The openness conjecture and complex Brunn-Minkowski inequalities

We discuss recent versions of the Brunn-Minkowski inequality in the complex setting, and use it to prove the openness conjecture of Demailly and Koll\'ar.Comment: This is an account of the results in arXiv:1305.5781 together with some background material. It is based on a lecture given at the Abel symposium in Trondheim, June 2013. 13 page

Pseudoconvex domains spread over complex homogeneous manifolds

Using the concept of inner integral curves defined by Hirschowitz we generalize a recent result by Kim, Levenberg and Yamaguchi concerning the obstruction of a pseudoconvex domain spread over a complex homogeneous manifold to be Stein. This is then applied to study the holomorphic reduction of pseudoconvex complex homogeneous manifolds X=G/H. Under the assumption that G is solvable or reductive we prove that X is the total space of a G-equivariant holomorphic fiber bundle over a Stein manifold such that all holomorphic functions on the fiber are constant.Comment: 21 page

Green Currents for Meromorphic Maps of Compact K\"ahler Manifolds

We consider the dynamics of meromorphic maps of compact K\"ahler manifolds. In this work, our goal is to locate the non-nef locus of invariant classes and provide necessary and sufficient conditions for existence of Green currents in codimension one.Comment: Statement of Theorem 1.5 is slightly improved. Proposition 5.2 and Theorem 5.3 are adde

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

Malantaŭ la kurteno: Esperanto kiel lingvoinventaĵo

Kiel konate, Zamenhof klarigis la esencon, la celon kaj la kialojn de Esperanto en multaj verkoj. Aliflanke, pli malofte oni povas trovi en liaj tekstoj spurojn de la metodo kiun li sekvis por inventi la lingvon. Kelkaj detaloj ja estas konataj, ekzemple la intuicio kiu venis al li kiam li komparis la rusajn vortojn ŝvejcarskaja (pordistejo) kaj konditorskaja (sukeraĵejo), aŭ la admiro kiun li ekhavis al la gramatiko de la angla, malpli riĉaj de gramatikaj formoj kompare al Latino aŭ la helena. Ĉi tie mi intencas kompari la sekretajn lingvojn inventitajn de infanoj en Montesori-bazlernejo, kiuj scias nenion pri Esperanto, kun la strukturo de Esperanto, por montri kiel la metodo kiun Zamenhof sekvis estas infanstila (ne ‘infaneca’, eĉ tute male) kaj, ke li fokusiĝis je la necesaj ecoj kiujn ĉiuj homaj lingvoj havas, dum nenecesaj trajtoj estis laŭble forĵetitaj