A gap theorem of Kahler manifolds with vanishing odd Betti numbers

Given a compact Kahler manifold M with vanishing odd Betti numbers, we add an additional condition, which is related to the Hirzebruch chi(y)-genus or the Chern number c(1)c(n-1) of M, to guarantee that M is pure type (i.e., the Hodge numbers h(p,q)(M) = 0 whenever p not equal q). We also present a sharp lower bound of the Chern number c(1)c(n-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 Kahler. Some remarks and related results are also discussed. (C) 2013 Elsevier B.V. All rights reserved.