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Deformations of elliptic Calabi--Yau manifolds

János Kollár

TL;DR

The paper develops a comprehensive framework for understanding deformations of elliptic Calabi--Yau fiber spaces, linking geometric and numerical criteria to the persistence of elliptic fibrations under deformations. It establishes a sharp nef-criterion for elliptic fibrations, derives detailed asymptotic cohomology formulas, and analyzes how morphisms and fibrations deform, including the emergence of Calabi--Yau orbibundles in the generically isotrivial case. It also provides smoothing results for highly singular CY fibrations and a rich supply of examples that illustrate both preservation and loss of elliptic structure under deformation. Collectively, these results culminate in conjectures about strong abundance for CY manifolds and a structural description of CY fibrations under deformation, with orbibundles playing a central role in the birational classification of generically isotrivial cases.

Abstract

The aim of this note is to investigate characterizations and deformations of elliptic Calabi--Yau manifolds, building on earlier works of Wilson and Oguiso. Version 2: References updated and small changes. Version 3: Smoothness conditions removed from several theorems, plus some reorganization. Version 4: Several references added.

Deformations of elliptic Calabi--Yau manifolds

TL;DR

The paper develops a comprehensive framework for understanding deformations of elliptic Calabi--Yau fiber spaces, linking geometric and numerical criteria to the persistence of elliptic fibrations under deformations. It establishes a sharp nef-criterion for elliptic fibrations, derives detailed asymptotic cohomology formulas, and analyzes how morphisms and fibrations deform, including the emergence of Calabi--Yau orbibundles in the generically isotrivial case. It also provides smoothing results for highly singular CY fibrations and a rich supply of examples that illustrate both preservation and loss of elliptic structure under deformation. Collectively, these results culminate in conjectures about strong abundance for CY manifolds and a structural description of CY fibrations under deformation, with orbibundles playing a central role in the birational classification of generically isotrivial cases.

Abstract

The aim of this note is to investigate characterizations and deformations of elliptic Calabi--Yau manifolds, building on earlier works of Wilson and Oguiso. Version 2: References updated and small changes. Version 3: Smoothness conditions removed from several theorems, plus some reorganization. Version 4: Several references added.

Paper Structure

This paper contains 10 sections, 25 theorems, 64 equations.

Table of Contents

  1. Calabi--Yau fiber spaces
  2. The main case
  3. Adjoint systems of large Kodaira dimension
  4. Asymptotic estimates for cohomology groups
  5. Deforming morphisms
  6. Smoothings of very singular varieties
  7. Calabi--Yau orbibundles
  8. Generically isotrivial Calabi--Yau fiber spaces
  9. Examples
  10. General conjectures

Key Result

Theorem 3

Let $X$ be an elliptic Calabi--Yau manifold such that $H^2(X, {\mathcal{O}}_X)=0$. Then every small deformation of $X$ is also an elliptic Calabi--Yau manifold.

Theorems & Definitions (38)

  • Theorem 3
  • Conjecture 4
  • Definition 5
  • Definition 7
  • Proposition 9
  • Theorem 10
  • Corollary 11
  • Definition 12
  • Theorem 14
  • Proposition 15
  • ...and 28 more