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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.

MSC:
 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
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References:
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