Biswas, Indranil ; Raghavendra, N.
(2005)
*Line bundles over a moduli space of logarithmic connections on a Riemann surface*
Geometric And Functional Analysis, 15
(4).
pp. 780-808.
ISSN 1016-443X

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Official URL: http://www.springerlink.com/content/n388253q66u195...

Related URL: http://dx.doi.org/10.1007/s00039-005-0523-x

## Abstract

We consider logarithmic connections, on rank n and degree d vector bundles over a compact Riemann surface X, singular over a fixed point x_{0} ∈ X with residue in the center of gl(n,C) the integers n and d are assumed to be mutually coprime. A necessary and sufficient condition is given for a vector bundle to admit such a logarithmic connection. We also compute the Picard group of the moduli space of all such logarithmic connections. Let N_{D}(L) denote the moduli space of all such logarithmic connections, with the underlying vector bundle being of fixed determinant L, and inducing a fixed logarithmic connection on the determinant line L. Let N'D(L)⊂N_{D}(L) be the Zariski open dense subset parametrizing all connections such that the underlying vector bundle is stable. The space of all global sections of certain line bundles N'_{D}(L) on are computed. In particular, there are no nonconstant algebraic functions on N'_{D}(L). Therefore, there are no nonconstant algebraic functions on N_{D}(L) although N_{D}(L) is biholomorphic to a representation space which admits nonconstant algebraic functions. The moduli space N'_{D}(L) admits a natural compactification by a smooth divisor. We investigate numerically effectiveness of this divisor at infinity. It turns out that the divisor is not numerically effective in general.

Item Type: | Article |
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Source: | Copyright of this article belongs to Birkhauser-Verlag. |

ID Code: | 3448 |

Deposited On: | 11 Oct 2010 10:02 |

Last Modified: | 11 Oct 2010 10:02 |

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