Kähler manifolds and cross quadratic bisectional curvature

  • Lei Ni

    University of California, San Diego, La Jolla, USA
  • Fangyang Zheng

    Chongqing Normal University, Chongqing, P.R. China
Kähler manifolds and cross quadratic bisectional curvature cover

A subscription is required to access this article.


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 (CQB) (which is a Hermitian form on maps between and ). We first show that compact Kähler manifolds with CQB (CQB is the restriction to rank 1 maps) or CQB are Fano, while nonnegative CQB or CQB 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 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 bundle. Finally, we give an example of a nonsymmetric, irreducible Kähler C-space with and positive CQB, as well as of compact non-locally-symmetric Kähler manifolds with and .

Cite this article

Lei Ni, Fangyang Zheng, Kähler manifolds and cross quadratic bisectional curvature. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1405