On Euler number of symplectic hyperbolic manifold

Abstract

In this article, we introduce a class of closed 2n-dimensional almost Kähler manifold X which called the special symplectic hyperbolic manifold. Those manifolds include Kähler hyperbolic manifolds. We study the spaces of L2-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)nχ(X)>0.