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Stable Degenerations of log Fano Fibration Germs

Jiyuan Han, Minghao Miao, Lu Qi, Linsheng Wang, Tong Zhang

Abstract

We prove the stable degeneration conjecture of log Fano fibration germs formulated by Sun-Zhang. Precisely, we introduce the $\mathbf{H}$-invariant for filtrations over a log Fano fibration germ, and show that there exists a unique quasi-monomial valuation $v_0$ minimizing the $\mathbf{H}$-invariant. Moreover, we prove that the associated graded ring of $v_0$ is finitely generated and induces a special degeneration to a K-semistable polarized log Fano fibration germ, which further admits a unique K-polystable special degeneration.

Stable Degenerations of log Fano Fibration Germs

Abstract

We prove the stable degeneration conjecture of log Fano fibration germs formulated by Sun-Zhang. Precisely, we introduce the -invariant for filtrations over a log Fano fibration germ, and show that there exists a unique quasi-monomial valuation minimizing the -invariant. Moreover, we prove that the associated graded ring of is finitely generated and induces a special degeneration to a K-semistable polarized log Fano fibration germ, which further admits a unique K-polystable special degeneration.
Paper Structure (40 sections, 81 theorems, 364 equations)

This paper contains 40 sections, 81 theorems, 364 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and conventions
  4. Fibrations
  5. Valuations
  6. Filtrations
  7. Birational invariants
  8. Log canonical threshold
  9. Log canonical slope
  10. Test configurations
  11. Valuations and filtrations induced by test configurations
  12. Relative cone construction
  13. Complements of the relative affine cones
  14. Test configurations of relative affine cones
  15. Torus actions and -twists
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $(X,\Delta)\to Z\ni o$ be a log Fano fibration germ. Then the following statements hold.

Theorems & Definitions (181)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Definition 2.3: Quasi-monomial valuations
  • Definition 2.4: Good valuations
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Remark 2.7
  • ...and 171 more