On flops and canonical metrics

TL;DR

This paper introduces a new explicit degeneration method using flops of the deformation to the normal cone to establish a broader notion of stability for proving non-existence of canonical Kähler metrics, with applications to the Calabi conjecture.

Contribution

It develops a widely applicable, explicit stability criterion based on flops of the deformation to the normal cone, extending previous approaches.

Findings

Provides a new method for proving non-existence of canonical Kähler metrics.

Applies the method to prove one direction of the Calabi conjecture for specific surfaces.

Offers a more general and computable stability notion for Kähler geometry.

Abstract

This article is concerned with an observation for proving non-existence of canonical Kahler metrics. The idea is to use a rather explicit type of degeneration that applies in many situations. Namely, in a variation on a theme introduced by Ross-Thomas, we consider flops of the deformation to the normal cone. This yields a rather widely applicable notion of stability that is still completely explicit and readily computable, but with wider scope. We describe some applications, among them, a proof of one direction of the Calabi conjecture for asymptotically logarithmic del Pezzo surfaces.