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79
Higher and derived stacks: a global overview
, 2005
"... These are expended notes of my talk at the summer institute in algebraic geometry (Seattle, JulyAugust 2005), whose main purpose is to present a global overview on the theory of higher and derived stacks. This text is far from being exhaustive but is intended to cover a rather large part of the sub ..."
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Cited by 60 (5 self)
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These are expended notes of my talk at the summer institute in algebraic geometry (Seattle, JulyAugust 2005), whose main purpose is to present a global overview on the theory of higher and derived stacks. This text is far from being exhaustive but is intended to cover a rather large part of the subject, starting from the motivations and the foundational material, passing through some examples and basic notions, and ending with some more recent developments and open questions.
Mirror symmetry, Langlands duality and Hitchin systems
 arXiv: math.AG/0205236 56 Hausel, T. and Sturmfels, B
"... Abstract. Among the major mathematical approaches to mirror symmetry are those of BatyrevBorisov and StromingerYauZaslow (SYZ). The first is explicit and amenable to computation but is not clearly related to the physical motivation; the second is the opposite. Furthermore, it is far from obvious ..."
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Cited by 58 (9 self)
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Abstract. Among the major mathematical approaches to mirror symmetry are those of BatyrevBorisov and StromingerYauZaslow (SYZ). The first is explicit and amenable to computation but is not clearly related to the physical motivation; the second is the opposite. Furthermore, it is far from obvious that mirror partners in one sense will also be mirror partners in the other. This paper concerns a class of examples that can be shown to satisfy the requirements of SYZ, but whose Hodge numbers are also equal. This provides significant evidence in support of SYZ. Moreover, the examples are of great interest in their own right: they are spaces of flat SLrconnections on a smooth curve. The mirror is the corresponding space for the Langlands dual group PGLr. These examples therefore throw a bridge from mirror symmetry to the duality theory of Lie groups and, more broadly, to the geometric Langlands program. When it emerged in the early 1990s, mirror symmetry was an aspect of theoretical physics, and specifically a duality between quantum field theories. Since then, many people have tried to place it on a mathematical foundation. Their labors have built up a fascinating but somewhat unruly subject. It describes some sort of relation between pairs of Calabi
Motivic degree zero Donaldson–Thomas invariants
, 2009
"... We refine the degree zero Donaldson–Thomas invariants of affine threespace C³ using the virtual motive on Hilb n (C³), the Hilbert scheme of n points on C³, defined via its description as a global degeneracy locus. Assuming a conjecture about vanishing cycles, we compute the motivic generating fu ..."
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Cited by 42 (2 self)
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We refine the degree zero Donaldson–Thomas invariants of affine threespace C³ using the virtual motive on Hilb n (C³), the Hilbert scheme of n points on C³, defined via its description as a global degeneracy locus. Assuming a conjecture about vanishing cycles, we compute the motivic generating function of C³ using a calculation involving the motive of the space of pairs of commuting matrices. We express the generating function of virtual motives of Hilbert schemes of points of an arbitrary smooth and quasiprojective threefold as a motivic exponential of the generating series of motives of projective spaces, obtaining an expression resembling Göttsche’s formula for an algebraic surface. As a consequence, we derive a formula for the generating series of the corresponding weight polynomials in terms of deformed MacMahon functions.
Asymptotic behaviour of tame harmonic bundles and an application to pure twistor Dmodules
"... We study the asymptotic behaviour of tame harmonic bundles. First of all, we prove a local freeness of the prolongation by an increasing order. Then we obtain the polarized mixed twistor structure. As one of the applications, we obtain the norm estimate of holomorphic or flat sections by weight filt ..."
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Cited by 40 (7 self)
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We study the asymptotic behaviour of tame harmonic bundles. First of all, we prove a local freeness of the prolongation by an increasing order. Then we obtain the polarized mixed twistor structure. As one of the applications, we obtain the norm estimate of holomorphic or flat sections by weight filtrations of the monodromies. As other application, we establish the correspondence of tame harmonic bundle and a pure twistor Dmodule of weight 0. Keywords: Higgs fields, harmonic bundle, variation of Hodge structure, mixed twistor structure, Dmodule.
Compactification of moduli of Higgs bundles
 J. Reine Angew. Math
, 1998
"... In this paper we consider a canonical compactification of M, the moduli space of stable Higgs bundles with fixed determinant of odd degree over a Riemann surface Σ, producing a projective variety ¯ M = M ∪ Z. We give a detailed study of the spaces ¯M, Z and M. In doing so we reprove some assertions ..."
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Cited by 34 (8 self)
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In this paper we consider a canonical compactification of M, the moduli space of stable Higgs bundles with fixed determinant of odd degree over a Riemann surface Σ, producing a projective variety ¯ M = M ∪ Z. We give a detailed study of the spaces ¯M, Z and M. In doing so we reprove some assertions of Laumon and Thaddeus on the nilpotent cone. 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
Geometric Realization of the SegalSugawara Construction, in Topology, geometry and quantum field theory
 Math. Soc. Lecture Note Ser. 308
, 2004
"... Abstract. We apply the technique of localization for vertex algebras to the SegalSugawara construction of an “internal ” action of the Virasoro algebra on affine KacMoody algebras. The result is a lifting of twisted differential operators from the moduli of curves to the moduli of curves with bund ..."
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Cited by 16 (5 self)
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Abstract. We apply the technique of localization for vertex algebras to the SegalSugawara construction of an “internal ” action of the Virasoro algebra on affine KacMoody algebras. The result is a lifting of twisted differential operators from the moduli of curves to the moduli of curves with bundles, with arbitrary decorations and complex twistings. This construction gives a uniform approach to a collection of phenomena describing the geometry of the moduli spaces of bundles over varying curves: the KZB equations and heat kernels on nonabelian theta functions, their critical level limit giving the quadratic parts of the BeilinsonDrinfeld quantization of the Hitchin system, and their infinite level limit giving a Hamiltonian description of the isomonodromy equations. 1. Introduction. 1.1. Uniformization. Let G be a complex connected simplyconnected simple algebraic group with Lie algebra g, and X a smooth projective curve over C. The geometry of G–bundles on X is intimately linked to representation theory of the affine KacMoody algebra ̂g, the universal central extension of the loop algebra Lg = g⊗K, where
Irregular connections and Kac–Moody root systems
"... Abstract. Some moduli spaces of irregular connections on the trivial bundle over the Riemann sphere will be identified with Nakajima quiver varieties. In particular this enables us to associate a Kac–Moody root system to such connections (yielding many isomorphisms between such moduli spaces, via th ..."
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Abstract. Some moduli spaces of irregular connections on the trivial bundle over the Riemann sphere will be identified with Nakajima quiver varieties. In particular this enables us to associate a Kac–Moody root system to such connections (yielding many isomorphisms between such moduli spaces, via the reflection functors for the corresponding Weyl group). The possibility of ‘reading ’ a quiver in different ways also yields numerous isomorphisms between such moduli spaces, often between spaces of connections on different rank bundles and with different polar divisors. Finally some results of CrawleyBoevey on the existence of stable connections will be extended to this more general context. 1.