Metric rigidity theorems on Hermitian locally

Proc. Nadl. Acad. Sci. USA Vol. 83, pp. 2288-2290, April 1986 Mathematics

Metric rigidity theorems on Hermitian locally symmetric spaces (seminegative line bundle/first Chern form/Borel-Weil theorem/Harish-Chandra embedding theorem/compact Kihler manifolds of semlpositive curvature)

NGAIMING MOK Department of Mathematics, Columbia University, New York, NY 10027

Communicated by Hyman Bass, November 4, 1985

ABSTRACT Let X = f/r be a compact quotient of an irreducible bounded symmetric domain t of rank >2 by a discrete group r of automorphisms without fixed points. It is well known that the Kihler-Einstein metric g on X carries seminegative curvature (in the sense of Griffiths). I show that any Hermitian metric h on X carrying seminegative curvature must be a constant multiple of g. This can be applied to prove rigidity theorems of holomorphic maps from X into Hermitian manifolds (Y, k) carrying seminegative curvature. These results are also generalized to the case of quotients of finite volume. On the other hand, let (X¢, g,) be an irreducible compact Hermitian symmetric manifold of rank >2. Then g, is KIlhler and carries semipositive holomorphic bisectional curvature. I prove that any Kiiher h on Xc carrying semipositive holomorphic bisectional curvature must be equal to gc up to a constant multiple and up to a biholomorphic transformation of X¢.

Kahiler manifolds of semipositive holomorphic bisectional curvature. THEOREM 1. Let (X, g) be a locally symmetric Hermitian space offinite volume uniformized by an irreducible bounded symmetric domain of rank :2. Suppose h is a Hermitian metric on X such that (X, h) carries seminegative curvature and h is dominated by a constant multiple of g. Then h = cg for some constant c > 0. Remark: By the Schwarz lemmas of Royden (4) [in case (X, h) is Kahiler] and Chen et al. (5), the condition that h is dominated by a constant multiple of g can be obtained by assuming that (X, h) is complete with holomorphic sectional curvature bounded above by a negative constant. Note however that in this hypothesis, (X, h) is not assumed to be complete. For (X, g) compact, in view of the work of Siu (6, 7), Theorem I can be reformulated as follows. THEOREM 1'. Let (M, h) be a compact Kahler manifold of seminegative holomorphic bisectional curvature homotopic to a locally symmetric Hermitian space (X, g) uniformized by an irreducible bounded symmetric domain of rank :2. Then (M, h) is biholomorphically or conjugate-biholomorphically isometric to (X, g). As an application of Theorem I we obtain the following. THEOREM 2. Let (X, g) be a compact locally symmetric Hermitian space uniformized by an irreducible bounded symmetric domain of rank 22. Let (N, h) be a Hermitian manifold carrying seminegative curvature. Suppose f: X -. N is a nonconstant holomorphic mapping. Then, up to a multiplicative constant, f is an isometry. In the case of compact Hermitian symmetric spaces (Xc, g) of positive Ricci curvature, we have the following. THEOREM 3. Let (X 0 and some 4) E Aut(Xc). (it) The analogue of Theorem 3 for Hermitian metrics h fails. Of principal importance to the Hermitian geometry of holomorphic vector bundles (V, h) is the fact that curvatures of Hermitian vector subbundles are smaller than or equal to those of the ambient bundle. From this one deduces the wellknown fact that sums of Hermitian metrics of seminegative curvature retain seminegativity, a fact crucial to the proofs of Theorems 1-3. Proofof Theorem 1: Denote by V the covariant differentiation on X defined by the Kahler-Einstein metric g. We can regard the metric h as a covariant tensor of type (1, 1). By the irreducibility of (X, g), to prove Theorem I it suffices to show that vh 0 on X. The Hermitian holomorphic vector bundle (T(X), g) corresponds to a Hermitian holomorphic line bundle (L, §) on PT(X) such that the first Chem form cl(L, g) is negative semidefinite (cf. ref. 3). Furthermore, cl(L, g) has a zero eigenvalue at (x, [n]) E PT, E if and only

