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) = (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 (X, {Y}, X, {Y}) where X X and Y 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.