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 Tr. Mat. Inst. Steklova, 2011, Volume 273, Pages 247–256 (Mi tm3279)

Self-correspondences of K3 surfaces via moduli of sheaves and arithmetic hyperbolic reflection groups

Viacheslav V. Nikulinab

a Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia
b Department of Pure Mathematics, University of Liverpool, Liverpool, UK

Abstract: In a series of our papers with Carlo Madonna (2002–2008), we described self-correspondences of a K3 surface over $\mathbb C$ via moduli of sheaves with primitive isotropic Mukai vectors for the Picard number 1 or 2 of the K3 surfaces. Here we give a natural and functorial answer to the same problem for an arbitrary Picard number. As an application, we characterize, in terms of self-correspondences via moduli of sheaves, K3 surfaces with reflective Picard lattice, that is, when the automorphism group of the lattice is generated by reflections up to finite index. It is known since 1981 that the number of reflective hyperbolic lattices is finite. We also formulate some natural unsolved related problems.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2011, 273, 229–237

Bibliographic databases:

Document Type: Article
UDC: 512.724+512.817.6

Citation: Viacheslav V. Nikulin, “Self-correspondences of K3 surfaces via moduli of sheaves and arithmetic hyperbolic reflection groups”, Modern problems of mathematics, Collected papers. In honor of the 75th anniversary of the Institute, Tr. Mat. Inst. Steklova, 273, MAIK Nauka/Interperiodica, Moscow, 2011, 247–256; Proc. Steklov Inst. Math., 273 (2011), 229–237

Citation in format AMSBIB
\Bibitem{Nik11} \by Viacheslav~V.~Nikulin \paper Self-correspondences of K3 surfaces via moduli of sheaves and arithmetic hyperbolic reflection groups \inbook Modern problems of mathematics \bookinfo Collected papers. In honor of the 75th anniversary of the Institute \serial Tr. Mat. Inst. Steklova \yr 2011 \vol 273 \pages 247--256 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3279} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2893549} \zmath{https://zbmath.org/?q=an:1226.14053} \elib{http://elibrary.ru/item.asp?id=16456349} \transl \jour Proc. Steklov Inst. Math. \yr 2011 \vol 273 \pages 229--237 \crossref{https://doi.org/10.1134/S0081543811040110} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000295982500011} 

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This publication is cited in the following articles:
1. Fuchs E., Meiri Ch., Sarnak P., “Hyperbolic Monodromy Groups For the Hypergeometric Equation and Cartan Involutions”, J. Eur. Math. Soc., 16:8 (2014), 1617–1671
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