no. 1
A Torelli theorem for moduli spaces of principal bundles over a curve
[Un théorème de Torelli pour les espaces des modules de fibrés principaux sur une courbe]
Biswas, Indranil1 ; Hoffmann, Norbert2
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
2 Freie Universität Berlin, Institut fûr Mathematik, Arnimallee 3, 14195 Berlin, Germany
Annales de l'Institut Fourier, Tome 62 (2012) no. 1, pp. 87-106.

Soient X et X ′ des surfaces de Riemann compactes de genre au moins 3, et G et G ′ des groupes complexes réductifs non abéliens. Si une composante ℳ G d (X) de l’espace des modules de G–fibrés principaux semi-stables sur X est isomorphe à une composante ℳ G ′ d ′ (X ′ ), alors X est isomorphe à X ′ .

Let X and X ′ be compact Riemann surfaces of genus ≥3, and let G and G ′ be nonabelian reductive complex groups. If one component ℳ G d (X) of the coarse moduli space for semistable principal G–bundles over X is isomorphic to another component ℳ G ′ d ′ (X ′ ), then X is isomorphic to X ′ .

DOI : 10.5802/aif.2700
Classification : 14D20, 14C34
Keywords: Principal bundle, moduli space, Torelli theorem
Mot clés : Fibré principal, espace des modules, théorème de Torelli

Biswas, Indranil 1 ; Hoffmann, Norbert 2

1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
2 Freie Universität Berlin, Institut fûr Mathematik, Arnimallee 3, 14195 Berlin, Germany 
@article{AIF_2012__62_1_87_0,
     author = {Biswas, Indranil and Hoffmann, Norbert},
     title = {A {Torelli} theorem for moduli spaces of principal bundles over a curve},
     journal = {Annales de l'Institut Fourier},
     pages = {87--106},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {62},
     number = {1},
     year = {2012},
     doi = {10.5802/aif.2700},
     mrnumber = {2986266},
     zbl = {1268.14010},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2700/}
}
TY  - JOUR
AU  - Biswas, Indranil
AU  - Hoffmann, Norbert
TI  - A Torelli theorem for moduli spaces of principal bundles over a curve
JO  - Annales de l'Institut Fourier
PY  - 2012
SP  - 87
EP  - 106
VL  - 62
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2700/
DO  - 10.5802/aif.2700
LA  - en
ID  - AIF_2012__62_1_87_0
ER  - 
%0 Journal Article
%A Biswas, Indranil
%A Hoffmann, Norbert
%T A Torelli theorem for moduli spaces of principal bundles over a curve
%J Annales de l'Institut Fourier
%D 2012
%P 87-106
%V 62
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2700/
%R 10.5802/aif.2700
%G en
%F AIF_2012__62_1_87_0
Biswas, Indranil; Hoffmann, Norbert. A Torelli theorem for moduli spaces of principal bundles over a curve. Annales de l'Institut Fourier, Tome 62 (2012) no. 1, pp. 87-106. doi : 10.5802/aif.2700. https://aif.centre-mersenne.org/articles/10.5802/aif.2700/

[1] Adams, S. Reduction of cocycles with hyperbolic targets, Ergodic Theory Dyn. Syst., Volume 16 (1996), pp. 1111-1145 | DOI | MR | Zbl

[2] Drezet, J.-M.; Narasimhan, M.S. Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. Math., Volume 97 (1989), pp. 53-94 | DOI | EuDML | MR | Zbl

[3] Gómez, T.L.; Sols, I. Moduli space of principal sheaves over projective varieties, Ann. of Math., Volume 161 (2005), pp. 1037-1092 | DOI | MR | Zbl

[4] Humphreys, J.E. Linear algebraic groups, Graduate Texts in Mathematics, 21, Springer-Verlag, New York - Heidelberg - Berlin, 1975 | MR | Zbl

[5] Hwang, J.-M.; Ramanan, S. Hecke curves and Hitchin discriminant, Ann. Sci. École Norm. Sup., Volume 37 (2004), pp. 801-817 | EuDML | Numdam | MR | Zbl

[6] Kouvidakis, A.; Pantev, T. The automorphism group of the moduli space of semi stable vector bundles, Math. Ann., Volume 302 (1995), pp. 225-268 | DOI | EuDML | MR | Zbl

[7] Kumar, S.; Narasimhan, M.S. Picard group of the moduli spaces of G–bundles, Math. Ann., Volume 308 (1997), pp. 155-173 | DOI | MR | Zbl

[8] Luna, D. Slices étalés, Mém. Soc. Math. Fr., Volume 33 (1973), pp. 81-105 | EuDML | Numdam | MR | Zbl

[9] Mumford, D. Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 34, Springer-Verlag, Berlin, 1965 | MR | Zbl

[10] Mumford, D.; Newstead, P. Periods of a moduli space of bundles on curves, Amer. Jour. Math., Volume 90 (1968), pp. 1200-1208 | DOI | MR | Zbl

[11] Narasimhan, M.S.; Ramanan, S. Moduli of vector bundles on a compact Riemann surface, Ann. of Math., Volume 89 (1969), pp. 14-51 | DOI | MR | Zbl

[12] Narasimhan, M.S.; Ramanan, S. Deformations of the moduli space of vector bundles over an algebraic curve, Ann. of Math., Volume 101 (1975), pp. 391-417 | DOI | MR | Zbl

[13] Pantev, T. Comparison of generalized theta functions, Duke Math. J., Volume 76 (1994), pp. 509-539 | DOI | MR | Zbl

[14] Ramanathan, A. Stable principal bundles on a compact Riemann surface, Math. Ann., Volume 213 (1975), pp. 129-152 | DOI | MR | Zbl

[15] Ramanathan, A. Moduli for principal bundles over algebraic curves, Proc. Indian Acad. Sci. (Math. Sci.), Volume 106 (1996), p. 301-328 and 421–449 | DOI | Zbl

[16] Schmitt, A.H.W. Singular principal bundles over higher–dimensional manifolds and their moduli spaces, Int. Math. Res. Not. (2002), pp. 1183-1209 | DOI | MR

[17] Serre, J.-P. On the fundamental group of a unirational variety, Jour. London Math. Soc., Volume 34 (1959), pp. 481-484 | DOI | MR | Zbl

[18] Sun, X. Minimal rational curves on moduli spaces of stable bundles, Math. Ann., Volume 331 (2005), pp. 925-937 | DOI | MR

[19] Tyurin, A.N. An analogue of the Torelli theorem for two-dimensional bundles over an algebraic curve of arbitrary genus (Russian), Izv. Akad. Nauk SSSR Ser. Mat., Volume 33 (1969), pp. 1149-1170 in English: Math. USSR, Izv. 3 (1971) | MR | Zbl

[20] Tyurin, A.N. Analogues of Torelli’s theorem for multidimensional vector bundles over an arbitrary algebraic curve (Russian), Izv. Akad. Nauk SSSR Ser. Mat., Volume 34 (1970), pp. 338-365 in English: Math. USSR, Izv. 4 (1971) | MR | Zbl

Cité par Sources :