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Néron models, minimal models, and birational group actions

János Kollár

TL;DR

The paper develops a framework for understanding how the Néron model of an Abelian variety interacts with minimal and log canonical models over a 1-dimensional base. It proves a general regularity criterion that extends birational group actions from the generic fiber to the total space, and uses it to show that the identity component of the Néron model acts regularly on minimal models under suitable hypotheses. A rigidity result for small modifications and the construction of equivariant completions in characteristic 0 play central roles, linking degeneration theory to birational geometry. The results illuminate when the Néron model sits as an open subset of a minimal model, relate semiabelian reduction to the structure of the central fiber, and extend symmetry considerations to log canonical models, with concrete examples illustrating both capabilities and limitations.

Abstract

Let $A_K$ be an Abelian variety over the quotient field of a Dedekind domain $R$. We show that the identity component of the Néron model of $A_K$ acts regularly on any minimal model of $A_K$ over $R$, and discuss when the Néron model is an open subset of a minimal model. The main technical result is the rigidity of small modifications, leading to a regularity criterion for birational group actions.

Néron models, minimal models, and birational group actions

TL;DR

The paper develops a framework for understanding how the Néron model of an Abelian variety interacts with minimal and log canonical models over a 1-dimensional base. It proves a general regularity criterion that extends birational group actions from the generic fiber to the total space, and uses it to show that the identity component of the Néron model acts regularly on minimal models under suitable hypotheses. A rigidity result for small modifications and the construction of equivariant completions in characteristic 0 play central roles, linking degeneration theory to birational geometry. The results illuminate when the Néron model sits as an open subset of a minimal model, relate semiabelian reduction to the structure of the central fiber, and extend symmetry considerations to log canonical models, with concrete examples illustrating both capabilities and limitations.

Abstract

Let be an Abelian variety over the quotient field of a Dedekind domain . We show that the identity component of the Néron model of acts regularly on any minimal model of over , and discuss when the Néron model is an open subset of a minimal model. The main technical result is the rigidity of small modifications, leading to a regularity criterion for birational group actions.

Paper Structure

This paper contains 7 sections, 24 theorems, 23 equations.

Table of Contents

  1. Main technical theorems
  2. Néron models
  3. Minimal models and Néron models
  4. Small modifications
  5. Equivariant completions
  6. Semiabelian reduction
  7. Log canonical models

Key Result

Theorem 1

Let $S$ be a 1-dimensional, irreducible, regular scheme with function field $K$, and $A_K$ an Abelian variety over $K$. Assume that $A_K$ has a projective, minimal model $\pi: A_S\to S$. Let $\operatorname{N\acute{e}r}^{\circ}(A_K)\subset \operatorname{N\acute{e}r}(A_K)$ be the identity component of ). Then the $A_K$-action on itself extends to a regular $\operatorname{N\acute{e}r}^{\circ}(A_K)$-a

Theorems & Definitions (35)

  • Theorem 1
  • Corollary 2
  • Corollary 4
  • Corollary 5
  • Theorem 7
  • Theorem 8
  • Example 10: Flopping a fiber of a product
  • Definition 12
  • Theorem 13: Néron models
  • Definition 14
  • ...and 25 more