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Valuative invariants for linearized line bundles on a spherical variety

Chenxi Yin

Abstract

We give formulas for various valuative invariants of linearized ample line bundles on a projective spherical variety. Then we show how to use these formulas to study $g$-solitons on a Fano spherical variety. As a corollary, we show that for a Fano horospherical manifold $X$, the corresponding cone $(K_{X})^{\times}$ always admits a Calabi-Yau cone metric.

Valuative invariants for linearized line bundles on a spherical variety

Abstract

We give formulas for various valuative invariants of linearized ample line bundles on a projective spherical variety. Then we show how to use these formulas to study -solitons on a Fano spherical variety. As a corollary, we show that for a Fano horospherical manifold , the corresponding cone always admits a Calabi-Yau cone metric.
Paper Structure (9 sections, 23 theorems, 77 equations)

This paper contains 9 sections, 23 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Invariants for filtrations
  3. Group action
  4. Preliminaries on spherical varieties
  5. Valuative invariants for linearized line bundles on a spherical variety
  6. Examples
  7. Valuative invariants for g-solitons
  8. Ding-stability For g-solitons
  9. Applications to Sasaki-Einstein metrics

Key Result

Theorem 1.1

tian1987kahler Let $X$ be a Fano manifold. If $\alpha(X,-K_{X}) > \frac{\mathrm{dim}(X)}{\mathrm{dim}(X)+1}$, then we have a Kähler–Einstein metric on $X$.

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.1
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 47 more