Pseudo-isomorphisms in dimension $3$ and applications to complex Monge-Ampere equation

Abstract

Let $X$ and $Y$ be compact Kähler manifolds of dimension $3$. A bimeromorphic map $f:X\rightarrow Y$ is pseudo-isomorphic if $f:X-I(f)\rightarrow Y-I(f^{-1})$ is an isomorphism. In this paper we investigate some properties of pseudo-isomorphisms. As an application, we associate to any pseudo-isomorphism in dimension $3$ and a smooth closed $(3,3)$ form $δ$ on $X\times X$ representing the cohomology class of the diagonal $Δ_X$, a Monge-Ampere operator $MA(f^*(θ),δ)=f^*(θ)\wedge f^*(θ)\wedge f^*(θ)$, here $θ$ is a smooth closed $(1,1)$ form on $Y$. We show that this Monge-Ampere operator is independent of the choice of $δ$, if the following cohomologous condition is satisfied: {\bf Condition.} For any curve $C\subset I(f^{-1})$, we have $\{θ\}.\{C\}=0$ in cohomology. We conclude the paper examining a simple pseudo-isomorphism in dimension $3$.