Quasi-potentials and regularization of currents, and applications

TL;DR

This paper demonstrates that a weak regularization technique for the diagonal in compact Kähler manifolds is compatible with wedge products, introduces an intersection product for positive currents, and explores quasi-potentials of positive closed currents.

Contribution

It establishes the compatibility of weak regularization with wedge products and defines a symmetric, local intersection product for positive currents.

Findings

Regularization $K_n$ is compatible with wedge product for positive $dd^c$-closed currents.

Introduces an intersection product for positive $dd^c$-closed currents.

Provides applications including a compatibility result for pullback operators.

Abstract

Let $Y$ be a compact K\"ahler manifold. We show that the weak regularization $K_n$ of Dinh and Sibony for the diagonal $\Delta_Y$ (see Section 2 for more detail) is compatible with wedge product in the following sense: If $T$ is a positive $dd^c$-closed $(p,p)$ current and $\theta$ is a smooth $(q,q)$ form then there is a sequence of positive $dd^c$-closed $(p+q,p+q)$ currents $S_n$ whose masses converge to 0 so that $-S_n\leq K_n(T\wedge \theta)-K_n(T)\wedge \theta \leq S_n$ for all $n$. We also prove a result concerning the quasi-potentials of positive closed currents. We give two applications of these results. First, we prove a corresponding compatibility with wedge product for the pullback operator defined in our previous paper. Second, we define an intersection product for positive $dd^c$-closed currents. This intersection is symmetric and has a local nature.