In the theory of locally symmetric Riemannian spaces (X, g) of negative Ricci curvature, uniqueness theorems for Riemannian metrics h of seminegative and bounded sectional curvature when X is of rank ,2 and (X, g) is of finite volume have been proved recently by Eberlein (1), Gromov (unpublished results), and Ballmann and Eberlein (2). In Hermitian geometry there is a more natural notion of negative curvature that applies to Hermitian holomorphic vector bundles. Let (V, h) be a Hermitian holomorphic vector bundle over a complex manifold M and let 0 = (Os) be the curvature form of (V, h), where 0e,, = v'Tij Q, 0iedzA dZ' in local holomorphic coordinates. Then, (V, h) is said to be negative [in the sense of Griffiths (3)] if e, =0jvaf-''If < 0 whenever v = (vy) and i7 = (Xq) are nonzero. We will say that a Hermitian manifold (X, h) carries negative curvature if and only if the holomorphic tangent bundle T(X) is negative in the induced Hermitian metric, also denoted by h. For KAfhler manifolds (X, h) this is equivalent to asserting that (X, h) carries negative holomorphic bisectional curvature, a condition weaker than the assertion that (X, h) carries negative Riemannian sectional curvature. The notions of seminegative, positive, and semipositive Hermitian holomorphic vector bundles are similarly defined. In this article I will state metric rigidity theorems on locally symmetric Hermitian spaces (X, g) of rank :2 in terms of the notions of seminegativity and semipositivity stated above. Sketches of proofs and applications of such uniqueness theorems and their proofs will be given. Unlike the situation of Riemannian geometry, I have developed a method that applies simultaneously to locally symmetric Hermitian spaces of negative or positive Ricci curvature. In the case of negative Ricci curvature, a major motivation is to study holomorphic mappings from X into seminegatively curved Hermitian manifolds while, in the case of positive Ricci curvature, I have in mind the classification problem of compact -

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Mathematics: Mok

Proc. Natl. Acad Scd USA 83 (1986)

if R"'Icf = 0 for some; 0 for the curvature tensor R' of g. Since X is of rank >2 at each x E X there exists such an q C Tx. Suppose Y is a k-dimensional complex submanifold of PT(X) such that at every point of Y there exist exactly s zero eigenvalues of ci(L1y, $) and let vm be a semipositive (m, m) form on Y. Then, { [-C1(L,

g)]k-S+1 Av31 = 0.

[1]

Since the integral is a topological invariant of the line bundle

L6, we have

{; [-c1(L, ,

+ /)Jk-s+1

Asv1

=

0.

[2]

