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 Izv. Akad. Nauk SSSR Ser. Mat., 1987, Volume 51, Issue 2, Pages 402–411 (Mi izv1300)

On correspondences between K3 surfaces

V. V. Nikulin

Abstract: Using arithmetic of integral quadratic forms and results of Mukai, it is proved among other things that an endomorphism over $\mathbf Q$ of the cohomology lattice of a $K3$ surface over $\mathbf C$ preserving the Hodge structure and the intersection form is induced by an algebraic cycle (as was conjectured in [2]) provided that the Picard lattice $S_X$ of the surface $X$ represents zero (in particular, this is so if $\operatorname{rg}S_X\geqslant5$). Previously this result was obtained by Mukai under the assumption that $\operatorname{rg}S_X\geqslant11$.
Bibliography: 7 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1988, 30:2, 375–383

Bibliographic databases:

UDC: 512.774+512.734+511.334
MSC: Primary 14J28; Secondary 14C30, 11E12

Citation: V. V. Nikulin, “On correspondences between K3 surfaces”, Izv. Akad. Nauk SSSR Ser. Mat., 51:2 (1987), 402–411; Math. USSR-Izv., 30:2 (1988), 375–383

Citation in format AMSBIB
\Bibitem{Nik87} \by V.~V.~Nikulin \paper On~correspondences between K3~surfaces \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1987 \vol 51 \issue 2 \pages 402--411 \mathnet{http://mi.mathnet.ru/izv1300} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=897004} \zmath{https://zbmath.org/?q=an:0653.14007} \transl \jour Math. USSR-Izv. \yr 1988 \vol 30 \issue 2 \pages 375--383 \crossref{https://doi.org/10.1070/IM1988v030n02ABEH001018} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. C. G. Madonna, V. V. Nikulin, “On a Classical Correspondence between K3 Surfaces”, Proc. Steklov Inst. Math., 241 (2003), 120–153
2. V. V. Nikulin, “On Correspondences of a K3 Surface with Itself. I”, Proc. Steklov Inst. Math., 246 (2004), 204–226
3. C. G. Madonna, V. V. Nikulin, “Explicit correspondences of a K3 surface with itself”, Izv. Math., 72:3 (2008), 497–508
4. Claudio Pedrini, “The Chow Motive of a K3 Surface”, Milan j math, 2009
5. Ulrich Schlickewei, “The Hodge conjecture for self-products of certain K3 surfaces”, Journal of Algebra, 324:3 (2010), 507
6. Viacheslav V. Nikulin, “Self-correspondences of K3 surfaces via moduli of sheaves and arithmetic hyperbolic reflection groups”, Proc. Steklov Inst. Math., 273 (2011), 229–237
7. Claudio Pedrini, “On the finite dimensionality of a K3 surface”, manuscripta math, 2011
8. Christian Liedtke, “Supersingular K3 surfaces are unirational”, Invent. math, 2014
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