Betti Numbers of the Moduli Space of Rank 3 Parabolic Higgs Bundles

By O. Garc'ia-Prada, Peter B. Gothen, V. Muñoz. 2004

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Parabolic Higgs bundles on a Riemann surface are of interest for many reasons, one of them being their importance in the study of representations of the fundamental group of the punctured surface in the complex general linear group. In this paper we calculate the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles with fixed and non-fixed determinant, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we carry out a careful analysis of them by studying their variation with this parameter. Thus we obtain in particular information about the topology of the moduli spaces of parabolic triples for the value of the parameter relevant to the study of parabolic Higgs bundles. The remaining critical submanifolds are also described: one of them is the moduli space of parabolic bundles, while the remaining ones have a description in terms of symmetric products of the Riemann surface. As another consequence of our Morse theoretic analysis, we obtain a proof of the parabolic version of a theorem of Laumon, which states that the nilpotent cone (the preimage of zero under the Hitchin map) is a Lagrangian subvariety of the moduli space of parabolic Higgs bundles.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b1203b3e3d0e3cd37b5030cfcd368b9301ce1a4d

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Citations in the corpus

Arithmetic harmonic analysis on character and quiver varieties II (2008, background): “homology as one crosses a wall of a certain chamber structure in the space of parameters. In hyperka¨hler situations, however, it has been observed that no change takes place; see for example [1] and [13]. Generalizing the argument of Corollary 2.2.4 of [23] here we prove that for a dense subset (in”
Hodge polynomials of SL (2,C)-character varieties for curves of small genus (2013, methodology): “In addition, the Poincaré polynomials have been computed in the context of parabolic Higgs bundles for G = SL(2,C) in [1] and for G = SL(3,C) and GL(3,C) in [13].”
Poincaré polynomials of character varieties, Macdonald polynomials and affine Springer fibers (2017, methodology): “y of this space was initiated by Hausel and Thaddeus for the case of rank 2 in [HT04] (without marked points). The case of rank 3 with marked points was studied by Garc´ıa-Prada, Gothen and Mun˜oz in [GPGMn07]. More partial results were obtained by Garc´ıa-Prada, Heinloth and Schmitt in [GPHS11]. In”