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55
Mixed Hodge polynomials of character varieties
"... We calculate the Epolynomials of certain twisted GL(n,C)character varietiesMn of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lietype GL(n,Fq) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geomet ..."
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Cited by 37 (9 self)
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We calculate the Epolynomials of certain twisted GL(n,C)character varietiesMn of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lietype GL(n,Fq) 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 Mn: 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
BETTI NUMBERS OF THE MODULI SPACE OF RANK 3 PARABOLIC HIGGS BUNDLES
, 2004
"... Abstract. 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 modul ..."
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Cited by 31 (8 self)
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Abstract. 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, 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. 1.
Mirror symmetry and Langlands duality in the nonabelian Hodge theory of a curve
 Geometric Methods in Algebra and Number Theory. Progress in Mathematics
, 2005
"... This is a survey of results and conjectures on mirror symmetry phenomena in the nonAbelian Hodge theory of a curve. We start with the conjecture of Hausel–Thaddeus which claims that certain Hodge numbers of moduli spaces of flat SL(n, C) and P GL(n, C)connections on a smooth projective algebraic cu ..."
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Cited by 22 (6 self)
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This is a survey of results and conjectures on mirror symmetry phenomena in the nonAbelian Hodge theory of a curve. We start with the conjecture of Hausel–Thaddeus which claims that certain Hodge numbers of moduli spaces of flat SL(n, C) and P GL(n, C)connections on a smooth projective algebraic curve agree. We then change our point of view in the nonAbelian Hodge theory of the curve, and concentrate on the SL(n, C) and P GL(n, C) character varieties of the curve. Here we discuss a recent conjecture of Hausel– RodriguezVillegas which claims, analogously to the above conjecture, that certain Hodge numbers of these character varieties also agree. We explain that for Hodge numbers of character varieties one can use arithmetic methods, and thus we end up explicitly calculating, in terms of Verlindetype formulas, the number of representations of the fundamental group into the finite groups SL(n, Fq) and P GL(n, Fq), by using the character tables of these finite groups of Lie type. Finally we explain a conjecture which enhances the previous result, and gives a simple formula for the mixed Hodge polynomials, and in particular for the Poincaré polynomials of these character varieties, and detail the relationship to results of Hitchin, Gothen, Garsia–Haiman and Earl–Kirwan. One consequence of this conjecture is a curious Poincaré duality type of symmetry, which leads to a conjecture, similar to Faber’s conjecture on the moduli space of curves, about a strong Hard Lefschetz theorem for the character variety, which can be considered as a generalization of both the Alvis– Curtis duality in the representation theory of finite groups of Lie type and a recent result of the author on the quaternionic geometry of matroids. 1
The construction problem in Kähler geometry
, 2004
"... One of the most surprising things in algebraic geometry is the fact that algebraic varieties over the complex numbers benefit from a collection of metric properties which strongly influence their topological and geometric shapes. The existence of a Kähler metric leads to all sorts of Hodge theoretic ..."
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Cited by 17 (2 self)
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One of the most surprising things in algebraic geometry is the fact that algebraic varieties over the complex numbers benefit from a collection of metric properties which strongly influence their topological and geometric shapes. The existence of a Kähler metric leads to all sorts of Hodge theoretical restrictions on the homotopy types of algebraic varieties. On the other hand, a sparse collection of examples shows that the remaining liberty is nontrivially large. Paradoxically, with all of this information, the research field remains as wide open as it was many decades ago, because the gap between the known restrictions, and the known examples of what can occur, only seems to grow wider and wider the more closely we look at it. In spite of the differentialgeometric nature of the questions and methods, the origins of the situation are very algebraic. We look at subvarieties of projective space over the complex numbers. The main overarching problem in algebraic geometry is to understand the classification of algebrogeometric objects. The topology of the usual complexvalued points of a variety plays
Quasi BPS Wilson loops, localization of loop equation by homology and exact beta function
 in the largeN limit of SU(N) YangMills theory”, [hepth/0809.4662
"... We localize the loop equation of largeN Y M theory in the ASD variables on a critical equation for an effective action by means of homological methods as opposed to the cohomological localization of equivariantly closed forms in local field theory. Our localization occurs for some special simple qu ..."
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Cited by 12 (4 self)
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We localize the loop equation of largeN Y M theory in the ASD variables on a critical equation for an effective action by means of homological methods as opposed to the cohomological localization of equivariantly closed forms in local field theory. Our localization occurs for some special simple quasi BPS Wilson loops, that have no perimeter divergence and no cusp anomaly for backtracking cusps, in a partial EguchiKawai reduction from four to two dimensions of the noncommutative theory in the limit of infinite noncommutativity and in a lattice regularization in which the ASD integration variables live at the points of the lattice, thus implying an embedding of parabolic Higgs bundles in the Y M functional integral. Homological localization is based on an analogy with cohomological localization. The analog of the invariance of the cohomological class of a closed form for the addition of a coboundary is the zigzag symmetry, i.e. the invariance of the holonomy class of a quasi BPS Wilson loop for the addition of the boundary ofa tiny strip, of size of the cutoff. The analog of the action being a closed
Mixed Hodge polynomials of character varieties  With an Appendix by Nicholas M. Katz
, 2008
"... We calculate the Epolynomials of certain twisted GL(n, C)character varieties Mn of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lietype GL(n, Fq) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geo ..."
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Cited by 10 (0 self)
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We calculate the Epolynomials of certain twisted GL(n, C)character varieties Mn of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lietype GL(n, Fq) 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 Mn: 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.
Degenerate Cohomological Hall algebra and quantized DonaldsonThomas invariants for mloop quivers
, 2011
"... We derive a combinatorial formula for quantized DonaldsonThomas invariants of the mloop quiver. Our main tools are the combinatorics of noncommutative Hilbert schemes and a degenerate version of the Cohomological Hall algebra of this quiver. 1 ..."
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Cited by 7 (2 self)
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We derive a combinatorial formula for quantized DonaldsonThomas invariants of the mloop quiver. Our main tools are the combinatorics of noncommutative Hilbert schemes and a degenerate version of the Cohomological Hall algebra of this quiver. 1
KATZ’S MIDDLE CONVOLUTION ALGORITHM
, 2006
"... Abstract. This is an expository account of Katz’s middle convolution operation on local systems over P 1 − {q1,...,qn}. We describe the Betti and de Rham versions, and point out that they give isomorphisms between different moduli spaces of local systems, following Völklein, DettweilerReiter, Harao ..."
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Cited by 7 (0 self)
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Abstract. This is an expository account of Katz’s middle convolution operation on local systems over P 1 − {q1,...,qn}. We describe the Betti and de Rham versions, and point out that they give isomorphisms between different moduli spaces of local systems, following Völklein, DettweilerReiter, HaraokaYokoyama. Kostov’s program for applying the Katz algorithm is to say that in the range where middle convolution no longer reduces the rank, one should give a direct construction of local systems. This has been done by Kostov and CrawleyBoevey. We describe here an alternative construction using the notion of cyclotomic harmonic bundles: these are like variations of Hodge structure except that the Hodge decomposition can go around in a circle. 1.