Positivity of direct images with a Poincaré type twist

New characterizations of plurisubharmonic functions and positivity of direct image sheaves

We give new characterizations of plurisubharmonic functions and Griffiths positivity of holomorphic vector bundles with singular Finsler metrics. As applications, we present a different method to prove plurisubharmonic variation of generalized Bergman kernel metrics and Griffiths positivity of the direct images of twisted relative canonical bundles associated to holomorphic families of Kähler manifolds.

A Takayama-type extension theorem

We prove a theorem on the extension of holomorphic sections of powers of adjoint bundles from submanifolds of complex codimension 1 having non-trivial normal bundle. The first such result, due to Takayama, considers the case where the canonical bundle is twisted by a line bundle that is a sum of a big and nef line bundle and a ${\mathbb Q}$-divisor that has kawamata log terminal singularites on the submanifold from which extension occurs. In this paper we weaken the positivity assumptions on the twisting line bundle to what we believe to be the minimal positivity hypotheses. The main new idea …

A converse of Hörmander's $L^2$-estimate and new positivity notions for vector bundles

We study conditions of Hörmander's $L^2$-estimate and the Ohsawa-Takegoshi extension theorem. Introducing a twisted version of Hörmander-type condition, we show a converse of Hörmander $L^2$-estimate under some regularity assumptions on an $n$-dimensional domain. This result is a partial generalization of the 1-dimensional result obtained by Berndtsson. We also define new positivity notions for vector bundles with singular Hermitian metrics by using these conditions. We investigate these positivity notions and compare them with classical positivity notions.

Griffiths正性的第一射流刻画

A first-jet characterization of Griffiths positivity

本文用第一射流的有限维凸条件刻画向量丛的Griffiths正性，并给出对偶障碍及满足第一级条件但具有微分核的显式例子。

A generalized Poincaré-Lelong formula

We prove a generalization of the classical Poincaré-Lelong formula. Given a holomorphic section $f$, with zero set $Z$, of a Hermitian vector bundle $E\to X$, let $S$ be the line bundle over $X\setminus Z$ spanned by $f$ and let $Q=E/S$. Then the Chern form $c(D_Q)$ is locally integrable and closed in $X$ and there is a current $W$ such that $dd^cW=c(D_E)-c(D_Q)-M,$ where $M$ is a current with support on $Z$. In particular, the top Bott-Chern class is represented by a current with support on $Z$. We discuss positivity of these currents, and we also reveal a close relation with principal value …

直像层的对偶Nakano正性与奇异Nakano正性

Dual Nakano positivity and singular Nakano positivity of direct image sheaves

本文研究了扭曲相对典范丛直像层上的典范度量，证明了其在光滑无形变条件下的对偶Nakano半正性及奇异度量下的局部Nakano半正性。