Jun-Muk Hwang

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

Base manifolds for fibrations of projective irreducible symplectic manifolds

Given a projective irreducible symplectic manifold M of dimension 2n, a projective manifold X and a surjective holomorphic map f:M→X with connected fibers of positive dimension, we prove that X is biholomorphic to the projective space of dimension n. The proof is obtained by exploiting two geometric structures at general points of X: the affine structure arising from the action variables of the Lagrangian fibration f and the structure defined by the variety of minimal rational tangents on the Fano manifold X.

Geometry of chains of minimal rational curves

Abstract Chains of minimal degree rational curves have been used as an important tool in the study of Fano manifolds. Their own geometric properties, however, have not been studied much. The goal of the paper is to introduce an infinitesimal method to study chains of minimal rational curves via varieties of minimal rational tangents and their higher secants. For many examples of Fano manifolds this method can be used to compute the minimal length of chains needed to join two general points. One consequence of our computation is a bound on the multiplicities of divisors at a general point of the moduli of stable bundles of rank two on a curve.

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

Volumes of complex analytic subvarieties of Hermitian symmetric spaces

We give lower bounds of volumes of k-dimensional complex analytic subvarieties of certain naturally defined domains in n-dimensional complex space forms of constant (positive, zero, or negative) holomorphic sectional curvature. For each 1 ≤ k ≤ n, the lower bounds are sharp in the sense that these bounds are attained by k-dimensional complete totally geodesic complex submanifolds. Such lower bounds are obtained by constructing singular potential functions corresponding to blow-ups of the Kähler metrics involved. Similar lower bounds are also obtained in the case of Hermitian symmetric spaces of noncompact type. In this case, the lower bounds are sharp for those values of k at which the Hermitian symmetric space contains k-dimensional complete totally geodesic complex submanifolds which are complex hyperbolic spaces of minimum holomorphic sectional curvature.

Characteristic foliation on the discriminant hypersurface of a holomorphic Lagrangian fibration

We give a Kodaira-type classification of general singular fibers of a holomorphic Lagrangian fibration in Fujikis class C. Our approach is based on the study of the characteristic vector field of the discriminant hypersurface, which naturally arises from the defining equation of the hypersurface via the symplectic form. As an application, we show that the characteristic foliation of the discriminant hypersurface has algebraic leaves which are either rational curves or smooth elliptic curves.

Holomorphic maps onto varieties of non-negative Kodaira dimension

A classical result in complex geometry says that the automorphism group of a manifold of general type is discrete. It is more generally true that there are only finitely many surjective morphisms between two fixed projective manifolds of general type. Rigidity of surjective morphisms, and the failure of a morphism to be rigid have been studied by a number of authors in the past. The main result of this paper states that surjective morphisms are always rigid, unless there is a clear geometric reason for it. More precisely, we can say the following. First, deformations of surjective morphisms between normal projective varieties are unobstructed unless the target variety is covered by rational curves. Second, if the target is not covered by rational curves, then surjective morphisms are infinitesimally rigid, unless the morphism factors via a variety with positive-dimensional automorphism group. In this case, the Hom-scheme can be completely described.

ON THE VOLUMES OF COMPLEX HYPERBOLIC MANIFOLDS WITH CUSPS

We study the problem of bounding the number of cusps of a complex hyperbolic manifold in terms of its volume. Applying algebra-geometric methods using Mumfords work on toroidal compactifications and its generalization due to N. Mok and W.-K. To, we get a bound which is considerably better than those obtained previously by methods of geometric topology.

Webs of Lagrangian tori in projective symplectic manifolds

For a Lagrangian torus A in a simply-connected projective symplectic manifold M, we prove that M has a hypersurface disjoint from a deformation of A. This implies that a Lagrangian torus in a compact hyperkähler manifold is a fiber of an almost holomorphic Lagrangian fibration, giving an affirmative answer to a question of Beauville’s. Our proof employs two different tools: the theory of action-angle variables for algebraically completely integrable Hamiltonian systems and Wielandt’s theory of subnormal subgroups.

On Seshadri Constants of Canonical Bundles of Compact Complex Hyperbolic Spaces

Upper and lower bounds for the Seshadri constants of canonical bundles of compact hyperbolic spaces are given in terms of metric invariants. The lower bound is obtained by carrying out the symplectic blow-up construction for the Poincaré metric, and the upper bound is obtained by a convexity-type argument.