K\"ahler manifolds and cross quadratic bisectional curvature

In this article we continue the study of the two curvature notions for K"ahler manifolds introduced by the first named author earlier: the so-called cross quadratic bisectional curvature (CQB) and its dual (${}^d$CQB). We first show that compact K"ahler manifolds with CQB${}_1>0$ or $\text{mbox}{}^d$CQB${}_1>0$ are Fano, while nonnegative CQB${}_1$ or $\text{mbox}{}^d$CQB${}_1$ leads to a Fano manifold as well, provided that the universal cover does not contain a flat de Rham factor. For the latter statement we employ the K"ahler-Ricci flow to deform the metric. We conjecture that all K"ahler C-spaces will have nonnegative CQB and positive ${}^d$CQB. By giving irreducible such examples with arbitrarily large second Betti numbers we show that the positivity of these two curvature put no restriction on the Betti number. A strengthened conjecture is that any K"ahler C-space will actually have positive CQB unless it is a ${\mathbb{P}}^1$ bundle. Finally we give an example of non-symmetric, irreducible K"ahler C-space with $b_2>1$ and positive CQB, as well as compact non-locally symmetric K"ahler manifolds with CQB$<0$ and ${}^d$CQB$<0$.