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42
Weighted projective embeddings, stability of orbifolds and constant scalar curvature Kähler metrics
, 2009
"... We embed polarised orbifolds with cyclic stabiliser groups into weighted projective space via a weighted form of Kodaira embedding. Dividing by the (nonreductive) automorphisms of weighted projective space then formally gives a moduli space of orbifolds. We show how to express this as a reductive q ..."
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Cited by 44 (2 self)
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We embed polarised orbifolds with cyclic stabiliser groups into weighted projective space via a weighted form of Kodaira embedding. Dividing by the (nonreductive) automorphisms of weighted projective space then formally gives a moduli space of orbifolds. We show how to express this as a reductive quotient and so a GIT problem, thus defining a notion of stability for orbifolds. We then prove an orbifold version of Donaldson’s theorem: the existence of an orbifold Kähler metric of constant scalar curvature implies Ksemistability. By extending the notion of slope stability to orbifolds we therefore get an explicit obstruction to the existence of constant scalar curvature orbifold Kähler metrics. We describe the manifold applications of this orbifold result, and show how many previously known results (Troyanov, GhigiKollár, RollinSinger, the AdS/CFT SasakiEinstein obstructions of GauntlettMartelliSparksYau) fit into this framework.
Homstacks and restriction of scalars
 Duke Math. J
"... Abstract. Fix an algebraic space S, and let X and Y be separated Artin stacks of finite presentation over S with finite diagonals (over S). We define a stack Hom S(X, Y) classifying morphisms between X and Y. Assume that X is proper and flat over S, and fppf–locally on S there exists a finite finite ..."
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Cited by 29 (3 self)
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Abstract. Fix an algebraic space S, and let X and Y be separated Artin stacks of finite presentation over S with finite diagonals (over S). We define a stack Hom S(X, Y) classifying morphisms between X and Y. Assume that X is proper and flat over S, and fppf–locally on S there exists a finite finitely presented flat cover Z → X with Z an algebraic space. Then we show that Hom S (X, Y) is an Artin stack with quasi–compact and separated diagonal. 1. Statements of results Fix an algebraic space S, let X and Y be separated Artin stacks of finite presentation over
REMARKS ON THE STACK OF COHERENT ALGEBRAS
, 2006
"... Abstract. We consider the stack of coherent algebras with proper support, a moduli problem generalizing Alexeev and Knutson’s stack of branchvarieties to the case of an Artin stack. The main results are proofs of the existence of Quot and Hom spaces in greater generality than is currently known and ..."
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Cited by 25 (6 self)
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Abstract. We consider the stack of coherent algebras with proper support, a moduli problem generalizing Alexeev and Knutson’s stack of branchvarieties to the case of an Artin stack. The main results are proofs of the existence of Quot and Hom spaces in greater generality than is currently known and several applications to Alexeev and Knutson’s original construction: a proof that the stack of branchvarieties is always algebraic, that limits of onedimensional families always exist, and that the connected components of the stack of branchvarieties are proper over the base under certain hypotheses on the ambient stack. 1.
GENERATING FUNCTIONS FOR COLORED 3D YOUNG DIAGRAMS AND THE DONALDSONTHOMAS INVARIANTS OF ORBIFOLDS
, 2008
"... We derive two multivariate generating functions for threedimensional Young diagrams (also called plane partitions). The variables correspond to a colouring of the boxes according to a finite Abelian subgroup G of SO(3). We use the vertex operator methods of Okounkov– Reshetikhin–Vafa for the easy ..."
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Cited by 25 (1 self)
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We derive two multivariate generating functions for threedimensional Young diagrams (also called plane partitions). The variables correspond to a colouring of the boxes according to a finite Abelian subgroup G of SO(3). We use the vertex operator methods of Okounkov– Reshetikhin–Vafa for the easy case G = Zn; to handle the considerably more difficult case G = Z2 × Z2, we will also use a refinement of the author’s recent q–enumeration of pyramid partitions. In the appendix, we relate the diagram generating functions to the DonaldsonThomas partition functions of the orbifold [C3/G]. We find a relationship between the DonaldsonThomas partition functions of the orbifold and its GHilbert scheme resolution. We formulate a crepant resolution conjecture for the DonaldsonThomas theory of local orbifolds satisfying the Hard Lefschetz condition.
