Bun Wong

On the holomorphic curvature of some intrinsic metrics

If G is a hyperbolic manifold in the sense of Kobayashi and the differential Kobayashi metric KG is of class C2, then the holomorphic curvature of KG is greater than or equal to -4. If G is Carath&odory-hyperbolic and the differential Carathxodory metric CG is of class C2, then the holomorphic curvature of CG is less than or equal to -4. With this result we obtain an intrinsic characterization of the unit ball.

Reduction of manifolds with semi-negative holomorphic sectional curvature

In this note, we continue the investigation of a projective Kähler manifold M of semi-negative holomorphic sectional curvature H. We introduce a new differential geometric numerical rank invariant which measures the number of linearly independent truly flat directions of H in the tangent spaces. We prove that this invariant is bounded above by the nef dimension and bounded below by the numerical Kodaira dimension of M. We also prove a splitting theorem for M in terms of the nef dimension and, under some additional hypotheses, in terms of the new rank invariant.

On a Domain in with generic piecewise smooth levi-flat boundary and non-compact automorphism group

In this paper, we study the problem of classification of piecewise smooth domains in with non-compact automorphism groups. We prove that a simply-connected domain in with generic piecewise smooth Levi-flat boundary and non-compact automorphism group is biholomorphic to a bidisc.

HERMITIAN METRIC WITH CONSTANT HOLOMORPHIC SECTIONAL CURVATURE ON CONVEX DOMAINS

Let D be a bounded convex domain in with a Hermitian metric of constant negative holomorphic sectional curvature such that all components blow up to infinity on the boundary of D. Then D is biholomorphic to the Euclidean ball.

Some remarks on the local moduli of tangent bundles over complex surfaces

Using the Hirzebruchs Riemann-Roch formula for endomorphism bundles over a compact complex two-fold we prove that the tangent bundle of a complex surface M of general type admits a nontrivial trace-free deformation, unless M is holomorphically covered by the euclidean ball. It follows that the tangent bundle of the Mostow-Siu surface, which is a Kähler surface with a negative definite curvature tensor, does have a nontrivial trace-free moduli. Among some other results we also point out a relationship between the Kuranishi obstruction and symmetric holomorphic two tensors on a complex surface.

Negative tangent bundles and hyperbolic manifolds

We construct a family of algebraic manifolds which are hyperbolic in the sense of Kobayashi, but whose tangent bundles are not negative.

On some chern polynomials of a compact complex manifold covered by a bounded domain

In this paper we proved that the Chern numbers of a compact complex manifold covered by a bounded domain have to satisfy certain conditions. For instance, for a compact complex two-fold covered by a bounded domain in the Chern number 2 1 – 2 must be non-negative, where c1 =1st Chern classC2 =2nd Chern class. Some corresponding inequalities for higher dimensional case are also true.