2009/12/31 by Josef G. Dorfmeister, Josef G Dorfmeister, Tianjun Li +1
Mathematics · #Algebraic Geometry and Number Theory #Connected sum #Geometric and Algebraic Topology #Geometry #Geometry and complex manifolds #Gromov–Witten invariant #Holomorphic function #Hypersurface #Manifold (fluid mechanics) #Mathematics #Moment map #Pure mathematics #Simply connected space #Symplectic geometry #Symplectic manifold #Symplectic representation #Symplectomorphism #Torus #math.AG #math.DG #math.SG
paper · pdf · doi:10.1142/s0219199712500629
published as Communications in Contemporary Mathematics Vol. 15, No. 1 (2013)
openalex publication_date 2012/11/08 · arxiv created 2013/02/12 · arxiv updated 2013/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We define relative Ruan invariants that count embedded connected symplectic submanifolds which contact a fixed symplectic hypersurface V in a symplectic 4-manifold (X, ω) at prescribed points with prescribed contact orders (in addition to insertions on X\V). We obtain invariants of the deformation class of (X, V, ω). Two large issues must be tackled to define such invariants: (1) curves lying in the hypersurface V and (2) genericity results for almost complex structures constrained to make V pseudo-holomorphic (or almost complex). Moreover, these invariants are refined to take into account rim-tori decompositions. In the latter part of the paper, we extend the definition to disconnected submanifolds and construct relative Gromov–Taubes invariants.