On $L^{2}$-harmonic forms of symplectic hyperbolic manifold.

In this article, we study the $L^{2}$-harmonic forms on the complete $2n$-dimensional symplectic hyperbolic manifold. We extend a vanishing of K\"{a}hler hyperbolic manifold proved by Gromov to symplectic case. Precisely, we prove that the spaces of $L^{2}$-harmonic forms of bi-degree $(p,q)$ vanish unless $p+q=n$.