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On the collapsing of Calabi-Yau manifolds and Kähler-Ricci flows

Yang Li, Valentino Tosatti

Abstract

We study the collapsing of Calabi-Yau metrics and of Kahler-Ricci flows on fiber spaces where the base is smooth. We identify the collapsed Gromov-Hausdorff limit of the Kahler-Ricci flow when the divisorial part of the discriminant locus has simple normal crossings. In either setting, we also obtain an explicit bound for the real codimension 2 Hausdorff measure of the Cheeger-Colding singular set, and identify a sufficient condition from birational geometry to understand the metric behavior of the limiting metric on the base.

On the collapsing of Calabi-Yau manifolds and Kähler-Ricci flows

Abstract

We study the collapsing of Calabi-Yau metrics and of Kahler-Ricci flows on fiber spaces where the base is smooth. We identify the collapsed Gromov-Hausdorff limit of the Kahler-Ricci flow when the divisorial part of the discriminant locus has simple normal crossings. In either setting, we also obtain an explicit bound for the real codimension 2 Hausdorff measure of the Cheeger-Colding singular set, and identify a sufficient condition from birational geometry to understand the metric behavior of the limiting metric on the base.

Paper Structure

This paper contains 18 sections, 12 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Calabi-Yau
  3. Kähler-Ricci flow
  4. Our results
  5. Acknowledgments
  6. Hausdorff measure bound for the singular set
  7. The approximation procedure
  8. The measure bound \ref{['volume']}
  9. Collapsing and the canonical bundle formula
  10. Volume form asymptotics
  11. The canonical bundle formula
  12. Vanishing orders
  13. A functorial divisor
  14. Collapsing
  15. Collapsing of the Kähler-Ricci flow
  16. ...and 3 more sections

Key Result

Theorem 1.3

In either the Calabi-Yau setup setupcy or the Kähler-Ricci flow setup setupkrf, assume that $N$ is smooth and $[\omega_N]\in H^2(N,\mathbb{Q})$, and let $\mathcal{H}^{2n-2}$ be the real $(2n-2)$-dimensional Hausdorff measure of the limit metric $d_Z$ on $N$. Then the Cheeger-Colding singular set $\m where $C_n$ is a dimensional constant.

Theorems & Definitions (26)

  • Conjecture 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • ...and 16 more