Line bundles over a moduli space of logarithmic connections on a Riemann surface

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 x0 ∈ 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 ND(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)⊂ND(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 ND(L) although ND(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
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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