Mabuchi geometry of big cohomology classes with prescribed singularities

Abstract

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

The contents of Section 4,5,6 will appear on Crelle's journal

References in corpus (5)

• On the constant scalar curvature Kähler metrics, general automorphism group
• A non-Archimedean approach to K-stability
• On the constant scalar curvature Kähler metrics, apriori estimates
• $L^1$ metric geometry of potentials with prescribed singularities on compact Kähler manifolds
• Integration by parts formula for non-pluripolar product

Cited by in corpus (3)

• $L^1$ metric geometry of potentials with prescribed singularities on compact Kähler manifolds
• The closures of test configurations and algebraic singularity types
• Relative pluripotential theory on compact Kähler manifolds