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

Abstract

It is shown that any affine toric variety $Y$, which is $\mathbb{Q}$-Gorenstein, admits a conical Ricci flat Kähler 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–Ampère equation in $\mathbb{R}^m$ with exponential right hand side and with prescribed target given by a proper convex cone, combined with transversal a priori estimates on $Y$.