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Generalized optimal degenerations of Fano varieties

Linsheng Wang

TL;DR

The paper generalizes Tian’s algebraic program by introducing the $\mathbf{H}^g$-invariant for log Fano pairs with a smooth strictly increasing weight $g$ whose logarithm is convex. It proves the existence and uniqueness of a minimizer $v_0$, which induces a $g'$-weighted semistable degeneration; this is refined by a unique $g'$-weighted K-polystable degeneration admitting a $g'$-soliton, completing a generalized semistable-to-polystable degeneration in the algebraic setting. The authors develop a comprehensive framework using valuations, filtrations, Okounkov bodies, and DH measures, and establish finite generation for the associated graded rings, linking the minimizer to weighted K-stability notions. They also provide concrete examples, including toric and Mori-Mukai cases, illustrating when optimal degenerations coincide with the original variety or yield nontrivial limits. Overall, the work broadens the algebraic approach to optimal degenerations and solitons, with implications for moduli, stability, and the interface with Kähler-Ricci flow dynamics.

Abstract

We prove a generalization of the algebraic version of Tian conjecture. Precisely, for any smooth strictly increasing function $g:\mathbb{R}\to\mathbb{R}_{>0}$ with ${\rm log}\circ g$ convex, we define the $\mathbf{H}^g$-invariant on a Fano variety $X$ generalizing the $\mathbf{H}$-invariant introduced by Tian-Zhang-Zhang-Zhu, and show that $\mathbf{H}^g$ admits a unique minimizer. Such a minimizer will induce the $g$-optimal degeneration of the Fano variety $X$, whose limit space admits a $g'$-soliton. We present an example of Fano threefold which has the same $g$-optimal degenerations for any $g$.

Generalized optimal degenerations of Fano varieties

TL;DR

The paper generalizes Tian’s algebraic program by introducing the -invariant for log Fano pairs with a smooth strictly increasing weight whose logarithm is convex. It proves the existence and uniqueness of a minimizer , which induces a -weighted semistable degeneration; this is refined by a unique -weighted K-polystable degeneration admitting a -soliton, completing a generalized semistable-to-polystable degeneration in the algebraic setting. The authors develop a comprehensive framework using valuations, filtrations, Okounkov bodies, and DH measures, and establish finite generation for the associated graded rings, linking the minimizer to weighted K-stability notions. They also provide concrete examples, including toric and Mori-Mukai cases, illustrating when optimal degenerations coincide with the original variety or yield nontrivial limits. Overall, the work broadens the algebraic approach to optimal degenerations and solitons, with implications for moduli, stability, and the interface with Kähler-Ricci flow dynamics.

Abstract

We prove a generalization of the algebraic version of Tian conjecture. Precisely, for any smooth strictly increasing function with convex, we define the -invariant on a Fano variety generalizing the -invariant introduced by Tian-Zhang-Zhang-Zhu, and show that admits a unique minimizer. Such a minimizer will induce the -optimal degeneration of the Fano variety , whose limit space admits a -soliton. We present an example of Fano threefold which has the same -optimal degenerations for any .
Paper Structure (20 sections, 31 theorems, 132 equations)

This paper contains 20 sections, 31 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Valuations
  4. Filtrations
  5. Okounkov bodies, concave transforms and DH measures
  6. Log canonical slopes and $\mathbf{L}$-functionals
  7. Higher rank finite generation
  8. Generalized $\mathbf{H}$-invariants
  9. Convexity
  10. Approximation by valuations
  11. Weighted delta invariants
  12. Existence of $\mathbf{H}^g$-minimizers and finite generation
  13. Approximation by test configurations
  14. Approximation by special test configurations
  15. Existence of $\mathbf{H}^g$-minimizer
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $(X,\Delta)$ be a log Fano pair, and $g:{\mathbb R}\to{\mathbb R}_{>0}$ be a smooth strictly increasing function with ${\rm log}\circ g$ convex. Then the $\mathbf{H}^g$-invariant (Definition Definition: H^g invariant) of $(X,\Delta)$ admits a unique minimizer $v_0$, which is a special valuation

Theorems & Definitions (64)

  • Theorem 1.1: Generalized Tian conjecture
  • Remark 1.2
  • Theorem 1.3: Theorem \ref{['Theorem: gHT: Weighted K-stability']}
  • Theorem 1.5
  • Remark 1.6
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 54 more