no. 6
Tensor product theorem for Hitchin pairs – An algebraic approach
[Théorème du produit tensoriel pour des paires de Hitchin – Une approche algébrique]
Balaji, V.1 ; Parameswaran, A.J.2
1 Chennai Mathematical Institute SIPCOT IT Park Siruseri-603103 (India)
2 Kerala School of Mathematics Kozhikode, Kerala and Tata Institute of Fundamental Research School of Mathematics Mumbai-400095 (India)
Annales de l'Institut Fourier, Tome 61 (2011) no. 6, pp. 2361-2403.

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 p, où p 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 p, with p satisfying some natural bounds. We also prove the corresponding theorem for polystable Hitchin pairs.

DOI : 10.5802/aif.2677
Classification : 14J60, 14D20
Keywords: Higgs semistable Hitchin pairs, Tannaka categories, group schemes, tensor products
Mot clés : paires de Hitchin semistables, catégories Tannakiennes, schémas en groupes, produit tensoriel

Balaji, V. 1 ; Parameswaran, A.J. 2

1 Chennai Mathematical Institute SIPCOT IT Park Siruseri-603103 (India)
2 Kerala School of Mathematics Kozhikode, Kerala and Tata Institute of Fundamental Research School of Mathematics Mumbai-400095 (India) 
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Balaji, V.; Parameswaran, A.J. Tensor product theorem for Hitchin pairs – An algebraic approach. Annales de l'Institut Fourier, Tome 61 (2011) no. 6, pp. 2361-2403. doi : 10.5802/aif.2677. https://aif.centre-mersenne.org/articles/10.5802/aif.2677/

[1] Balaji, V.; Parameswaran, A. J. Semistable principal bundles. II. Positive characteristics, Transform. Groups, Volume 8 (2003) no. 1, pp. 3-36 | DOI | MR | Zbl

[2] Bardsley, Peter; Richardson, R. W. Étale slices for algebraic transformation groups in characteristic p, Proc. London Math. Soc. (3), Volume 51 (1985) no. 2, pp. 295-317 | DOI | MR | Zbl

[3] Beilinson, Alexander; Drinfeld, Vladimir Chiral algebras, American Mathematical Society Colloquium Publications, 51, American Mathematical Society, Providence, RI, 2004 | MR | Zbl

[4] Biswas, Indranil; Schumacher, Georg Yang-Mills equation for stable Higgs sheaves, Internat. J. Math., Volume 20 (2009) no. 5, pp. 541-556 | DOI | MR | Zbl

[5] Bogomolov, F. A. Holomorphic tensors and vector bundles on projective manifolds, Izv. Akad. Nauk SSSR Ser. Mat., Volume 42 (1978) no. 6, pp. 1227-1287 | MR | Zbl

[6] Deligne, P.; Milne, J. Tannaka categories, Springer Lecture Notes in Mathematics, Volume 900 (1982), pp. 101-228 | DOI | Zbl

[7] Gieseker, D. On a theorem of Bogomolov on Chern classes of stable bundles, Amer. J. Math., Volume 101 (1979) no. 1, pp. 77-85 | DOI | MR | Zbl

[8] Hesselink, Wim H. Uniform instability in reductive groups, J. Reine Angew. Math., Volume 303/304 (1978), pp. 74-96 | MR | Zbl

[9] Hitchin, N. J. The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3), Volume 55 (1987) no. 1, pp. 59-126 | DOI | MR | Zbl

[10] Hitchin, Nigel Stable bundles and integrable systems, Duke Math. J., Volume 54 (1987) no. 1, pp. 91-114 | DOI | MR | Zbl

[11] Ilangovan, S.; Mehta, V. B.; Parameswaran, A. J. Semistability and semisimplicity in representations of low height in positive characteristic, A tribute to C. S. Seshadri (Chennai, 2002) (Trends Math.), Birkhäuser, Basel, 2003, pp. 271-282 | MR | Zbl

[12] Kempf, George R. Instability in invariant theory, Ann. of Math. (2), Volume 108 (1978) no. 2, pp. 299-316 | DOI | MR | Zbl

[13] Kirwan, Frances Clare Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, 31, Princeton University Press, Princeton, NJ, 1984 | MR | Zbl

[14] Laszlo, Y.; Pauly, C. The action of the Frobenius maps on rank 2 vector bundles in characteristic 2, J. Algebraic Geom., Volume 11 (2002) no. 2, pp. 219-243 | DOI | MR | Zbl

[15] Mehta, V. B.; Parameswaran, A. J. Geometry of low height representations, Algebra, arithmetic and geometry, Part I, II (Mumbai, 2000) (Tata Inst. Fund. Res. Stud. Math.), Volume 16, Tata Inst. Fund. Res., Bombay, 2002, pp. 417-426 | MR | Zbl

[16] Mehta, Vikram Bhagvandas Representations of algebraic groups and principal bundles on algebraic varieties, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) (2002), pp. 629-635 | MR | Zbl

[17] Milne, J.S. Semisimple Lie algebras, algebraic groups, and tensor categories (2007) (Milne’s home page)

[18] Ngô, Bao Châu Fibration de Hitchin et endoscopie, Invent. Math., Volume 164 (2006) no. 2, pp. 399-453 | DOI | MR | Zbl

[19] Nori, Madhav V. The fundamental group-scheme, Proc. Indian Acad. Sci. Math. Sci., Volume 91 (1982) no. 2, pp. 73-122 | DOI | MR | Zbl

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

[21] Ramanathan, A. Stable principal bundles on a compact Riemann surface - Construction of moduli space, Bombay University (1976) (Ph. D. Thesis) | MR

[22] Rousseau, G. Instabilité dans les fibrés vectoriels (d’après Bogomolov), Algebraic surfaces (Orsay, 1976–78) (Lecture Notes in Math.), Volume 868, Springer, Berlin, 1981, pp. 277-292 | MR | Zbl

[23] Serre, Jean-Pierre Sur la semi-simplicité des produits tensoriels de représentations de groupes, Invent. Math., Volume 116 (1994) no. 1-3, pp. 513-530 | DOI | EuDML | MR | Zbl

[24] Serre, Jean-Pierre Moursund Lectures (1998) (University of Oregon, Mathematics Department)

[25] Serre, Jean-Pierre Complète réductibilité, Astérisque (2005) no. 299, pp. 195-217 (Séminaire Bourbaki. Vol. 2003/2004, Exp. No. 932, viii) | EuDML | Numdam | MR | Zbl

[26] Simpson, Carlos T. Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. (1992) no. 75, pp. 5-95 | DOI | EuDML | Numdam | MR | Zbl

[27] Simpson, Carlos T. 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 | DOI | EuDML | Numdam | MR | Zbl

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