ON COMPACT K HLER MANIFOLDS OF NONNEGATIVE BISECTIONAL CURVATURE, I

ON COMPACT K HLER MANIFOLDS OF NONNEGATIVE BISECTIONAL CURVATURE, I

BY ALAN HOWARD, BRIAN SMYTH

University of Notre Dame Notre Dame, Indiana, U.S.A.

and H. WUQ) University of California Berkeley, California, U.S.A.

This is the first of two papers devoted to the study of compact Kghler manifolds of nonnegative bisectional curvature (see also [Wu2]). The main result of this paper is the following theorem; together with its corollaries below, this theorem shows t h a t such manifolds possess a rigid internal structure. For their statements, recall from [Wul] t h a t the Rieei curvature is quasi-positive iff it is everywhere nonnegative and is positive in all directions at a point; an equivalent definition is t h a t the Ricci tensor Ric is everywhere positive semi-definite and is positive definite at a point.

T I - i ~ o R ~ r . Let M be an n-dimensional compact Ki~hler mani]old with nonnegative bi- sectional curvature and let the m a x i m u m rank o~ Ric on M be n - k (0 <~ k <. n). Then:

(A) The universal covering o] M is holomorphieaUy isometric to a direct product M ' • C ~, where M ' is an ( n - k)-dimensional compact KShler mani]old with quasi-positive Ricei curva- ture and C ~ is equipped with the fiat metric.

(B) M ' is algebraic, possesses no nonzero holomorphic q./orms /or q>~ 1, and is holo- morphieaUy isometric to a direct product o/ compact Kiihler mani/olds M 1 • ... • M~, where each M ~ has quasi-positive Rieci curvature and satis/ies H 2 ( M t, Z) -~ Z.

(C) There is a fiat, compact complex mani/old B and a holomorphic, locally isometrically trivial/ibration p: M ~ B whose/ibre is M ' .

(D) There exists a compact Kahler mani/old M*, a fiat complex torus T, and a com- mutative diagram:

M * , T

i 1

M - , B

(1) Research partially supported by the National Science Foundation.

5 2 A . H O W A R D E T A L .

where the horizontal maps are holomorphic, locally isometrically trivial/ibrations with fibre M ' , and the vertical maps a r e / i n i t e coverings. Furthermore, M* is globally diffeomorphic to M ' •

I n particular, ~I(M) is either trivial or an infinite crystallographic group.

C o R 0 L L A R ~ 1. Let M be a compact KShler manifold o/nonnegative bisectional curvature.

T h e n the following are equivalent:

(A) M is simply connected.

(B) The first Betti number is zero.

(C) M has quasi-positive Ricci curvature.

C o R 0 L L A R u 2. A simply connected compact K~hler manifold o/nonnegative bisectional curvature is irreducible (in the sense o / t h e de R h a m decomposition theorem) i/f its second Betti number is one.

A theorem slightly weaker t h a n the one above was first announced without proof b y the first two authors at the end of fitS] in 1971. Unaware of this result in [HS], but mo- tivated by his a t t e m p t to extend the argument of [SY] from positive bisectional curvature to nonnegative bisectional curvature, the third author independently arrived at a theorem also slightly weaker t h a n the one above. The present paper is roughly patterned after the arguments of the third author which rely on the structure theorem of Cheeger-Gromoll ([CG1], [CG2]) on compact Riemannian manifolds of nonnegative Ricci curvature.

Section 1

We summarize the preliminary material in this section. First recall the structure theorem of Cheeger-Gromoll ([CG1], [CG2]) specialized to the K/~hlerian case. Let M be a compact K/~hler manifold with nonnegative Ricci curvature. Then its universal covering manifold is holomorphically isometric to a direct product M ' • (~k, where M ' is a compact K~thler manifold and both the flat metric on C k and the product metric on M ' • C z are understood (here the Ki~hlerian deRham decomposition theorem is needed as well; see [KN], p. 171). Moreover, there is a finite covering M* of M such t h a t M* is diffeomorphie to M# • T ~, where T ~ is a complex k-dimensional torus and M# is a compact Kghler manifold covered b y M'. This implies t h a t ~I(M#) is finite. As a consequence, if the Ricei curvature of M is quasi-positive, so is t h a t of M ' • C k and hence k = 0 and zl(M) is itself finite (cf. the comments in [Wul] on this fact).

Next we review the basic Bochner technique needed for the purpose at hand (el.

[GK] or [L], pp. 3-6). Let ~ be a real (1,])-form on a compact Ki~hler manifold and let

O:N COMPACT K~.HLER -:~IANIFOLDS O F I~ONNEGATIVE BISECTIONAL CURVATURE. I 53 R~B, R~BCD be respectively the components of the Rieci and Riemannian curvature tensors (sign convention: R ~ is a positive multiple of the sectional curvature). Define

F($) = 2RAs $AC$~ __ R~CD $A~$CD.

