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203
Localization of virtual classes
"... We prove a localization formula for the virtual fundamental class in the general context of C∗equivariant perfect obstruction theories. Let X be an algebraic scheme with a C∗action and a C∗equivariant perfect obstruction theory. The virtual fundamental class [X] vir in ..."
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Cited by 173 (25 self)
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We prove a localization formula for the virtual fundamental class in the general context of C∗equivariant perfect obstruction theories. Let X be an algebraic scheme with a C∗action and a C∗equivariant perfect obstruction theory. The virtual fundamental class [X] vir in
GromovWitten invariants in algebraic geometry, preprint
, 1996
"... GromovWitten invariants for arbitrary projective varieties and arbitrary genus are constructed using the techniques from [K. Behrend, B. Fantechi. The Intrinsic Normal Cone.] ..."
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Cited by 147 (2 self)
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GromovWitten invariants for arbitrary projective varieties and arbitrary genus are constructed using the techniques from [K. Behrend, B. Fantechi. The Intrinsic Normal Cone.]
Hodge integrals and GromovWitten theory
 Invent. Math
"... Let Mg,n be the nonsingular moduli stack of genus g, npointed, DeligneMumford stable curves. For each marking i, there is an associated cotangent line bundle Li → Mg,n with fiber T ∗ C,pi over the moduli point [C, p1,...,pn]. Let ψi = c1(Li) ∈ H ∗ (Mg,n, Q). The integrals of products of the ψ cla ..."
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Cited by 118 (12 self)
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Let Mg,n be the nonsingular moduli stack of genus g, npointed, DeligneMumford stable curves. For each marking i, there is an associated cotangent line bundle Li → Mg,n with fiber T ∗ C,pi over the moduli point [C, p1,...,pn]. Let ψi = c1(Li) ∈ H ∗ (Mg,n, Q). The integrals of products of the ψ classes
A holomorphic Casson invariant for CalabiYau 3folds, and bundles on K3 fibrations
 J. Differential Geom
, 2000
"... We briefly review the formal picture in which a CalabiYau nfold is the complex analogue of an oriented real nmanifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a CalabiYau 3fol ..."
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Cited by 105 (5 self)
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We briefly review the formal picture in which a CalabiYau nfold is the complex analogue of an oriented real nmanifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a CalabiYau 3fold. We develop the deformation theory necessary to obtain the virtual moduli cycles of [LT], [BF] in moduli spaces of stable sheaves whose higher obstruction groups vanish. This gives, for instance, virtual moduli cycles in Hilbert schemes of curves in P 3, and Donaldson – and GromovWitten – like invariants of Fano 3folds. It also allows us to define the holomorphic Casson invariant of a CalabiYau 3fold X, prove it is deformation invariant, and compute it explicitly in some examples. Then we calculate moduli spaces of sheaves on a general K3 fibration X, enabling us to compute the invariant for some ranks and Chern classes, and equate it to GromovWitten invariants of the “Mukaidual ” 3fold for others. As an example the invariant is shown to distinguish Gross ’ diffeomorphic 3folds. Finally the Mukaidual 3fold is shown to be CalabiYau and its cohomology is related to that of X. 1
GromovWitten invariants and quantization of quadratic Hamiltonians
, 2001
"... We describe a formalism based on quantization of quadratic hamiltonians and symplectic actions of loop groups which provides a convenient home for most of known general results and conjectures about GromovWitten invariants of compact symplectic manifolds and, more generally, Frobenius structures at ..."
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Cited by 87 (3 self)
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We describe a formalism based on quantization of quadratic hamiltonians and symplectic actions of loop groups which provides a convenient home for most of known general results and conjectures about GromovWitten invariants of compact symplectic manifolds and, more generally, Frobenius structures at higher genus. We state several results illustrating the formalism and its use. In particular, we establish Virasoro constraints for semisimple Frobenius structures and outline a proof of the Virasoro conjecture for Gromov – Witten invariants of complex projective spaces and other Fano toric manifolds. Details will be published elsewhere.
GromovWitten theory and DonaldsonThomas theory I
"... The GromovWitten theory of a 3fold X is defined via integrals over the moduli space of stable maps. The DonaldsonThomas theory of X is defined ..."
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Cited by 77 (15 self)
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The GromovWitten theory of a 3fold X is defined via integrals over the moduli space of stable maps. The DonaldsonThomas theory of X is defined
Intersection theory on M1,4 and elliptic GromovWitten invariants
, 1997
"... We find a new relation among codimension 2 algebraic cycles in the moduli space M1,4, and use this to calculate the elliptic GromovWitten invariants of projective spaces CP² and CP³. ..."
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Cited by 63 (5 self)
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We find a new relation among codimension 2 algebraic cycles in the moduli space M1,4, and use this to calculate the elliptic GromovWitten invariants of projective spaces CP² and CP³.
STABLE MORPHISMS TO SINGULAR SCHEMES AND RELATIVE STABLE MORPHISMS
"... Let W/C be a degeneration of smooth varieties so that the special fiber has normal crossing singularity. In this paper, we first construct the stack of expanded degenerations of W. We then construct the moduli space of stable morphisms to this stack, which provides a degeneration of the moduli spa ..."
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Cited by 50 (4 self)
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Let W/C be a degeneration of smooth varieties so that the special fiber has normal crossing singularity. In this paper, we first construct the stack of expanded degenerations of W. We then construct the moduli space of stable morphisms to this stack, which provides a degeneration of the moduli spaces of stable morphisms associated to W/C. Using a similar technique, for a pair (Z, D) of smooth variety and a smooth divisor, we construct the stack of expanded relative pairs and then the moduli spaces of relative stable morphisms to (Z, D). This is the algebrogeometric analogue of DonaldsonFloer theory in gauge theory. The construction of relative GromovWitten invariants and the degeneration formula of GromovWitten invariants will be treated in the subsequent paper.