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On Abel–Jacobi maps of Lagrangian families

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Abstract

We study in this article the cohomological properties of Lagrangian families on projective hyper-Kähler manifolds. First, we give a criterion for the vanishing of Abel–Jacobi maps of Lagrangian families. Using this criterion, we show that under a natural condition, if the variation of Hodge structures on the degree 1 cohomology of the fibers of the Lagrangian family is maximal, its Abel–Jacobi map is trivial. We also construct Lagrangian families on generalized Kummer varieties whose Abel–Jacobi map is not trivial, showing that our criterion is optimal.

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Acknowledgements

I would like to thank Claire Voisin for sharing her ideas with me. This paper would not be finished without her insight, instructions and encouragement. Many constructions and examples in the article are fruits of friendly and interesting discussions with her. I would also like to thank Fabrizio Anella, Peter Yi Wei, Mauro Varesco and Tim Ryan for helpful discussions. I thank the referee for his/her wonderful refereeing work and for numerous constructive comments and suggestions. This work was done during the preparation of my PhD thesis. I would like to thank the Institut de Mathématiques de Jussieu - Paris Rive Gauche for marvelous research environment. The thesis is supported by the ERC Synergy Grant HyperK (Grant agreement No. 854361).

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  1. Sorbonne Université and Université de Paris, CNRS, IMJ-PRG, 75005, Paris, France

    Chenyu Bai

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  1. Chenyu Bai
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Bai, C. On Abel–Jacobi maps of Lagrangian families. Math. Z. 304, 34 (2023). https://doi.org/10.1007/s00209-023-03292-9

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