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65
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.
SMOOTH TORIC DELIGNEMUMFORD STACKS
, 2009
"... We give a geometric definition of smooth toric DeligneMumford stacks using the action of a “torus”. We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In particular, we give a geometric interpretation of the combinatorial data contained in a sta ..."
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Cited by 27 (0 self)
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We give a geometric definition of smooth toric DeligneMumford stacks using the action of a “torus”. We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In particular, we give a geometric interpretation of the combinatorial data contained in a stacky fan. We also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks.
A relation between the parabolic Chern characters of the de Rham bundles
, 2006
"... In this paper, we consider the weight i de Rham–Gauss–Manin bundles on a smooth variety arising from a smooth projective morphism f: XU − → U for i ≥ 0. We associate to each weight i de Rham bundle, a certain parabolic bundle on S and consider their parabolic Chern characters in the rational Chow gr ..."
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Cited by 25 (6 self)
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In this paper, we consider the weight i de Rham–Gauss–Manin bundles on a smooth variety arising from a smooth projective morphism f: XU − → U for i ≥ 0. We associate to each weight i de Rham bundle, a certain parabolic bundle on S and consider their parabolic Chern characters in the rational Chow groups, for a good compactification S of U. We show the triviality of the alternating sum of these parabolic bundles in the (positive degree) rational Chow groups. This removes the hypothesis of semistable reduction in the original result of this kind due to Esnault and Viehweg.
Stable twisted curves and their rspin structures
, 2007
"... The object of this paper is the notion of rspin structure: a line bundle whose rth power is isomorphic to the canonical bundle. Over the moduli functor Mg of smooth genusg curves, rspin structures form a finite torsor under the group of rtorsion line bundles. Over the moduli functor Mg of stable ..."
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Cited by 21 (7 self)
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The object of this paper is the notion of rspin structure: a line bundle whose rth power is isomorphic to the canonical bundle. Over the moduli functor Mg of smooth genusg curves, rspin structures form a finite torsor under the group of rtorsion line bundles. Over the moduli functor Mg of stable curves, rspin structures form an étale stack, but the finiteness and the torsor structure are lost. In the present work, we show how this bad picture can be definitely improved simply by placing the problem in the category of Abramovich and Vistoli’s twisted curves. First, we find that within such category there exist several different compactifications of Mg; each one corresponds to a different multiindex ⃗ l = (l0, l1,...) identifying a notion of stability: ⃗ lstability. Then, we determine the suitable choices of ⃗ l for which rspin structures form a finite torsor over the moduli of ⃗ lstable curves.
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
Towards an enumerative geometry of the moduli space of twisted curves and rth roots
, 2008
"... ..."
Boundedness of families of canonically polarized manifolds: A higher dimensional analogue of Shafarevich’s conjecture
"... ABSTRACT. We show that the number of deformation types of canonically polarized manifolds over an arbitrary variety with proper singular locus is finite, and that this number is uniformly bounded in any finite type family of base varieties. As a corollary we show that a direct generalization of the ..."
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Cited by 13 (6 self)
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ABSTRACT. We show that the number of deformation types of canonically polarized manifolds over an arbitrary variety with proper singular locus is finite, and that this number is uniformly bounded in any finite type family of base varieties. As a corollary we show that a direct generalization of the geometric version of Shafarevich’s original conjecture holds for infinitesimally rigid families of canonically polarized varieties. CONTENTS
MODULI SPACES OF SEMISTABLE SHEAVES ON PROJECTIVE DELIGNEMUMFORD STACKS
, 811
"... Abstract. In this paper we introduce a notion of Gieseker stability for coherent sheaves on tame DeligneMumford stacks with projective moduli scheme and some chosen generating sheaf on the stack in the sense of Olsson and Starr [OS03]. We prove that this stability condition is open, and pure dimens ..."
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Cited by 12 (1 self)
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Abstract. In this paper we introduce a notion of Gieseker stability for coherent sheaves on tame DeligneMumford stacks with projective moduli scheme and some chosen generating sheaf on the stack in the sense of Olsson and Starr [OS03]. We prove that this stability condition is open, and pure dimensional semistable sheaves form a bounded family. We explicitly construct the moduli stack of semistable sheaves as a finite type global quotient, and study the moduli scheme of stable sheaves and its natural compactification in the same spirit as the seminal paper of Simpson [Sim94]. With this general machinery we are able to retrieve, as special cases, results of Lieblich [Lie07] and Yoshioka [Yos06] about moduli of twisted sheaves and results of MaruyamaYokogawa [MY92] about moduli of parabolic bundles. Overview We define a notion of stability for coherent sheaves on stacks, and construct a moduli stack of semistable sheaves. The class of stacks that is suitable to approach this problem is the class of projective stacks: tame stacks (for instance DeligneMumford stacks in characteristic zero) with projective moduli scheme and a locally free sheaf that is “very ample ” with respect to