Kahler manifolds and cross quadratic bisectional curvature


Abstract in English

In this article we continue the study of the two curvature notions for Kahler 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 Kahler manifolds with CQB$_1>0$ or $mbox{}^d$CQB$_1>0$ are Fano, while nonnegative CQB$_1$ or $mbox{}^d$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 Kahler-Ricci flow to deform the metric. We conjecture that all Kahler 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 Kahler C-space will actually have positive CQB unless it is a ${mathbb P}^1$ bundle. Finally we give an example of non-symmetric, irreducible Kahler C-space with $b_2>1$ and positive CQB, as well as compact non-locally symmetric Kahler manifolds with CQB$<0$ and $^d$CQB$<0$.

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