Continuity of the Weil-Petersson potential

Abstract

Let ${\mathcal{M}}_{\mathrm{KSB}}$ (resp. ${\mathcal{M}}_{\mathrm{KSB}}^{\prime }$) be the the moduli space of $n$-dimensional Kähler-Einstein manifolds (resp. varieties) $X$ with $K_X$ ample. We prove that the Weil-Petersson metric on ${\mathcal{M}}_{\mathrm{KSB}}$ extends uniquely to the projective variety ${\mathcal{M}}_{\mathrm{KSB}}^{\prime }$, as a closed positive current with continuous local potentials. This generalizes a theorem of Wolpert which treats the case $n=1$, and also confirms a conjecture of Berman-Guenancia. In addition, we derive uniform estimates for the volumes of sublevel sets of Kähler-Einstein potentials

References in corpus (2)

• Degeneration of Kahler-Einstein manifolds of negative scalar curvature
• Riemannian geometry of Kahler-Einstein currents III: compactness of Kahler-Einstein manifolds of negative scalar curvature

Cited by in corpus (1)

• Kahler-Einstein metric near an isolated log canonical singularity