Shigeharu Takayama

HODGE METRICS AND THE CURVATURE OF HIGHER DIRECT IMAGES

Using the harmonic theory developed by Takegoshi for representation of relative cohomology and the framework of computation of curvature of direct images bundles by Berndtsson, we prove that the higher direct images by a smooth morphism of the relative canonical bundle twisted by a semi-positive vector bundle are locally free and semi-positively curved, when endowed with a suitable Hodge type metric.

Local Simple Connectedness of Resolutions of Log-Terminal Singularities

We study fundamental groups related with log-terminal singularities, and show that fundamental groups are preserved by a resolution of singularities. As corollaries, we show that fundamental groups are invariant under various fundamental operations in the Minimal Model Program, for example, by contractions of extremal rays, by flips, by pluricanonical morphisms of minimal varieties of general type.

Hodge metrics and positivity of direct images

Abstract Building on Fujita-Griffiths method of computing metrics on Hodge bundles, we show that the direct image of an adjoint semi-ample line bundle by a projective submersion has a continuous metric with Griffiths semi-positive curvature. This shows that for every holomorphic semi-ample vector bundle E on a complex manifold, and every positive integer k, the vector bundle SkE ⊗ det E has a continuous metric with Griffiths semi-positive curvature. If E is ample on a projective manifold, the metric can be made smooth and Griffiths positive.

Iitaka’s fibrations via multiplier ideals

We give a new characterization of Iitakas fibration of algebraic varieties associated to line bundles. Introducing an intersection number of line bundles and curves by using the notion of multiplier ideal sheaves, Iitakas fibration can be regarded as a numerically trivial fibration in terms of this intersection theory.

On the ∂u∂-equation over pseudoconvex Kahler manifolds

Abstract:The Picard variety Pic0(?n) of a complex n-dimensional torus?n is the group of all holomorphic equivalence classes of topologically trivial holomorphic (principal) line bundles on ?n. The total space of a topologically trivial holomorphic (principal) line bundle on a compact Kähler manifold is weakly pseudoconvex. Thus we can regard Pic0(?n) as a family of weakly pseudoconvex Kähler manifolds. We consider a problem whether the Kodairas -Lemma holds on a total space of holomorphic line bundle belonging to Pic0(?n). We get a criterion for this problem using a dynamical system of translations on Pic0(?n). We also discuss the problem whether the -Lemma holds on strongly pseudoconvex Kähler manifolds or not. Using the result of ColColţoiu, we find a 1-convex complete Kähler manifold on which the -Lemma does not hold.

On relative base point freeness of adjoint bundle

We give an effective result on the relative base point freeness of an adjoint bundle for a pair of a projective morphism and a relatively ample line bundle.

\(\partial \bar \partial \)-Lemma on Noncompact and Kähler Manifolds

A complex manifold X is called weakly 1-complete if there exists a C ∞-plurisubharmonic exhaustive function on X. On weakly 1-complete Kahler manifolds we consider whether ∂∂-Lemma holds or not.