57 research outputs found
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
- …