Period and index in the Brauer group of an arithmetic surface (with an appendix by Daniel Krashen
, 2006
"... ABSTRACT. This paper consists of two parts: in the first, we use the deformation theory of twisted sheaves on stacks to generalize results of de Jong and Saltman on the periodindex problem for the Brauer group, yielding various new cases of the standard conjecture. In the second part, we use the ge ..."
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Cited by 19 (5 self)
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ABSTRACT. This paper consists of two parts: in the first, we use the deformation theory of twisted sheaves on stacks to generalize results of de Jong and Saltman on the periodindex problem for the Brauer group, yielding various new cases of the standard conjecture. In the second part, we use the geometric techniques of the first part to relate the Hasse principle for certain smooth projective geometrically rational varieties over global fields to the periodindex problem for Brauer groups of arithmetic surfaces. We also include an appendix by D. Krashen showing that the periodindex bounds of the first part are
ON THE GLOBAL QUOTIENT STRUCTURE OF THE SPACE OF TWISTED STABLE MAPS TO A QUOTIENT STACK
"... Abstract. Let X be a tame proper DeligneMumford stack of the form [M/G] where M is a scheme and G is an algebraic group. We prove that the stack Kg,n(X, d) of twisted stable maps is a quotient stack and can be embedded into a smooth DeligneMumford stack. When G is finite, we give a more precise co ..."
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Cited by 11 (3 self)
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Abstract. Let X be a tame proper DeligneMumford stack of the form [M/G] where M is a scheme and G is an algebraic group. We prove that the stack Kg,n(X, d) of twisted stable maps is a quotient stack and can be embedded into a smooth DeligneMumford stack. When G is finite, we give a more precise construction of Kg,n(X, d) using Hilbert schemes and admissible Gcovers. We fix a base scheme B. 1.
GENERATING FUNCTIONS FOR COLOURED 3D YOUNG DIAGRAMS AND THE DONALDSONTHOMAS INVARIANTS OF ORBIFOLDS
, 2008
"... We derive two multivariate generating functions for threedimensional Young diagrams (also called plane partitions). The variables correspond to a colouring of the boxes according to a finite abelian subgroup G of SO(3). These generating functions turn out to be orbifold Donaldson–Thomas partition fu ..."
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Cited by 10 (1 self)
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We derive two multivariate generating functions for threedimensional Young diagrams (also called plane partitions). The variables correspond to a colouring of the boxes according to a finite abelian subgroup G of SO(3). These generating functions turn out to be orbifold Donaldson–Thomas partition functions for the orbifold [C 3 /G]. We need only the vertex operator methods of Okounkov–Reshetikhin– Vafa for the easy case G = Zn; to handle the considerably more difficult case G = Z2 × Z2, we will also use a refinement of the author’s recent q–enumeration of pyramid partitions. In the appendix, we relate the diagram generating functions to the DonaldsonThomas partition functions of the orbifold [C 3 /G]. We find a relationship between the DonaldsonThomas partition functions of the orbifold and its GHilbert scheme resolution. We formulate a crepant resolution conjecture for the DonaldsonThomas theory of local orbifolds satisfying the Hard Lefschetz condition.
GROTHENDIECK DUALITY FOR DeligneMumford Stacks
, 2009
"... We prove the existence of the dualizing functor for a separated morphism of algebraic stacks with affine diagonal; then we explicitly develop duality for compact DeligneMumford stacks focusing in particular on the morphism from a stack to its coarse moduli space and on representable morphisms. We ..."
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Cited by 9 (0 self)
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We prove the existence of the dualizing functor for a separated morphism of algebraic stacks with affine diagonal; then we explicitly develop duality for compact DeligneMumford stacks focusing in particular on the morphism from a stack to its coarse moduli space and on representable morphisms. We explicitly compute the dualizing complex for a smooth stack over an algebraically closed field and prove that Serre duality holds for smooth compact DeligneMumford stacks in its usual form. We prove also that a proper CohenMacaulay stack has a dualizing sheaf and it is an invertible sheaf when it is Gorenstein. As an application of this general machinery we compute the dualizing sheaf of a tame nodal curve.