On donne une approche algébrique à l’étude des paires de Hitchin et on démontre le théorème du produit tensoriel pour des paires de Hitchin semistables sur les courbes projectives lisses définies sur un corps algébrique clos de caractéristique nulle ou bien de caractéristique , où désigne un nombre premier borné. On démontre aussi un théorème similaire pour des paires de Hitchin polystables.
We give an algebraic approach to the study of Hitchin pairs and prove the tensor product theorem for Higgs semistable Hitchin pairs over smooth projective curves defined over algebraically closed fields of characteristic zero and characteristic , with satisfying some natural bounds. We also prove the corresponding theorem for polystable Hitchin pairs.
@article{AIF_2011__61_6_2361_0, author = {Balaji, V. and Parameswaran, A.J.}, title = {Tensor product theorem for Hitchin pairs -- An algebraic approach}, journal = {Annales de l'Institut Fourier}, volume = {61}, year = {2011}, pages = {2361-2403}, doi = {10.5802/aif.2677}, zbl = {1248.14046}, mrnumber = {2976315}, language = {en}, url = {http://dml.mathdoc.fr/item/AIF_2011__61_6_2361_0} }
Balaji, V.; Parameswaran, A.J. Tensor product theorem for Hitchin pairs – An algebraic approach. Annales de l'Institut Fourier, Tome 61 (2011) pp. 2361-2403. doi : 10.5802/aif.2677. http://gdmltest.u-ga.fr/item/AIF_2011__61_6_2361_0/
[1] Semistable principal bundles. II. Positive characteristics, Transform. Groups, Tome 8 (2003) no. 1, pp. 3-36 | Article | MR 1959761 | Zbl 1084.14013
[2] Étale slices for algebraic transformation groups in characteristic , Proc. London Math. Soc. (3), Tome 51 (1985) no. 2, pp. 295-317 | Article | MR 794118 | Zbl 0604.14037
[3] Chiral algebras, American Mathematical Society, Providence, RI, American Mathematical Society Colloquium Publications, Tome 51 (2004) | MR 2058353 | Zbl 1138.17300
[4] Yang-Mills equation for stable Higgs sheaves, Internat. J. Math., Tome 20 (2009) no. 5, pp. 541-556 | Article | MR 2526306 | Zbl 1169.53017
[5] Holomorphic tensors and vector bundles on projective manifolds, Izv. Akad. Nauk SSSR Ser. Mat., Tome 42 (1978) no. 6, pp. 1227-1287 | MR 522939 | Zbl 0439.14002
[6] Tannaka categories, Springer Lecture Notes in Mathematics, Tome 900 (1982), pp. 101-228 | Article | Zbl 0477.14004
[7] On a theorem of Bogomolov on Chern classes of stable bundles, Amer. J. Math., Tome 101 (1979) no. 1, pp. 77-85 | Article | MR 527826 | Zbl 0431.14005
[8] Uniform instability in reductive groups, J. Reine Angew. Math., Tome 303/304 (1978), pp. 74-96 | MR 514673 | Zbl 0386.20020
[9] The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3), Tome 55 (1987) no. 1, pp. 59-126 | Article | MR 887284 | Zbl 0634.53045
[10] Stable bundles and integrable systems, Duke Math. J., Tome 54 (1987) no. 1, pp. 91-114 | Article | MR 885778 | Zbl 0627.14024
[11] Semistability and semisimplicity in representations of low height in positive characteristic, A tribute to C. S. Seshadri (Chennai, 2002), Birkhäuser, Basel (Trends Math.) (2003), pp. 271-282 | MR 2017588 | Zbl 1067.20061
[12] Instability in invariant theory, Ann. of Math. (2), Tome 108 (1978) no. 2, pp. 299-316 | Article | MR 506989 | Zbl 0406.14031
[13] Cohomology of quotients in symplectic and algebraic geometry, Princeton University Press, Princeton, NJ, Mathematical Notes, Tome 31 (1984) | MR 766741 | Zbl 0553.14020
[14] The action of the Frobenius maps on rank vector bundles in characteristic , J. Algebraic Geom., Tome 11 (2002) no. 2, pp. 219-243 | Article | MR 1874113 | Zbl 1080.14527
[15] Geometry of low height representations, Algebra, arithmetic and geometry, Part I, II (Mumbai, 2000), Tata Inst. Fund. Res., Bombay (Tata Inst. Fund. Res. Stud. Math.) Tome 16 (2002), pp. 417-426 | MR 1940675 | Zbl 1060.20039
[16] Representations of algebraic groups and principal bundles on algebraic varieties, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing (2002), pp. 629-635 | MR 1957070 | Zbl 1007.22020
[17] Semisimple Lie algebras, algebraic groups, and tensor categories (2007) (Milne’s home page)
[18] Fibration de Hitchin et endoscopie, Invent. Math., Tome 164 (2006) no. 2, pp. 399-453 | Article | MR 2218781 | Zbl 1098.14023
[19] The fundamental group-scheme, Proc. Indian Acad. Sci. Math. Sci., Tome 91 (1982) no. 2, pp. 73-122 | Article | MR 682517 | Zbl 0586.14006
[20] Some remarks on the instability flag, Tohoku Math. J. (2), Tome 36 (1984) no. 2, pp. 269-291 | Article | MR 742599 | Zbl 0567.14027
[21] Stable principal bundles on a compact Riemann surface - Construction of moduli space, Bombay University (1976) (Ph. D. Thesis) | MR 369747
[22] Instabilité dans les fibrés vectoriels (d’après Bogomolov), Algebraic surfaces (Orsay, 1976–78), Springer, Berlin (Lecture Notes in Math.) Tome 868 (1981), pp. 277-292 | MR 638604 | Zbl 0477.14012
[23] Sur la semi-simplicité des produits tensoriels de représentations de groupes, Invent. Math., Tome 116 (1994) no. 1-3, pp. 513-530 | Article | MR 1253203 | Zbl 0816.20014
[24] Moursund Lectures (1998) (University of Oregon, Mathematics Department)
[25] Complète réductibilité, Astérisque (2005) no. 299, pp. 195-217 (Séminaire Bourbaki. Vol. 2003/2004, Exp. No. 932, viii) | Numdam | MR 2167207 | Zbl 1156.20313
[26] Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. (1992) no. 75, pp. 5-95 | Article | Numdam | MR 1179076 | Zbl 0814.32003
[27] Moduli of representations of the fundamental group of a smooth projective variety. I, Inst. Hautes Études Sci. Publ. Math. (1994) no. 79, pp. 47-129 | Article | Numdam | MR 1307297 | Zbl 0891.14005