Periodic Reporting for period 1 - CanMetCplxGeom (Finding canonical metrics in complex differential geometry)
Berichtszeitraum: 2021-09-01 bis 2023-08-31
A central problem in complex geometry is the search for canonical Kähler metrics, which are optimal notions of distance on a Kähler manifold. This goes back to the Uniformisation theorem in complex dimension one, and is a highly active area of research to the present day. In higher dimensions, Aubin and Yau's resolution of the Calabi conjecture showed that one always obtains such metrics for canonically polarised and Calabi-Yau manifolds. However, in general, higher dimensional complex manifolds may or may not admit a canonical metric. A key goal in complex differential geometry is to understand whether or not a given complex manifold admits a canonical Kähler metric, such as those of constant scalar curvature (cscK), or more generally, extremal metrics. This is a highly non-trivial question, and has surprising links to algebraic geometry. The Yau-Tian-Donaldson (YTD) conjecture is central to the field. This predicts that an algebraic notion of stability should be able to detect whether or not a given Kähler manifold admits a canonical metric. It is similar in spirit to the Hitchin-Kobayashi correspondence for vector bundles, which characterises the existence of Hermite-Einstein metrics on vector bundles via the notion of slope stability. This was established in the 1980's by Uhlenbeck-Yau and Donaldson.
A key goal in the project is to produce such metrics. Another is to study how these metrics behave in families. Together with Dervan, I started a research programme on this theme. We introduced an equation, called the Optimal Symplectic Connection equation, for good families of canonical metrics on fibrations, and a notion of stability for fibrations. For projectivised vector bundles, this recovers the notions of Hermite-Einstein metrics and slope stability as in the Hitchin-Kobayashi correspondence for vector bundles. An overall goal in this research programme is to establish a Hitchin-Kobayashi/YTD type conjecture. This requires the development of many differential and algebro-geometric tools.
A key goal in the project is to produce such metrics. Another is to study how these metrics behave in families. Together with Dervan, I started a research programme on this theme. We introduced an equation, called the Optimal Symplectic Connection equation, for good families of canonical metrics on fibrations, and a notion of stability for fibrations. For projectivised vector bundles, this recovers the notions of Hermite-Einstein metrics and slope stability as in the Hitchin-Kobayashi correspondence for vector bundles. An overall goal in this research programme is to establish a Hitchin-Kobayashi/YTD type conjecture. This requires the development of many differential and algebro-geometric tools.
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.
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.
As mentioned above, one clear and successful theme for the project has been the development of techniques to address semistable perturbation problems in complex geometry. The project has resulted in a very solid technique for attacking such problems, which was not available before, and which should be applicable in many problems, not just the ones related to the project. This makes it useful not only for the proposed project, but for the wider complex differential geometry community.
There was also made progress on the algebro-geometric aspects of the theory, which involved learning and developing techniques that are further away from my training in complex differential geometry. These will be important for one of the main overarching themes of the project, namely the YTD type correspondence for the OSC equation. To address issues related to this it is important with strong differential and algebro-geometric foundations. Further, these techniques should be important for the understanding of specific examples - the PDEs on the differential geometry side are very hard to solve, but in certain situations, the algebro-geometric quantities can be calculated, allowing for an understanding of the equation from the algebro-geometric side.
The project resolved many of the more ambitious aspects of the proposal. Me and my collaborators also resolved some problems that were not included in the original proposal, but which were found to be interesting and relevant results given the progress made. Some of the problems addressed during the course of the project were not finalised by the end of the project, but will continue to be studied and resolved in the time afterwards.
Overall, the techniques and results of the project should be impactful for the community doing research in complex differential and algebraic geometry.
Being a project in pure mathematics, there is so far to me no clear potential for commercial exploitation of the results obtained.
There was also made progress on the algebro-geometric aspects of the theory, which involved learning and developing techniques that are further away from my training in complex differential geometry. These will be important for one of the main overarching themes of the project, namely the YTD type correspondence for the OSC equation. To address issues related to this it is important with strong differential and algebro-geometric foundations. Further, these techniques should be important for the understanding of specific examples - the PDEs on the differential geometry side are very hard to solve, but in certain situations, the algebro-geometric quantities can be calculated, allowing for an understanding of the equation from the algebro-geometric side.
The project resolved many of the more ambitious aspects of the proposal. Me and my collaborators also resolved some problems that were not included in the original proposal, but which were found to be interesting and relevant results given the progress made. Some of the problems addressed during the course of the project were not finalised by the end of the project, but will continue to be studied and resolved in the time afterwards.
Overall, the techniques and results of the project should be impactful for the community doing research in complex differential and algebraic geometry.
Being a project in pure mathematics, there is so far to me no clear potential for commercial exploitation of the results obtained.