Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the birational isotriviality of the Albanese morphism of a log Calabi-Yau pair with a torus action

Linus Rösler

Abstract

Let $(X,Δ)$ be a projective, log canonical, $K$-trivial pair over the complex numbers. Let $Z$ be a minimal log canonical center of $(X,Δ)$ and suppose that there exists a torus $\mathbb{T}\subseteq\operatorname{Aut}(X)$ preserving $Δ$ and such that $\dim\mathbb{T}=\operatorname{codim}_X Z$. Then we show that two general fibers of the Albanese morphism $\operatorname{alb}_X$ are birationally equivalent. In particular, the pathological example of a projective, log canonical, $K$-trivial variety whose Albanese morphism is not generically birationally isotrivial, recently constructed by Bernasconi, Filipazzi, Patakfalvi and Tsakanikas, can be avoided under the additional hypothesis that there exists a torus of large enough dimension in the automorphism group of the given pair.

On the birational isotriviality of the Albanese morphism of a log Calabi-Yau pair with a torus action

Abstract

Let be a projective, log canonical, -trivial pair over the complex numbers. Let be a minimal log canonical center of and suppose that there exists a torus preserving and such that . Then we show that two general fibers of the Albanese morphism are birationally equivalent. In particular, the pathological example of a projective, log canonical, -trivial variety whose Albanese morphism is not generically birationally isotrivial, recently constructed by Bernasconi, Filipazzi, Patakfalvi and Tsakanikas, can be avoided under the additional hypothesis that there exists a torus of large enough dimension in the automorphism group of the given pair.
Paper Structure (12 sections, 16 theorems, 42 equations)

This paper contains 12 sections, 16 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Strategy of proof
  3. Notation
  4. Acknowledgements
  5. Preliminaries
  6. Torus actions
  7. Minimal model program
  8. MMP and torus actions
  9. Albanese morphism of open varieties
  10. The Biał ynicki--Birula decomposition
  11. Proof of main theorem
  12. Examples

Key Result

Theorem 1

Let $(X,\Delta)$ be a log Calabi--Yau pair and let $Z$ be a minimal log canonical center of $(X,\Delta)$. Assume that there exists a torus $\mathbb{T}\subseteq\mathop{\mathrm{Aut}}\nolimits(X)$ of dimension $\mathop{\mathrm{codim}}\nolimits_X Z$ that preserves $\Delta$. Then the Albanese morphism $\

Theorems & Definitions (50)

  • Theorem 1
  • Remark 1
  • Definition 1
  • Remark 1
  • Definition 2
  • Lemma 1
  • proof
  • Definition 3: Fogarty_Fixed_point_schemes
  • Lemma 2
  • proof
  • ...and 40 more