Mabuchi geometry of big cohomology classes with prescribed singularities

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.