Cohomology of the moduli of parabolic vector bundles |
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Authors: | Nitin Nitsure |
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Institution: | (1) School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, 400 005 Bombay, India |
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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 semi-stable 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 co-compact).
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 Harder-Narasimhan 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 Harder-Narasimhan
strata.
If in addition to ‘stable = semistable’ the rank and degree are coprime, then the moduli space turns out to be torsion-free
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. |
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Keywords: | Cohomology parabolic vector bundles moduli space Betti numbers algebraic family Sobolev spaces |
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