Ngaiming Mok

Uniruled projective manifolds with irreducible reductive G-structures

We will call a G-structure modeled after a compact irreducible Hermitian Symmetrie space S of rank ^ 2, an S-structure. (See section 3 or [KO] for a precise definition. Note that our S-structure is called G(S)-structure in [KO].) Such structures were studied by many authors in the 60s (see [Oc] and the references there). From the 80s, they were studied by people working on twistor theory (see [Ba], [Ma] and the references there). When one studies these works, what is rather amazing, at least to the authors, is the lack of a nonflat example among compact manifolds. One may even expect that 5-structures are always flat under mild conditions. One result along this line is

Uniqueness Theorems of Hermitian Metrics of Seminegative Curvature on Quotients of Bounded Symmetric Domains

In the theory of locally symmetric spaces of negative Ricci curvature it has been a classical problem to study the extent to which the topology of the manifold determines the geometry, and, in the Hermitian case, the complex structure. The rigidity theorem of Mostow [26] asserts that for a compact locally symmetric Riemannian manifold of negative Ricci curvature, the manifold is determined up to isometry and normalizing constants by its fundamental group among the class of such manifolds, with the obvious exceptions involving compact Riemann surfaces. The same theorem for locally symmetric Riemannian manifolds of finite volume and negative Ricci curvature was proved by Prasad [27] in case of rank 1 and included in the super-rigidity theorem of Margulis [13] in the case of rank ? 2. In the class of compact locally symmetric Hermitian manifolds of negative Ricci curvature, Calabi-Vesentini [5] and Borel [3] proved vanishing theorems of certain cohomology groups which imply in particular that for X locally irreducible of complex dimension> 2, there exists no non-trivial deformation of X as a complex manifold. In this direction Siu ([32], [33]) proved the strong rigidity of the Kdhler manifold X in the sense that any compact Kkhler manifold M homotopic to X is necessarily biholomorphic or conjugate-biholomorphic to X. If M is also locally symmetric by assumption then it follows from the uniqueness of Kihler-Einstein metrics of negative Ricci curvature (Yau [37]) that the (conjugate-) biholomorphism is in fact an isometry up to a normalizing constant. It is thus natural to ask

Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles

Let X be a Fano manifold of Picard number 1 admitting a rational curve with trivial normal bundle and f : X′ → X be a generically finite surjective holomorphic map from a projective manifold X′ onto X. When the domain manifold X′ is fixed and the target manifold X is a priori allowed to deform we prove that the holomorphic map f : X′ → X is locally rigid up to biholomorphisms of target manifolds. This result complements, with a completely different method of proof, an earlier local rigidity theorem of ours (see J. Math. Pures Appl. 80 (2001), 563– 575) for the analogous situation where the target manifold X is a Fano manifold of Picard number 1 on which there is no rational curve with trivial normal bundle. In another direction, given a Fano manifold X′ of Picard number 1, we prove a finiteness result for generically finite surjective holomorphic maps of X′ onto Fano manifolds (necessarily of Picard number 1) admitting rational curves with trivial normal bundles. As a consequence, any 3-dimensional Fano manifold of Picard number 1 can only dominate a finite number of isomorphism classes of projective

On Fano manifolds with Nef tangent bundles admitting 1-dimensional varieties of minimal rational tangents

Let X be a Fano manifold of Picard number 1 with numerically effective tangent bundle. According to the principal case of a conjecture of Campana-Peternells, X should be biholomorphic to a rational homogeneous manifold G/P, where G is a simple Lie group, and P ⊂ G is a maximal parabolic subgroup. In our opinion there is no overriding evidence for the Campana-Peternell Conjecture for the case of Picard number 1 to be valid in its full generality. As part of a general programme that the author has undertaken with Jun-Muk Hwang to study uniruled projective manifolds via their varieties of minimal rational tangents, a new geometric approach is adopted in the current article in a special case, consisting of (a) recovering the generic variety of minimal rational tangents C x , and (b) recovering the structure of a rational homogeneous manifold from C x . The author proves that, when b 4 (X) = 1 and the generic variety of minimal rational tangents is 1-dimensional, X is biholomorphic to the projective plane P 2 , the 3-dimensional hyperquadric Q 3 , or the 5-dimensional Fano homogeneous contact manifold of type G 2 , to be denoted by K(G 2 ). The principal difficulty is part (a) of the scheme. We prove that C x C PT x (X) is a rational curve of degrees < 3, and show that d = 1 resp. 2 resp. 3 corresponds precisely to the cases of X = P 2 resp. Q 3 resp. K(G 2 ). Let κ be the normalization of a choice of a Chow component of minimal rational curves on X. Nefness of the tangent bundle implies that κ is smooth. Furthermore, it implies that at any point x ∈ X, the normalization κ x of the corresponding Chow space of minimal rational curves marked at is smooth. After proving that κ x is a rational curve, our principal object of study is the universal family u of κ, giving a double fibration p: u → κ, μ: u → X, which gives P 1 -bundles. There is a rank-2 holomorphic vector bundle V on κ whose projectivization is isomorphic to p: u → κ. We prove that V is stable, and deduce the inequality d < 4 from the inequality c 2 1 (V) < 4c 2 (V) resulting from stability and the existence theorem on Hermitian-Einstein metrics. The case of d = 4 is ruled out by studying the structure of the curvature tensor of the Hermitian-Einstein metric on V in the special case where c 2 1 (V) = 4c 2 (V).

