CONICAL CALABI-YAU METRICS ON TORIC AFFINE VARIETIES AND CONVEX CONES

Journal article, 2023

It is shown that any affine toric variety Y , which is 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 Rm with exponential right hand side and with prescribed target given by a proper convex cone, combined with transversal a priori estimates on Y .