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Enriques surfaces with trivial Brauer map and involutions on hyperkähler manifolds

Fabian Reede

Abstract

Let $X$ be an Enriques surface. Using Beauville's result about the triviality of the Brauer map of $X$, we define a new involution on the category of coherent sheaves on the canonically covering K3 surface $\overline{X}$. We relate the fixed locus of this involution to certain Picard schemes of the noncommutative pair $(X,\mathcal{A})$, where $\mathcal{A}$ is an Azumaya algebra on $X$ defined by the nontrivial element in the Brauer group of $X$.

Enriques surfaces with trivial Brauer map and involutions on hyperkähler manifolds

Abstract

Let be an Enriques surface. Using Beauville's result about the triviality of the Brauer map of , we define a new involution on the category of coherent sheaves on the canonically covering K3 surface . We relate the fixed locus of this involution to certain Picard schemes of the noncommutative pair , where is an Azumaya algebra on defined by the nontrivial element in the Brauer group of .
Paper Structure (4 sections, 22 theorems, 82 equations)

This paper contains 4 sections, 22 theorems, 82 equations.

Table of Contents

  1. Enriques surfaces with trivial Brauer map
  2. Stable sheaves and involutions on hyperkähler manifolds
  3. Twisted Picard schemes: smooth cases
  4. Twisted Picard schemes: singular cases

Key Result

Theorem 1.1

Let $X$ be an Enriques surface. The Brauer map $q^{*}: \mathop{\mathrm{Br}}\nolimits(X)\rightarrow \mathop{\mathrm{Br}}\nolimits(\overline{X})$ is trivial if and only if there is $L=\mathop{\mathrm{\mathcal{O}}}\nolimits_{\overline{X}}(\ell)\in \mathop{\mathrm{Pic}}\nolimits(\overline{X})$ with $\io

Theorems & Definitions (47)

  • Theorem 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 1.5
  • proof
  • Corollary 1.6
  • Remark 1.7
  • Lemma 2.1
  • proof
  • ...and 37 more