On Euler number of symplectic hyperbolic manifold

In this article, we introduce a class of closed 2n-dimensional almost Kahler manifold X which called the special symplectic hyperbolic manifold. Those manifolds include Kahler 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.(c) 2023 Elsevier Inc. All rights reserved.