A large focus during the course of the whole project period has been on developing techniques for semistable perturbation problems, both on manifolds and on vector bundles. These are significantly more involved than their counterparts for polystable objects and throughout the project period these techniques have been honed and improved upon, gradually removing various technical assumptions to land on a solid technique that will not only be useful to solve the problems related to the project, but other problems for canonical metrics in complex geometry.
For bundles, this was first focused on for pullback bundles on fibrations in work with Tipler. Under some technical assumptions, we were able to give a very explicit proof that show how a necessary algebro-geometric criterion comes in, in order to solve the classical HYM equation in the setting of pulled back semistable bundles. This was further extended in work with Dervan and McCarthy. A key idea there was to introduce a geometric flow in the local setting, to be able to give a local Kempf-Ness theorem for the finite dimensional problem we reduce to via our analytic construction.
On manifolds, this was taken up in two contexts. The first was that of blowups in joint work with Dervan. Here we manage to push a previously well-studied question further by a different approach involving tools from semistable perturbation problems. The second is work with Tipler. Here we study the problem of changing the Kähler class for certain strictly semistable manifolds, in order to determine when the new Kähler classes admits cscK metrics. This generalises a classical result of LeBrun and Simanca to the strictly semistable situation. Here we also developed a quite different approach to this question, using a different technique than in the other semistable perturbation problems.
During the project period, it also became apparent that it is important to not only make progress on the theoretical study of the OSC equation, but also to produce more examples of fibrations where one has a solution to the OSC equation, or where one can determine that a solution cannot exist. One such construction was produced in joint work with Spotti. Here we produced examples of solutions to the OSC equation in a very specific, but non-trivial, setting. This again relied on perturbative techniques. Further, while researching ways to produce non-examples using algebro-geometric techniques, Tipler and I discovered that we could deduce some surprising results about the behaviour of the Futaki invariant on the Kähler cone of K-polystable Fano threefolds.
The above works resulted in many conference invitations, in Cambridge, at the Mittag-Leffler Institute, at Oberwolfach, at the NCM in Aalto, at a workshop in Le Croisic and at the 2021 Several Complex Variables meeting (Oslo). It also resulted in many invitations for seminars/visits, in-person and virtually, in Paris, Chicago, Northwestern, Lisbon, Nantes, University of Maryland, Bangalore, Potsdam and Trieste. I was also asked to give a lecture series on my work virtually for Tsinghua university.
I co-supervised a PhD student, Engberg, during the project. He defended his thesis in 2022, and we will have a joint paper that resulted from work we did during this period. This develops the beta invariant in this setting. Through this, we in particular recover the relation with slope stability for vector bundles in a very clear and explicit manner.
