TAILIEUCHUNG - Đề tài " Relative GromovWitten invariants "

We define relative Gromov-Witten invariants of a symplectic manifold relative to a codimension-two 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 ‘V -stable’ maps. Simple special cases include the Hurwitz numbers for algebraic curves and the enumerative invariants of Caporaso and Harris. Gromov-Witten invariants are invariants of a closed symplectic manifold (X, ω). | Annals of Mathematics Relative Gromov- Witten invariants By Eleny-Nicoleta lonel and Thomas H. Parker Annals of Mathematics 157 2003 45 96 Relative Gromov-Witten invariants By ELENy-NicoLETA Ionel and Thomas H. Parker Abstract We define relative Gromov-Witten invariants of a symplectic manifold relative to a codimension-two 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 V-stable maps. Simple special cases include the Hurwitz numbers for algebraic curves and the enumerative invariants of Caporaso and Harris. Gromov-Witten invariants are invariants of a closed symplectic manifold X w . To define them one introduces a compatible almost complex structure J and a perturbation term V and considers the maps f C X from a genus g complex curve C with n marked points which satisfy the pseudo-holomorphic map equation df V and represent a class A f G H2 X . The set of such maps together with their limits forms the compact space of stable maps Mgn X A . For each stable map the domain determines a point in the Deligne-Mumford space Mg n of curves and evaluation at each marked point determines a point in X. Thus there is a map Mg n X A Mg n X Xn. The Gromov-Witten invariant of X w is the homology class of the image for generic J V . It depends only on the isotopy class of the symplectic structure. By choosing bases of the cohomologies of Mgn and Xn 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 Gromov-Witten invariants for a symplec-tic manifold X w 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 . The first author was

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