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 Kähler manifold if the lifted Lee form $\tilde{\theta }$ on the universal covering space of $M$ is $d$(bounded). We shown 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. $>$) $0$.

arXiv comment10 Pages, to appear in Math. Z