Kahler-Einstein metrics with edge singularities

TL;DR

This paper investigates the existence and regularity of Kahler-Einstein metrics with edge singularities on compact Kahler manifolds, establishing existence for various cone angles and showing solutions have detailed asymptotic expansions.

Contribution

It proves the existence of Kahler-Einstein metrics with edge singularities for all cone angles up to 2π and demonstrates their polyhomogeneity, extending known results to singular settings.

Findings

Existence of Kahler-Einstein metrics with edge singularities for all cone angles ≤ 2π.

Solutions are polyhomogeneous with smooth asymptotic expansions along the divisor.

Results extend smooth case theories to singular edge cases.

Abstract

This article considers the existence and regularity of Kahler-Einstein metrics on a compact Kahler manifold $M$ with edge singularities with cone angle $2\pi\beta$ along a smooth divisor $D$. We prove existence of such metrics with negative, zero and some positive cases for all cone angles $2\pi\beta\leq 2\pi$. The results in the positive case parallel those in the smooth case. We also establish that solutions of this problem are polyhomogeneous, i.e., have a complete asymptotic expansion with smooth coefficients along $D$ for all $2\pi\beta < 2\pi$.