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

Limits on the production of scalar leptoquarks from Z (0) decays at LEP

A search has been made for pairs and for single production of scalar leptoquarks of the first and second generations using a data sample of 392000 Z0 decays from the DELPHI detector at LEP 1. No signal was found and limits on the leptoquark mass, production cross section and branching ratio were set. A mass limit at 95% confidence level of 45.5 GeV/c2 was obtained for leptoquark pair production. The search for the production of a single leptoquark probed the mass region above this limit and its results exclude first and second generation leptoquarks D0 with masses below 65 GeV/c2 and 73 GeV/c2 respectively, at 95% confidence level, assuming that the D0lq Yukawa coupling alpha(lambda) is equal to the electromagnetic one. An upper limit is also given on the coupling alpha(lambda) as a function of the leptoquark mass m(D0)