A birational version of K-stability for big classes

TL;DR

This paper develops a new theory of uniform K-stability for big line bundles on smooth projective varieties, extending existing concepts and providing a valuative characterization that is stable under certain geometric transformations.

Contribution

It introduces a birational version of K-stability for big classes, generalizing previous theories and establishing its invariance under pullbacks and pushforwards.

Findings

Provides a valuative criterion for uniform K-stability.

Shows stability is preserved under pullbacks and certain pushforwards.

Defines K-stability at the level of b-divisors.

Abstract

We introduce a theory of uniform K-stability for big line bundles on smooth projective varieties. This extends the existing theory both for varieties with ample line bundles, and for varieties with big anticanonical class. Our main result gives a valuative characterisation of uniform K-stability, through finite collections of divisorial valuations. We further prove that uniform K-stability is preserved under pullbacks and certain pushforwards, which implies that uniform K-stability is well-defined at the level of b-divisors.