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 $L^2$-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 $(-1{)}^n\chi (X)>0$.

arXiv comment26 pages, appear in Adv. Math