Invariants of Varieties and Singularities inspired by Kahler-Einstein problems

TL;DR

This paper generalizes K-stability to broader algebraic contexts, linking volume function concavity with invariants' behavior during the Minimal Model Program, and explores new connections in algebraic geometry.

Contribution

It extends K-stability framework to singularities, families over higher dimensions, and surfaces, revealing new insights into invariants and the Minimal Model Program.

Findings

Volume function concavity implies invariants decrease along MMP

Generalized K-stability applies to singularities and higher-dimensional families

New results connect MMP theory with invariants in algebraic geometry

Abstract

We extend the framework of K-stability (Tian, Donaldson) to more general algebro-geometric setting, such as partial desingularisations of (fixed) singularities, (not necessarily flat) families over higher dimensional base and the classical birational geometry of surfaces. We also observe that "concavity" of the volume function implies decrease of the (generalised) Donaldson-Futaki invariants along the Minimal Model Program, in our generalised settings. Several related results on the connection with the MMP theory, some of which are new even in the original setting of families over curves, are also proved.