TBA

The Chern-Ricci flow is one of several proposed generalisations of the Kahler-Ricci flow in the Hermitian setting. The aim of this talk is twofold. We first outline the behaviour of the Chern-Ricci flow on complex minimal surfaces. Then, motivated by several results on minimal surfaces, we show that the curvature type is independent of the starting metric in its 'class' for long-time solutions.  This demonstrates a curvature type independence result that holds for the Kahler case.  

Since Calabi's original paper, the Calabi Ansatz has been central for constructions in Kähler geometry. Calabi himself used it to construct complete Ricci-flat metrics on the total space of the canonical bundle of a Kähler-Einstein Fano manifold (B, \omega_B), generalizing some well-known examples coming from physics. Over the years, work of Koiso, Cao, Feldman-Ilmanen-Knopf, Futaki-Wang, Chi Li, and others have shown that the Calabi Ansatz can be used to produce complete Kähler-Ricci solitons, important singularity models for the Kähler-Ricci flow, on certain line bundles over (B, \omega_B). In this talk, I'll explain a generalization of these results to the total space of some higher-rank direct-sum vector bundles over (B, \omega_B). In our case the Calabi Ansatz is not suitable, and we instead use the theory of hamiltonian 2-forms, introduced by Apostolov-Calderbank-Gauduchon-Tønneson-Friedman. The construction produces new examples of complete shrinkers, steadies, and Calabi-Yau metrics, as well as recovering some known ones.

Ricci-flat metrics are fundamental in differential geometry, and they are easier to study when they have additional structures. I will describe my works on 4d non-trivially conformally Kähler Ricci-flat metrics, which actually is a very natural class of 4d Ricci-flat metrics. This leads to a classification of asymptotic geometries of such metrics at infinity and a classification of such gravitational instantons

In this joint work with Mihai Paun, we show that a so-called Q-sheaf on a log terminal Kahler threefold satisfies a suitable Bogomolov-Gieseker inequality as soon as it is stable with respect to a Kahler class. I will discuss the strategy of the proof as well as some applications if time permits.