Abstract
We revisit generic vanishing results for perverse sheaves with any field coefficients on a complex semi-abelian variety, and indicate several topological applications. In particular, we obtain finiteness properties for the integral Alexander modules of complex algebraic varieties mapping to semi-abelian varieties. Similar results were recently derived by the authors by using Morse-theoretic arguments.
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Bhatt, B., Schnell, S., Scholze, P.: Vanishing theorems for perverse sheaves on abelian varieties, revisited. Sel. Math. (N.S.) 24(1), 63–84 (2018)
Budur, N., Liu, Y., Wang, B.: The monodromy theorem for compact Kähler manifolds and smooth quasi-projective varieties. Math. Ann. 371(3–4), 1069–1086 (2018)
Budur, N., Wang, B.: Cohomology jump loci of quasi-projective varieties. Ann. Sci. Éc Norm. Supér. (4) 48(1), 227–236 (2015)
Budur, N., Wang, B.: Cohomology jump loci of differential graded Lie algebras. Compos. Math. 151(8), 1499–1528 (2015)
Dimca, A.: Sheaves in Topology. Universitext, Springer, Berlin (2004)
Eisenbud, D.: Commutative Algebra. With a View Toward Algebraic Geometry. Graduate Texts in Mathematics, 150. Springer, New York (1995)
Gabber, O., Loeser, F.: Faisceaux pervers \(\ell \)-adiques sur un tore. Duke Math. J. 83(3), 501–606 (1996)
Iitaka, S.: Logarithmic forms of algebraic varieties. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23(3), 525–544 (1976)
Katz, N., Laumon, G.: Transformation de Fourier et majoration de sommes exponentielles. Inst. Hautes Etudes Sci. Publ. Math. 62, 361–418 (1985)
Krämer, T.: Perverse sheaves on semiabelian varieties. Rend. Semin. Mat. Univ. Padova 132, 83–102 (2014)
Lazarsfeld, R.: Positivity in algebraic geometry. I. Classical setting: line bundles and linear series. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], pp. 48. Springer, Berlin (2004)
Liu, Y., Maxim, L.: Characteristic varieties of hypersurface complements. Adv. Math. 306, 451–493 (2017)
Liu, Y., Maxim, L., Wang, B.: Topology of subvarieties of complex semi-abelian varieties. arXiv:1706.07491
Milnor, J.: Infinite cyclic coverings. In: 1968 Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967), pp. 115–133, Prindle, Weber and Schmidt, Boston, Mass (1967)
Papadima, S., Suciu, A.: Bieri-Neumann-Strebel-Renz invariants and homology jumping loci. Proc. Lond. Math. Soc. (3) 100(3), 795–834 (2010)
Schürmann, J.: Topology of Singular Spaces and Constructible Sheaves, Monografie Matematyczne 63. Birkhäuser Verlag, Basel (2003)
Shafarevich, I.R.: Basic algebraic geometry. 2. Schemes and complex manifolds, 2nd edn. Springer, Berlin (1994)
Wang, B.: Algebraic surfaces with zero-dimensional cohomology support locus. Taiwan. J. Math. 22(3), 607–614 (2018)
Acknowledgements
We are grateful to Zhixian Zhu for useful discussions. The authors thank the Mathematics Departments at East China Normal University (Shanghai, China) and University of Science and Technology of China (Hefei, China) for hospitality during the preparation of this work. The first author is partially supported by 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, grant PN-III-P4-ID-PCE-2016-0030. The third author is partially supported by NSF grant DMS-1701305.
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Liu, Y., Maxim, L. & Wang, B. Generic vanishing for semi-abelian varieties and integral Alexander modules. Math. Z. 293, 629–645 (2019). https://doi.org/10.1007/s00209-018-2194-y
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DOI: https://doi.org/10.1007/s00209-018-2194-y