A note on Euler number of locally conformally Kähler manifolds

Abstract

Let $$M^{2n}$$ be a compact Riemannian manifold of non-positive (resp. negative) sectional curvature. We call $$(M,J,\theta )$$ a d(bounded) locally conformally Kahler manifold if the lifted Lee form $${\tilde{\theta }}$$ on the universal covering space of M is d(bounded). We show that if $$M^{2n}$$ is homeomorphic to a d(bounded) LCK manifold, then its Euler number satisfies the inequality $$(-1)^{n}\chi (M^{2n})\ge $$ (resp. >) 0.