Kähler manifolds and cross quadratic bisectional curvature

Abstract

We continue the study of the two curvature notions for Kähler 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 $T^{\prime }M$ and $T^{\prime \prime }M$). We first show that compact Kähler manifolds with CQB${}_1>0$ (CQB${}_1$ is the restriction to rank 1 maps) or ${}^d$CQB${}_1>0$ 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 Kähler–Ricci flow to deform the metric. We conjecture that all Kähler 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 Kähler C-space actually has positive CQB unless it is a ${\mathbb{P}}^1$ bundle. Finally, we give an example of a nonsymmetric, irreducible Kähler C-space with $b_2>1$ and positive CQB, as well as of compact non-locally-symmetric Kähler manifolds with $\text{CQB}<0$ and ${}^d\text{CQB}<0$.