symplectic. Perhaps the first step is to study the behavior of extremal rays under symplectic deformations.

\[thm:extremal\] Let $R$ be an extremal ray of a smooth projective $4$-fold $X$. Assume one of the followings:

1.  The extremal contraction does not contract a divisor to a point.

2.  The contraction contracts a divisor to a point. And the divisor is isomorphic to ${{\mathbb}{P}}^3$ with normal bundle ${{\mathcal}{O}}_{{{\mathbb}{P}}^3}(-a), a=1, 2, 3$, or a smooth quadric hypersurface $Q$ in ${{\mathbb}{P}}^4$ with normal bundle ${{\mathcal}{O}}_{Q}(-a), a=1, 2$, or a del Pezzo threefold with at worst rational singularities or isolated singularites.

Then there is a curve class $[\alpha]$, spanning $R$, such that there is a non-zero Gromov-Witten invariant $\langle A_1, \ldots, A_n \rangle^X_{0, [\alpha]}$.

Previous results for surfaces and threefolds are obtained by Ruan [@RuanExtremal]. Indeed, Ruan observed that this type of results has many interesting consequences regarding the symplectic topology of the underlying symplectic manifolds. In the following, we denote the corresponding symplectic manifold of a smooth projective variety by $(V, \omega)$. First recall some definitions from [@RuanExtremal].

Let $J$ be an almost complex structure on $V$, compatible with $\omega$. Define the $J$-effective cone $NEJ(V)$ to be the subset of $H_2(V, {{\mathbb}{R}})$ given by

