The weigthed Monge-Ampère energy of quasiplurisubharmonic functions

On a mixed Monge-Ampère operator for quasiplurisubharmonic functions with analytic singularities

We consider mixed Monge-Ampère products of quasiplurisubharmonic functions with analytic singularities, and show that such products may be regularized as explicit one parameter limits of mixed Monge-Ampère products of smooth functions, generalizing results of Andersson, Błocki and the last author in the case of non-mixed Monge-Ampère products. Connections to the theory of residue currents, going back to Coleff-Herrera, Passare and others, play an important role in the proof. As a consequence we get an approximation of Chern and Segre currents of certain singular hermitian metrics on vector bun…

Quasiplurisubharmonic Green functions

Given a compact Kähler manifold $X$, a quasiplurisubharmonic function is called a Green function with pole at $p\in X$ if its Monge-Ampère measure is supported at $p$. We study in this paper the existence and properties of such functions, in connection to their singularity at $p$. A full characterization is obtained in concrete cases, such as (multi)projective spaces.

On the extension of quasiplurisubharmonic functions

Let $(V,ω)$ be a compact Kähler manifold such that $V$ admits a cover by Zariski-open Stein sets with the property that $ω$ has a strictly plurisubharmonic exhaustive potential on each element of the cover. If $X\subset V$ is an analytic subvariety, we prove that any $ω|_X$-plurisubharmonic function on $X$ extends to a $ω$-plurisubharmonic function on $V$. This result generalizes a previous result of ours on the extension of singular metrics of ample line bundles. It allows one to show that any transcendental Kähler class in the real Neron-Severi space $NS_{\mathbb R}(V)$ has this extension pr…

关于广义 Monge-Ampère 乘积的 Lelong 数

On Lelong Numbers of Generalized Monge-Ampère Products

本文研究了拟多次下调和函数的广义 Monge-Ampère 乘积，给出了其在正常全纯浸没下前推的 Lelong 数估计，并将其应用于伪有效向量丛的 Chern 和 Segre 流，推广了 X. Wu 关于非 nef 轨迹的近期结果。