On Euler number of symplectic hyperbolic manifold

In this article, we introduce a class of closed $2n$-dimensional almost K\"{a}hler manifold $X$ which called the special symplectic hyperbolic manifold. Those manifolds include K\"{a}hler hyperbolic manifolds. We study the spaces of $L^{2}$-harmonic forms on the universal covering space of $X$. We then prove the Singer conjecture on special symplectic hyperbolic case. As an application, we can show that the Euler number of a special symplectic manifold satisfies the inequality $(-1)^{n}\chi(X)>0$.