Since the integrand is pointwise nonnegative it must vanish identically on $9. For a suitable choice of 9 and vs-, we can show then that c1(Lly, g + /) has exactly s zero eigenvalues everywhere. This forces the vanishing of many bisectional curvature Ra- of (X, g + h) for which Ra,;,, = 0. Comparing the two expressions we can conclude that vh,- = 0 for any r E T(X) at the point. With a sufficiently large set of (cr, A) we can show that vh 0 on X, proving Theorem 1. To construct $9 let A., C PTX be defined by unit vectors a E Tx such that Raa-aa0I = supETr,,jjI=j I|Rfejj and define Y = ht(X) = UXx At, For X of rank >2 and locally irreducible the Hermitian form Ha(4, C) = ReF has exactly s = p(Xo) 1 zero eigenvalues, where p(Xo) is the degree of (strong) nondegeneracy of bisectional curvatures of the covering domain X. computed by Siu (7) and by Zhong (8). Write X. = GO/K where Go is semisimple and denote by o the coset eK. We can prove that [a] E Ato if and only if a is a dominant weight vector of the irreducible K-representation space To, so that Ato is a homogeneous complex submanifold of PTO by the Borel-Weil theorem (10). Let XO C X, be the Borel embedding of XO into its compact dual X, = GIK. Then the complexified Lie group GC acts on PT(XC). Using the complex analyticity of AO and the Borel embedding theorem, we can show that At(XO) = U. Ex A., is precisely GC([aoJ) n PT(Xo) for any [a0] E At, soothat 1l(X0) and hence $9 = A4(X) is a complex manifold. Furthermore, we can check by using the Harish-Chandra embedding XO C C' that the zero eigenvectors of cl(L, §) at [a] E Al(X) are tangent to $9 = At(X). Choosing v,_1 to be the (s - 1)th exterior power of the pullback of the Kahler form of (X, g), we can show that vch- = 0 for all [a] E A., and all ; E T, such that RT = 0, r C Tx arbitrary. Finally, by expanding a and T we can verify the identity vh 0, thus proving Theorem 1. Proof of Theorem 2: One simply applies Theorem 1 to (X, g + f*h). Proof of Theorem 3: For (Xc, gc) a compact Hermitian symmetric space, the cotangent bundle (T*(Xc), g*) is a Hermitian vector bundle of seminegative curvature. Let (A, g*) be the corresponding Hermitian line bundle on PT*(X). Then c1(A, gc*) is negative semidefinite everywhere. Let 44(Xc) be defined similar to Al(X) in Theorem 1. In terms of the Borel embedding Xoc-* Xc one can choose gc so that (X0, go)=-+ (X, gc) is an isometry at the origin o and the curvature tensor of (Xc, gc) at o is simply opposite to that of (X0, go). One can then prove that A(XO) = A(Xc) n PT(Xo). For every X E Tx we denote by Ee T* the lifting of ; by the contravariant metric tensor (gu). Let Mt*(XC) C PT*(X) be thus defined by lifting. Applying the argument of Theorem I to A and $9 = Al*(Xc), we can prove that vcha* = 0 for all a* E 9,naacc = Rc*-*;; = 0 for the curvature tensor Rc of (Xc, ge), r* E T* arbitrary. The C-linear subspace W C Tx ® T* spanned by t® a* is the proper subspace of zero eigenvectors of the quadratic form P on Txe T* defined by P(rh ® Xf ;

2289

7t 0 X2) = * To prove Theorem 3 by the results of Mok and Zhong (9), it suffices to show that (Xc, h) is Einstein. For this we combine the equations vcha* - 0, vahc*r* = 0 (obtained similarly) and global geometric considerations. We cross define a characteristic sphere of Xc to be an embedded Riemann sphere S such that at each x E S we have TX(S) = CQ(x) for some [a(x)] E Ax. Through every x E Xc and for every [a(x)] E8 Ax there exists a characteristic sphere passing through x with TX(S) = a. We can proceed to prove that S is totally geodesic with respect to any (Xc, h) in Theorem 3. Using complex geodesic coordinates (Zk) at the origin o given by the Harish-Chandra embedding, it suffices to show that raa, = 0 for f I a (Riemann-Christoffol symbols). According to Mok and Zhong (9), we have an orthogonal decomposition To = Ca d@ Xea @ X0 such that f E ea if and only if Ra-Zx = '/2R,0;-a and t E X. if and only if R,, = R;*-,-*aa=0 . We can then choose an orthonormal basis {ei: i = 1, i = p E Ha or i = q E N0} accordingly and denoting aha/azk by h-7k, etc., we have = r= > i

hqJ"7h,

= > j

hqJlh -= 0.

By expanding a and ; in the equations ha*A, c= = Oat o, we also have hP",I + h',e,= 0 for some 8E S e Since (Xc, h) is Kihler we obtain

rFll,

=F = E - -I

[3]

8

kh'h,l = -I hPJ',Ihu= EhUh

h~Jhcj,7= - hc'hj= 2 h;',1hJ= 0.

[4]

