Kähler parabolicity and the Euler number of compact manifolds of non-positive sectional curvature

Abstract.

Let $M^{2n}$ be a compact Riemannian manifold of non-positive sectional curvature. It is shown that if $M^{2n}$ is homeomorphic to a Kähler manifold, then its Euler number satisfies the inequality $(-1)^n \chi(M^{2n})\geq 0$.