Quasi-potentials and regularization of currents, and applications

Let $Y$ be a compact Kähler manifold. We show that the weak regularization $K_n$ of Dinh and Sibony for the diagonal $Δ_Y$ (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 $θ$ 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 $-S_n\leq K_n(T\wedge θ)-K_n(T)\wedge θ\leq S_n$ 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.

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