Kähler manifolds and cross quadratic bisectional curvature

We continue the study of the two curvature notions for K & auml;hler manifolds introduced by the first named author earlier: the so-called cross quadratic bisectional curvature (CQB) and its dual ((d)CQB) (which is a Hermitian form on maps between T ' M and T '' M). We first show that compact K & auml;hler manifolds with CQB(1)>0 (CQB(1) is the restriction to rank 1 maps) or dCQB(1 )> 0 are Fano, while nonnegative CQB(1) or (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 & auml;hler-Ricci flow to deform the metric. We conjecture that all K & auml;hler C-spaces 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 curvatures puts no restriction on the Betti number. A strengthened conjecture is that any K & auml;hler C-space actually has positive CQB unless it is a P1 bundle. Finally, we give an example of a nonsymmetric, irreducible K & auml;hler C-space with b(2)>1 and positive CQB, as well as of compact non-locally-symmetric K & auml;hler manifolds with CQB<0 and dCQB<0.