Volume minimization and obstructions to solving some problems in K\"ahler geometry

TL;DR

This paper surveys obstructions to solving existence problems for special K"ahler metrics, showing they are linked to volume functionals and parametrized by vector fields, aiding in identifying solutions.

Contribution

It extends the understanding of obstructions in K"ahler geometry to K"ahler-Ricci solitons, Sasaki-Einstein, and Einstein-Maxwell metrics, unifying their analysis.

Findings

Obstructions are derived as derivatives of volume functionals.

Obstructions are parametrized by vector fields.

The approach guides where to seek solutions.

Abstract

There is an obstruction to the existence of K\"ahler -Einstein metrics which is used to define the GIT weight for K-stability, and it has been extended to various geometric problems. This survey paper considers such extended obstructions to the existence problem of K\"ahler -Ricci solitons, Sasaki-Einstein metrics and (conformally) Einstein-Maxwell K\"ahler metrics. These three cases have a common feature that the obstructions are parametrized by a space of vector fields. We see, in these three cases, the obstructions are obtained as the derivative of suitable volume functionals. This tells us for which vector fields we should try to solve the existence problems.