Kaehler manifolds of constant curvature with conical singularities

Constant curvature metrics surround us, we live in Euclidean space of zero curvature, little soap bubbles have positive constant curvature. Objects of constant negative curvature are less familiar, but they do appear in Nature in the shape of corals and leaves. Not surprisingly, constant curvature metrics play an important role in geometric topology, which studies manifolds, i.e. higher dimensional generalisations of surfaces. It is a geometer's dream to find a canonical metric on a given manifold so that its topology, i.e. its shape up to stretching and squeezing, will be captured by its geometry. One famous incarnation of this idea is Thurston's geometrization conjecture solved by Grigori Perelman. This conjecture gives topological criteria for a compact 3-manifold to admit a constant curvature metric.

The goal of this project is to study a generalisation of constant curvature manifolds, namely constant curvature manifolds with conical singularities. Here, a prototypical example is the surface of a regular tetrahedron (which is topologically a sphere). This surface has conical singularities of angle 180 degrees at the vertices of the tetrahedron and is flat elsewhere. More generally, surfaces of all polyhedra are flat surfaces with conical singularities. One of the central objects of this project consists of higher-dimensional generalisations of polyhedral surfaces, namely polyhedral Kaehler manifolds.

Higher-dimensional polyhedral Kaehler manifolds are connected to rich mathematical structures and exhibit a lot of rigidity, this can be illustrated by the following example. Hirzebruch conjectured that any collection of 3n lines in the complex projective plane with each line intersecting others in n+1 points, is a collection of mirrors of a complex reflection group (a complex analogue of a crystallographic group). It turns out that any such collection of lines is the singular locus of a polyhedral Kaehler metric on the complex plane. This result gives a plausible approach for settling the Hirzebruch conjecture. Looking for various restrictions that the existence of a polyhedral Kaehler metric imposes on the underlying manifold and its singular locus is one of the main goals of this project.

Coming back to surfaces, we note that flat surfaces with conical singularities are quite well understood. Surprisingly, this is not at all the case for curvature one (i.e. spherical) surfaces with conical singularities. The study of this topic can be traced back to the beginning of 20th century and the work of Felix Klein, however it is full of open questions. For example, the following simple question was settled only in 2018.
Question: what are all possible collections of conical angles that a spherical surface with conical singularities can have? The answer to this question required a number of involved tools, such as parabolic bundles and gluing techniques.

An important feature of spherical surfaces with conical singularities is that the spaces of such metrics are interesting geometric objects in their own right. Investigation of such moduli spaces is a second theme of this project. We plan to give a first full description of such moduli spaces of low dimensions, we will study the topology of higher-dimensional moduli spaces and investigate their natural maps to the space of Riemann surfaces. It is worth noting that in contrast to moduli spaces of spherical surfaces, the current knowledge of moduli spaces of Riemann surfaces is extremely vast and this topic is connected to virtually all geometric disciplines from integrable systems to string theory. We hope that the moduli spaces of spherical metrics could have a similar fate.

Academic impact. The main impact of this project will be on the mathematical community. In particular, by using mostly geometric and synthetic tools we will be able to answer questions on spheres with conical points which are of considerable interest to the PDE community. I hope that these results and methods will be then understood and translated into PDE methods which will find further applications.

Possible scientific impact. The moduli space of Riemann surfaces is an extremely important object both for Mathematics in general and for some branches of Theoretical Physics. The moduli space of spherical metrics with conical singularities seem to be quite close cousins of the former, but they are not properly understood. I believe that being such close relatives of marked Riemann surfaces, spherical surfaces should also find a multitude of applications.

Dissemination and training. The travel support provided by the fellowship will help me to disseminate my results at various conferences and seminars, in particular, at the institutions where co-investigators work. Additional impact of the project will come from the training of a PDRA, and the visits to the UK of my collaborators. In particular, the impact will be enhanced by my involvement with the EPSRC funded CDT, LSGNT. Various spin-offs of the project will serve as PhD themes for students at this centre. Finally, the impact will be facilitated by the workshop that I plan to organise at KCL.