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95
Gromov–Witten theory of Deligne–Mumford stacks
, 2006
"... 2. Chow rings, cohomology and homology of stacks 5 3. The cyclotomic inertia stack and its rigidification 10 4. Twisted curves and their maps 18 ..."
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Cited by 127 (9 self)
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2. Chow rings, cohomology and homology of stacks 5 3. The cyclotomic inertia stack and its rigidification 10 4. Twisted curves and their maps 18
A DEGENERATION FORMULA OF GWINVARIANTS
, 2001
"... This is the sequel to the paper [Li]. In this paper, we construct the virtual moduli cycles of the degeneration of the moduli of stable morphisms constructed in [Li]. We also construct the virtual moduli cycles of the moduli of relative stable morphisms of a pair of a smooth divisor in a smooth var ..."
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Cited by 80 (4 self)
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This is the sequel to the paper [Li]. In this paper, we construct the virtual moduli cycles of the degeneration of the moduli of stable morphisms constructed in [Li]. We also construct the virtual moduli cycles of the moduli of relative stable morphisms of a pair of a smooth divisor in a smooth variety. Based on these, we prove a degeneration formula of the GromovWitten invariants.
DonaldsonThomas type invariants via microlocal geometry
"... We prove that DonaldsonThomas type invariants are equal to weighted Euler characteristics of their moduli spaces. In particular, such invariants depend only on the scheme structure of the moduli space, not the symmetric obstruction theory used to define them. We also introduce new invariants genera ..."
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Cited by 58 (1 self)
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We prove that DonaldsonThomas type invariants are equal to weighted Euler characteristics of their moduli spaces. In particular, such invariants depend only on the scheme structure of the moduli space, not the symmetric obstruction theory used to define them. We also introduce new invariants generalizing DonaldsonThomas type invariants to moduli problems with open moduli space. These are useful for computing DonaldsonThomas type invariants over stratifications.
Moduli of Twisted Sheaves
, 2004
"... Abstract. We study moduli of semistable twisted sheaves on smooth proper morphisms of algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For curves, they are essentially isomorphic to spaces of semistable vector bundles. In the case of su ..."
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Cited by 48 (9 self)
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Abstract. We study moduli of semistable twisted sheaves on smooth proper morphisms of algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For curves, they are essentially isomorphic to spaces of semistable vector bundles. In the case of surfaces, we show (under a mild hypothesis on the twisting class) that the spaces are asympotically geometrically irreducible, normal, generically smooth, and l.c.i. over the base. We also develop general tools necessary for these results: the theory of associated points and purity of sheaves on Artin stacks, twisted Bogomolov inequalities,
Algebraic orbifold quantum products
"... The purpose of this note is to give an overview of our work on defining algebraic counterparts for W. Chen and Y. Ruan’s GromovWitten Theory of orbifolds. This work will be described in detail in a subsequent paper. The presentation here is generally based on lectures given by two of us at the Orbi ..."
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Cited by 41 (1 self)
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The purpose of this note is to give an overview of our work on defining algebraic counterparts for W. Chen and Y. Ruan’s GromovWitten Theory of orbifolds. This work will be described in detail in a subsequent paper. The presentation here is generally based on lectures given by two of us at the Orbifold Workshop
HODGE INTEGRALS AND HURWITZ NUMBERS VIA VIRTUAL LOCALIZATION
, 2000
"... Abstract. Ekedahl, Lando, Shapiro, and Vainshtein announced a remarkable formula ([ELSV]) expressing Hurwitz numbers (counting covers of P 1 with specified simple branch points, and specified branching over one other point) in terms of Hodge integrals. We give a proof of this formula using virtual l ..."
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Cited by 39 (5 self)
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Abstract. Ekedahl, Lando, Shapiro, and Vainshtein announced a remarkable formula ([ELSV]) expressing Hurwitz numbers (counting covers of P 1 with specified simple branch points, and specified branching over one other point) in terms of Hodge integrals. We give a proof of this formula using virtual localization on the moduli space of stable maps, and describe how the proof could be simplified by the proper algebrogeometric definition of a “relative
Logarithmic GromovWitten invariants
 Journal of the American Mathematical Society
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Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee
, 2001
"... A recent paper by Cox, Katz, and Lee [3] states the following conjecture on virtual moduli cycles. Let X be a nonsingular complex projective variety; a vector bundle V on X is convex if, for every genus zero stable map ' : C ! X, we have H 1 (C; f V ) = 0. Over the moduli stack of genus ..."
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Cited by 33 (3 self)
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A recent paper by Cox, Katz, and Lee [3] states the following conjecture on virtual moduli cycles. Let X be a nonsingular complex projective variety; a vector bundle V on X is convex if, for every genus zero stable map ' : C ! X, we have H 1 (C; f V ) = 0. Over the moduli stack of genus zero stable maps M 0;n (X; ), we have the universal curve n+1 : M 0;n+1 (X; ) ! M 0;n (X; ) and evaluation morphism e n+1 : M 0;n+1 (X; ) ! X. If V is convex, then V ;n := (<F9.79