logo ANNALES DE L'INSTITUT FOURIER

With cedram.org
Table of contents for this issue | Previous article | Next article
Pierre-Henri Chaudouard; Gérard Laumon
Un théorème du support pour la fibration de Hitchin
(A support theorem for the Hitchin fibration)
Annales de l'institut Fourier, 66 no. 2 (2016), p. 711-727, doi: 10.5802/aif.3023
Article PDF
Class. Math.: 14F20, 14D20, 14D24
Keywords: Hitchin fibration, perverse sheaves, decomposition theorem, relative cohomology

Résumé - Abstract

The main tool in Ngô Bao Châu’s proof of the Langlands-Shelstad fundamental lemma is a support theorem on the relative cohomology of the elliptic part of the Hitchin fibration. In the case of $\mathrm{GL}(n)$ and a divisor of degree $>2g-2$, the theorem states that this relative cohomology is completely determined by its restriction to any open dense subset of the base of the Hitchin fibration. In this article, we prove that the theorem is true in this particular case for the whole Hitchin fibration, including the global nilpotent cone.

Bibliography

[1] Allen B. Altman & Steven L. Kleiman, “Compactifying the Picard scheme”, Adv. in Math. 35 (1980) no. 1, p. 50-112 Article |  MR 555258 |  Zbl 0427.14015
[2] Arnaud Beauville, M. S. Narasimhan & S. Ramanan, “Spectral curves and the generalised theta divisor”, J. Reine Angew. Math. 398 (1989), p. 169-179 Article |  MR 998478 |  Zbl 0666.14015
[3] A. A. Beĭlinson, J. Bernstein & P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque 100, Soc. Math. France, Paris, 1982, p. 5–171  MR 751966
[4] Siegfried Bosch, Werner Lütkebohmert & Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 21, Springer-Verlag, Berlin, 1990 Article |  MR 1045822 |  Zbl 0705.14001
[5] Pierre-Henri Chaudouard, “Sur la contribution unipotente dans la formule des traces d’Arthur pour les groupes généraux linéaires”, http://arxiv.org/abs/1411.3005
[6] Pierre-Henri Chaudouard & Gérard Laumon, “Le lemme fondamental pondéré. II. Énoncés cohomologiques”, Ann. of Math. (2) 176 (2012) no. 3, p. 1647-1781 Article |  MR 2979859 |  Zbl 1264.11043
[7] Steven Diaz & Joe Harris, “Ideals associated to deformations of singular plane curves”, Trans. Amer. Math. Soc. 309 (1988) no. 2, p. 433-468 Article |  MR 961600 |  Zbl 0707.14022
[8] Eduardo Esteves, “Compactifying the relative Jacobian over families of reduced curves”, Trans. Amer. Math. Soc. 353 (2001) no. 8, p. 3045-3095 (electronic) Article |  MR 1828599 |  Zbl 0974.14009
[9] Oscar García-Prada, Jochen Heinloth & Alexander Schmitt, “On the motives of moduli of chains and Higgs bundles”, J. Eur. Math. Soc. (JEMS) 16 (2014) no. 12, p. 2617-2668 Article |  MR 3293805
[10] Jochen Heinloth, “The intersection form on moduli spaces of twisted $PGL_n$-Higgs bundles vanishes”, http://arxiv.org/abs/1412.2232
[11] Nigel Hitchin, “Stable bundles and integrable systems”, Duke Math. J. 54 (1987) no. 1, p. 91-114 Article |  MR 885778 |  Zbl 0627.14024
[12] François Loeser, “Déformations de courbes planes (d’après Severi et Harris)”, Astérisque (1987) no. 152-153, p. 4, 187-205 (1988), Séminaire Bourbaki, Vol. 1986/87 Numdam |  MR 936855 |  Zbl 0636.14008
[13] Sergey Mozgovoy & Olivier Schiffmann, “Counting Higgs bundles”, http://arxiv.org/abs/1411.2101
[14] Bao Châu Ngô, “Le lemme fondamental pour les algèbres de Lie”, Publ. Math. Inst. Hautes Études Sci. (2010) no. 111, p. 1-169 Numdam |  Zbl 1200.22011
[15] Nitin Nitsure, “Moduli space of semistable pairs on a curve”, Proc. London Math. Soc. (3) 62 (1991) no. 2, p. 275-300 Article |  MR 1085642 |  Zbl 0733.14005
[16] Daniel Schaub, “Courbes spectrales et compactifications de jacobiennes”, Math. Z. 227 (1998) no. 2, p. 295-312 Article |  MR 1609069 |  Zbl 0932.14016
[17] Olivier Schiffmann, “Indecomposable vector bundles and stable Higgs bundles over smooth projective curves”, http://arxiv.org/abs/1406.3839
top