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 Kähler manifolds of dimension $3$, and $f:X\dashrightarrow Y$ be a pseudo-isomorphism. Let $α_2,α_3$ 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 $T_1$, $f^*(α_2)$ and $f^*(α_3)$ satisfies a Bedford-Taylor's type monotone convergence depends only on the cohomology classes of $α_2,α_3$. Special attention is given to the case where $T_1=f^*(α_1)$ where $α_1$ 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 $α_1$, $α_2$ and $α_3$, but in contrast the resulting signed measure is symmetric in $α_1$, $α_2$ and $α_3$. 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 $α_1$, $α_2$, $α_3$ are more singular. Dynamics of pseudo-isomorphisms in dimension $3$ are essential in proving these results.