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Perverse sheaves on semi-abelian varieties

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We give a complete (global) characterization of \({\mathbb {C}}\)-perverse sheaves on semi-abelian varieties in terms of their cohomology jump loci. Our results generalize Schnell’s work on perverse sheaves on complex abelian varieties, as well as Gabber–Loeser’s results on perverse sheaves on complex affine tori. We apply our results to the study of cohomology jump loci of smooth quasi-projective varieties, to the topology of the Albanese map, and in the context of homological duality properties of complex algebraic varieties.

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Acknowledgements

We would like to thank Jörg Schürmann, Dima Arinkin and Nero Budur for valuable discussions. We are also grateful to Rainer Weissauer and Thomas Krämer for bringing related works and questions to our attention. The first author thanks the Mathematics Department at the University of Wisconsin-Madison for hospitality during the preparation of this work. The first author is partially supported by the starting Grant KY2340000123 from University of Science and Technology of China, the project “Analysis and Geometry on Bundles” of Ministry of Science and Technology of the People’s Republic of China and Nero Budur’s research project G0B2115N from the Research Foundation of Flanders. The second author is partially supported by the Simons Foundation (Collaboration Grant #567077) and by the Romanian Ministry of National Education (CNCS-UEFISCDI Grants PN-III-P4-ID-PCE-2016-0030 and PN-III-P4-ID-PCE-2020-0029). The third author is partially supported by the NSF Grant DMS-1701305 and a Sloan Fellowship.

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Liu, Y., Maxim, L. & Wang, B. Perverse sheaves on semi-abelian varieties. Sel. Math. New Ser. 27, 30 (2021). https://doi.org/10.1007/s00029-021-00635-4

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