Vanishing theorems for constructible sheaves on abelian varieties
Authors:
Thomas Krämer and Rainer Weissauer
Journal:
J. Algebraic Geom. 24 (2015), 531-568
DOI:
https://doi.org/10.1090/jag/645
Published electronically:
April 20, 2015
MathSciNet review:
3344764
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Abstract |
References |
Additional Information
Abstract: We show that the hypercohomology of most character twists of perverse sheaves on a complex abelian variety vanishes in all non-zero degrees. As a consequence we obtain a vanishing theorem for constructible sheaves and a relative vanishing theorem for a homomorphism between abelian varieties. Our proof relies on a Tannakian description for convolution products of perverse sheaves, and with future applications in mind we discuss the basic properties of the arising Tannaka groups.
References
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- Thomas Krämer and Rainer Weissauer, On the Tannaka group attached to the theta divisor of a generic principally polarized abelian variety, Math. Z. (to appear).
- Laurent Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002), no. 1, 1–241 (French, with English and French summaries). MR 1875184, DOI https://doi.org/10.1007/s002220100174
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- James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
- Takuro Mochizuki, Asymptotic behaviour of tame harmonic bundles and an application to pure twistor $D$-modules. I, Mem. Amer. Math. Soc. 185 (2007), no. 869, xii+324. MR 2281877, DOI https://doi.org/10.1090/memo/0869
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- Claude Sabbah, Polarizable twistor $\scr D$-modules, Astérisque 300 (2005), vi+208 (English, with English and French summaries). MR 2156523
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- Christian Schnell, Holonomic D-modules on abelian varieties, Inst. Hautes Études Sci. Publ. Math. (to appear).
- Jean-Pierre Serre, Sur la topologie des variétés algébriques en caractéristique $p$, Symposium internacional de topología algebraica International symposium on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 24–53 (French). MR 0098097
- Jean-Pierre Serre, Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, Springer-Verlag, New York, 1988. Translated from the French. MR 918564
- Rainer Weissauer, Brill-Noether sheaves. arXiv:math/0610923.
- Rainer Weissauer, Tannakian categories attached to abelian varieties, Modular forms on Schiermonnikoog, Cambridge Univ. Press, Cambridge, 2008, pp. 267–274. MR 2512364, DOI https://doi.org/10.1017/CBO9780511543371.014
- Rainer Weissauer, Semisimple algebraic tensor categories. arXiv:0909.1793.
- Rainer Weissauer, A remark on rigidity of BN-sheaves. arXiv:1111.6095.
- Rainer Weissauer, Degenerate perverse sheaves on abelian varieties. arXiv:1204.2247.
- Rainer Weissauer, Why certain Tannaka groups attached to abelian varieties are almost connected. arXiv:1207.4039.
- Rainer Weissauer, Vanishing theorems for constructible sheaves on abelian varieties over finite fields. arXiv:1407.0844.
References
- Yves André and Bruno Kahn, Nilpotence, radicaux et structures monoïdales, Rend. Sem. Mat. Univ. Padova 108 (2002), 107–291 (French, with English summary). With an appendix by Peter O’Sullivan. MR 1956434 (2003m:18009)
- Yves André and Bruno Kahn, Erratum: “Nilpotency, radicals and monoidal structures” [Rend. Sem. Mat. Univ. Padova 108 (2002), 107–291; MR1956434], Rend. Sem. Mat. Univ. Padova 113 (2005), 125–128 (French). MR 2168984 (2006g:18008)
- A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966 (86g:32015)
- Edward Bierstone and Pierre D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), no. 2, 207–302. MR 1440306 (98e:14010), DOI https://doi.org/10.1007/s002220050141
- Jürgen Bingener and Hubert Flenner, Constructible and quasiconstructible sheaves on analytic spaces, Abh. Math. Sem. Univ. Hamburg 54 (1984), 119–139. MR 780245 (87f:32025), DOI https://doi.org/10.1007/BF02941447
- Gebhard Böckle and Chandrashekhar Khare, Mod $l$ representations of arithmetic fundamental groups. II. A conjecture of A. J. de Jong, Compos. Math. 142 (2006), no. 2, 271–294. MR 2218896 (2007a:11072), DOI https://doi.org/10.1112/S0010437X05002022
- Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252 (French). MR 601520 (83c:14017)
- The Grothendieck Festschrift. Vol. II, A collection of articles written in honor of the 60th birthday of Alexander Grothendieck, edited by P. Cartier, L. Illusie, N. M. Katz, G. Laumon, Y. Manin and K. A. Ribet. Progress in Mathematics, vol. 87, Birkhäuser Boston Inc., Boston, MA, 1990. MR 1106895 (92b:00051)
- P. Deligne, Catégories tensorielles, Mosc. Math. J. 2 (2002), no. 2, 227–248 (French, with English summary). Dedicated to Yuri I. Manin on the occasion of his 65th birthday. MR 1944506 (2003k:18010)
- P. Deligne and N. Katz (eds.), Séminaire de Géométrie Algébrique du Bois Marie 1967-1969. Groupes de monodromie en géométrie algebrique II (SGA 7), Lecture Notes in Math., 340, Springer, Berlin, 1973.
