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Perverse sheaves on semi-abelian varieties—a survey of properties and applications

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Abstract

We survey recent developments in the study of perverse sheaves on semi-abelian varieties. As concrete applications, we discuss various restrictions on the homotopy type of complex algebraic manifolds (expressed in terms of their cohomology jump loci), homological duality properties of complex algebraic manifolds, as well as new topological characterizations of semi-abelian varieties.

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Notes

  1. The statement remains true without any assumption on \(H_1(X,\mathbb {Z})\), but in this generality one needs to consider all characters of \(\pi _1(X)\), i.e., all elements of \(\mathrm{Hom}(\pi _1(X), \mathbb {C}^*)\), and the corresponding cohomology jump loci; see [31, Theorem 4.6] for details.

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Acknowledgements

We would like to thank the organizers of the conference Topology and Geometry: A conference in memory of Ştefan Papadima (Bucharest, Romania, May 2018), where part of this work was presented.

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Authors and Affiliations

  1. BCAM - Basque Center for Applied Mathematics, Mazarredo, 14, 48009, Bilbao, Basque Country, Spain

    Yongqiang Liu

  2. Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI, 53706-1388, USA

    Laurentiu Maxim & Botong Wang

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  1. Yongqiang Liu
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  2. Laurentiu Maxim
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Correspondence to Laurentiu Maxim.

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Dedicated to the memory of Professor Ştefan Papadima

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Y. Liu was supported by the ERCEA 615655 NMST Consolidator Grant and also by the Basque Government through the BERC 2018-2021 program and by the Spanish Ministry of Science, Innovation and Universities: BCAM Severo Ochoa accreditation SEV-2017-0718. L. Maxim was partially supported by the Simons Foundation Collaboration Grant # 567077 and by CNCS-UEFISCDI Grant PN-III-P4-ID-PCE-2016-0030. B. Wang was partially supported by the NSF Grant DMS-1701305.

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Liu, Y., Maxim, L. & Wang, B. Perverse sheaves on semi-abelian varieties—a survey of properties and applications. European Journal of Mathematics 6, 977–997 (2020). https://doi.org/10.1007/s40879-019-00340-9

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