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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
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
RiemannRoch for algebraic stacks: I
 COMPOSITIO MATH
"... In this paper we establish RiemannRoch and LefschtezRiemannRoch theorems for arbitrary proper maps between algebraic stacks in the sense of Artin. The RiemannRoch theorem is established as a natural transformation between the Gtheory of algebraic stacks and topological Gtheory for stacks: we d ..."
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Cited by 4 (1 self)
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In this paper we establish RiemannRoch and LefschtezRiemannRoch theorems for arbitrary proper maps between algebraic stacks in the sense of Artin. The RiemannRoch theorem is established as a natural transformation between the Gtheory of algebraic stacks and topological Gtheory for stacks: we define the latter as the localization of Gtheory by topological Khomology. The LefschtezRiemannRoch is an extension of this including the action of a torus for DeligneMumford stacks. This generalizes the corresponding RiemannRoch theorem (LefschetzRiemannRoch theorem) for proper maps between schemes (that are also equivariant for the action of a torus, respectively) making use of some fundamental results due to Vistoli and Toen. A key result established here is that topological Gtheory (as well as rational Gtheory) has cohomological descent on the isovariant étale site of an algebraic stack. This extends cohomological descent for topological Gtheory on schemes as proved by Thomason.
Representability for some moduli stacks of framed sheaves
 MANUSCRIPTA MATH. 109, 85–91 (2002)
, 2002
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MODULI SPACES OF FRAMED SHEAVES ON CERTAIN RULED SURFACES OVER ELLIPTIC CURVES
"... Abstract. Fix a ruled surface S obtained as the projective completion of a line bundle L on a complex elliptic curve C; we study the moduli problem of parametrizing certain pairs consisting of a sheaf E on S and a map of E to a fixed reference sheaf on S. We prove that the full moduli stack for this ..."
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Cited by 1 (1 self)
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Abstract. Fix a ruled surface S obtained as the projective completion of a line bundle L on a complex elliptic curve C; we study the moduli problem of parametrizing certain pairs consisting of a sheaf E on S and a map of E to a fixed reference sheaf on S. We prove that the full moduli stack for this problem is representable by a scheme in some cases. Moreover, the moduli stack admits an action by the group C ∗ , and we determine its fixedpoint set, which leads to explicit formulas for the rational homology of the moduli space. 1.
RiemannRoch for Algebraic Stacks: III . . .
"... In this paper we apply the RiemannRoch and LefschtezRiemannRoch theorems proved in our earlier papers to define virtual fundamental classes for the moduli stacks of stable curves in great generality and establish various formulae for them. ..."
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In this paper we apply the RiemannRoch and LefschtezRiemannRoch theorems proved in our earlier papers to define virtual fundamental classes for the moduli stacks of stable curves in great generality and establish various formulae for them.
RiemannRoch for Algebraic Stacks: II
"... In this paper we establish RiemannRoch and LefschtezRiemannRoch theorems for arbitrary proper maps of finite cohomological dimension between algebraic DGstacks for which coarsemodulispaces exist as quasiprojective schemes over a Noetherian excellent base scheme. (Observe that this includes al ..."
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In this paper we establish RiemannRoch and LefschtezRiemannRoch theorems for arbitrary proper maps of finite cohomological dimension between algebraic DGstacks for which coarsemodulispaces exist as quasiprojective schemes over a Noetherian excellent base scheme. (Observe that this includes also Artin stacks with finite diagonal.) The RiemannRoch theorem is established as a natural transformation between the Gtheory of algebraic stacks and Bredonstyle homology theories for stacks. These homology theories are defined in the spirit of the classical Bredonhomology theories defined for the action of compact topological groups on spaces: they reduce to the usual homology theories for algebraic spaces and yet capture information from the residual gerbes when applied to algebraic stacks. The LefschtezRiemannRoch is an extension of this including the action of tori. Applications include various formulae for virtual fundamental classes for moduli stacks of stable curves which are discussed in detail in the sequel to this paper.