- Pierre Deligne, James S. Milne et al., Tannakian categories, Hodge Cycles, Motives and Shimura Varieties, Lecture Notes in Math., 900, Springer-Verlag, Berlin-New York, 1982, pp. 101–228. MR 0654325 (84m:14046)
- Vladimir Drinfeld, On a conjecture of Kashiwara, Math. Res. Lett. 8 (2001), no. 5-6, 713–728. MR 1879815 (2003c:14022), DOI https://doi.org/10.4310/MRL.2001.v8.n6.a3
- J. Franecki and M. Kapranov, The Gauss map and a noncompact Riemann-Roch formula for constructible sheaves on semiabelian varieties, Duke Math. J. 104 (2000), no. 1, 171–180. MR 1769729 (2002b:14013), DOI https://doi.org/10.1215/S0012-7094-00-10417-6
- Ofer Gabber and François Loeser, Faisceaux pervers $l$-adiques sur un tore, Duke Math. J. 83 (1996), no. 3, 501–606 (French). MR 1390656 (97i:14016), DOI https://doi.org/10.1215/S0012-7094-96-08317-9
- Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448 (French). MR 0232821 (38 \#1144)
- D. Gaitsgory, On de Jong’s conjecture, Israel J. Math. 157 (2007), 155–191. MR 2342444 (2008j:14021), DOI https://doi.org/10.1007/s11856-006-0006-2
- V. Ginsburg, Characteristic varieties and vanishing cycles, Invent. Math. 84 (1986), no. 2, 327–402. MR 833194 (87j:32030), DOI https://doi.org/10.1007/BF01388811
- Mark Green and Robert Lazarsfeld, Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville, Invent. Math. 90 (1987), no. 2, 389–407. MR 910207 (89b:32025), DOI https://doi.org/10.1007/BF01388711
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725 (80b:14001)
- Revêtements étales et groupe fondamental (SGA 1), Séminaire de géométrie algébrique du Bois Marie 1960–61. [Algebraic Geometry Seminar of Bois Marie 1960-61]. Directed by A. Grothendieck. With two papers by M. Raynaud. Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin; MR0354651 (50 #7129)]. Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 3, Société Mathématique de France, Paris, 2003 (French). MR 2017446 (2004g:14017)
- Heisuke Hironaka, Introduction to the theory of infinitely near singular points, Memorias de Matematica del Instituto “Jorge Juan”, No. 28. Consejo Superior de Investigaciones Científicas, Madrid, 1974. MR 0399505 (53 \#3349)
- Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki, $D$-modules, perverse sheaves, and representation theory, Progress in Mathematics, vol. 236, Birkhäuser Boston Inc., Boston, MA, 2008. Translated from the 1995 Japanese edition by Takeuchi. MR 2357361 (2008k:32022)
- Luc Illusie, Théorie de Brauer et caractéristique d’Euler-Poincaré (d’après P. Deligne), The Euler-Poincaré characteristic (French), Astérisque, vol. 82, Soc. Math. France, Paris, 1981, pp. 161–172. MR 629127 (83m:14014)
- Luc Illusie, Autour du théorème du monodromie locale. Astérisque 223 (1994), 9-57. MR 1293970 (95k:14032)
- Reinhardt Kiehl and Rainer Weissauer, Weil conjectures, perverse sheaves and $l$’adic Fourier transform, 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], vol. 42, Springer-Verlag, Berlin, 2001. MR 1855066 (2002k:14026)
- Thomas Krämer and Rainer Weissauer, On the Tannaka group attached to the theta divisor of a generic principally polarized abelian variety, Math. Z. (to appear).
