UI). We have

(~p,q -1- i

0~n)x~

= 0xu~ + oz (J~n~o ~ ) ~

= (1 + f,'n o ~)Ozff~ + f"on ~ 8zaP 0t~P 9

Hence, the matrix of the metric ~p,q "-[-i 0-8q9 is of the form A + T where A is positive definite n and T has rank one and positive trace. So A + T is positive definite and we get the result. [] L e m m a 3.6.

Let n in N* and r a positive real number. Then

fllXIl<r dVx(Mn(C)) _ Idet Xl 2

532

Julien Grivaux

Proof.

We can write

OO

fll

dVx(Mn(C)) _ ~ , s XIl<_r [det X] 2 Ic=0 dVx(Mn(C))

k dVx(Mn(C)) Idet Xl 2

We put Y = 2~X, so

f ~ k+l <IIXII <_r/2k

Idet Xl 2

_

i/2<llyll< dVy(Mn(C))l r- det El 2

The terms in the series are strictly positive and independent of k. The sum is therefore infinite. [] We can now prove that

Ol(~p,q)is upper bounded by

1. Suppose t h a t ol(~p,q) > 1. Then

there exists a positive C such that for every integer n, [ e -~" _< C. Using Lemma 3.5 and Jzr (U1) monotonous convergence, f~

(UI)

FI < C. Since rr(U1)Chas zero measure, fG

p,q (C)

FI < C.

Let [ in Z be such that I f) I = 0 (this is possible since p < q). We have Remark that

_

ml(P[) = mz(Zi).

_

Thus,

IziU < r, det(tpiFi) < M, so that ~

P[{ml (Pi)} -1 = PI. det(id+tZiZi) = det(tp~pi)Idetml(Z[)l -a. For dVzi(Mq,p(C)) < + o o . Integrating o v e r t h e Zill<-r [ d e t m l ( Z i ) [ 2

< + c ~ , w h i c h is in c o n t r a d i c t i o n

r e m a i n i n g v a r i a b l e s (ZiJ)i -[cc~icyields f dllZll_ d V zdet p ( C ) ) <r I( M ZI a w i t h the r e s u l t o f L e m m a 3.6. T h u s , w e o b t a i n ot (Gp,q) < 1.

References

[1] Aubin, T. l~quations du type Monge-Amp~re sur les vari6t6s kahl6riennes compactes, Bull. Sci. Math. 102, 63-95, (1978). [2] Aubin, T. R6duction du cas positif de l'6quation de Monge-Amp~re sur les vari6t6s Kahl6riennes compactes ~ la d6monstration d'une in6galit6, J. Funct. Anal. 57, 143-153, (1984). [3] Aubin, T. M6triques d'Einstein-Kahleret exponentiel de fonctions admissibles, Z Func. Anal. 88,385-394, (1990). [4] Aubin, T. Some Nonlinear Problems in Riemannian Geometry, Springer-Verlag, Berlin, (1998). [5] Ben Abdesselen, A. Lower bound of admissible functions on sphere, Bull. Sci. Math 126, 675~580, (2002). [6] Ben Abdesselen, A. Enveloppes inf6rieures de fonctions admissibles sur l'espace projectif complexe. Cas sym6trique, to appear in Differential Geom. Appl. [7] Ben Abdesselen, A. and Cherrier, P. Einstein-Kahler metrics on a class of bundles involving integral weights, J. Math. Pures Appl. 3(81), 259-281, (2002). [8] BenAbdesselen, A. andCherrier, P. Estimations of Ricci tensor on certainFanomanifolds, Math. Z. 233, 481-505, (2000). [9] Futaki, A. An obstruction to the existence of Kahler-Einstein metrics, Invent. Math. 73, 437-443, (1983). [10] [11] [12] [13] [14] Kobayashi, S. and Nomizu, K. Foundations of Differential Geometry, Vol. II, John Wiley & Sons, (1969). Lichnerowicz, A. Sur les transformations analytiques des vari&6s k'~hl6riennes, Cr. Acac. Sci. 244, 3011-3014, (1957). Matsushima, Y. Sur la structure du groupe d'hom6omorphismes analytiques d'une certaine vari6t6 k~hl6rienne,

Nagoya Math. J. 11, 145-150, (1957).

Real, C. M6triques d'Einstein-K~ihler sur des variStSs ~ premiere classe de Chern positive, J. Func. Anal. 106, 145-188, (1992). Tian, G. On Kahler-Einstein metrics on certain Kahler manifolds with C l (M) > 0, Invent. Math. 89, 225-246, (1987).

Tian's Invariant of the Grassmann Manifold [15]

533

Yau, S.T. On the Ricci curvature of a compact Kahler manifold and the complex Monge-Amp&e equations, I, Comm. Pure Appl. Math. 31, 339411, (1978).

Received December 14, 2005 Universit6 Pierre et Marie Curie e-mail: julien.grivanx @free.fr

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