Mabuchi geometry of big cohomology classes with prescribed singularities

TL;DR

This paper develops a metric space framework for finite energy K"ahler potentials with prescribed singularities on compact K"ahler spaces, generalizing existing metrics for ample classes.

Contribution

It introduces a new metric $d_p$ on spaces of potentials with prescribed singularities, proving completeness and extending previous work beyond ample classes.

Findings

Defined non-pluripolar products for quasi-psh functions on complex spaces

Constructed and proved completeness of the $d_p$ metric space for potentials with prescribed singularities

Generalized the $d_p$ metric from ample classes to big cohomology classes.

Abstract

Let $X$ be a compact K\"ahler unibranch complex analytic space of pure dimension. Fix a big class $\alpha$ with smooth representative $\theta$ and a model potential $\phi$ with positive mass. We define and the study non-pluripolar products of quasi-plurisubharmonic functions on $X$. We study the spaces $\mathcal{E}^p(X,\theta;[\phi])$ of finite energy K\"ahler potentials with prescribed singularities for each $p\geq 1$. We define a metric $d_p$ and show that $(\mathcal{E}^p(X,\theta;[\phi]),d_p)$ is a complete metric space. This construction generalizes the usual $d_p$-metric defined for an ample class.