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Equivariant CM minimization for extremal manifolds

Gabriel Frey

Abstract

We prove an equivariant version of the CM minimization conjecture for extremal Kähler manifolds. This involves proving that, given an equivariant punctured family of polarized varieties, a relative version of the CM degree is strictly minimized by an extremal filling. This generalizes a result by Hattori for cscK manifolds with discrete automorphism group by allowing automorphisms and extremal metrics. As a main tool, we extend results by Székelyhidi on asymptotic filtration Chow stability of cscK manifolds with discrete automorphism group to the extremal setting.

Equivariant CM minimization for extremal manifolds

Abstract

We prove an equivariant version of the CM minimization conjecture for extremal Kähler manifolds. This involves proving that, given an equivariant punctured family of polarized varieties, a relative version of the CM degree is strictly minimized by an extremal filling. This generalizes a result by Hattori for cscK manifolds with discrete automorphism group by allowing automorphisms and extremal metrics. As a main tool, we extend results by Székelyhidi on asymptotic filtration Chow stability of cscK manifolds with discrete automorphism group to the extremal setting.

Paper Structure

This paper contains 23 sections, 27 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Strategy of the proof
  4. Structure of the paper
  5. Acknowledgments
  6. Preliminaries
  7. The CM line bundle and models
  8. Models over DVRs
  9. Equivariant and twisted models
  10. Invariant, twisted and good filtrations.
  11. Okounkov bodies and filtrations.
  12. Invariant and twisted filtrations.
  13. Torus-invariant filtrations
  14. Good filtrations
  15. Relative stability of extremal polarized manifolds
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $C$ be a smooth projective curve with a special point $0$, and let $(\mathcal{X},\mathcal{L}) \to C$ and $(\mathcal{X}',\mathcal{L}') \to C$ be fillings of a flat family $(X,L) \to C \setminus \{ 0 \}$ such that the special fiber $(\mathcal{X}_0, \mathcal{L}_0)$ is smooth, admits a cscK metric a where the inequality is strict if and only if the fillings are not isomorphic.

Theorems & Definitions (68)

  • Theorem 1.1: hattori2024minimizing
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Remark 2.5
  • ...and 58 more