  Let $(M,\bar{g})$ be a K\"ahler surface with a constant holomorphic sectional
curvature $k>0$, and $\Sigma$ an immersed symplectic surface in $M$. Suppose
$\Sigma$ evolves along the mean curvature flow in $M$. In this paper, we show
that the symplectic mean curvature flow exists for long time and converges to a
holomorphic curve if the initial surface satisfies $|A|^2\leq 2/3|H|^2+1/2 k$
and $\cos\alpha\geq \frac{\sqrt{30}}{6}$ or $|A|^2\leq 2/3 |H|^2+4/5
k\cos\alpha$ and $\cos\alpha\ge251/265$.

Differential Geometry