From the total geodesy of S in (Xc, h) we can conclude that, for each [a] 8E A4t, a ® a is an eigenvector of the quadratic form P on T, T, defined by P(7h ( X1, 77 0 X2) = Rc and from this the fact that a E T, is an eigenvector of the Ricci form of h at x, hence showing that (Xc, h) is KAhlerEinstein and proving Theorem 3 by the results of Mok and Zhong (9). Other Results. I collect here results on the topology and geometry of Hermitian locally symmetric spaces that can be obtained as consequences of the main theorems (Theorems 1-3) or can be derived by similar methods. Concerning the situation of bounded symmetric domains and their compact quotients we have the following. THEOREM 4. Let r be the fundamental group of an irreducible compact Hermitian locally symmetric space X of rank :2. Let F' be the fundamental group of a compact Kahler manifold Y of strongly negative curvature in the sense of Siu (6). Then, there exists no nontrivial group homomorphism p:r -Fr'. THEOREM 5. Let [1 be a bounded symmetric domain equipped with a complete Kahler-Einstein metric g. Suppose there exists on Q a complete Hermitian metric h of bounded torsion such that the curvature of the Hermitian manifold (fQ, h) is bounded between two negative constants. Then fi is of rank 1. Remarks: (i) Theorem 4 for the case in which r' is the fundamental group of a compact quotient of the unit ball in Cn is a consequence of the super-rigidity theorem of Margulis (11). (ii) Theorem 5 for h a Kahler metric is a result of Yang (12). Proof of Theorem 4: Let p: i1(X) -> i1T(Y) be a nontrivial group homomorphism. Since X is a K(ir, 1), there exists a smooth map f:X-+ Y such that (p is the map induced by f on fundamental groups by the results of Eells and Sampson (13), we may assume f to be harmonic. Thus, by the results of Siu (6), f is either holomorphic or antiholomorphic unless real rank (f ) s2. When real rank(f ) = 2 by the foliation tech-

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Mathematics: Mok

niques of Jost and Yau (14) and Mok (15), we can also obtain a nontrivial holomorphic map f':X 4-* S into some compact Riemann surface of genus ,O. By Theorem 2 (and an easy generalization) we obtain a contradiction unless real rank(f ) = 1. But then f would map X onto a closed geodesic of Y, yielding a nontrivial map of r1(X)/[X, X1 = H1(X) into Z. This contradicts the well-known fact that the first Betti number X vanishes. Proof of Theorem 5: By the results of Borel (16), there exists a compact quotient X = fl/F of fQ. The Hermitian metric h on fi induces an upper-semicontinuous Finsler metric so on X by the covering map iril --÷ X by taking suprema over preimages. so can be smoothed to yield on X a smooth Finsler metric s of negative curvature. Recall that L denotes the holomorphic line bundle on PT(X) induced by the holomorphic tangent bundle T(X) ++ X. s corresponds to a Hermitian metric of negative curvature on L +-> PT(X). The integral formula (Eq. 2) on Al(X) with s replacing g + h given in the proof of Theorem I yields immediately a contradiction unless fl is of rank 1. Regarding the geometry of compact Kdhler manifolds of semipositive holomorphic bisectional curvature, there is the following general conjecture. Conjecture. Let (X, h) be a compact Kahler manifold of semipositive holomorphic bisectional curvature and positive Ricci curvature. Then (X, h) is biholomorphically isometric to (X1, hl)x x(Xm, hm), where Xi, 1 i m, is the underlying complex manifold of some irreducible Hermitian symmetric space and hi is a Hermitian symmetric metric on Xi unless Xi is a projective space. Based on the statement and the method of proof of Theorem 3, we obtain the following. THEOREM 6. Let X = [email protected] be a compact Hermitian symmetric space and h be a Kahler metric on X carrying semipositive holomorphic bisectional curvature. Then (X, h) splits isometrically into (X1, hl)x.x(Xm, hm). THEOREM 7. The conjecture above is truefor X ofcomplex dimension n 3. Proof of Theorem 6: It suffices to take X = X' x X'. Let (z.)1SXil and (Zk),+1,_k,-n be local holomorphic coordinates at x? E X', x" E X". By an integral formula similar to but much more elementary than the one used in the proof of Theorem 3, we can deduce that Whl/zk = ah'Z/azi = 0 at (x'x") for 1 s i 1, I + 1 k n and 1 jjs n. Theorem 6 asserts that ahglazk = ahkglazi = 0 at (x', x"). This is obtained from the preceding equations, the Kahler condition 0hah/0zz = ahv/a4z (1 S a, /1, -yS n), and the equation 0/0z4 (Xjh~ihyJ = 0/0z"( j h3Jhyj) = 0. Proof of Theorem 7: Using the results of Howard and Smyth (17), Mori (18), Siu and Yau (19), Siu (20), and Bando (21), Bando has proven that, for n 3, X is biholomorphic to a compact Hermitian symmetric space. The conjecture for n S 3 then follows from Theorems 3 and 6. Finally, I record a theorem on the geometry of projective submanifolds that is motivated by study of the curvature tensor of Hermitian symmetric spaces. Let X be an n-dimensional compact Hermitian symmetric space. Recall that A., C PTX(X) is the set of all [a] E PTX(X) such that a realizes the maximum holomorphic sectional curvature at x. Ig-

