2002/10/17 by Chiu-Chu Melissa Liu, Liu, Chiu-Chu Melissa · 4 citations
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Symplectic Geometry (math.SG) #math.SG
paper · pdf · doi:10.48550/arxiv.math/0210257
85 pages, 20 figures; Section 3 and 4 corrected and refined; notation of the rest of the paper changed accordingly
openalex publication_date 2002/10/17 · arxiv created 2004/12/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (X,ω) be a symplectic manifold, J be an ω-tame almost complex structure, and L be a Lagrangian submanifold. The stable compactification of the moduli space of parametrized J-holomorphic curves in X with boundary in L (with prescribed topological data) is compact and Hausdorff in Gromov's C^∞-topology. We construct a Kuranishi structure with corners in the sense of Fukaya and Ono. This Kuranishi structure is orientable if L is spin. In the special case where the expected dimension of the moduli space is zero, and there is an S1 action on the pair (X,L) which preserves J and acts freely on L, we define the Euler number for this S1 equivariant pair and the prescribed topological data. We conjecture that this rational number is the one computed by localization techniques using the given S1 action.