Using ideas of Donaldson it was shown by Auroux and myself that every compact symplectic 4-manifold $(X,\omega)$ can be realized as an approximately holomorphic branched covering of $\mathbb{CP}^2$ whose branch curve is a symplectic curve in $\mathbb{CP}^2$ with cusps and nodes as only singularities (however the nodes may have reversed orientation). Such a covering is obtained by constructing a suitable triple of sections of the line bundle $L^{\otimes k}$, where $L$ is a line bundle obtained from the symplectic form (its Chern class is given by $c_1(L)=\frac{1}{2\pi}[\omega]$ when this class is integral), and where $k$ is a large enough integer.
Moreover, it was shown in that the braid monodromy techniques introduced by Moishezon and Teicher in algebraic geometry can be used in this situation to derive, for each large enough value of the degree $k$, monodromy invariants which completely describe the symplectic 4-manifold $(X,\omega)$ up to symplectomorphism. We first show that these are only homological invariants.
After that using braid monodromy we introduce and compute new invariant Fukaya category of a Symplectic Lefschetz pencil (Seidel's version) which as mirror symmetry suggests is very promising.
