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Vanishing theorems for perverse sheaves on abelian varieties, revisited

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We revisit some of the basic results of generic vanishing theory, as pioneered by Green and Lazarsfeld, in the context of constructible sheaves. Using the language of perverse sheaves, we give new proofs of some of the basic results of this theory. Our approach is topological/arithmetic, and avoids Hodge theory.

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References

  1. Beĭlinson, A.A., Bernstein, J., Deligne, P.: Faisceaux Pervers. Analysis and Topology on Singular Spaces. I (Luminy, 1981). Astérisque, pp. 5–71. Soc. Math. France, Paris (1982)

    Google Scholar 

  2. Bernšteĭn, I.N., Gel’fand, I.M., Gel’fand, S.I.: Algebraic vector bundles on \({ P}^{n}\) and problems of linear algebra. Funktsional. Anal. i Prilozhen. 12(3), 66–67 (1978)

    MathSciNet  Google Scholar 

  3. Böckle, G., Khare, C.: Mod \(l\) representations of arithmetic fundamental groups. II. A conjecture of A. J. de Jong. Compos. Math. 142(2), 271–294 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Deligne, P.: La conjecture de Weil. II. Inst. Hautes Études Sci. Publ. Math. 52, 137–252 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dwyer, W.G., Greenlees, J.P.C.: Complete modules and torsion modules. Am. J. Math. 124(1), 199–220 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Deligne, P., Griffiths, P., Morgan, J., Sullivan, D.: Real homotopy theory of Kähler manifolds. Invent. Math. 29(3), 245–274 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dimca, A.: Sheaves in Topology. Universitext. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  8. de Jong, A.J.: A conjecture on arithmetic fundamental groups. Isr. J. Math. 121, 61–84 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Drinfeld, V.: On a conjecture of Kashiwara. Math. Res. Lett. 8(5–6), 713–728 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gaitsgory, D.: On de Jong’s conjecture. Isr. J. Math. 157, 155–191 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Green, M., Lazarsfeld, R.: Deformation theory, generic vanishing theorems, and some conjectures of Enriques. Catanese and Beauville. Invent. Math. 90(2), 389–407 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  12. Green, M., Lazarsfeld, R.: Higher obstructions to deforming cohomology groups of line bundles. J. Am. Math. Soc. 4(1), 87–103 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gabber, O., Loeser, F.: Faisceaux pervers \(l\)-adiques sur un tore. Duke Math. J. 83(3), 501–606 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Groth, M.: A short course on \(\infty \)-categories. https://arxiv.org/abs/1007.2925

  15. Hotta, R., Kiyoshi, T., Toshiyuki, T.: \(D\)-Modules, Perverse Sheaves, and Representation Theory (Translated from the 1995 Japanese Edition by Takeuchi). Progress in Mathematics, vol. 236. Birkhäuser Boston Inc., Boston (2008)

    MATH  Google Scholar 

  16. Kashiwara, M.: Semisimple holonomic \(D\)-modules. In: Kashiwara, A., Matsuo, A., Saito, K., Satake, I. (eds.) Topological Field Theory, Primitive Forms and Related Topics (Kyoto, 1996), Progress in Mathematics, vol. 160, pp. 267–271. Birkhäuser Boston, Boston (1998)

  17. 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)

  18. Kashiwara, M., Schapira, P.: Sheaves on Manifolds (With a Chapter in French by Christian Houzel). rundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292. Springer, Berlin (1990)

    MATH  Google Scholar 

  19. Krämer, T., Weissauer, R.: Vanishing theorems for constructible sheaves on abelian varieties. J. Algebraic Geom. 24(3), 531–568 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lafforgue, L.: Chtoucas de Drinfeld et correspondance de Langlands. Invent. Math. 147(1), 1–241 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lurie, J.: Higher algebra. http://www.math.harvard.edu/~lurie/. Accessed 2 Dec 2017

  22. Lurie, J.: Higher Topos Theory. Annals of Mathematics Studies, vol. 170. Princeton University Press, Princeton (2009)

    MATH  Google Scholar 

  23. Mochizuki, T.: Wild harmonic bundles and wild pure twistor \(D\)-modules. Astérisque 340, x+607 (2011)

    MathSciNet  MATH  Google Scholar 

  24. Popa, M., Schnell, C.: Generic vanishing theory via mixed Hodge modules. Forum Math. Sigma 1, e1, 60 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sabbah, C.: Polarizable twistor \(D\)-modules. Astérisque 300, vi+208 (2005)

    MathSciNet  MATH  Google Scholar 

  26. Saito, M.: Modules de Hodge polarisables. Publ. Res. Inst. Math. Sci. 24(6), 849–995 (1989), 1988

  27. Schnell, C.: Holonomic D-modules on abelian varieties. Inst. Hautes. Études Sci. Publ. Math. 121(1), 1–55 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Scholze, P.: On torsion in the cohomology of locally symmetric varieties. Ann. Math. (2) 182(3), 945–1066 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Simpson, C.: Subspaces of moduli spaces of rank one local systems. Ann. Sci. École Norm. Sup. (4) 26(3), 361–401 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  30. The Stacks Project Authors: Stacks project. http://stacks.math.columbia.edu (2016). Accessed 2 Dec 2017

  31. Weissauer, R.: Vanishing theorems for constructible sheaves on abelian varieties over finite fields. Math. Ann. 365(1–2), 559–578 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Bhargav Bhatt.

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Bhatt, B., Schnell, C. & Scholze, P. Vanishing theorems for perverse sheaves on abelian varieties, revisited. Sel. Math. New Ser. 24, 63–84 (2018). https://doi.org/10.1007/s00029-017-0377-8

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