A gap theorem of Kähler manifolds with vanishing odd Betti numbers

Given a compact Kähler manifold M with vanishing odd Betti numbers, we add an additional condition, which is related to the Hirzebruch χy-genus or the Chern number c1cn−1 of M, to guarantee that M is pure type (i.e., the Hodge numbers hp,q(M)=0 whenever p≠q). We also present a sharp lower bound of the Chern number c1cn−1[M] in terms of Betti numbers. As an application, we give a more neat proof of a result due to Wright, which links some much earlier works of Frankel and Kosniowski. Using our observation, we can generalize the concept of “pure type” for any general compact symplectic manifold and it coincides with the original one when this symplectic manifold is Kähler. Some remarks and related results are also discussed.