I f ~ is harmonic and F(~)>~0, then F ( ~ ) = 0 and ~ is parallel. This is the basic observation.

L e t {X t ... X~, J X 1 .... , JX~} be a local frame field t h a t diagonMizes ~, i.e. $ , . ~-$(X~, JX~) are the only nonzero components, then

F(~) = 2 ~. R~,.z.(~,, - ~j~.)~ w h e r e R . , z . =- (Rxdx~Xj, J X j ) . (-~ )

Since /~..jj. is the bisectional curvature defined b y spanR { X . J X , } and spanR {Xj, J X j } , we have from ( ~e):

LEMMA 1. I / a compact Kdhler maul/old M has nonnegative bisectional curvature, then all harmonic/orms o/type (1,1) are parallel.

F r o m now on assume M is compact K~hler with nonnegative bisectional curvature.

L e t ~o be the K~hler form of M. I f ~ is a harmonic (1,1)-form distinct from (o, define a tensor field S, one-fold contravariant and one-fold covariant, b y the equation ~(X, Y ) = (co(X), Y) for all vector fields X and Y. To be more precise, let ~' ~nd G be respectively the 2-fold covariant H e r m i t i a n tensor fields associated with $ and eo (i.e., G is the Ks metric); we then define S by

~'(x, y ) = G(s(x), r), vx, y.

I t is clear t h a t S is self-adjoint relative to the K~hler metric G; consequently S defines a diagonalizable linear transformation at each t a n g e n t space M~ of M. On the other hand, since ~' and G are both parallel tensor fields (Lamina 1), a straightforward reasoning shows t h a t S is also parallel. Thus the linear transformation S~: M ~ M x has the same set of eigenvalues {ai ... ~k} (a~ER, {~} distinct) for all x E M and moreover, if V~(x) ... Vk(x) are the corresponding eigenspaces at Mx, then the { V~(x)} are m u t u a l l y orthogonal compIcx subspaces of Mx and the distribution x~->V~(x) is a parallel distribution on M for each i.

Since we assume t h a t ~:#(o, k ) 2 . Thus invoking the del~ham decomposition theorem for K~hler manifolds, we have proved:

LEMM A 2. ,Let M be a simply connected compact K~hler mani/old o/nonnegative bisec- tional curvature. I / h 1' I(M) > 1, then M splits holomorphieally and isometrically into a direct

54 A. ~owAm) v.T AL.

product o/compact Kahler mani/olds M 1 • M2, where dim M~ >t 1 ]or

i = 1, 2

(we have used the standard notation: hP'q(M) ~the dimension of the space o[ harmonic forms o/type (p, q) ).

Section 2

We now prove the theorem. Thus suppose M is an n-dimensional compact K~hler manifold of nonnegative bisectional curvature such t h a t the maximum rank of Rie on M is (n-/~). Since M has nonnegative Ricci curvature, the theorem of Cheeger-Gromoll states t h a t the universal covering of M is holomorphically isometric to

M ' • C z,

where 0 ~ 1 ~< n and M ' is a simply connected compact K~hler manifold of nonnegative bisectional curva- ture. We first prove t h a t l = k. Let

h l ' l ( M ' ) = s .

B y repeated applications of L e m m a 2,

M '

is holomorphically isometric to a direct product of compact K~hler manifolds M~ • • M~, where hl'~(Mt) = 1 for each i. We claim t h a t each M~ must have quasi-positive Ricci curva- ture. To prove this claim, we need the following lemmas.

L E M ~ A 3. Let M be a compact Kghler mani/old. I / ~ is a positive semi-de/inite /orm o/

type

(1,1)

on 2/i such that its harmonic component H~ is parallel, then the rank o] H~ equals the maximum rank of $.

Proo].

Let o) be the Kithler form of M and let dim M = n . Then for a n y k,

fM~k A co~-k= fM(H~)k A eo~-~. ( ~e ~ )

Let r = m a x rank ~ and let t = r a n k HE. The positive semi-definiteness of ~ implies t h a t the left side of (~r ~-) is positive when k = r. Thus (H$)r=~ 0, thereby proving t ~> r. On the other hand, since H~ and ~o are both parallel 2-forms, (H~)kA oJ n-k is also parallel and hence equals a constant multiple of the volume form of M. Since, by the definition oft, (H~) ~ A ~o n-t is nonzero at each point of M, it follows t h a t the right side of (~- ~-) is nonzero when

k=t.

Thus ~t is not identically zero and t ~<r. Q.E.D.

LEMM), 4. Let M be a simply connected compact Kdhler mani/old o/ nonnegative bisec- tional curvature. I / ~ is its Ricci /orm, then q) has a nonzero harmonic component H T.

Proo/.

