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The Intrinsic Normal Cone
 Invent. Math
, 1997
"... We suggest a construction of virtual fundamental classes of certain types of moduli spaces. Contents 0 ..."
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Cited by 353 (9 self)
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We suggest a construction of virtual fundamental classes of certain types of moduli spaces. Contents 0
Algebraic (geometric) nstacks
"... In the introduction of LaumonMoretBailly ([LMB] p. 2) they refer to a possible theory of algebraic nstacks: Signalons au passage que Grothendieck propose d’élargir à son tour le cadre précédent en remplaçant les 1champs par des nchamps (grosso modo, des faisceaux en ncatégories sur (Aff) ou su ..."
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Cited by 15 (4 self)
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In the introduction of LaumonMoretBailly ([LMB] p. 2) they refer to a possible theory of algebraic nstacks: Signalons au passage que Grothendieck propose d’élargir à son tour le cadre précédent en remplaçant les 1champs par des nchamps (grosso modo, des faisceaux en ncatégories sur (Aff) ou sur un site arbitraire) et il ne fait guère de doute qu’il existe une notion utile de nchamps algébriques.... The purpose of this paper is to propose such a theory. I guess that the main reason why Laumon and MoretBailly didn’t want to get into this theory was for fear of getting caught up in a horribly technical discussion of nstacks of groupoids over a general site. In this paper we simply assume that a theory of nstacks of groupoids exists. This is not an unreasonable assumption, first of all because there is a relatively good substitute—the theory of simplicial presheaves or presheaves of spaces ([Bro] [BG] [Jo] [Ja] [Si3] [Si2])— which should be equivalent, in an appropriate sense, to any eventual theory of nstacks; and second of all because it seems likely that a real theory of nstacks of ngroupoids could be developped in the near future ([Br2], [Ta]).
GromovWitten theory of étale gerbes I: root gerbes, in preparation
"... Abstract. Let X be a smooth complex projective algebraic variety. Given a line bundle L over X and an integer r> 1 we study the GromovWitten theory of the stack rp L/X of rth root of L. We prove an exact formula expressing genus 0 GromovWitten invariants of rp L/X in terms of those of X. Assum ..."
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Abstract. Let X be a smooth complex projective algebraic variety. Given a line bundle L over X and an integer r> 1 we study the GromovWitten theory of the stack rp L/X of rth root of L. We prove an exact formula expressing genus 0 GromovWitten invariants of rp L/X in terms of those of X. Assuming that either rp L/X or X has semisimple quantum cohomology, we prove an exact formula between higher genus invariants. We also present constructions of moduli stacks of twisted stable maps to rp L/X starting from moduli stack of stable maps to X.
DEFORMATIONS AND AUTOMORPHISMS: A FRAMEWORK FOR GLOBALIZING LOCAL TANGENT AND OBSTRUCTION SPACES
, 805
"... Abstract. Building on Schlessinger’s work, we define a framework for studying geometric deformation problems which allows us to systematize the relationship between the local and global tangent and obstruction spaces of a deformation problem. Starting from Schlessinger’s functors of Artin rings, we ..."
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Abstract. Building on Schlessinger’s work, we define a framework for studying geometric deformation problems which allows us to systematize the relationship between the local and global tangent and obstruction spaces of a deformation problem. Starting from Schlessinger’s functors of Artin rings, we proceed in two steps: we replace functors to sets by categories fibered in groupoids, allowing us to keep track of automorphisms, and we work with deformation problems naturally associated to a scheme X, and which naturally localize on X, so that we can formalize the local behavior. The first step is already carried out by Rim in the context of his homogeneous groupoids, but we develop the theory substantially further. In this setting, many statements known for a range of specific deformation problems can be proved in
CHARACTERIZING ARTIN STACKS
"... Abstract. We study properties of morphisms of stacks in the context of the homotopy theory of presheaves of groupoids on a small site C. There is a natural method for extending a property P of morphisms of sheaves on C to a property P of morphisms of presheaves of groupoids. We prove that the proper ..."
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Abstract. We study properties of morphisms of stacks in the context of the homotopy theory of presheaves of groupoids on a small site C. There is a natural method for extending a property P of morphisms of sheaves on C to a property P of morphisms of presheaves of groupoids. We prove that the property P is homotopy invariant in the local model structure on P (C, Grpd)L when P is stable under pullback and local on the target. Using the homotopy invariance of the properties of being a representable morphism, representable in algebraic spaces, and of being a cover, we obtain homotopy theoretic characterizations of algebraic and Artin stacks as those which are equivalent to simplicial objects in C satisfying certain analogues of the Kan conditions. The definition of Artin stack can naturally be placed within a hierarchy which roughly measures how far a stack is from being representable. We call the higher analogues of Artin stacks nalgebraic stacks, and provide a characterization of these in terms of simplicial objects. A consequence of this characterization is that, for presheaves of groupoids, nalgebraic is the same as 3algebraic for all n ≥ 3. As an application of these results we show that a stack is nalgebraic if and only if the homotopy orbits of a group action on it is. 1.