A refined derived Torelli theorem for Enriques surfaces
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Mathematische Annalen
A refined derived Torelli theorem for Enriques surfaces Chunyi Li1 · Howard Nuer2 · Paolo Stellari3
· Xiaolei Zhao4
Received: 29 December 2019 / Revised: 20 October 2020 / Accepted: 26 October 2020 © The Author(s) 2020
Abstract We prove that two general Enriques surfaces defined over an algebraically closed field of characteristic different from 2 are isomorphic if their Kuznetsov components are equivalent. We apply the same techniques to give a new simple proof of a conjecture by Ingalls and Kuznetsov relating the derived categories of the blow-up of general Artin–Mumford quartic double solids and of the associated Enriques surfaces. Mathematics Subject Classification 18E30 · 14J28 · 14F05
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 2 Semiorthogonal decompositions and an extension result 2.1 Generalities . . . . . . . . . . . . . . . . . . . . . 2.2 Extending Fourier–Mukai functors . . . . . . . . . 2.3 A side remark: a dg category approach . . . . . . . 3 The geometric setting . . . . . . . . . . . . . . . . . . 3.1 The case of Enriques surfaces . . . . . . . . . . . . 3.2 Artin–Mumford quartic double solids . . . . . . . . 4 Spherical objects in Enriques categories . . . . . . . . . 5 Proof of the main results . . . . . . . . . . . . . . . . . 5.1 A general extension result . . . . . . . . . . . . . .
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Communicated by Vasudevan Srinivas. CL was an Early Career Fellow supported by the Leverhulme Trust, and a University Research Fellow supported by the Royal Society. HN was partially supported by the NSF postdoctoral fellowship DMS-1606283, by the NSF RTG Grant DMS-1246844, and by the NSF FRG Grant DMS-1664215. PS was partially supported by the ERC Consolidator Grant ERC-2017-CoG-771507-StabCondEn, by the research projects PRIN 2017 “Moduli and Lie Theory” and FARE 2018 HighCaSt (Grant number R18YA3ESPJ), and by the International Associated Laboratory LIA LYSM (in cooperation with AMU, CNRS, ECM and INdAM). XZ was partially supported by the Simons Collaborative Grant 636187.
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Paolo Stellari [email protected] https://sites.unimi.it/stellari
Extended author information available on the last page of the article
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C. Li et al. 5.2 Proof of Theorems A and B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction An Enriques surface is a smooth projective surface X with
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