Least negative intersections of positive closed currents on compact K\"ahler manifolds

Abstract

Let $X$ be a compact K\"ahler manifold of dimension $k$. Let $R$ be a positive closed $(p,p)$ current on $X$, and $T_1,\ldots ,T_{k-p}$ be positive closed $(1,1)$ currents on $X$. We define a so-called least negative intersection of the currents $T_1,T_2,\ldots ,T_{k-p}$ and $R$, as a sublinear bounded operator \begin{eqnarray*} \bigwedge (T_1,\ldots ,T_{k-p},R):~C^0(X)\rightarrow \mathbb{R}. \end{eqnarray*} This operator is {\bf symmetric} in $T_1,\ldots ,T_{k-p}$. It is {\bf independent} of the choice of a quasi-potential $u_i$ of $T_i$, of the choice of a smooth closed $(1,1)$ form $\theta _i$ in the cohomology class of $T_i$, and of the choice of a K\"ahler form on $X$. Its total mass $<\bigwedge (T_1,\ldots ,T_{k-p},R),1>$ is the intersection in cohomology $\{T_1\}\{T_2\}\ldots \{T_{k-p}\}.\{R\}$. It has a semi-continuous property concerning approximating $T_i$ by appropriate smooth closed $(1,1)$ forms, plus some other good properties. If $p=0$ and $T_1=\ldots =T_k=T$, we have a least negative Monge-Ampere operator $MA(T)=\bigwedge (T,\ldots ,T)$. If the set where $T$ has positive Lelong numbers does not contain any curve, then $MA(T)$ is positive. Several examples are given.