Special biconformal changes of K\"ahler surface metrics

TL;DR

This paper investigates special biconformal changes in K"ahler surface metrics, establishing their existence for various canonical metrics on Del Pezzo surfaces, including K"ahler-Einstein and K"ahler-Ricci solitons.

Contribution

It demonstrates the existence of nontrivial special biconformal changes for several classes of canonical K"ahler metrics on Del Pezzo surfaces.

Findings

Existence of nontrivial special biconformal changes for K"ahler-Einstein metrics with holomorphic vector fields.

Existence of such changes for non-Einstein K"ahler-Ricci solitons.

Existence for K"ahler metrics with nonconstant Killing potentials and geodesic gradients.

Abstract

The term "special biconformal change" refers, basically, to the situation where a given nontrivial real-holomorphic vector field on a complex manifold is a gradient relative to two K\"ahler metrics, and, simultaneously, an eigenvector of one of the metrics treated, with the aid of the other, as an endomorphism of the tangent bundle. A special biconformal change is called nontrivial if the two metrics are not each other's constant multiples. For instance, according to a 1995 result of LeBrun, a nontrivial special biconformal change exists for the conformally-Einstein K\"ahler metric on the two-point blow-up of the complex projective plane, recently discovered by Chen, LeBrun and Weber; the real-holomorphic vector field involved is the gradient of its scalar curvature. The present paper establishes the existence of nontrivial special biconformal changes for some canonical metrics on Del Pezzo surfaces, viz. K\"ahler-Einstein metrics (when a nontrivial holomorphic vector field exists), non-Einstein K\"ahler-Ricci solitons, and K\"ahler metrics admitting nonconstant Killing potentials with geodesic gradients.