Quasi-potentials and regularization of currents, and applications

Abstract

Let $Y$ be a compact Kähler 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 $d{d}^c$-closed $(p,p)$ current and $\theta$ is a smooth $(q,q)$ form then there is a sequence of positive $d{d}^c$-closed $(p+q,p+q)$ currents $S_n$ whose masses converge to 0 so that $-S_n\le K_n(T\wedge \theta )-K_n(T)\wedge \theta \le S_n$ 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 $d{d}^c$-closed currents. This intersection is symmetric and has a local nature.

11 pages

References in corpus (2)

• Value distribution for sequences of rational mappings and complex dynamics
• Pull-back of currents by meromorphic maps