On the S-fundamental group scheme
[Sur le schéma en groupes S-fondamental]
Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 2077-2119.

Nous introduisons un nouveau schéma en groupes fondamental pour les variétés définies sur un corps algébriquement clos (ou simplement parfait) de caractéristique positive. Nous utilisons ce schéma en groupes pour étudier des généralisations en caractéristique positive des résultats de C. Simpson. Nous étudions également quelques propriétés de ce schéma en groupes fondamental, en particulier nous obtenons des résultats de type “Lefschetz”.

We introduce a new fundamental group scheme for varieties defined over an algebraically closed (or just perfect) field of positive characteristic and we use it to study generalization of C. Simpson’s results to positive characteristic. We also study the properties of this group and we prove Lefschetz type theorems.

DOI : https://doi.org/10.5802/aif.2667
Classification : 14J60,  14F05,  14F35,  14L15
Mots clés : groupe fondamental, caractéristique positive, fibres numériquement plate, résultats type “Lefschetz”
@article{AIF_2011__61_5_2077_0,
     author = {Langer, Adrian},
     title = {On the S-fundamental group scheme},
     journal = {Annales de l'Institut Fourier},
     pages = {2077--2119},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {5},
     year = {2011},
     doi = {10.5802/aif.2667},
     mrnumber = {2961849},
     zbl = {1247.14019},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2667/}
}
Langer, Adrian. On the S-fundamental group scheme. Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 2077-2119. doi : 10.5802/aif.2667. http://www.numdam.org/articles/10.5802/aif.2667/

[1] Balaji, V.; Parameswaran, A. J. An analogue of the Narasimhan-Seshadri theorem and some application (2009) (preprint, arXiv:0809.3765)

[2] Barton, C. Tensor products of ample vector bundles in characteristic p, Amer. J. Math., Volume 93 (1971), pp. 429-438 | Article | MR 289525 | Zbl 0221.14011

[3] Biswas, I.; Holla, Y. Comparison of fundamental group schemes of a projective variety and an ample hypersurface, J. Algebraic Geom., Volume 16 (2007) no. 3, pp. 547-597 | Article | MR 2306280 | Zbl 1120.14038

[4] Biswas, I.; Parameswaran, A. J.; Subramanian, S. Monodromy group for a strongly semistable principal bundle over a curve, Duke Math. J., Volume 132 (2006) no. 1, pp. 1-48 | Article | MR 2219253 | Zbl 1106.14032

[5] Biswas, I.; Subramanian, S. Numerically flat principal bundles, Tohoku Math. J. (2), Volume 57 (2005) no. 1, pp. 53-63 http://projecteuclid.org/getRecord?id=euclid.tmj/1113234834 | Article | MR 2113990 | Zbl 1072.32010

[6] Brenner, H. There is no Bogomolov type restriction theorem for strong semistability in positive characteristic, Proc. Amer. Math. Soc., Volume 133 (2005) no. 7, pp. 1941-1947 | Article | MR 2137859 | Zbl 1083.14050

[7] Brenner, H.; Kaid, A. On deep Frobenius descent and flat bundles, Math. Res. Lett., Volume 15 (2008) no. 6, pp. 1101-1115 | MR 2470387 | Zbl 1200.14061

[8] Deligne, P. Le groupe fondamental de la droite projective moins trois points, Galois groups over Q (Berkeley, CA, 1987) (Math. Sci. Res. Inst. Publ.), Volume 16, Springer, New York, 1989, pp. 79-297 | MR 1012168 | Zbl 0742.14022

[9] Deligne, P.; Illusie, L. Relèvements modulo p 2 et décomposition du complexe de de Rham, Invent. Math., Volume 89 (1987) no. 2, pp. 247-270 | Article | MR 894379 | Zbl 0632.14017

[10] Deligne, P.; Milne, J. S. Tannakian categories, Lecture Notes in Mathematics, 900 (1982), pp. ii+414 | Zbl 0477.14004

[11] Groupes de monodromie en géométrie algébrique. II (Lecture Notes in Mathematics, Vol. 340), 1973 Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II), Dirigé par P. Deligne et N. Katz | MR 354657 | Zbl 0258.00005

[12] Demailly, J.-P.; Peternell, T.; Schneider, M. Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom., Volume 3 (1994) no. 2, pp. 295-345 | MR 1257325 | Zbl 0827.14027

[13] Diaz, S.; Harbater, D. Strong Bertini theorems, Trans. Amer. Math. Soc., Volume 324 (1991) no. 1, pp. 73-86 | Article | MR 986689 | Zbl 0744.14004

[14] Fulton, W. Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 2, Springer-Verlag, Berlin, 1984 | MR 732620 | Zbl 0885.14002

[15] Fulton, W.; Lazarsfeld, R. Positive polynomials for ample vector bundles, Ann. of Math. (2), Volume 118 (1983) no. 1, pp. 35-60 | Article | MR 707160 | Zbl 0537.14009

[16] Gieseker, D. Stable vector bundles and the Frobenius morphism, Ann. Sci. École Norm. Sup. (4), Volume 6 (1973), pp. 95-101 | EuDML 81915 | Numdam | MR 325616 | Zbl 0281.14013

[17] Gieseker, D. Flat vector bundles and the fundamental group in non-zero characteristics, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 2 (1975) no. 1, pp. 1-31 | EuDML 83683 | Numdam | MR 382271 | Zbl 0322.14009

