14 research outputs found
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