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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 282 (36 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
"... Research partially supported by THproject This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in [EGH]. We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic Field Theory. The theorems generalize Gromov’s compactness ..."
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Cited by 161 (8 self)
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Research partially supported by THproject This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in [EGH]. We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic Field Theory. The theorems generalize Gromov’s compactness theorem in [Gr] as well as compactness theorems in Floer homology theory, [F1, F2], and in contact geometry, [H, HWZ8]. Contents
Relative GromovWitten invariants
 Ann. of Math
, 2003
"... We define relative GromovWitten invariants of a symplectic manifold relative to a codimensiontwo symplectic submanifold. These invariants are the key ingredients in the symplectic sum formula of [IP4]. The main step is the construction of a compact space of ‘Vstable ’ maps. Simple special cases i ..."
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Cited by 123 (10 self)
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We define relative GromovWitten invariants of a symplectic manifold relative to a codimensiontwo symplectic submanifold. These invariants are the key ingredients in the symplectic sum formula of [IP4]. The main step is the construction of a compact space of ‘Vstable ’ maps. Simple special cases include the Hurwitz numbers for algebraic curves and the enumerative invariants of Caporaso and Harris. GromovWitten invariants are invariants of a closed symplectic manifold (X, ω). To define them, one introduces a compatible almost complex structure J and a perturbation term ν, and considers the maps f: C → X from a genus g complex curve C with n marked points which satisfy the pseudoholomorphic map equation ∂f = ν and represent a class A =[f] ∈ H2(X). The set of such maps, together with their limits, forms the compact space of stable maps Mg,n(X, A). For each stable map, the domain determines a point in the DeligneMumford space Mg,n of curves, and evaluation at each marked point determines a point in X. Thus there is a map (0.1) Mg,n(X, A) → Mg,n × X n. The GromovWitten invariant of (X, ω)isthe homology class of the image for generic (J, ν). It depends only on the isotopy class of the symplectic structure. By choosing bases of the cohomologies of Mg,n and X n, the GW invariant can be viewed as a collection of numbers that count the number of stable maps satisfying constraints. In important cases these numbers are equal to enumerative invariants defined by algebraic geometry. In this article we construct GromovWitten invariants for a symplectic manifold (X, ω) relative to a codimension two symplectic submanifold V. These invariants are designed for use in formulas describing how GW invariants The research of both authors was partially supported by the N.S.F. The first author was also
Curve counting via stable pairs in the derived category
"... Abstract. 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 resu ..."
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Cited by 116 (22 self)
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Abstract. 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 101 (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 84 (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.