[18] Grothendieck, A. Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 4, Société Mathématique de France, Paris, 2005 (Séminaire de Géométrie Algébrique du Bois Marie, 1962, Augmenté d’un exposé de Michèle Raynaud. [With an exposé by Michèle Raynaud], With a preface and edited by Yves Laszlo, Revised reprint of the 1968 French original) | MR 2171939 | Zbl 0197.47202

[19] Hartshorne, R. Algebraic geometry, Springer-Verlag, New York, 1977 (Graduate Texts in Mathematics, No. 52) | MR 463157 | Zbl 0531.14001

[20] Huybrechts, D.; Lehn, M. The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997 | MR 1450870 | Zbl 0872.14002

[21] Ishimura, S. A descent problem of vector bundles and its applications, J. Math. Kyoto Univ., Volume 23 (1983) no. 1, pp. 73-83 | MR 692730 | Zbl 0523.14017

[22] Jantzen, J. C. Representations of algebraic groups, Mathematical Surveys and Monographs, 107, American Mathematical Society, Providence, RI, 2003 | MR 2015057 | Zbl 1034.20041

[23] Kollár, J. Rational curves on algebraic varieties, 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], 32, Springer-Verlag, Berlin, 1996 | MR 1440180 | Zbl 0877.14012

[24] Langer, A. Semistable sheaves in positive characteristic, Ann. of Math. (2), Volume 159 (2004) no. 1, pp. 251-276 Addendum, Ann. of Math. (2), 160(3):1211–1213 | Article | MR 2051393 | Zbl 1080.14014

[25] Langer, A. Semistable principal G-bundles in positive characteristic, Duke Math. J., Volume 128 (2005) no. 3, pp. 511-540 | Article | MR 2145742 | Zbl 1081.14018

[26] Langer, A. Moduli spaces of sheaves and principal G-bundles, Algebraic geometry—Seattle 2005. Part 1 (Proc. Sympos. Pure Math.), Volume 80, Amer. Math. Soc., Providence, RI, 2009, pp. 273-308 | MR 2483939 | Zbl 1179.14010

[27] Manivel, L. Vanishing theorems for ample vector bundles, Invent. Math., Volume 127 (1997), pp. 401-416 | Article | MR 1427625 | Zbl 0906.14011

[28] Mehta, V. B. Some remarks on the local fundamental group scheme and the big fundamental group scheme, preprint (2008), pp. 14pp | MR 2423233 | Zbl 1157.14006

[29] Mehta, V. B.; Nori, M. V. Semistable sheaves on homogeneous spaces and abelian varieties, Proc. Indian Acad. Sci. Math. Sci., Volume 93 (1984), pp. 1-12 | Article | MR 796768 | Zbl 0592.14017

[30] Mehta, V. B.; Ramanathan, A. Restriction of stable sheaves and representations of the fundamental group, Invent. Math., Volume 77 (1984) no. 1, pp. 163-172 | Article | EuDML 143143 | MR 751136 | Zbl 0525.55012

[31] Mehta, V. B.; Subramanian, S. On the fundamental group scheme, Invent. Math., Volume 148 (2002) no. 1, pp. 143-150 | Article | MR 1892846 | Zbl 1020.14006

[32] Nori, M. V. The fundamental group-scheme, Proc. Indian Acad. Sci. Math. Sci., Volume 91 (1982) no. 2, pp. 73-122 | Article | MR 682517 | Zbl 0586.14006

[33] Okonek, C.; Schneider, M.; Spindler, H. Vector bundles on complex projective spaces, Progress in Mathematics, 3, Birkhäuser Boston, Mass., 1980 | MR 561910 | Zbl 0438.32016

[34] Pauly, C. A smooth counterexample to Nori’s conjecture on the fundamental group scheme, Proc. Amer. Math. Soc., Volume 135 (2007) no. 9, p. 2707-2711 (electronic) | Article | MR 2317943 | Zbl 1115.14026

[35] Ramanan, S.; Ramanathan, A. Some remarks on the instability flag, Tohoku Math. J. (2), Volume 36 (1984) no. 2, pp. 269-291 | Article | MR 742599 | Zbl 0567.14027

[36] dos Santos, J. P. P. Fundamental group schemes for stratified sheaves, J. Algebra, Volume 317 (2007) no. 2, pp. 691-713 | Article | MR 2362937 | Zbl 1130.14032

[37] Shiho, A. Crystalline fundamental groups. I. Isocrystals on log crystalline site and log convergent site, J. Math. Sci. Univ. Tokyo, Volume 7 (2000) no. 4, pp. 509-656 | MR 1800845 | Zbl 0984.14009

[38] Simpson, C. T. Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math., Volume 75 (1992), pp. 5-95 | Article | EuDML 104080 | Numdam | MR 1179076 | Zbl 0814.32003

[39] Szpiro, L. Sur le théorème de rigidité de Parsin et Arakelov, Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. II (Astérisque), Volume 64, Soc. Math. France, Paris, 1979, pp. 169-202 | MR 563470 | Zbl 0425.14005

[40] Włodarczyk, Jarosław Toroidal varieties and the weak factorization theorem, Invent. Math., Volume 154 (2003) no. 2, pp. 223-331 | Article | MR 2013783 | Zbl 1130.14014