K\"ahler-Ricci Flow preserves negative anti-bisectional curvature

Abstract

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 K\"ahler 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 K\"ahler 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 K\"ahler-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.