The Monge-Ampere operator of some singular (1,1) currents coming from pseudo-isomorphisms in dimension $3$

Abstract

A wide and natural class of closed currents - which are differences of positive closed currents - can be constructed by pulling back smooth closed forms using rational maps. These currents are very singular in general, and hence defining intersections between them is challenging. In this paper, we use our previous results to investigate this question in the case where the rational maps in question are pseudo-isomorphisms (i.e. bimeromorphic maps which, along with their inverses, have no exceptional divisors) in dimension $3$. Our main result, to be described in a more concrete form later in the paper, is as follows. {\bf Theorem.} Let $X,Y$ be compact K\"ahler manifolds of dimension $3$, and $f:X\dashrightarrow Y$ be a pseudo-isomorphism. Let $\alpha _2,\alpha _3$ be smooth closed $(1,1)$ forms on $Y$, and $T_1$ a difference of two positive closed $(1,1)$ currents on $X$. Then, whether the intersection of the currents $T_1$, $f^*(\alpha _2)$ and $f^*(\alpha _3)$ satisfies a Bedford-Taylor's type monotone convergence depends only on the cohomology classes of $\alpha _2,\alpha _3$. Special attention is given to the case where $T_1=f^*(\alpha _1)$ where $\alpha _1$ is a smooth closed $(1,1)$ form on $Y$. It is then shown that satisfying the above mentioned Bedford-Taylor's type monotone convergence is asymmetric in $\alpha _1$, $\alpha _2$ and $\alpha _3$, but in contrast the resulting signed measure is symmetric in $\alpha _1$, $\alpha _2$ and $\alpha _3$. We relate this Bedford-Taylor's type monotone convergence to the least-negative intersection we defined previously. These results can be extended to the case where $\alpha _1$, $\alpha _2$, $\alpha _3$ are more singular. Dynamics of pseudo-isomorphisms in dimension $3$ are essential in proving these results.