On the regularity of conical Calabi-Yau potentials

TL;DR

This paper proves that locally bounded conical Calabi-Yau potentials on Fano cones are smooth on the regular locus using pluripotential theory, extending previous results to more general, non-toric cases.

Contribution

It establishes the regularity of conical Calabi-Yau potentials on Fano cones without relying on symmetry assumptions, broadening the understanding of their geometric properties.

Findings

Locally bounded potentials are smooth on the regular locus.

The proof is purely pluripotential and does not depend on symmetry.

Extends previous toric results to general Fano cones.

Abstract

Using pluripotential theory on degenerate Sasakian manifolds, we show that a locally bounded conical Calabi-Yau potential on a Fano cone is actually smooth on the regular locus. This work is motivated by a similar result obtained by R. Berman in the case where the cone is toric. Our proof is purely pluripotential and independent of any extra symmetry imposed on the cone.