-

-

-

-

-

noring references to x, we can call At C Pn-1 (At with the induced metric) the characteristic projective submanifold associated to X. Our result is as follows. THEOREM 8. Let A--> Pn-l be the characteristic projective submanifold associated to an n-dimensional irreducible compact Hermitian symmetric space. Then, the second fundamental form of A in pn-l is parallel. Moreover, every full isometric immerison of a Kahler manifold N into a projective space Pm (carrying a standard metric) with parallel secondfundamentalform is equivalent to some A---> Pn-l thus obtained. Proof: By a pinching theorem of Ros (22), it suffices for the first statement to show that the second fundamental form of the embedding AMc-> P'"- has length bounded by 1/2 assuming pn-l is of constant holomorphic sectional curvature 1. This is obtained from some sharp curvature inequalities used by Mok and Zhong (9). The converse is obtained from the classification theorem of Nakagawa and Takagi (23). Details of the proofs of Theorems 1-8 will appear elsewhere. Note Added in Proof. Recently, I have affirmed the conjecture stated before Theorem 6 characterizing compact Kahler manifolds of semipositive holomorphic bisectional curvature (24).

I thank Profs. H. Bass, M. Kuranishi, D. Phong, Y.-T. Siu, and S.-T. Yau for important discussions on the present research work. I thank Prof. Bass in particular for presenting this work to the National Academy of Sciences. 1. Eberlein, P. (1983) Ergod. Th. Dynam. Sys. 3, 47-85. 2. Ballmann, W. & Eberlein, P. (1985) Preprint. 3. Griffiths, P. (1969) Global Analysis (Univ. of Tokyo Press, Tokyo), pp. 185-252. 4. Royden, H. L. (1980) Math. Helvetici 55, 547-558. 5. Chen, Z.-H., Cheng, S.-Y. & Lu, Q.-K. (1979) Sci. Sinica 11, 1238-1247. 6. Siu, Y.-T. (1980) Ann. Math. 112, 73-111. 7. Siu, Y.-T. (1981) Duke Math. J. 48, 857-871. 8. Zhong, J.-Q. (1984) Proceedings of the 1981 Hangzhou Conference (Birkhauser, Boston), pp. 127-140. 9. Mok, N. & Zhong, J.-Q., J. Differ. Geom., in press. 10. Serre, J.-P. (1954) Seminaire Bourbaki, Expose 100. 11. Margulis, G. A. (1977) Am. Math. Soc. Transl. Ser. 2 109, 3345. 12. Yang, P. (1976) Duke Math. J. 43, 871-874. 13. Eells, J. & Sampson, J. H. (1964) Am. J. Math. 86, 109-160. 14. Jost, J. & Yau, S.-T. (1983) Math. Ann. 262, 145-166. 15. Mok, N. (1985) Math. Ann. 272, 197-216. 16. Borel, A. (1963) Topology 2, 111-122. 17. Howard, A. & Smyth, B. (1971) J. Differ. Geom. 5, 491-502. 18. Mori, S. (1979) Ann. Math. 110, 593-606. 19. Siu, Y.-T. & Yau, S.-T. (1980) Invent. Math. 59, 189-204. 20. Siu, Y.-T. (1980) Duke Math. J. 47, 641-654. 21. Bando, S. (1984) J. Differ. Geom. 19, 283-297. 22. Ros, A. (1984) Preprint. 23. Nakagawa, H. & Takagi, R. (1967) J. Math. Soc. Jpn. 28, 638667. 24. Mok, N. (1986) Preprint.