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176
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 143 (4 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
Conformal blocks and generalized theta functions
 Comm. Math. Phys
, 1994
"... The aim of this paper is to construct a canonical isomorphism between two vector spaces associated to a Riemann surface X. The first of these spaces is the space of conformal blocks Bc(r) (also called the space of vacua), which plays an important role in conformal field theory. It is defined as foll ..."
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Cited by 141 (8 self)
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The aim of this paper is to construct a canonical isomorphism between two vector spaces associated to a Riemann surface X. The first of these spaces is the space of conformal blocks Bc(r) (also called the space of vacua), which plays an important role in conformal field theory. It is defined as follows: choose a point p ∈ X, and let AX be the
Stable pairs, linear systems and the Verlinde formula
 Invent. Math
, 1994
"... Let X be a smooth projective complex curve of genus g ≥ 2, let Λ→X be a line bundle of degree d> 0, and let (E, φ) be a pair consisting of a vector bundle E →X such that Λ 2 E = Λ and a section φ ∈ H 0 (E) − 0. This paper will study the moduli theory of such pairs. However, it is by no means a r ..."
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Cited by 108 (5 self)
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Let X be a smooth projective complex curve of genus g ≥ 2, let Λ→X be a line bundle of degree d> 0, and let (E, φ) be a pair consisting of a vector bundle E →X such that Λ 2 E = Λ and a section φ ∈ H 0 (E) − 0. This paper will study the moduli theory of such pairs. However, it is by no means a routine generalization of the wellknown theory of stable
The line bundles on the moduli of parabolic Gbundles over curves and their sections
, 1996
"... ..."
Infinite Grassmannians and moduli spaces of Gbundles
 Math. Annalen
, 1994
"... These are notes for my eight lectures given at the C.I.M.E. session on “Vector bundles on curves. New directions ” held at Cetraro (Italy) in June 1995. The work presented here was done in collaboration with M.S. Narasimhan and A. Ramanathan and appeared in [KNR]. These notes differ from [KNR] in th ..."
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Cited by 68 (4 self)
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These are notes for my eight lectures given at the C.I.M.E. session on “Vector bundles on curves. New directions ” held at Cetraro (Italy) in June 1995. The work presented here was done in collaboration with M.S. Narasimhan and A. Ramanathan and appeared in [KNR]. These notes differ from [KNR] in that we have
Vector bundles on curves and generalized theta functions: recent results and open problems
 CAMBRIDGE UNIVERSITY PRESS
, 1995
"... The moduli spaces of vector bundles on a compact Riemann surface carry a natural line bundle, the determinant bundle. The sections of this line bundle and its multiples constitute a nonabelian generalization of the classical theta functions. New ideas coming from mathematical physics have shed a ..."
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Cited by 58 (3 self)
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The moduli spaces of vector bundles on a compact Riemann surface carry a natural line bundle, the determinant bundle. The sections of this line bundle and its multiples constitute a nonabelian generalization of the classical theta functions. New ideas coming from mathematical physics have shed a new light on these spaces of sections—allowing notably to compute their dimension (Verlinde’s formula). This survey paper is devoted to giving an overview of these ideas and of the most important recent results on the subject.
Asymptotic faithfulness of the quantum SU(n) representations of the mapping class groups in the singular case
 In preparation
"... Abstract. We prove that the sequence of projective quantum SU(n) representations of the mapping class group of a closed oriented surface, obtained from the projective flat SU(n)Verlinde bundles over Teichmüller space, is asymptotically faithful, that is the intersection over all levels of the kerne ..."
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Cited by 44 (14 self)
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Abstract. We prove that the sequence of projective quantum SU(n) representations of the mapping class group of a closed oriented surface, obtained from the projective flat SU(n)Verlinde bundles over Teichmüller space, is asymptotically faithful, that is the intersection over all levels of the kernels of these representations is trivial, whenever the genus is at least 3. For the genus 2 case, this intersection is exactly the order two subgroup, generated by the hyperelliptic involution, in the case of even degree and n = 2. Otherwise the intersection is also trivial in the genus 2 case. 1.
The Picard group of the moduli of Gbundles on a curve
 Compositio Math. 112
, 1998
"... This paper is concerned with the moduli space of principal Gbundles on an algebraic curve, for G a complex semisimple group. While the case G = SLr, which corresponds to vector bundles, has been extensively studied in algebraic geometry, the general case has attracted much less attention until rec ..."
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Cited by 34 (3 self)
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This paper is concerned with the moduli space of principal Gbundles on an algebraic curve, for G a complex semisimple group. While the case G = SLr, which corresponds to vector bundles, has been extensively studied in algebraic geometry, the general case has attracted much less attention until recently, when it
GVSHEAVES, FOURIERMUKAI TRANSFORM, AND GENERIC VANISHING
, 2010
"... We prove a formal criterion for generic vanishing, in the sense originated by Green and Lazarsfeld and pursued further by Hacon, but in the context of an arbitrary FourierMukai correspondence. For smooth projective varieties we apply this to deduce a Kodairatype generic vanishing theorem for adjo ..."
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Cited by 30 (15 self)
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We prove a formal criterion for generic vanishing, in the sense originated by Green and Lazarsfeld and pursued further by Hacon, but in the context of an arbitrary FourierMukai correspondence. For smooth projective varieties we apply this to deduce a Kodairatype generic vanishing theorem for adjoint bundles associated to nef line bundles, and in fact a more general generic Nadeltype vanishing theorem for multiplier ideal sheaves. Still in the context of the Picard variety, the same method gives various other generic vanishing results, by reduction to standard vanishing theorems. We further use our criterion in order to address some examples related to generic vanishing on higher rank moduli spaces.