Fangyang Zheng

In recent work (Pure Appl. Anal. 2 (2020), 397-426), the first named author and J. Zhang found a connection between the regularity theory of optimal transport and the curvature of Kähler manifolds. In particular, we showed that the MTW tensor for a cost function $c(x,y)=\Psi (x-y)$ can be understood as the anti-bisectional curvature of an associated Kähler metric defined on a tube domain. Here, the anti-bisectional curvature is defined as $R(\mathcal{X},\overline{\mathcal{Y}},\mathcal{X},\overline{\mathcal{Y}})$ where $\mathcal{X}$ and $\mathcal{Y}$ are polarized $(1,0)$ vectors and $R$ is the curvature tensor. The correspondence between the anti-bisectional curvature and the MTW tensor provides a meaningful sense in which the anti-bisectional curvature can have a sign (i.e., be positive or negative). In this paper, we study the behavior of the anti-bisectional curvature under Kähler-Ricci flow. We find that non-positive anti-bisectional curvature is preserved under the flow. In complex dimension two, we also show that non-negative orthogonal anti-bisectional curvature (i.e., the MTW(0) condition) is preserved under the flow. We provide several applications of these results – in complex geometry, optimal transport, and affine geometry.

In 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 (${}^d$CQB). We first show that compact Kähler 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 Kähler-Ricci flow to deform the metric. We conjecture that all Kähler 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 Kähler C-space will actually have positive CQB unless it is a ${\mathbb{P}}^1$ bundle. Finally we give an example of non-symmetric, irreducible Kähler C-space with $b_2>1$ and positive CQB, as well as compact non-locally symmetric Kähler manifolds with CQB$<0$ and ${}^d$CQB$<0$.

This paper is the first step in a systematic project to study examples of Kähler manifolds with positive holomorphic sectional curvature ($H>0$). Previously Hitchin proved that any compact Kähler surface with $H>0$ must be rational and he constructed such examples on Hirzebruch surfaces $M_{2,k}=\mathbb{P}(H^k\oplus 1_{{\mathbb{CP}}^1})$. We generalize Hitchin's construction and prove that any Hirzebruch manifold $M_{n,k}=\mathbb{P}(H^k\oplus 1_{{\mathbb{CP}}^{n-1}})$ admits a Kähler metric of $H>0$ in each of its Kähler classes. We demonstrate that the pinching behaviors of holomorphic sectional curvatures of new examples differ from those of Hitchin's which were studied in the recent work of Alvarez-Chaturvedi-Heier. Some connections to recent works on the Kähler-Ricci flow on Hirzebruch manifolds are also discussed. It seems interesting to study the space of all Kähler metrics of $H>0$ on a given Kähler manifold. We give higher dimensional examples such that some Kähler classes admit Kähler metrics with $H>0$ and some do not.