On the limit behavior of metrics in continuity method to Kahler-Einstein problem in toric Fano case

TL;DR

This paper investigates the limit behavior of metrics solving a continuity family of complex Monge-Ampere equations on toric Fano manifolds, revealing that the limit metric exhibits conic singularities determined by the moment polytope.

Contribution

It demonstrates that the limit metric in the continuity method for Kahler-Einstein problems on toric Fano manifolds develops conic singularities, linking geometric and algebraic properties.

Findings

Limit metric satisfies a singular complex Monge-Ampere equation.

Limit metric exhibits conic type singularities.

Singularities are characterized by the geometry of the moment polytope.

Abstract

This is a continuation of paper \cite{Li}. On any toric Fano manifold, we discuss the behavior of limit metric of a sequence of metrics, which are solutions to a continuity family of complex Monge-Ampere equations in Kahler-Einstein problem. We show that the limit metric satisfies a singular complex Monge-Ampere equation. This shows the conic type singularity for the limit metric. The information of conic type singularities can be read from the geometry of the moment polytope.