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

TL;DR

This paper studies pseudo-isomorphisms between compact Kähler threefolds, introducing a Monge-Ampère operator associated with these maps and exploring conditions for its independence from auxiliary choices.

Contribution

It defines a Monge-Ampère operator for pseudo-isomorphisms in dimension 3 and establishes conditions under which this operator is well-defined and independent of certain choices.

Findings

The Monge-Ampère operator is well-defined under specific cohomological conditions.

The operator's independence from the choice of the form extdelta ext is proven under the cohomologous condition.

Analysis of a simple pseudo-isomorphism illustrates the theoretical results.

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. 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 $\delta$ on $X\times X$ representing the cohomology class of the diagonal $\Delta_X$, a Monge-Ampere operator $MA(f^*(\theta),\delta)=f^*(\theta)\wedge f^*(\theta)\wedge f^*(\theta)$, here $\theta$ is a smooth closed $(1,1)$ form on $Y$. We show that this Monge-Ampere operator is independent of the choice of $\delta$, if the following cohomologous condition is satisfied: {\bf Condition.} For any curve $C\subset I(f^{-1})$, we have $\{\theta \}.\{C\}=0$ in cohomology. We conclude the paper examining a simple pseudo-isomorphism in dimension $3$.