Local moduli of holomorphic bundles

We study moduli of holomorphic vector bundles on non-compact varieties. We discuss filtrability and algebraicity of bundles and calculate dimensions of local moduli. As particularly interesting examples, we describe numerical invariants of bundles on some local Calabi-Yau threefolds.Comment: 18 pages. Revision history: v1: As submitted for publication. v2: minor corrections, as publishe

On the cohomology of pseudoeffective line bundles

The goal of this survey is to present various results concerning the cohomology of pseudoeffective line bundles on compact K{\"a}hler manifolds, and related properties of their multiplier ideal sheaves. In case the curvature is strictly positive, the prototype is the well known Nadel vanishing theorem, which is itself a generalized analytic version of the fundamental Kawamata-Viehweg vanishing theorem of algebraic geometry. We are interested here in the case where the curvature is merely semipositive in the sense of currents, and the base manifold is not necessarily projective. In this situation, one can still obtain interesting information on cohomology, e.g. a Hard Lefschetz theorem with pseudoeffective coefficients, in the form of a surjectivity statement for the Lefschetz map. More recently, Junyan Cao, in his PhD thesis defended in Grenoble, obtained a general K{\"a}hler vanishing theorem that depends on the concept of numerical dimension of a given pseudoeffective line bundle. The proof of these results depends in a crucial way on a general approximation result for closed (1,1)-currents, based on the use of Bergman kernels, and the related intersection theory of currents. Another important ingredient is the recent proof by Guan and Zhou of the strong openness conjecture. As an application, we discuss a structure theorem for compact K{\"a}hler threefolds without nontrivial subvarieties, following a joint work with F.Campana and M.Verbitsky. We hope that these notes will serve as a useful guide to the more detailed and more technical papers in the literature; in some cases, we provide here substantially simplified proofs and unifying viewpoints.Comment: 39 pages. This survey is a written account of a lecture given at the Abel Symposium, Trondheim, July 201

Surfaces of general type with pg=1, q=0, K2=6 and grassmannians

We construct examples of surfaces of general type with = 1, = 0 and 2 = 6. We use as key varieties Fano fourfolds and Calabi–Yau threefolds that are zero section of some special homogeneous vector bundle on Grassmannians. We link as well our construction to a classical Campedelli surface, using the Pfaffian–Grassmannian correspondence

Hough Transform and Laguerre Geometry for the Recognition and Reconstruction of Special 3D Shapes

We put the Hough transform, a method from Image Processing, into relation to Laguerre geometry, a concept of classical geometry, and study both concepts in the 3D case. It is shown how Laguerre geometry, which works in the set of oriented planes, is used in the detection of special shapes such as planes, spheres, rotational cones and cylinders, general cones and cylinders, and general developable surfaces. We perform shape recognition tasks by principal component analysis on a set of points in the so-called Blaschke model of Laguerre geometry. These points are Blaschke image points of estimated tangent planes at the given data points. Finally we present examples and show how the implementation also takes advantage of mathematical morphology on images, which are defined on meshes