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Diameter bounds for degenerating Calabi-Yau metrics

Yang Li, Valentino Tosatti

Abstract

We obtain sharp upper and lower bounds for the diameter of Ricci-flat Kahler metrics on polarized Calabi-Yau degeneration families, as conjectured by Kontsevich-Soibelman.

Diameter bounds for degenerating Calabi-Yau metrics

Abstract

We obtain sharp upper and lower bounds for the diameter of Ricci-flat Kahler metrics on polarized Calabi-Yau degeneration families, as conjectured by Kontsevich-Soibelman.

Paper Structure

This paper contains 4 sections, 2 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Volume form asymptotics
  3. Diameter lower bound
  4. Diameter upper bound

Key Result

Theorem 1.1

Let $\pi:X\to\Delta^*$ be a polarized Calabi-Yau degeneration family, suppose that the dimension $m$ of the essential skeleton $\mathrm{Sk}(X)$ is positive, and let $\omega_t$ be the Ricci-flat Kähler metric on $X_t$ in the class $\frac{1}{|\log|t||}c_1(L)|_{X_t}$, for $t\in\Delta^*.$ Then there is for all $t\in\Delta^*$ with $|t|$ sufficiently small.

Theorems & Definitions (5)

  • Theorem 1.1
  • Proposition 3.1
  • proof
  • proof : Proof of the diameter lower bound in Theorem \ref{['th1']}
  • proof : Proof of the diameter upper bound in Theorem \ref{['th1']}