The Betti Numbers of the Moduli Space of Stable Rank 3 Higgs Bundles on a Riemann Surface

Gothen, Peter B.

doi:10.1142/s0129167x94000449

Cited by 76 publications

(109 citation statements)

References 14 publications

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“…These results are related to the results of Domic and Toledo [4,26,27] and, as being pointed out to the author recently, are also related to the work of Gothen [8] which computed the Poincaré polynomials for the components of Hom(π 1 (X), PSL(3, C))/ PSL(3, C), where deg(E) is coprime to 3. Most of this research was carried out while the author was at the University of Maryland at College Park.…”

Section: Introductionsupporting

confidence: 75%

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The moduli of flat PU(2,1) structures on Riemann surfaces

Xia¹

2000

Pacific J. Math.

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For a compact Riemann surface X of genus g > 1, Hom(π 1 (X), U(p, 1))/ U(p, 1) is the moduli space of flat U(p, 1)connections on X. There is an integer invariant, τ , the Toledo invariant associated with each element in Hom(π 1 (X), U(p, 1))/ U(p, 1). If q = 1, then −2(g − 1) ≤ τ ≤ 2(g − 1). This paper shows that Hom(π 1 (X), U(p, 1))/ U(p, 1) has one connected component corresponding to each τ ∈ 2Z with −2(g − 1) ≤ τ ≤ 2(g − 1). Therefore the total number of connected components is 2(g − 1) + 1.

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Section: Introductionsupporting

confidence: 75%

“…When τ is not an integer, these are the type (2,1) and (1,1,1) spaces in [8]. Note the (1,2) types give τ < 0 and therefore need not be considered here.…”

Section: Hodge Bundles and Deformationmentioning

confidence: 99%

The moduli of flat PU(2,1) structures on Riemann surfaces

Xia¹

2000

Pacific J. Math.

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“…It is known that the middle cohomology of M 2 has dimension g = 1 2 (2g −2)+1 by [41] and that the middle Betti number of M 3 is 2g 2 − g = 1 2 (2g − 2) 2 + 3 2 (2g − 2) + 1 dimensional [25]. For n ≥ 4 the middle Betti number of M n is not known.…”

Section: Purity Conjecturementioning

confidence: 99%

Mixed Hodge polynomials of character varieties

Hausel¹,

2008

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We calculate the E-polynomials of certain twisted GL(n, C)-character varieties M n of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type GL(n, F q ) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n, C)-character variety. The calculation also leads to several conjectures about the cohomology of M n : an explicit conjecture for its mixed Hodge polynomial; a conjectured curious Hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver. We prove these conjectures for n = 2.

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“…This allows us to bring in the tools of Morse-Bott theory, as was done by Hitchin in [35] and Gothen in [26,27]. The corresponding action is rotation of the Higgs field through θ:…”

Section: The Circle Action and Complex Variations Of Hodge Structurementioning

confidence: 99%

“…Historically, after Hitchin's calculation in [35], the Betti numbers for rank 3 were computed by Gothen [26,27] and the Betti numbers for rank 4 were calculated by García-Prada, Heinloth, and Schmitt [24]. All of these calculations stem from Hitchin's Morse-theoretic localization technique [35], although the rank 4 calculation is motivic in nature.…”

mentioning

confidence: 99%