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Action of correspondences on filtrations on cohomology and 0-cycles of Abelian varieties

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Abstract

We prove that, given a symmetrically distinguished correspondence of a suitable complex abelian variety (which includes any abelian variety of dimension at most 5, powers of complex elliptic curves, etc.) that vanishes as a morphism on a certain quotient of its middle singular cohomology, then it vanishes as a morphism on the deepest part of a particular filtration on the Chow group of 0-cycles of the abelian variety. As a consequence, we prove that an automorphism of such an abelian variety that acts as the identity on a certain quotient of its middle singular cohomology acts as the identity on the deepest part of this filtration on the Chow group of 0-cycles of the abelian variety. As an application, we prove that for the generalized Kummer variety associated to a complex abelian surface and the automorphism induced from a symplectic automorphism of the complex abelian surface, the automorphism of the generalized Kummer variety acts as the identity on a certain subgroup of its Chow group of 0-cycles.

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Acknowledgements

The author would like to express gratitude to his Ph.D. thesis advisor Prof. V. Srinivas for introducing him to the subject and suggesting the problem as well as constant guidance and encouragement that led to this paper. The author would also like to thank Prof. N. Fakhruddin for bringing [26] to his attention and subsequent suggestions during the work. The author would like to thank both for pointing out errors in the earlier version, and suggestions which immensely improved the exposition of the paper. The author would like to thank D. Huybrechts, H.-Y. Lin for pointing out an error in the earlier version. The author also thanks the referee for helpful suggestions and corrections.

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  1. Fakultät für Mathematik, Universität Duisburg-Essen, 45127, Essen, Germany

    Rakesh R. Pawar

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  1. Rakesh R. Pawar
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Correspondence to Rakesh R. Pawar.

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The author was supported by Tata Institute of Fundamental Research, Mumbai and “Dr. Shyama Prasad Mukherjee (SPM) fellowship” funded by Council of Scientific & Industrial Research, Government of India (SPM-07/858(0139)/2012).

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Pawar, R.R. Action of correspondences on filtrations on cohomology and 0-cycles of Abelian varieties. Math. Z. 292, 655–675 (2019). https://doi.org/10.1007/s00209-018-2151-9

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