Complex Geometrics Flows

In the tentative to move from the Kähler to the non-Kähler setting, one can formulate several problems concerning Hermitian metrics on complex manifolds with special curvature properties. Among these problems, we mention the existence of Hermitian metrics with constant scalar curvature with respect to the Chern connection, and the generalizations of the Kähler-Einstein condition to the non-Kähler setting. They are usually translated and attacked as analytic pdes.

In this context, the Chern-Ricci flow plays an useful role. The Chern-Ricci flow is a parabolic evolution equation for Hermitian metrics that extends the Kähler-Ricci flow to Hermitian manifolds. It is expected that the behavior of solutions of the Chern-Ricci flow deeply reflects the underlying complex structure. In particular, understanding the behaviour of the Chern-Ricci flow on non-Kähler compact complex surfaces is particularly interesting, due to the fact that minimal class VII surfaces are not yet completely classified.

In this talk, we study the problem of uniform convergence of the normalized Chern-Ricci flow on Inoue-Bombieri surfaces with Gauduchon metrics.

The talk is based on a joint work with Valentino Tosatti, and on collaborations and discussions with Simone Calamai, Francesco Pediconi, Cristiano Spotti.

In this talk I will overview joint work with J. Jordan and J. Streets, in arXiv:2106.13716, about Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form. These metrics give a natural extension of Calabi-Yau metrics to the setting of complex, non-Kähler manifolds, and arise independently in mathematical physics. We reinterpret this condition in terms of the Hermitian-Einstein equation on an associated holomorphic Courant algebroid, and thus refer to solutions as Bismut Hermitian-Einstein. This implies Mumford-Takemoto slope stability obstructions, and using these we exhibit infinitely many topologically distinct complex manifolds in every dimension with vanishing first Chern class which do not admit Bismut Hermitian-Einstein metrics. This reformulation also leads to a new description of pluriclosed flow, as introduced by Streets and Tian, implying new global existence results. In particular, on all complex non-Kähler surfaces of nonnegative Kodaira dimension. On complex manifolds which admit Bismut-flat metrics we show global existence and convergence of pluriclosed flow to a Bismut-flat metric.

We will survey the moving-bracket/algebraic soliton approach to study complex geometric flows of homogeneous almost-Hermitian structures.

The first half of this talk will be a historical survey of geometric flows and their applications in various subfields of differential geometry. We will then specialize to complex non-Kähler geometry, and discuss links from complex geometry to heterotic string theory. Finally, we will present some results on the Anomaly flow, which is a geometric flow inspired from a system of equations in heterotic string theory developed in joint works with T. Fei, D.H. Phong and X.-W. Zhang.

We will give an overview of our recent works regarding classification of ancient solutions in the Mean Curvature Flow and the Ricci flow. We will briefly discuss the methods we used to achieve those results in both flows and the importance of such results in singularity analysis.

Generalized Ricci flow is a natural flow coupling the Ricci flow with an evolution equation for a closed 3-form. This flow emerges in two different settings in complex geometry expanding the applicability of the Ricci flow beyond the world of Kähler manifolds. The purpose of the talk is to explain how the steady soliton equations for this flow give rise to the notions of canonical metrics in non-Kähler and bi-Hermitian complex geometry. The corresponding differential equations have been long known in physics, so the description and classification of compact/complete solutions is of great interest. We show how the rich complex-geometric structure of the underlying manifolds allows for the adaptation of many methods of the Kähler geometry to the study of the Generalized Ricci flow and its solitons.