- Laurent Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002), no. 1, 1–241 (French, with English and French summaries). MR 1875184 (2002m:11039), DOI https://doi.org/10.1007/s002220100174
- J. S. Milne, Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 103–150. MR 861974
- James S. Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531 (81j:14002)
- Takuro Mochizuki, Asymptotic behaviour of tame harmonic bundles and an application to pure twistor $D$-modules. I, II, Mem. Amer. Math. Soc. 185 (2007), nos. 869-870. MR 2281877 (2007j:32028a), MR2283665 (2007j:32028b)
- Neantro Saavedra Rivano, Catégories Tannakiennes, Lecture Notes in Mathematics, Vol. 265, Springer-Verlag, Berlin, 1972 (French). MR 0338002 (49 \#2769)
- Claude Sabbah, Polarizable twistor $\mathcal {D}$-modules, Astérisque 300 (2005), vi+208 (English, with English and French summaries). MR 2156523 (2006d:32009)
- Morihiko Saito, Introduction to mixed Hodge modules, Actes du Colloque de Théorie de Hodge (Luminy, 1987). Astérisque 179-180 (1989), 10, 145–162. MR 1042805 (91j:32041)
- Morihiko Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, 849–995 (1989) (French). MR 1000123 (90k:32038), DOI https://doi.org/10.2977/prims/1195173930
- Morihiko Saito, Decomposition theorem for proper Kähler morphisms, Tôhoku Math. J. 42 (1990) 127-148. MR 1053945 (91j:32042)
- Manfred Scheunert, The theory of Lie superalgebras, An introduction, Lecture Notes in Mathematics, vol. 716, Springer, Berlin, 1979. MR 537441 (80i:17005)
- Christian Schnell, Holonomic D-modules on abelian varieties, Inst. Hautes Études Sci. Publ. Math. (to appear).
- Jean-Pierre Serre, Sur la topologie des variétés algébriques en caractéristique $p$, Symposium internacional de topología algebraica International symposium on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 24–53 (French). MR 0098097 (20 \#4559)
- Jean-Pierre Serre, Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, Springer-Verlag, New York, 1988. Translated from the French. MR 918564 (88i:14041)
- Rainer Weissauer, Brill-Noether sheaves. arXiv:math/0610923.
- Rainer Weissauer, Tannakian categories attached to abelian varieties, Modular forms on Schiermonnikoog, Cambridge Univ. Press, Cambridge, 2008, pp. 267–274. MR 2512364 (2011a:18006), DOI https://doi.org/10.1017/CBO9780511543371.014
- Rainer Weissauer, Semisimple algebraic tensor categories. arXiv:0909.1793.
- Rainer Weissauer, A remark on rigidity of BN-sheaves. arXiv:1111.6095.
- Rainer Weissauer, Degenerate perverse sheaves on abelian varieties. arXiv:1204.2247.
- Rainer Weissauer, Why certain Tannaka groups attached to abelian varieties are almost connected. arXiv:1207.4039.
- Rainer Weissauer, Vanishing theorems for constructible sheaves on abelian varieties over finite fields. arXiv:1407.0844.
Additional Information
Thomas Krämer
Affiliation:
Mathematisches Institut, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 288, D-69120 Heidelberg, Germany
Address at time of publication:
Centre de Mathématiques Laurent Schwartz, École Polytechnique, F-91128 Palaiseau cedex, France
Email:
tkraemer@mathi.uni-heidelberg.de, thomas.kraemer@polytechnique.edu
Rainer Weissauer
Affiliation:
Mathematisches Institut, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 288, D-69120 Heidelberg, Germany
MR Author ID:
181615
Email:
weissauer@mathi.uni-heidelberg.de
Received by editor(s):
September 12, 2012
Received by editor(s) in revised form:
June 9, 2013
Published electronically:
April 20, 2015
Article copyright:
© Copyright 2015
University Press, Inc.
The copyright for this article reverts to public domain 28 years after publication.