Relative methods in symplectic topology.

Abstract

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 $(X,\omega)$ at prescribed points with prescribed contact orders (in addition to insertions on $X\backslash V$) for generic $V$. We obtain invariants of the deformation class of $(X,V,\omega)$. 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 $\mathcal C_M^V$. As an application, we determine the {\it symplectic cone $\mathcal C_M$} of certain $T^2$-fibrations. In particular, for some elliptic surfaces we verify the conjecture in \cite{TJL1}: If $M$ underlies a minimal K\"ahler surface with $p_g>0$, the symplectic cone $\mathcal C_M$ is equal to $\mathcal P^{c_1(M)}\cup \mathcal P^{-c_1(M)}$, where $\mathcal P^{\alpha}=\{e\in H^2(M;\mathbb R)|e\cdot e>0 \hbox{ and } e\cdot \alpha>0\}$ for nonzero $\alpha \in H^2(M;\mathbb R)$ and $\mathcal P^{0}=\{e\in H^2(M;\mathbb R)|e\cdot e>0 \}$.