Big and nef classes, Futaki Invariant and resolutions of cubic threefolds

TL;DR

This paper explores the use of the Futaki invariant in relation to constant scalar curvature Kähler metrics on algebraic manifolds, focusing on resolutions of singularities and extending classical results inspired by Ding and Tian.

Contribution

It extends classical and recent results on the Futaki invariant, particularly in the context of resolutions of singularities for algebraic manifolds.

Findings

Extended the definition of Futaki invariant for resolved singularities

Connected Futaki invariant with existence of constant scalar curvature Kähler metrics

Built upon Ding and Tian's foundational work in the field

Abstract

In this note we revisit and extend few classical and recent results on the definition and use of the Futaki invariant in connection with the existence problem for Kaehler constant scalar curvature metrics on polarized algebraic manifolds, especially in the case of resolution of singularities. The general inspiration behind this work is no doubt the beautiful 1992 paper by Ding and Tian which contains the germs of a huge amount of the successive developments in this fundamental problem, and it is a great pleasure to dedicate this to Professor G. Tian on the occasion of his birthday!