Conical Calabi-Yau metrics on toric affine varieties and convex cones

TL;DR

This paper proves that any Q-Gorenstein affine toric variety admits a unique conical Ricci-flat Kähler metric, with the Reeb vector minimizing volume, extending previous results to more general singularities.

Contribution

It establishes the existence of conical Calabi-Yau metrics on all Q-Gorenstein affine toric varieties, generalizing prior work on isolated singularities.

Findings

Existence of Ricci-flat Kähler metrics on Q-Gorenstein affine toric varieties.

The Reeb vector minimizes the volume functional on the Reeb cone.

Extension of previous results to non-isolated singularities.

Abstract

It is shown that any affine toric variety Y, which is Q-Gorenstein, admits a conical Ricci flat Kahler metric, which is smooth on the regular locus of Y. The corresponding Reeb vector is the unique minimizer of the volume functional on the Reeb cone of Y. The case when the vertex point of Y is an isolated singularity was previously shown by Futaki-Ono-Wang. The proof is based on an existence result for the inhomogeneous Monge-Ampere equation in real Euclidean space with exponential right hand side and prescribed target given by a proper convex convex, combined with transversal a priori estimates on Y.