Relative methods in symplectic topology

In the first part we define relative Ruan invariants that count embedded connected symplectic submanifolds which contact a fixed symplectic hypersurface V in a symplectic 4-manifold (X, o) at prescribed points with prescribed contact orders (in addition to insertions on XV ) for generic V. We obtain invariants of the deformation class of (X, V, o). Two large issues must be tackled to define such invariants: (1) Curves lying in the divisor V; and (2) genericity results for almost complex structures constrained to make V symplectic. 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.;In the second part we introduce the notion of the relative symplectic cone CVM . As an application, we determine the symplectic cone C M of certain T2-fibrations. In particular, for some elliptic surfaces we verify the conjecture in [32]: If M underlies a minimal Kahler surface with pg > 0, the symplectic cone CM is equal to Pc1&parl0;M&parr0; ∪P-c1&parl0; M&parr0; , where Pa=&cubl0;e∈H 2&parl0;M;R&parr0;&vbm0;e˙ e>0ande˙a >0 for nonzero alpha ∈ H2(M; R ) and Pa=&cubl0;e∈H 2&parl0;M;R&parr0;&vbm0;e˙ e>0&cubr0; .