Hermitian Curvature Flow and Curvature Positivity Conditions

Abstract

In the present thesis, we study metric flows on, not necessarily Kahler, complex Hermitian manifolds. Using the framework of the Hermitian curvature flows, due to Streets and Tian, we find a distinguished metric flow (further referred to as the HCF), which shares many features of the Ricci flow. For a large family of convex sets of Chern curvature tensors, we prove its invariance under the HCF.

Varying these convex sets, we demonstrate that the HCF preserves many natural curvature (semi)positivity conditions in complex geometry: Griffiths/dual-Nakano/m-dual positivity, positivity of the holomorphic orthogonal bisectional curvature, lower bounds on the second scalar curvature. The key ingredient in the proof of these results is a very special form of the evolution equation for the Chern curvature tensor, which we were able to obtain by introducing a torsion-twisted connection. Motivated by these results, we formulate a differential-geometric version of Campana-Peternell conjecture, which characterizes the rational homogeneous manifolds by certain curvature semipositivity properties. We propose a metric flow approach based on the HCF and make an initial progress towards the conjecture. Specifically, we characterize complex manifolds admitting a metric of quasipositive Griffiths curvature, and find obstructions on the torsion-twisted holonomy group of an Hermitian manifold with a semipositive dual-Nakano curvature. We illustrate the behavior of the HCF by explicitly computing it on all complex homogeneous manifold, equipped with submersion metrics.