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119
Holomorphic Disks and Topological Invariants for Closed ThreeManifolds
 ANN. OF MATH
, 2000
"... The aim of this article is to introduce certain topological invariants for closed, oriented threemanifolds Y, equipped with a Spin c structure t. Given a Heegaard splitting of Y  U0 tie U1, these theories are variants of the Lagrangian Floer homology for the gfold symmetric product of Y relat ..."
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Cited by 274 (37 self)
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The aim of this article is to introduce certain topological invariants for closed, oriented threemanifolds Y, equipped with a Spin c structure t. Given a Heegaard splitting of Y  U0 tie U1, these theories are variants of the Lagrangian Floer homology for the gfold symmetric product of Y relative to certain totally real subspaces associated to U0 and U1.
Compactness results in Symplectic Field Theory
, 2003
"... This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in [4]. We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic Field Theory. The theorems generalize Gromov’s compactness theorem in [8] as well as compactness theorem ..."
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Cited by 159 (9 self)
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This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in [4]. We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic Field Theory. The theorems generalize Gromov’s compactness theorem in [8] as well as compactness theorems in Floer homology theory, [6, 7], and in contact geometry, [9, 19].
Curve counting via stable pairs in the derived category
, 2009
"... For a nonsingular projective 3fold X, we define integer invariants virtually enumerating pairs (C,D) where C ⊂ X is an embedded curve and D ⊂ C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting in ..."
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Cited by 114 (21 self)
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For a nonsingular projective 3fold X, we define integer invariants virtually enumerating pairs (C,D) where C ⊂ X is an embedded curve and D ⊂ C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting invariants are conjecturally equivalent, after universal transformations, to both the GromovWitten and DT theories of X. For CalabiYau 3folds, the latter equivalence should be viewed as a wallcrossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric CalabiYau case, a completely new form of the topological vertex is described. The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We
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 98 (5 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.
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 81 (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.
Relative virtual localization and vanishing of tautological classes on moduli spaces of curves
 Duke Math. J
"... ABSTRACT. We prove a localization formula for the moduli space of stable relative maps. As an application, we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i − g + 1 genus 0 components. As ..."
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Cited by 68 (6 self)
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ABSTRACT. We prove a localization formula for the moduli space of stable relative maps. As an application, we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i − g + 1 genus 0 components. As consequences, we prove and generalize various conjectures and theorems about various moduli spaces of curves (due to Getzler, Ionel, Faber, Looijenga, Pandharipande, Diaz, and others). This theorem appears to be the geometric content behind these results; the rest is straightforward graph combinatorics. The theorem also suggests the importance of the stratification of the moduli space by number of rational components. CONTENTS
Toric degenerations of toric varieties and tropical curves
 DUKE MATH. J
, 2004
"... We show that the counting of rational curves on a complete toric variety that are in general position to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraicgeometric and relies on degeneration techniques and log deformation theory. ..."
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Cited by 63 (5 self)
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We show that the counting of rational curves on a complete toric variety that are in general position to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraicgeometric and relies on degeneration techniques and log deformation theory.
GromovWitten/DonaldsonThomas correspondence for toric 3folds
, 2008
"... We prove the equivariant GromovWitten theory of a nonsingular toric 3fold X with primary insertions is equivalent to the equivariant DonaldsonThomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Mariño, and Vafa of the GromovWitten theory of local CalabiYau ..."
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Cited by 61 (17 self)
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We prove the equivariant GromovWitten theory of a nonsingular toric 3fold X with primary insertions is equivalent to the equivariant DonaldsonThomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Mariño, and Vafa of the GromovWitten theory of local CalabiYau toric 3folds are proven to be correct in the full 3leg setting.