Geodesic
On~correspondences between K3~surfaces
Izvestiya. Mathematics , Tome 30 (1988) no. 2, pp. 375-383.

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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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V. V. Nikulin. On~correspondences between K3~surfaces. Izvestiya. Mathematics , Tome 30 (1988) no. 2, pp. 375-383. http://geodesic.mathdoc.fr/item/IM2_1988_30_2_a9/

[1] Algebraicheskie poverkhnosti, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 75, 1965, 215 pp. | MR | Zbl

[2] Shafarevitch I. R., “Le theoreme de Torelli pour les surfaces algebraique de type K3”, Acta Congres int. Mathematiciens, 1970, 413–417 | MR

[3] Pyatetskii-Shapiro I. I., Shafarevich I. R., “Teorema Teorelli dlya algebraicheskikh poverkhnostei tipa K3”, Izv. AN SSSR. Ser. matem., 35:3 (1971), 530–572

[4] Mukai Sh., On the moduli of bundles on K3 surfaces, I, preprint, 1984 | MR

[5] Nikulin V. V., “Tselochislennye simmetricheskie bilineinye formy i nekotorye ikh geometricheskie prilozheniya”, Izv. AN SSSR. Ser. matem., 43:1 (1979), 111–177 | MR | Zbl

[6] Kassels Dzh., Ratsionalnye kvadratichnye formy, Mir, M., 1982 | MR

[7] Kneser M., “Kjassenzahlen indefiniter quadratischer Formen in drei oder mehr Veränderlichen”, Arch. Math. (Basel), 7 (1956), 323–332 | MR | Zbl