On Euler number of symplectic hyperbolic manifold

TL;DR

This paper introduces a new class of symplectic hyperbolic manifolds, studies their harmonic forms, proves a conjecture, and establishes an inequality for their Euler number, advancing understanding in symplectic geometry.

Contribution

It defines special symplectic hyperbolic manifolds, proves the Singer conjecture for this class, and derives an Euler number inequality.

Findings

Proved the Singer conjecture for special symplectic hyperbolic manifolds.

Established that the Euler number satisfies $(-1)^n imes ext{Euler characteristic} > 0$.

Analyzed the space of $L^2$-harmonic forms on universal covers.

Abstract

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