Relative Ruan and Gromov-Taubes Invariants of Symplectic 4-Manifolds

TL;DR

This paper introduces relative Ruan and Gromov-Taubes invariants for symplectic 4-manifolds, counting embedded symplectic submanifolds with contact conditions, extending existing invariants to include hypersurface contact and rim tori considerations.

Contribution

It defines new relative invariants for symplectic 4-manifolds that incorporate contact conditions with hypersurfaces and rim tori, extending prior invariants to more complex configurations.

Findings

Defined relative Ruan invariants with contact conditions

Extended invariants to disconnected submanifolds

Constructed relative Gromov-Taubes invariants

Abstract

We define relative Ruan invariants that count embedded connected symplectic submanifolds which contact a fixed stable symplectic hypersurface V in a symplectic 4-manifold (X,w) at prescribed points with prescribed contact orders (in addition to insertions on X\V) for stable V. We obtain invariants of the deformation class of (X,V,w). 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.