On some natural torsors over moduli spaces of parabolic bundles. (English) Zbl 1521.14063
The article under review deals with isomorphism between two natural torsors over moduli spaces of stable parabolic bundles.
Let \(X\) be a compact connected Riemann surface and \(S\) a finite subset. For a rank \(r\) vector bundle we fix parabolic data on \(S\) such that the total parabolic weight for entire \(S\) is an integer. Let \(\mathcal{N}_P(r)\) denote the moduli space of stable parabolic bundles of rank \(r\) and parabolic degree zero with parabolic structure of the given type over the points of \(S\).
Given any stable parabolic bundle \(E_{\ast} \in \mathcal{N}_P(r)\) there is a unique parabolic connection on \(E_{\ast}\) such that the corresponding monodromy homomorphism \(\pi_1 (X \setminus S, x_0 ) \to GL_r(\mathbb{C})\) has its image contained in \(U(r)\) (due to Mehta and Seshadri). The construction of this unitary connection is not algebro-geometric/complex-analytic. Let \(\mathcal{N}C_P(r)\) be the moduli space of pairs of the form \((E_{\ast}, D_E )\), where \(E_{\ast} \in \mathcal{N}_P(r)\) and \(D_E\) is a parabolic connection on \(E_{\ast}\). The projection map defined by \((E_{\ast}, D_E) \mapsto E_{\ast}\) makes \(\mathcal{N}C_P(r)\) a holomorphic torsor over \(\mathcal{N}_P(r)\) for \(T^{\ast}\mathcal{N}_P(r)\). Assigning to every \(E_{\ast} \in \mathcal{N}_P(r)\) the unique parabolic connection \(D_E\) on \(E_{\ast}\) with unitary monodromy, we obtain a section \(\tau_{U P} : \mathcal{N}_P(r) \to \mathcal{N}C_{P}(r)\). This section \( \tau_{U P}\) is not holomorphic.
Let \(\Theta\) be the theta line bundle on \(\mathcal{N}_P(r)\) and let \(\mathcal{U}_P \to \mathcal{N}_P(r)\) be the algebraic fiber bundle defined by the sheaf of holomorphic connections on \(\Theta\). This \(\mathcal{U}_P\) is a holomorphic torsor over \(\mathcal{N}_P(r)\) for \(T^{\ast}\mathcal{N}_P(r)\). We have a Hermitian structure on the line bundle \(\Theta\) due to a construction of Quillen. The corresponding Chern connection on \(\Theta\) defines a \(C^{\infty}\) section \(\tau_{QP} : \mathcal{N}_P(r) \to \mathcal{U}_P\). This section \(\tau_{QP}\) is not holomorphic.
The main result of the article is that there is a natural holomorphic isomorphism between the two \(T^{\ast}\mathcal{N}_P(r)\)-torsors \(\mathcal{N}C_{P}(r)\) and \(\mathcal{U}_P\) that takes the section \(\tau_{U P}\) to the section \(\tau_{QP}\). It is deduced that this isomorphism is in fact algebraic. This result is a parabolic analog of results for vector bundles (see [I. Biswas and J. Hurtubise, Adv. Math. 389, Article ID 107918, 29 p. (2021; Zbl 1472.14038)]).
Let \(X\) be a compact connected Riemann surface and \(S\) a finite subset. For a rank \(r\) vector bundle we fix parabolic data on \(S\) such that the total parabolic weight for entire \(S\) is an integer. Let \(\mathcal{N}_P(r)\) denote the moduli space of stable parabolic bundles of rank \(r\) and parabolic degree zero with parabolic structure of the given type over the points of \(S\).
