Poincaré polynomial of the moduli spaces of parabolic bundles

Holla, Yogish I.

doi:10.1007/bf02878682

Cited by 13 publications

(16 citation statements)

References 8 publications

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“…In case (a) the corresponding critical subvariety can obviously be identified with the moduli space of ordinary parabolic bundles. Its Betti numbers can be computed from a formula given by Nitsure [34] and Holla [25]. In Section 10 below we work out explicitly what their formula gives for the Poincaré polynomial in our situation of rank three parabolic bundles.…”

Section: Morse Indicesmentioning

confidence: 99%

“…The Betti numbers of the moduli space of parabolic vector bundles were computed by Nitsure [34] and Holla [25]. Here we work out Holla's formula for the special case when the rank is 3 and all flags at the parabolic points are full.…”

Section: Betti Numbers Of the Moduli Space Of Rank Three Parabolic Bumentioning

confidence: 99%

“…So we can choose the degree coprime with the rank and the weights as small as convenient, to facilitate our computations. In Section 10, based on the computations by Nitsure [34] and Holla [25] of the Betti numbers of the moduli space of parabolic bundles, we work out the formula for the rank 3 case. In Section 11 we collect all the computations, to give the Poincaré polynomial of the rank 3 moduli space of parabolic Higgs bundles.…”

Section: Introductionmentioning

confidence: 99%

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Betti numbers of the moduli space of rank 3 parabolic Higgs bundles

Garcı́a-Prada¹,

Gothen²,

Muñoz³

2007

Memoirs of the AMS

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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.

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Section: Morse Indicesmentioning

confidence: 99%

Section: Betti Numbers Of the Moduli Space Of Rank Three Parabolic Bumentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Betti numbers of the moduli space of rank 3 parabolic Higgs bundles

Garcı́a-Prada¹,

Gothen²,

Muñoz³

2007

Memoirs of the AMS

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“…First of all, the topology of V s s / Iso 0 has been extensively studied, and in particular it is known to be connected, see [8,3] or [9]. But V s s / Iso 0 = C s s,δ /G C A,δ and G C A,δ is connected (since π 2 (SL(2, C)) is trivial), so it follows that the set C s s,δ of stable connections is connected.…”

Section: The Second Kempf-ness-type Theoremmentioning

confidence: 99%

Around Polygons in ℝ3 and S3

Millson

Poritz

2001

Commun. Math. Phys.

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We survey certain moduli spaces in low dimensions and some of the geometric structures that they carry, and then construct identifications among all of these spaces. In particular, we identify the moduli spaces of polygons in R 3 and S 3 , the moduli space of restricted representations of the fundamental group of a punctured 2-sphere, the moduli space of flat connections on a punctured sphere, the moduli space of parabolic bundles on a sphere, the moduli space of weighted points on CP 1 and the symplectic quotient of SO(3) acting diagonally on (S 2 ) n . All of these spaces depend on parameters and some the above identifications require the parameters to be small. One consequence of this work is that these spaces are all biholomorphic with respect to the most natural complex structures they can each be given.

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“…Their proof uses the Morse theory of the energy functional on a space homotopy equivalent to M(X − ). Inductive formulas for the Poincaré polynomials of the moduli spaces of parabolic bundles using these techniques have been given by Nitsure [25] and Holla [14]. is the marking corresponding to holonomy −1 around a puncture.…”

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confidence: 99%