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72
The Verlinde Algebra and the Cohomology of the Grassmannian, preprint IASSNSHEP 93/41
, 1993
"... The article is devoted to a quantum field theory explanation of the relationship between the Verlinde algebra of the group U(k) at level N −k and the “quantum” cohomology of the Grassmannian of complex k planes in N space. In §2, I explain the relation between the Verlinde algebra and the gauged WZW ..."
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Cited by 110 (3 self)
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The article is devoted to a quantum field theory explanation of the relationship between the Verlinde algebra of the group U(k) at level N −k and the “quantum” cohomology of the Grassmannian of complex k planes in N space. In §2, I explain the relation between the Verlinde algebra and the gauged WZW model of G/G; in §3, I describe the quantum cohomology and its origin in a quantum field theory; and in §4, I present a path integral argument for mapping between them. My main goal in these lecture notes will be to elucidate a formula of Doron Gepner [1], which relates two mathematical objects, one rather old and one rather new. Along the way we will consider a few other matters as well. The old structure is the cohomology ring of the Grassmannian G(k, N) of
Geometric invariant theory and flips
 Jour. AMS
, 1996
"... Ever since the invention of geometric invariant theory, it has been understood that the quotient it constructs is not entirely canonical, but depends on a choice: the choice of a linearization of the group action. However, the founders of the subject never made a systematic study of this dependence. ..."
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Cited by 90 (3 self)
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Ever since the invention of geometric invariant theory, it has been understood that the quotient it constructs is not entirely canonical, but depends on a choice: the choice of a linearization of the group action. However, the founders of the subject never made a systematic study of this dependence. In light of its fundamental and elementary nature, this is a rather surprising gap, and this paper will attempt to fill it. In a way, this neglect is understandable, because the different quotients must be related by birational transformations, whose structure in higher dimensions is poorly understood. However, it has been considerably clarified in the last dozen years with the advent of Mori theory. In particular, the birational transformations that Mori called flips are ubiquitous in geometric invariant theory; indeed, one of our main results (3.3) describes the mild conditions under which the transformation between two quotients is given by a flip. This paper will not use any of the deep results of Mori theory, but the notion of a flip is central to it. The definition of a flip does not describe the birational transformation explicitly, but in the general case there is not much more to say. So to obtain more concrete results, hypotheses
The line bundles on the moduli of parabolic Gbundles over curves and their sections
, 1997
"... : Let X be a complex, smooth, complete and connected curve and G be a complex simple and simply connected algebraic group. We compute the Picard group of the stack of quasiparabolic Gbundles over X, describe explicitly its generators in case for classical G and G 2 and then identify the correspond ..."
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Cited by 69 (4 self)
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: Let X be a complex, smooth, complete and connected curve and G be a complex simple and simply connected algebraic group. We compute the Picard group of the stack of quasiparabolic Gbundles over X, describe explicitly its generators in case for classical G and G 2 and then identify the corresponding spaces of global sections with the vacua spaces of Tsuchiya, Ueno and Yamada. The method uses the uniformization theorem which describes these stacks as double quotients of certain infinite dimensional algebraic groups. We describe also the dualizing bundle of the stack of Gbundles and show that it admits a unique square root, which we construct explicitly. If G is not simply connected, the square root depends on the choice of a thetacharacteristic. These results about stacks allow to recover the DrezetNarasimhan theorem (for the coarse moduli space) and to show an analogous statement when G = Sp 2r . We prove also that the coarse moduli spaces of semistable SOr bundles are not loca...
Gauge theory for embedded surfaces
 I, Topology
, 1993
"... (i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly ..."
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Cited by 68 (6 self)
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(i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly
GEOMETRIC PROOFS OF HORN AND SATURATION CONJECTURES
, 2005
"... We provide a geometric proof of the Schubert calculus interpretation of the Horn conjecture, and show how the saturation conjecture follows from it. The geometric proof gives a strengthening of Horn and saturation conjectures. We also establish transversality theorems for Schubert calculus in non ..."
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Cited by 30 (4 self)
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We provide a geometric proof of the Schubert calculus interpretation of the Horn conjecture, and show how the saturation conjecture follows from it. The geometric proof gives a strengthening of Horn and saturation conjectures. We also establish transversality theorems for Schubert calculus in nonzero characteristic.
Kähler spaces, nilpotent orbits, and singular reduction
, 2001
"... For a stratified symplectic space, a suitable concept of stratified Kähler polarization, defined in terms of an appropriate LieRinehart algebra, encapsulates Kähler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified Kähler spac ..."