First observe t h a t ~ is not identically zero. Otherwise the bisectional curvature, being nonnegative, would be identically zero and hence the curvature tensor is itself identically zero. B y the assumption of simple connectivity, M would then be isometric to

ON COMPACT KAHLEI~ MANIFOLDS OF NONNEGATIVE BISECTIONAL CURVATURE. ][ 5 5

complex euclidean space. This contradicts compactness. I f n = d i m M, then ~0 A o) ~-1 (o) = Ks form of M) is everywhere nonnegative and is positive somewhere. Hence

O< ~MCp A COn-l= /M(HCf) A qgn-1 ,

thereby proving t h a t H ~ is n o t zero. Q.E.D.

We now return to the proof of the theorem. L e t ~ , eo~ be respectively the Ricci form and K~hler form of M~. Since h l ' l ( M ~ ) = 1 , the harmonic component of ~ is equal to c~o~

for some cER. Since M~ has nonnegative bisectional curvature, L e m m a 4 implies c~=0. B y L e m m a 1 and L e m m a 3, the m a x i m u m r a n k of ~ equals the r a n k of co), which is equal to dim M~. Thus M~ has quasi-positive Ricci curvature for each

i=l

... s. I t follows t h a t M ' itself has quasi-positive Ricci curvature. Thus n - 1 ( = dim M ' ) is equal to the m a x i m u m r a n k of the Ricci tensor of M ' , which equals t h a t of

M'

• C ~ (because C z has the flat metric), which in turn equals t h a t of M (because

M'

• C~-~M is a local isometry); hence n - l = n - k , i.e., l = It. This proves p a r t (A) of the theorem.

To prove p a r t (B), the fact t h a t M ' has no holomorphic q-forms for all q >~ 1 follows from the quasi-positivity of the Ricci curvature and a simple generalization of the K o d a i r a vanishing theorem (Theorem 6 of [R]; see also Theorem B of [Wu2]). :Now we also know t h a t M ' is holomorphically isometric to M 1 • ... • M~, where

hl"l(M~)=1

for each i. B y K o d a i r a ' s embedding theorem, each M~ is therefore algebraic and hence so is M itself.

Finally to prove

H2(M~,

Z) ~ Z, observe t h a t

h2'~ ~0

so t h a t

H2(M~,

R) ~ R . Since M~ is simply connected, the universal coefficient theorem for cohomology now gives

H2(M~,

Z) ~ Z.

To prove (C), let ~ be the fundamental group of M; ~ consists of isometries acting freely on M ' x C ~. Introduce the notation:

I(N)

denotes the group of isometries of a n y Riemannian manifold N. Consider the n a t u r a l projection ~0:

I(M')• I(~)~I(Ck).

Since

I(M')

is a compact Lie group, the kernel of the restriction of ~0 to ~ is a finite group to be denoted b y ker ~. The quotient space M ' / k e r ~0 is a compact Ki~hler manifold with quasi- positive Ricci curvature and hence, b y p a r t (A) of Theorem B of [Wu2], m u s t be itself simply connected. Thus ker ~0 is trivial, which is equivalent to saying t h a t ~: g - ~ I ( C k) is an isomorphism onto a crystallograph subgroup F ~ I(C k) (cf. [Wo], Chapter 3). Now let B = C~/F. Since ~ acts as holomorphic isometries on M ' • C k respecting the product metric, it is straightforward to verify t h a t p: M = M ' x

Cz/z-.'B

which is defined b y projecting on the second factor is a locally isometrically trivial holomorphic fibration with fibre M ' . This concludes the proof of p a r t (C).

Finally to prove (D), let ~0 be a free abelian subgroup of r a n k 2k with finite index in

5 6 A. HOWARD ET AL.

t h e f u n d a m e n t a l g r o u p ~. Define F ~ ~ 0 ( ~ ~ T - Ck/F ~ a n d M *=- M ' • (~k/~0. T h e n t h e com- m u t a t i v e d i a g r a m in (D) i m m e d i a t e l y follows. T h e o n l y n o n t r i v i a l a s s e r t i o n in (D) is t h a t c o n c e r n i n g M* being g l o b a l l y d i f f e o m o r p h i c to M ' • T; t h i s i n v o l v e s a careful choice of

~z ~ in ~ a n d h a s a l r e a d y b e e n d o n e in [CG2], p. 440. Q . E . D . B o t h corollaries a r e s t r a i g h t f o r w a r d consequences of t h e t h e o r e m e x c e p t for t h e i m p l i c a t i o n ( B ) ~ ( A ) in C o r o l l a r y 1. To p r o v e this, one i n v o k e s t h e C h e e g e r - G r o m o l l s t r u c t u r e t h e o r e m t o show t h a t if t h e f i r s t B e t t i n u m b e r of M is zero, t h e n g l ( M ) m u s t be f i n i t e (see T h e o r e m A of [Wu2]). B y t h e a b o v e t h e o r e m , if z l ( M ) is finite, i t is t r i v i a l .

B i b l i o g r a p h y

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[R] ~:~IEMENSCHNEIDER, O., Characterizing Moi~ezon spaces b y almost positive coherent analytic sheaves. 1~lath. Z., 123 (1971), 263-284.

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Received November 7, 1979