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A birational version of K-stability for big classes

Ruadhaí Dervan, Rémi Reboulet

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.

A birational version of K-stability for big classes

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.

Paper Structure

This paper contains 25 sections, 35 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. K-stability for big line bundles
  3. Preliminaries on positivity and volume
  4. Positivity of line bundles.
  5. Positive intersection theory.
  6. Restricted volumes and differentiability of volume
  7. Fujita approximation.
  8. K-stability
  9. Divisorial stability
  10. A Calabi--Yau theorem
  11. The Monge--Ampère measure of a test configuration
  12. The Calabi--Yau theorem
  13. A variational characterisation.
  14. Divisorial stability and K-stability
  15. Filtrations on the algebra of sections of a big line bundle
  16. ...and 10 more sections

Key Result

Theorem 1.1

Suppose $X$ is a smooth projective variety, and $L$ is a big line bundle on $X$. Then $(X,L)$ is uniformly K-stable if and only if it is divisorially stable.

Theorems & Definitions (92)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 82 more