Abstract
Let Z be a finite subset of a compact connected Riemann Surface X. Let \({\fancyscript{M}_X^{lc}}\) denote the moduli space of pairs (L, D) where L is a line bundle on X and D is a logarithmic connection on L singular along Z. Then \({\fancyscript{M}_X^{lc}}\) has a natural symplectic structure [ω X ]. We show that the pair \({(\fancyscript{M}_X^{lc},[\omega_X])}\) determines X and there are no nonconstant algebraic functions on \({\fancyscript{M}_X^{lc}}\). We also prove a Torelli type theorem for the moduli space of parabolic bundles.
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Balaji V., Biswas I., del Bano Rollin S.: A Torelli type theorem for the moduli space of parabolic vector bundles over curves. Math. Proc. Camb. Philos. Soc. 130(2), 269–280 (2001)
Bhosle U.: Picard groups of the moduli spaces of vector bundles. Math. Ann. 314(2), 245–263 (1999)
Biswas I., Guruprasad K.: Principal bundles on open surfaces and invariant functions on Lie groups. Int. J. Math. 4(4), 535–544 (1993)
Biswas I., Munoz V.: The Torelli theorem for the moduli spaces of connections on a Riemann surface. Topology 46(3), 295–317 (2007)
Deligne, P.: Thorie de Hodge. II. Inst. Hautes tudes Sci. Publ. Math. no. 40, pp. 5–57 (1971)
Esnault, H., Viehweg, E.: Deligne-Beilinson cohomology. Beilinson’s conjectures on special values of L-functions. Perspect. Math., vol. 4, pp. 43–91. Academic Press, Boston, MA (1988)
Kouvidakis A., Pantev T.: The automorphism group of the moduli space of semistable vector bundles. Math. Ann. 302(2), 225–268 (1995)
Mumford D.: Curves and Their Jacobians. The University of Michigan Press, Ann Arbor, MI (1975)
Mumford D., Newstead P.: Periods of a moduli space of bundles on curves. Am. J. Math. 90, 1200–1208 (1968)
Narasimhan M.S., Ramanan S.: Deformations of the moduli space of vector bundles over an algebraic curve. Ann. Math. (2) 101, 391–417 (1975)
Narasimhan M.S., Ramanan S.: Moduli of vector bundles on a compact Riemann surface. Ann. Math. (2) 89, 14–51 (1969)
Narasimhan M.S., Ramdas T.R.: Factorisation of generalised theta functions. I. Invent. Math. 114(3), 565–623 (1993)
Ramanan S.: The moduli spaces of vector bundles over an algebraic curve. Math. Ann. 200, 69–84 (1973)
Simpson, C.T.: Higgs bundles and local systems. Inst. Hautes tudes Sci. Publ. Math. No. 75, pp. 5–95 (1992)
Simpson, C.T.: Moduli of representations of the fundamental group of a smooth projective variety. I. Inst. Hautes tudes Sci. Publ. Math. No. 79, pp. 47–129 (1994)
Simpson, C.T.: Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Etudes Sci. Publ. Math. No. 80 (1994), pp. 5–79 (1995)
Sun X.: Degeneration of moduli spaces and generalized theta functions. J. Algebr. Geom. 9(3), 459–527 (2000)
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Sebastian, R. Torelli theorems for moduli of logarithmic connections and parabolic bundles. manuscripta math. 136, 249–271 (2011). https://doi.org/10.1007/s00229-011-0446-9
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DOI: https://doi.org/10.1007/s00229-011-0446-9