Given any stable parabolic bundle \(E_{\ast} \in \mathcal{N}_P(r)\) there is a unique parabolic connection on \(E_{\ast}\) such that the corresponding monodromy homomorphism \(\pi_1 (X \setminus S, x_0 ) \to GL_r(\mathbb{C})\) has its image contained in \(U(r)\) (due to Mehta and Seshadri). The construction of this unitary connection is not algebro-geometric/complex-analytic. Let \(\mathcal{N}C_P(r)\) be the moduli space of pairs of the form \((E_{\ast}, D_E )\), where \(E_{\ast} \in \mathcal{N}_P(r)\) and \(D_E\) is a parabolic connection on \(E_{\ast}\). The projection map defined by \((E_{\ast}, D_E) \mapsto E_{\ast}\) makes \(\mathcal{N}C_P(r)\) a holomorphic torsor over \(\mathcal{N}_P(r)\) for \(T^{\ast}\mathcal{N}_P(r)\). Assigning to every \(E_{\ast} \in \mathcal{N}_P(r)\) the unique parabolic connection \(D_E\) on \(E_{\ast}\) with unitary monodromy, we obtain a section \(\tau_{U P} : \mathcal{N}_P(r) \to \mathcal{N}C_{P}(r)\). This section \( \tau_{U P}\) is not holomorphic.
Let \(\Theta\) be the theta line bundle on \(\mathcal{N}_P(r)\) and let \(\mathcal{U}_P \to \mathcal{N}_P(r)\) be the algebraic fiber bundle defined by the sheaf of holomorphic connections on \(\Theta\). This \(\mathcal{U}_P\) is a holomorphic torsor over \(\mathcal{N}_P(r)\) for \(T^{\ast}\mathcal{N}_P(r)\). We have a Hermitian structure on the line bundle \(\Theta\) due to a construction of Quillen. The corresponding Chern connection on \(\Theta\) defines a \(C^{\infty}\) section \(\tau_{QP} : \mathcal{N}_P(r) \to \mathcal{U}_P\). This section \(\tau_{QP}\) is not holomorphic.
The main result of the article is that there is a natural holomorphic isomorphism between the two \(T^{\ast}\mathcal{N}_P(r)\)-torsors \(\mathcal{N}C_{P}(r)\) and \(\mathcal{U}_P\) that takes the section \(\tau_{U P}\) to the section \(\tau_{QP}\). It is deduced that this isomorphism is in fact algebraic. This result is a parabolic analog of results for vector bundles (see [I. Biswas and J. Hurtubise, Adv. Math. 389, Article ID 107918, 29 p. (2021; Zbl 1472.14038)]).
Reviewer: Souradeep Majumder (Tirupati)
MSC:
| 14H60 | Vector bundles on curves and their moduli |
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
Citations:
Zbl 1472.14038References:
| [1] | Atiyah, M. F., Complex analytic connections in fibre bundles, Trans. Am. Math. Soc., 85, 181-207 (1957) · Zbl 0078.16002 |
| [2] | Atiyah, M. F., Riemann surfaces and spin structures, Ann. Sci. Éc. Norm. Supér., 4, 47-62 (1971) · Zbl 0212.56402 |
| [3] | Atiyah, M. F.; Bott, R., The Yang-Mills equations over Riemann surfaces, Philos. Trans. R. Soc. Lond., 308, 523-615 (1983) · Zbl 0509.14014 |
| [4] | Beauville, A.; Narasimhan, M. S.; Ramanan, S., Spectral curves and the generalised theta divisor, J. Reine Angew. Math., 398, 169-179 (1989) · Zbl 0666.14015 |
| [5] | Ben-Zvi, D.; Biswas, I., Theta functions and Szegő kernels, Int. Math. Res. Not., 24, 1305-1340 (2003) · Zbl 1026.14008 |
| [6] | Biquard, O., Fibrés paraboliques stables et connexions singulières plates, Bull. Soc. Math. Fr., 119, 231-257 (1991) · Zbl 0769.53013 |
| [7] | Biswas, I., Parabolic bundles as orbifold bundles, Duke Math. J., 88, 305-325 (1997) · Zbl 0955.14010 |
| [8] | Biswas, I., Connections on a parabolic principal bundle over a curve, Can. J. Math., 58, 262-281 (2006) · Zbl 1103.53015 |
