On the curvature of conic Kaehler-Einstein metrics

TL;DR

This paper establishes regularity results for degenerate Monge-Ampère equations on Kähler manifolds and applies them to analyze the curvature properties of conical Kähler-Einstein metrics, providing geometric conditions for their boundedness.

Contribution

It introduces a new regularity theorem for degenerate Monge-Ampère equations and uses it to study curvature bounds of conical Kähler-Einstein metrics with divisors.

Findings

Proved regularity of Monge-Ampère equations degenerate along divisors.

Derived geometric conditions ensuring boundedness of conical Kähler-Einstein metrics.

Enhanced understanding of curvature behavior near divisors in Kähler geometry.

Abstract

We prove a regularity result for Monge-Amp\`ere equations degenerate along smooth divisor on Kaehler manifolds in Donaldson's spaces of $\beta$-weighted functions. We apply this result to study the curvature of Kaehler metrics with conical singularities along divisors and give a geometric sufficient condition on the divisor for its boundedness.