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Ricci-flat metrics on Calabi-Yau manifolds

Valentino Tosatti

TL;DR

This work analyzes how Ricci-flat Kähler metrics on Calabi–Yau manifolds degenerate as their Kähler classes approach the boundary of the Kähler cone. It blends the Calabi–Yau theorem, pluripotential theory, and fibration geometry to separate non-collapsing and collapsing regimes, proving smooth convergence away from null loci for nef and big classes, and developing a detailed asymptotic expansion and higher-order estimates in semiample, torus-fibered, and isotrivial collapsing settings. It also investigates the limiting metric spaces under Gromov–Hausdorff convergence, establishing sharp results for the base spaces in semiample cases and constructing intricate counterexamples in the non-semiample case. The work highlights both the robust structure of Ricci-flat degenerations and the delicate phenomena that arise when nef classes fail to be big or semiample, with broad implications for complex geometry and moduli of CY manifolds.

Abstract

We study the space of Ricci-flat Kahler metrics on a given Calabi-Yau manifold, pose a number of questions about their possible degenerations, and survey some recent results on these questions.

Ricci-flat metrics on Calabi-Yau manifolds

TL;DR

This work analyzes how Ricci-flat Kähler metrics on Calabi–Yau manifolds degenerate as their Kähler classes approach the boundary of the Kähler cone. It blends the Calabi–Yau theorem, pluripotential theory, and fibration geometry to separate non-collapsing and collapsing regimes, proving smooth convergence away from null loci for nef and big classes, and developing a detailed asymptotic expansion and higher-order estimates in semiample, torus-fibered, and isotrivial collapsing settings. It also investigates the limiting metric spaces under Gromov–Hausdorff convergence, establishing sharp results for the base spaces in semiample cases and constructing intricate counterexamples in the non-semiample case. The work highlights both the robust structure of Ricci-flat degenerations and the delicate phenomena that arise when nef classes fail to be big or semiample, with broad implications for complex geometry and moduli of CY manifolds.

Abstract

We study the space of Ricci-flat Kahler metrics on a given Calabi-Yau manifold, pose a number of questions about their possible degenerations, and survey some recent results on these questions.

Paper Structure

This paper contains 23 sections, 10 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Kähler metrics and Ricci curvature
  3. Calabi-Yau manifolds and Ricci-flat metrics
  4. Degenerations of Ricci-flat metrics
  5. The non-collapsing case
  6. The boundary of the Kähler cone
  7. Nef and big classes
  8. The collapsing case
  9. Nef classes which are not big
  10. Collapsing setup
  11. Semiample classes
  12. No singular fibers
  13. Quasi-isometry
  14. Uniform convergence
  15. Higher order estimates in the case of torus fibers
  16. ...and 8 more sections

Key Result

Theorem 1.1

If $(X,\alpha)$ is a compact Kähler manifold, then the map in bij is a bijection.

Theorems & Definitions (22)

  • Theorem 1.1: Calabi-Yau Theorem CaYa
  • Example 1.2
  • Example 1.4
  • Example 1.6
  • Theorem 2.1: Demailly-Păun DP
  • Theorem 2.2: Collins-T. CT
  • Theorem 2.3: T. To, Boucksom-Eyssidieux-Guedj-Zeriahi BEGZ, Collins-T. CT
  • Example 2.4
  • Example 2.5
  • Conjecture 2.7: Filip-T., Conj. 1.2 in FT
  • ...and 12 more