Nitsure, Nitin (1986) Cohomology of the moduli of parabolic vector bundles Proceedings of the Indian Academy of Sciences  Mathematical Sciences, 95 (1). pp. 6177. ISSN 02534142

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Official URL: http://www.ias.ac.in/j_archive/mathsci/95/1/6177/...
Related URL: http://dx.doi.org/10.1007/BF02837250
Abstract
The purpose of this paper is to compute the Betti numbers of the moduli space ofparabolic vector bundles on a curve (see Seshadri [7], [8] and Mehta & Seshadri [4]), in the case where every semistable parabolic bundle is necessarily stable. We do this by generalizing the method of Atiyah and Bott [1] in the case of moduli of ordinary vector bundles. Recall that (see Seshadri [7]) the underlying topological space of the moduli of parabolic vector bundles is the space of equivalence classes of certain unitary representations of a discrete subgroup Γ which is a lattice in PSL (2,R). (The lattice Γ need not necessarily be cocompact). While the structure of the proof is essentially the same as that of Atiyah and Bott, there are some difficulties of a technical nature in the parabolic case. For instance the HarderNarasimhan stratification has to be further refined in order to get the connected strata. These connected strata turn out to have different codimensions even when they are part of the same HarderNarasimhan strata. If in addition to 'stable = semistable' the rank and degree are coprime, then the moduli space turns out to be torsionfree in its cohomology. The arrangement of the paper is as follows. In § 1 we prove the necessary basic results about algebraic families of parabolic bundles. These are generalizations of the corresponding results proved by Shatz [9]. Following this, in § 2 we generalize the analytical part of the argument of Atiyah and Bott (§ 14 of [1]). Finally in § 3 we show how to obtain an inductive formula for the Betti numbers of the moduli space. We illustrate our method by computing explicitly the Betti numbers in the special case of rank = 2, and one parabolic point.
Item Type:  Article 

Source:  Copyright of this article belongs to Indian Academy of Sciences. 
Keywords:  Cohomology; Parabolic Vector Bundles; Moduli Space; Betti Numbers; Algebraic Family; Sobolev Spaces 
ID Code:  24795 
Deposited On:  30 Nov 2010 09:12 
Last Modified:  17 May 2016 08:24 
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