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The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians. (English) Zbl 1523.14011

Summary: We compute the subgroup of the monodromy group of a generalized Kummer variety associated to equivalences of derived categories of abelian surfaces. The result was announced by the author and S. Mehrotra [Math. Nachr. 290, No. 5–6, 876–884 (2017; Zbl 1400.14096)]. G. Mongardi [Algebr. Geom. 3, No. 3, 385–391 (2016; Zbl 1373.14011)] showed that the subgroup constructed here is in fact the whole monodromy group. As an application we prove the Hodge conjecture for the generic abelian fourfold of Weil type with complex multiplication by an arbitrary imaginary quadratic number field \(K\), but with trivial discriminant invariant in \(\mathbb{Q}^\ast / \mathrm{Nm} ( K^\ast )\). The latter result is inspired by a recent observation of O’Grady that the third intermediate Jacobians of smooth projective varieties of generalized Kummer deformation type form complete families of abelian fourfolds of Weil type. Finally, we prove the surjectivity of the Abel-Jacobi map from the Chow group \(\mathrm{CH}^2 ( Y )_0\) of codimension 2 algebraic cycles homologous to zero on every projective irreducible holomorphic symplectic manifold \(Y\) of Kummer type onto the third intermediate Jacobian of \(Y\), as predicted by the generalized Hodge Conjecture.

MSC:

14C25 Algebraic cycles
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
14G05 Rational points
14J42 Holomorphic symplectic varieties, hyper-Kähler varieties

References:

[1] Acknowledgements. I am grateful to Kieran O’Grady for sharing with me an early draft of his insightful paper [43]; the insight it provided was crucial in the proof of Theorem 1.5. I thank Misha Verbitsky for reference [40].
[2] I am grateful to Claire Voisin for her suggestion of Theorem 1.10. I thank the referees for their insightful comments and suggestions. I thank Bert van Geemen for pointing out a typo in a previous version of equation (4.29).
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