Compactifying complete Kähler-Einstein manifolds of finite topological type and bounded curvature

Siu-Yau [16] studied the compactification of complete Kaihler manifolds of finite volume and of Riemannian sectional curvature pinched between two negative constants. In [12] the first author of the present article started a systematic study of the more general problem of compactifying complete Kwhler manifolds of finite volume and of bounded curvature. A number of conjectures were formulated, which can all be regarded as conjectural generalizations of the compactification of arithmetic varieties (i.e. quotients of bounded symmetric domains Q by torsion-free arithmetic subgroups of Aut(Q)) in a differentialgeometric setting. Here and henceforth all discrete groups of automorphisms will be assumed torsion-free. In the same article we treated the case of Kaihler surfaces and proved:

Germs of measure-preserving holomorphic maps from bounded symmetric domains to their Cartesian products

Abstract Let X be the quotient of an irreducible bounded symmetric domain Ω by a lattice. In order to characterize algebraic correspondences on X commuting with exterior Hecke correspondences, Clozel–Ullmo studied certain germs of measure-preserving maps from (Ω; 0) into its Cartesian products, proving that such maps are totally geodesic when dim(X) = 1. Here we prove total geodesy when dim(Ω) ≧ 2 by methods of analytic continuation. For Bn, n ≧ 2, total geodesy follows then from Alexanders theorem. When rank(Ω) ≧ 2, we deduce total geodesy from Alexander-type theorems, especially from a new Alexander-type theorem involving Reg(∂Ω) in place of the Shilov boundary.

Characterization of Certain Holomorphic Geodesic Cycles on Quotients of Bounded Symmetric Domains in terms of Tangent Subspaces

Let Ω be an irreducible bounded symmetric domain and Γ ⊂ Aut(Ω) be a torsion-free discrete group of automorphisms, X ≔ Ω/Γ. We study the problem of algebro-geometric and differential-geometric characterizations of certain compact holomorphic geodesic cycles S ⊂ X. We treat special cases of the problem, pertaining to a situation in which S is a compact holomorphic curve, and to the case where Ω is a classical domain dual to the hyperquadric. In both cases we consider algebro-geometric characterizations in terms of tangent subspaces. As a consequence we derive effective pinching theorems where certain complex submanifolds S ⊂ X are proven to be totally geodesic whenever their scalar curvatures are pinched between certain computed universal constants, independent of the volume of the submanifold S, giving new examples of the gap phenomenon for the characterization of compact holomorphic geodesic cycles.

Projective Algebraicity of Minimal Compactifications of Complex-Hyperbolic Space Forms of Finite Volume

Let Ωbe a bounded symmetric domain and Γ⊂ Aut(Ω) be an irreducible nonuniform torsion-free discrete subgroup. When Γis of rank ≥ 2, Γis necessarily arithmetic, and X :=Ω/Γ?admits a Satake-Baily-Borel compactification. When Ω?is of rank 1, i.e., the complex unit ball Bn of dimension n≥ 1,Γmay be nonarithmetic.When n≥2, by a general result of Siu and Yau, X is pseudoconcave and it can be compactified to a Moishezon space by adding a finite number of normal isolated singularities. In this article we show that for X := Bn/Γ the latter compactification is in fact projective-algebraic.We do this by showing that, just as in the arithmetic case of rank-1, X admits a smooth toroidal compactification \(\bar{X}\)M obtained by adjoining an Abelian variety to each of its finitely many ends, and \(\bar{X}\)M can be blown down to a normal projective-algebraic variety \(\bar{X}\)Min by solving \(\bar{\partial}\)with L2-estimates with respect to the canonical Kahler-Einstein metric and by normalization. As an application, we give an alternative proof of results of Koziarz-Mok on the submersion problem in the case of complex-hyperbolic space forms of finite volume by adapting the cohomological arguments in the compact case to general hyperplane sections of the minimal projective-algebraic compactifications which avoid the isolated singularities.

Compactification of complete Kähler surfaces of finite volume satisfying certain curvature conditions

On etudie systematiquement la compactification des varietes de Kahler completes de volume fini et de courbure sectionnelle bornee

Characterization of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1

In a series of works one of the authors has developed with J.-M. Hwang a geometric theory of uniruled projective manifolds basing on the study of varieties of minimal rational tangents, and the geometric theory has especially been applied to rational homogeneous manifolds of Picard number 1. In Mok [Astérisque 322, pp. 151–205] and Hong-Mok [J. Diff. Geom. 86 (2010), pp. 539–567] the authors have started the study of uniruled projective subvarieties, and a method was developed for characterizing certain subvarieties of rational homogeneous manifolds. The method relies on non-equidimensional Cartan-Fubini extension and a notion of parallel transport of varieties of minimal rational tangents. In the current article we apply the notion of parallel transport to a characterization of smooth Schubert varieties of rational homogeneous manifolds of Picard number 1. Given a pair (S, S0) consisting of a rational homogeneous manifold S of Picard number 1 and a smooth Schubert variety S0 of S, where no restrictions are placed on S0 when S = G/P is associated to a long root (while necessarily some cases have to be excluded when S is associated to a short root), we prove that any subvariety of S having the same homology class as S0 must be gS0 for some g ∈ Aut(S). We reduce the problem first of all to a characterization of local deformations St of S0 as a subvariety of S. By Kodaira stability, St is uniruled by minimal rational curves of S lying on St. We establish a biholomorphism between St and S0 which extends to a global automorphism by reconstructing St by means of a repeated use of parallel transport of varieties of minimal rational tangents along minimal rational curves issuing from a general base point. Our method is applicable also to the case of singular Schubert varieties provided that there exists a minimal rational curve on the smooth locus of the variety. Received August 6, 2010, January 7, 2011, February 3, 2011, June 2, 2011 and, in revised form, July 31, 2011. The first author’s work was supported by Research Settlement Fund for the new faculty of College of Natural Sciences in SNU 301-20070006. The second author’s research was partially supported by GRF grant HK7039/06P of the Research Grants Council of Hong Kong.