Ohsawa-Takegoshi extension theorem for compact K\"ahler manifolds and applications

Our main goal in this article is to prove a new extension theorem for sections of the canonical bundle of a weakly pseudoconvex K\"ahler manifold with values in a line bundle endowed with a possibly singular metric. We also give some applications of our result.Comment: 16 pages. There was a serious error in the first version. It is now fixed by using Guan-Zhou's result on the strong openness conjectur

Additivity of the approximation functional of currents induced by Bergman kernels

In this note, we give a positive answer to a question raised by Jean-Pierre Demailly in 2013, and show the additivity of the approximation functional of closed positive $(1,1)$-currents induced by Bergman kernels.Comment: 4 page

A decomposition theorem for projective manifolds with nef anticanonical bundle

Let X be a simply connected projective manifold with nef anticanonical bundle. We prove that X is a product of a rationally connected manifold and a manifold with trivial canonical bundle. As an application we describe the MRC fibration of any projective manifold with nef anticanonical bundle

Manifolds with nef anticanonical bundle

Let X be a compact K\"ahler manifold such that the anticanonical bundle $-K_X$ is nef. A classical conjecture claims that the Albanese map is submersive. We prove this conjecture if the general fibre is a weak Fano manifold. If X is projective we prove the conjecture also for fibres of dimension at most two.Comment: 39 pages, changed metadat

∂∂ˉ\partial\bar\partial-lemmas and a conjecture of O. Fujino

It contains the proof of a very general $\partial\bar\partial$-lemma, together with a decomposition theorem for currents with values in a (singular) Hermitian line bundle. As a corollary, we establish the K\"ahler version on an injectivity theorem due to O. Fujino in the projective case.Comment: Comments welcome! We have decided to split into two parts our previous work arXiv:2012.05063. This is the second par