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

TL;DR

This paper introduces a least negative intersection operator for positive closed currents on compact K"ahler manifolds, generalizing intersection theory with desirable symmetry, independence, and semi-continuity properties.

Contribution

It defines a novel least negative intersection operator for positive closed currents, extending intersection theory with key invariance and continuity features.

Findings

The operator is symmetric in the currents.

It is independent of quasi-potential and cohomology class choices.

When currents have no positive Lelong numbers on curves, the operator yields positive measures.

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.