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Real multiplication on \(K3\) surfaces and Kuga-Satake varieties. (English) Zbl 1161.14027

A \(K3\)-type Hodge structure is a simple, rational, polarized weight 2 Hodge structure \(V\) with \(\dim V^{2,0}=1\). Yu. G. Zarhin [J. Reine Angew. Math. 341, 193–220 (1983; Zbl 0506.14034)] proved that the endomorphism algebra \(F\) of a \(K3\)-type Hodge structure is either a totally real field or a CM field. The author concentrates on the case that \(F\) is totally real. The CM case is already covered in B. van Geemen [J. Math. Soc. Japan 53, No. 4, 813–833 (2001; Zbl 1074.14509)].
The first result is an existence result: Given a totally real field \(F\) and an integer \(m>2\) then there exist a \(K3\)-type Hodge structure \(V\) with endomorphism algebra \(F\) and \(\dim_F V=m\).
Secondly, the author studies the following construction: Given a \(K3\)-type Hodge structure \(V\) with a totally positive endomorphism, the author constructs a new polarization on \(V\). (This construction is similar to a well-known construction for weight 1 Hodge structures coming from abelian Varieties.) Assume that the \(K3\)-type Hodge structure \(V\) is a Hodge substructure of \(H^2(S)\) for some smooth \(K3\) surface \(S\), where the polarization is induced by the cupproduct. The author shows that under a condition on \(\dim V\) the newly obtained Hodge structure is a Hodge substructure of a \(K3\) surface \(H^2(S_a)\). I.e., the Hodge structures are isomorphic, but with different polarizations.
Finally, the author considers the Kuga-Satake variety of \(K3\)-type Hodge structures with real multiplication. The author relates the Kuga-Sutake construction to the corestriction of a Clifford algebra. This generalizes work by Mumford and F. Galluzzi [Indag. Math., New Ser. 11, No. 4, 547–560 (2000; Zbl 1034.14018)].

MSC:

14J28 \(K3\) surfaces and Enriques surfaces
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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References:

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