| [9] | Biswas, I., A criterion for the existence of a flat connection on a parabolic vector bundle, Adv. Geom., 2, 231-241 (2002) · Zbl 1013.14010 |
| [10] | Biswas, I.; Hurtubise, J., A canonical connection on bundles on Riemann surfaces and Quillen connection on the theta bundle, Adv. Math., 389, Article 107918 pp. (2021) · Zbl 1472.14038 |
| [11] | Biswas, I.; Logares, M., Connection on parabolic vector bundles over curves, Int. J. Math., 22, 593-602 (2011) · Zbl 1221.14043 |
| [12] | Biswas, I.; Raghavendra, N., Determinants of parabolic bundles on Riemann surfaces, Proc. Indian Acad. Sci. Math. Sci., 103, 41-71 (1993) · Zbl 0779.32021 |
| [13] | Biswas, I.; Raina, A. K., Projective structures on a Riemann surface. II, Int. Math. Res. Not., 13, 685-716 (1999) · Zbl 0939.14014 |
| [14] | Biswas, I.; Favale, F. F.; Pirola, G. P.; Torelli, S., Quillen connection and the uniformization of Riemann surfaces, Ann. Mat. Pura Appl., 201, 2825-2835 (2022) · Zbl 1504.14056 |
| [15] | Borne, N., Fibrés paraboliques et champ des racines, Int. Math. Res. Not., 16, Article rnm049 pp. (2007) · Zbl 1197.14035 |
| [16] | Borne, N., Sur les représentations du groupe fondamental d’une variété privée d’un diviseur à croisements normaux simples, Indiana Univ. Math. J., 58, 137-180 (2009) · Zbl 1186.14016 |
| [17] | Deligne, P., Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, vol. 163 (1970), Springer-Verlag: Springer-Verlag Berlin-New York · Zbl 0244.14004 |
| [18] | Drézet, J.-M.; Narasimhan, M. S., Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. Math., 97, 53-94 (1989) · Zbl 0689.14012 |
| [19] | Griffiths, P.; Harris, J., Principles of Algebraic Geometry, Pure and Applied Mathematics (1978), Wiley-Interscience: Wiley-Interscience New York · Zbl 0408.14001 |
| [20] | Laszlo, Y.; Sorger, C., The line bundles on the moduli of parabolic G-bundles over curves and their sections, Ann. Sci. Éc. Norm. Supér., 30, 499-525 (1997) · Zbl 0918.14004 |
| [21] | Maruyama, M.; Yokogawa, K., Moduli of parabolic stable sheaves, Math. Ann., 293, 77-99 (1992) · Zbl 0735.14008 |
| [22] | Mehta, V. B.; Seshadri, C. S., Moduli of vector bundles on curves with parabolic structures, Math. Ann., 248, 205-239 (1980) · Zbl 0454.14006 |
| [23] | Narasimhan, M. S.; Ramadas, T. R., Factorisation of generalised theta functions. I, Invent. Math., 114, 565-623 (1993) · Zbl 0815.14014 |
| [24] | Narasimhan, M. S.; Seshadri, C. S., Stable and unitary vector bundles on a compact Riemann surface, Ann. Math., 82, 540-567 (1965) · Zbl 0171.04803 |
| [25] | Quillen, D. G., Determinants of Cauchy-Riemann operators on Riemann surfaces, Funct. Anal. Appl., 19, 37-41 (1985) · Zbl 0603.32016 |
| [26] | Simpson, C. T., Harmonic bundles on noncompact curves, J. Am. Math. Soc., 3, 713-770 (1990) · Zbl 0713.58012 |
| [27] | Sun, X., Degeneration of moduli spaces and generalized theta functions, J. Algebraic Geom., 9, 459-527 (2000) · Zbl 0971.14030 |
| [28] | Weil, A., Généralisation des fonctions abéliennes, J. Math. Pures Appl., 17, 47-87 (1938) · JFM 64.0361.02 |
| [29] | Yokogawa, K., Infinitesimal deformation of parabolic Higgs sheaves, Int. J. Math., 6, 125-148 (1995) · Zbl 0830.14007 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
