Complex Monge-Ampere operators via pseudo-isomorphisms: the well-defined cases

Abstract

Let $X$ and $Y$ be compact K\"ahler 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. Let $T=T^+-T^-$ be a current on $Y$, where $T^{\pm}$ are positive closed $(1,1)$ currents which are smooth outside a finite number of points. We assume that the following condition is satisfied: {\bf Condition 1.} For every curve $C$ in $I(f^{-1})$, then in cohomology $\{T\}.\{C\}=0$. Then, we define a natural push-forward $f_*(\varphi dd^cu\wedge f^*(T))$ for a quasi-psh function $u$ and a smooth function $\varphi$ on $Y$. We show that this pushforward satisfies a Bedford-Taylor's monotone convergence type. Assume moreover that the following two conditions are satisfied {\bf Condition 2.} The signed measure $T\wedge T\wedge T$ has no mass on $I(f^{-1})$. {\bf Condition 3.} For every curve $C$ in $I(f^{-1})$, the measure $T\wedge [C]$ has no Dirac mass. Then, we define a Monge-Ampere operator $MA(f^*(T))=f^*(T)\wedge f^*(T)\wedge f^*(T)$ for $f^*(T)$. We show that this Monge-Ampere operator satisfies several continuous properties, including a Bedford-Taylor's monotone convergence type when $T$ is positive. The measures $MA(f^*(T))$ are in general quite singular. Also, note that it may be not possible to define $f^*(T^{\pm})\wedge f^*(T^{\pm})\wedge f^*(T^{\pm})$.