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Abelian varieties associated to certain K3 surfaces. (English) Zbl 0715.14037
Let Y be a K3 surface obtained by resolution of singularities of the two- sheeted covering of \({\mathbb{P}}^ 2_{{\mathbb{C}}}\) branched along six lines. The Schoen construction allows one to describe Y on the other hand as the quotient (by a finite group) of the symmetric square of a curve C of genus 5 having an automorphism J of order 4 such that \(C/J=E\) is an elliptic curve. One proves the
Theorem: Let P be the Prym variety of the four-sheeted covering \(C\to C/J=E\), i.e., the connected component of the unity in ker(Jac(C)\(\to Jac(E))\) \((\dim (P)=4)\). Then there exists an algebraic cycle \(\gamma \in CH^ 2(P\times Y)\) such that the corresponding homomorphism \(H^ 2(Y,{\mathbb{Q}})\to H^ 2(P,{\mathbb{Q}})\) is a monomorphism on the lattice of transcendental cycles.
The algebraic correspondence \(\gamma\) is the Kuga-Satake-Deligne correspondence for K3 surfaces of the considered type up to isogenies and copies of P as a direct summand.

14K30 Picard schemes, higher Jacobians
14J28 \(K3\) surfaces and Enriques surfaces
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14H52 Elliptic curves
14C25 Algebraic cycles
Full Text: Numdam EuDML
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