The basepoint-freeness threshold of a very general abelian surface

For abelian surfaces of Picard rank 1, we perform explicit computations of the cohomological rank functions of the ideal sheaf of one point, and in particular of the basepoint-freeness threshold. Our main tool is the relation between cohomological rank functions and Bridgeland stability. In virtue of recent results of Caucci and Ito, these computations provide new information on the syzygies of polarized abelian surfaces.

Some applications of the theory of harmonic integrals

In this survey, we present recent techniques on the theory of harmonic integrals to study the cohomology groups of the adjoint bundle with the multiplier ideal sheaf of singular metrics. As an application, we give an analytic version of the injectivity theorem.

GREEN FUNCTIONS WITH SINGULARITIES ALONG COMPLEX SPACES

We study properties of a Green function GA with singularities along a complex subspace A of a complex manifold X. It is defined as the largest negative plurisubharmonic function u satisfying locally u≤ log |ψ|+C, where ψ=(ψ1,…,ψm), ψ1,…,ψm are local generators for the ideal sheaf ℐA of A, and C is a constant depending on the function u and the generators. A motivation for this study is to estimate global bounded functions from the sheaf ℐA and thus proving a "Schwarz lemma" for ℐA.

PSEUDO-EFFECTIVE LINE BUNDLES ON COMPACT KÄHLER MANIFOLDS

The goal of this work is to pursue the study of pseudo-effective line bundles and vector bundles. Our first result is a generalization of the Hard Lefschetz theorem for cohomology with values in a pseudo-effective line bundle. The Lefschetz map is shown to be surjective when (and, in general, only when) the pseudo-effective line bundle is twisted by its multiplier ideal sheaf. This result has several geometric applications, e.g. to the study of compact Kähler manifolds with pseudo-effective canonical or anti-canonical line bundles. Another concern is to understand pseudo-effectivity in more algebraic terms. In this direction, we introduce the concept of an "almost" nef line bundle, and mean by this that the degree of the bundle is nonnegative on sufficiently generic curves. It can be shown that pseudo-effective line bundles are almost nef, and our hope is that the converse also holds true. This can be checked in some cases, e.g. for the canonical bundle of a projective 3-fold. From this, we derive some geometric properties of the Albanese map of compact Kähler 3-folds.

On the structure of local cohomology modules for monomial curves in

Our setting for this paper is projective 3-space over an algebraically closed field K. By a curve C ⊂ is meant a 1-dimensional, equidimensional projective algebraic set, which is locally Cohen-Macaulay. Let be the Hartshorne-Rao module of finite length (cf. [R]). Here Z is the set of integers and ℐc the ideal sheaf of C. In [GMV] it is shown that , where is the homogeneous ideal of C, is the first local cohomology module of the R-module M with respect to . Thus there exists a smallest nonnegative integer k ∊ N such that , (see also the discussion on the 1-st local cohomology module in [GW]). Also in [GMV] it is shown that k = 0 if and only if C is arithmetically Cohen-Macaulay and C is arithmetically Buchsbaum if and only if k ≤ 1. We therefore have the following natural definition.