On the existence of K\"ahler metrics of constant scalar curvature

TL;DR

This paper characterizes the K"ahler classes with constant scalar curvature on certain Fano manifolds, showing the existence of infinitely many such classes without K"ahler-Einstein metrics.

Contribution

It identifies the analytic subvariety of the second cohomology where constant scalar curvature K"ahler metrics exist, revealing new existence results for these metrics.

Findings

Infinitely many nonhomothetic K"ahler classes with constant scalar curvature

No K"ahler-Einstein metric exists on the studied manifolds

Characterization of the subvariety where the Bando-Calabi-Futaki character vanishes

Abstract

For certain compact complex Fano manifolds $M$ with reductive Lie algebras of holomorphic vector fields, we determine the analytic subvariety of the second cohomology group of $M$ consisting of K\"ahler classes whose Bando-Calabi-Futaki character vanishes. Then a K\"ahler class contains a K\"ahler metric of constant scalar curvature if and only if the K\"ahler class is contained in the analytic subvariety. On examination of the analytic subvariety, it is shown that $M$ admits infinitely many nonhomothetic K\"ahler classes containing K\"ahler metrics of constant scalar curvature but does not admit any K\"ahler-Einstein metric.