Kähler-Ricci flow preserves negative anti-bisectional curvature

In recent work, the first named author and Zhang found a connection between the regularity theory of optimal transport and the curvature of Kähler manifolds. In particular, we showed that the Ma-Trudinger-Wang (MTW) tensor for a cost function c ( x , y ) = Ψ ( x − y ) 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 ( X , Y ¯ , X , Y ¯ ) R(\mathcal {X}, \overline {\mathcal {Y}},\mathcal {X},\overline {\mathcal {Y}}) where X \mathcal {X} and Y \mathcal {Y} are polarized ( 1 , 0 ) (1,0) vectors and R R is the curvature tensor. The polarization 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. Somewhat unexpectedly, 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.