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Generic regularity of intermediate complex structure limits

Yang Li, Valentino Tosatti

TL;DR

The paper analyzes collapsing Ricci-flat Kähler metrics on Calabi–Yau manifolds near intermediate complex structure limits ($0<m<n$). It constructs ansatz reference metrics from a non-Archimedean Monge–Ampère framework and proves a trimmed Harnack inequality together with a De Giorgi iteration to obtain $C^0$ convergence of the Ricci-flat metric $\omega_{CY,t}$ to the ansatz $\omega_t$ on the generic region, with $C^\infty$ control after rescaling by $|\log|t||^{1/2}$. This extends the large complex structure results ($m=n$) to the intermediate case by modeling the geometry as a $Z$-fibration over a Calabi–Yau base and unwrapping $T^m$-fibers, handling the two-scale degenerate geometry. The methods open avenues for later construction of special Lagrangian or coisotropic fibrations on the generic region and have potential applicability to collapsing settings of HT-type, where similar two-scale analysis arises.

Abstract

We study certain polarized degenerations of Calabi-Yau manifolds near an intermediate complex structure limit, and improve the potential $C^0$-convergence to a metric convergence result on the generic region for the corresponding collapsing Ricci-flat Kähler metrics.

Generic regularity of intermediate complex structure limits

TL;DR

The paper analyzes collapsing Ricci-flat Kähler metrics on Calabi–Yau manifolds near intermediate complex structure limits (). It constructs ansatz reference metrics from a non-Archimedean Monge–Ampère framework and proves a trimmed Harnack inequality together with a De Giorgi iteration to obtain convergence of the Ricci-flat metric to the ansatz on the generic region, with control after rescaling by . This extends the large complex structure results () to the intermediate case by modeling the geometry as a -fibration over a Calabi–Yau base and unwrapping -fibers, handling the two-scale degenerate geometry. The methods open avenues for later construction of special Lagrangian or coisotropic fibrations on the generic region and have potential applicability to collapsing settings of HT-type, where similar two-scale analysis arises.

Abstract

We study certain polarized degenerations of Calabi-Yau manifolds near an intermediate complex structure limit, and improve the potential -convergence to a metric convergence result on the generic region for the corresponding collapsing Ricci-flat Kähler metrics.

Paper Structure

This paper contains 10 sections, 6 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Setup
  3. The Calabi-Yau family
  4. Ricci-flat metrics
  5. Ansatz reference metrics
  6. Local coordinates.
  7. Model case
  8. Harnack Inequality
  9. De Giorgi type blow up argument
  10. Higher order estimates on the generic region

Key Result

Theorem 1.1

On the generic region of $X_t$, the Calabi-Yau metric $\omega_{{\rm CY},t}$ converges in $C^0$-sense to the ansatz metric $\omega_t$ as $t\to 0$, namely

Theorems & Definitions (18)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 8 more