Calabi–Yau Manifolds with Torsion and Geometric Flows

Abstract

The main theme of these lectures is the study of Hermitian metrics in non-Kahler complex geometry. We will specialize to a certain class of Hermitian metrics which generalize Kahler Ricci-flat metrics to the non-Kahler setting. These non-Kahler Calabi–Yau manifolds have their origins in theoretical physics, where they were introduced in the works of C. Hull and A. Strominger. We will introduce tools from geometric analysis, namely geometric flows, to study this non-Kahler Calabi–Yau geometry. More specifically, we will discuss the Anomaly flow, which is a version of the Ricci flow customized to this particular geometric setting. This flow was introduced in joint works with Duong Phong and Xiangwen Zhang. Section 2.1 contains a review of Hermitian metrics, connections, and curvature. Section 2.2 is dedicated to the geometry of Calabi–Yau manifolds equipped with a conformally balanced metric. Section 2.3 introduces the Anomaly flow in the simplest case of zero slope, where the flow can be understood as a deformation path connecting non-Kahler to Kahler geometry. Section 2.4 concerns the Anomaly flow with α′ corrections, which is motivated from theoretical physics and canonical metrics in non-Kahler geometry.