Riemannian geometry of Kahler-Einstein currents

TL;DR

This paper investigates the Riemannian geometry of canonical Kahler-Einstein currents on complex varieties, establishing metric properties, tangent cones, and degeneration behaviors relevant to moduli space compactification.

Contribution

It introduces a detailed geometric analysis of Kahler-Einstein currents on singular varieties, including metric completion and tangent cone structure, and explores their degenerations.

Findings

The metric completion of the regular part is a compact length space homeomorphic to the original variety.

Established a special degeneration for Kahler-Einstein manifolds of general type.

Provided applications to Calabi-Yau degenerations and Kahler-Ricci flow on minimal models.

Abstract

We study Riemannian geometry of canonical Kahler-Einstein currents on projective Calabi-Yau varieties and canonical models of general type with crepant singularities. We prove that the metric completion of the regular part of such a canonical current is a compact metric length space homeomorphic to the original projective variety, with well-defined tangent cones. We also prove a special degeneration for Kahler-Einstein manifolds of general type as an approach to establish the compactification of the moduli space of Kahler-Einstein manifolds of general type. A number of applications are given for degeneration of Calabi-Yau manifolds and the Kahler-Ricci flow on smooth minimal models of general type.