Abstract
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.
Similar content being viewed by others
References
Arinkin, D., Bezrukavnikov, R.: Perverse coherent sheaves. Mosc. Math. J. 10(1), 3–29, 271 (2010)
Baues, O., Cortés, V.: Aspherical Kähler manifolds with solvable fundamental group. Geom. Dedicata 122, 215–229 (2006)
Beilinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. Astérisque 100, Paris, Soc. Math. Fr. (1982)
Bhatt, B., Schnell, S., Scholze, P.: Vanishing theorems for perverse sheaves on abelian varieties revisited. Sel. Math. (N.S.) 24(1), 63–84 (2018)
Bieri, R., Eckmann, B.: Groups with homological duality generalizing Poincaré duality. Invent. Math. 20, 103–124 (1973)
Bruns, W., Gubeladze, J.: Polytopes, Rings, and K-Theory. Springer Monographs in Mathematics. Springer, Dordrecht (2009)
Budur, N., Wang, B.: Cohomology jump loci of differential graded Lie algebras. Compos. Math. 151(8), 1499–1528 (2015)
Budur, N., Wang, B.: Absolute sets and the decomposition theorem. Ann. Sci. École Norm. Sup. 4(53), 469–536 (2020)
de Cataldo, M.A.A., Migliorini, L.: The decomposition theorem, perverse sheaves and the topology of algebraic maps. Bull. Am. Math. Soc. (N.S.) 46(4), 535–633 (2009)
Denham, G., Suciu, A.: Local systems on arrangements of smooth, complex algebraic hypersurfaces. Forum Math. Sigma 6, e6 (2018)
Denham, G., Suciu, A., Yuzvinsky, S.: Abelian duality and propagation of resonance. Sel. Math. 23(4), 2331–2367 (2017)
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)
Franecki, J., Kapranov, M.: The Gauss map and a noncompact Riemann–Roch formula for constructible sheaves on semiabelian varieties. Duke Math. J. 104(1), 171–180 (2000)
Gabber, O., Loeser, F.: Faisceaux pervers \(\ell \)-adiques sur un tore. Duke Math. J. 83(3), 501–606 (1996)
Gelfand, S., MacPherson, R., Vilonen, K.: Perverse sheaves and quivers. Duke Math. J. 83(3), 621–643 (1996)
Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, no. 52. Springer, New York (1977)
Iitaka, S.: Logarithmic forms of algebraic varieties. J. Fac. Sci. Sect. Univ. Tokyo Math. 23(3), 525–544 (1976)
Kashiwara, M.: \(t\)-structures on the derived categories of holonomic \({\cal{D}}\)-modules and coherent \({\cal{O}}\)-modules. Mosc. Math. J. 4(4), 847–868, 981 (2004)
Kiehl, R., Weissauer, R.: Weil conjectures, perverse sheaves and l’adic Fourier transform. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 42. Springer, Berlin (2001)
Krämer, T.: Perverse sheaves on semiabelian varieties. Rend. Semin. Mat. Univ. Padova 132, 83–102 (2014)
Krämer, T., Weissauer, R.: Vanishing theorems for constructible sheaves on abelian varieties. J. Algebr. Geom. 24(3), 531–568 (2015)
Liu, Y., Maxim, L., Wang, B.: Generic vanishing for semi-abelian varieties and integral Alexander modules. Math. Z. 293(1–2), 629–645 (2019)
Liu, Y., Maxim, L., Wang, B.: Mellin transformation, propagation, and abelian duality spaces. Adv. Math. 335, 231–260 (2018)
Liu, Y., Maxim, L., Wang, B.: Perverse sheaves on semi-abelian varieties—a survey of properties and applications. Eur. J. Math. 6(3), 977–997 (2020)
Liu, Y., Maxim, L., Wang, B.: Aspherical manifolds, Mellin transformation and a question of Bobadilla-Kollár, arXiv:2006.09295
MacPherson, R.: Global questions in the topology of singular spaces. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), 213–235, PWN, Warsaw (1984)
MacPherson, R., Vilonen, K.: Elementary construction of perverse sheaves. Invent. Math. 84(2), 403–435 (1986)
Mochizuki, T.: Wild harmonic bundles and wild pure twistor D-modules. Astérisque No. 340, (2011)
Pareschi, G., Popa, M.: GV-sheaves, Fourier–Mukai transform, and generic vanishing. Am. J. Math. 133(1), 235–271 (2011)
Popa, M., Schnell, C.: Generic vanishing theory via mixed Hodge modules. Forum Math. Sigma 1 (2013), Paper No. e1
Saito, M.: Modules de Hodge polarisables. Publ. Res. Inst. Math. Sci. 24(6), 849–995 (1988)
Saito, M.: Mixed Hodge Modules. Publ. Res. Inst. Math. Sci. 26(2), 221–333 (1990)
Schnell, C.: Holonomic \({\cal{D}}\)-modules on abelian varieties. Publ. Math. Inst. Hautes Études Sci. 121, 1–55 (2015)
Schürmann, J.: Topology of Singular Spaces and Constructible Sheaves, Monografie Matematyczne 63. Birkhäuser Verlag, Basel (2003)
Wang, B.: Algebraic surfaces with zero-dimensional cohomology support locus. Taiwan. J. Math. 22(3), 607–614 (2018)
Weissauer, R.: Degenerate Perverse Sheaves on Abelian Varieties, arXiv:1204.2247
Weissauer, R.: Vanishing theorems for constructible sheaves on abelian varieties over finite fields. Math. Ann. 365(1–2), 559–578 (2016)
Weissauer, R.: Remarks on the nonvanishing of cohomology groups for perverse sheaves on abelian varieties, arXiv:1612.01500
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Accepted:
Published:
DOI: https://doi.org/10.1007/s00029-021-00635-4
Keywords
- Semi-abelian variety
- Perverse sheaf
- Mellin transformation
- Cohomology jump loci
- Albanese map
- Generic vanishing
- Abelian duality space