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Cited by 21 (16 self)
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For a stratified symplectic space, a suitable concept of stratified Kähler polarization, defined in terms of an appropriate LieRinehart algebra, encapsulates Kähler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified Kähler space. This notion establishes an intimate relationship between nilpotent orbits, singular reduction, invariant theory, reductive dual pairs, Jordan triple systems, symmetric domains, and prehomogeneous spaces; in particular, in the world of singular Poisson geometry, the closures of principal holomorphic nilpotent orbits, positive definite hermitian JTS’s, and certain prehomogeneous spaces appear as different incarnations of the same structure. The space of twisted representations of the fundamental group of a closed surface in a compact Lie group or, equivalently, a moduli space of central YangMills connections on a principal bundle over a surface, inherits a (positive) normal (stratified) Kähler structure, as does the closure of a holomorphic nilpotent orbit in a semisimple Lie algebra of hermitian type or, equivalently, the closure of the stratum of the associated prehomogeneous space of parabolic type arising from the corresponding
Moduli spaces of parabolic Higgs bundles and parabolic K(D) pairs over smooth curves: I
, 1998
"... This paper concerns the moduli spaces of rank two parabolic Higgs bundles and parabolic K(D) pairs over a smooth curve. Precisely which parabolic bundles occur in stable K(D) pairs and stable Higgs bundles is determined. Using Morse theory, the moduli space of parabolic Higgs bundles is shown to b ..."
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Cited by 19 (2 self)
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This paper concerns the moduli spaces of rank two parabolic Higgs bundles and parabolic K(D) pairs over a smooth curve. Precisely which parabolic bundles occur in stable K(D) pairs and stable Higgs bundles is determined. Using Morse theory, the moduli space of parabolic Higgs bundles is shown to be a noncompact, connected, simply connected manifold, and a computation of its Poincaré polynomial is given.
The quantization conjecture revisited
 Ann. of Math
"... Version: 8/5/98 ABSTRACT: I prove the following strong version of the quantization conjecture of Guillemin and Sternberg: for a reductive group action on a smooth, compact, polarized variety ( X, ℓ), the cohomologies of ℓ over the GIT quotient are equal to the invariant parts of the cohomologies ove ..."
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Cited by 18 (4 self)
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Version: 8/5/98 ABSTRACT: I prove the following strong version of the quantization conjecture of Guillemin and Sternberg: for a reductive group action on a smooth, compact, polarized variety ( X, ℓ), the cohomologies of ℓ over the GIT quotient are equal to the invariant parts of the cohomologies over X. This generalizes the theorem of [GS], which concerned the spaces of global sections, and strengthens its subsequent extensions ([M], [V]) to RiemannRoch numbers. A remarkable byproduct is the rigidity of cohomology of certain vector bundles over the GIT quotient under a small change in the defining polarization. One application is a new proof of the BorelWeilBott theorem of [T] for the moduli stack M of Gbundles over a curve, and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures. Also studied are equivariant holomorphic forms and the equivariant Hodgetode Rham spectral sequences for X and its strata, whose collapse at E1 is shown. Collapse of the Hodgetode Rham sequence for M is hence deduced.
Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers
 Adv. Math
, 2009
"... Abstract. The space of unitary local systems of rank one on the complement of an arbitrary divisor in a complex projective algebraic variety can be described in terms of parabolic line bundles. We show that multiplier ideals provide natural stratifications of this space. We prove a structure theorem ..."
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Cited by 18 (2 self)
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Abstract. The space of unitary local systems of rank one on the complement of an arbitrary divisor in a complex projective algebraic variety can be described in terms of parabolic line bundles. We show that multiplier ideals provide natural stratifications of this space. We prove a structure theorem for these stratifications in terms of complex tori and convex rational polytopes, generalizing to the quasiprojective case results of GreenLazarsfeld and Simpson. As an application we show the polynomial periodicity of Hodge numbers h q,0 of congruence covers in any dimension, generalizing results of E. Hironaka and Sakuma. We extend the structure theorem and polynomial periodicity to the setting of cohomology of unitary local systems. In particular, we obtain a generalization of the polynomial periodicity of Betti numbers of unbranched congruence covers due to SarnakAdams. We derive a geometric characterization of finite abelian covers, which recovers the classic one and the one of Pardini. We use this, for example, to prove a conjecture of Libgober about Hodge numbers of abelian covers. 1.
Chern–Simons states at genus one
 Commun. Math. Phys
, 1994
"... We present a rigorous analysis of the Schrödinger picture quantization for the SU(2) ChernSimons theory on 3manifold torus×line, with insertions of Wilson lines. The quantum states, defined as gauge covariant holomorphic functionals of smooth su(2)connections on the torus, are expressed by degree ..."
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Cited by 18 (5 self)
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We present a rigorous analysis of the Schrödinger picture quantization for the SU(2) ChernSimons theory on 3manifold torus×line, with insertions of Wilson lines. The quantum states, defined as gauge covariant holomorphic functionals of smooth su(2)connections on the torus, are expressed by degree 2k thetafunctions satisfying additional conditions. The conditions are obtained by splitting the space of semistable su(2)connections into nine submanifolds and by analyzing the behavior of states at four codimension 1 strata. We construct the KnizhnikZamolodchikovBernard connection allowing to compare the states for different complex structures of the torus and different positions of the Wilson lines. By letting two Wilson lines come together, we prove a recursion relation for the dimensions of the spaces of states which, together with the (unproven) absence of states for spins> 1 2level implies the Verlinde dimension formula. 1.