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Autoequivalences of Derived Categories of Bielliptic Surfaces

Yuki Tochitani

Abstract

We determine the generators of the autoequivalence group of the derived category of coherent sheaves on a bielliptic surface over an algebraically closed field of arbitrary characteristic. As a consequence, we prove that any algebraic variety derived equivalent to such a surface is isomorphic to the surface itself.

Autoequivalences of Derived Categories of Bielliptic Surfaces

Abstract

We determine the generators of the autoequivalence group of the derived category of coherent sheaves on a bielliptic surface over an algebraically closed field of arbitrary characteristic. As a consequence, we prove that any algebraic variety derived equivalent to such a surface is isomorphic to the surface itself.

Paper Structure

This paper contains 15 sections, 30 theorems, 136 equations, 1 table.

Table of Contents

  1. Introduction
  2. Motivations and results
  3. Notation and conventions
  4. Acknowledgements.
  5. Preliminaries
  6. Bielliptic surfaces
  7. The Euler form on surfaces
  8. Action of autoequivalences on $\mathbb{Z} \oplus \mathop{\mathrm{Num}}\nolimits (X) \oplus \mathbb{Z}$
  9. Canonical covers
  10. Relative Fourier--Mukai transforms
  11. The proof of \ref{['MT1']}
  12. Auxiliary results
  13. The proof of \ref{['Prop:rank']}
  14. The proof of \ref{['Prop:frac']} and \ref{['Prop:sheaf']}
  15. Consequences of \ref{['section3']}

Key Result

Theorem 1.1

Let $X$ be a bielliptic surface over $k$ of arbitrary characteristic. Then we have a short exact sequence where $\pi$ is defined by $\pi (\Phi) = \Phi_{M}$ for $\Phi \in \mathop{\mathrm{Auteq}}\nolimits D(X)$.

Theorems & Definitions (60)

  • Theorem 1.1: =\ref{['ESC']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Remark 2.6
  • ...and 50 more