Differential Geometry Seminar (2024)

Constant scalar curvature Kähler metrics and semistable vector bundles

A central question in Kähler geometry is if a Kähler manifold admits a canonical metric, such as a Kähler-Einstein metric or more generally a constant scalar curvature Kähler (cscK) metric, in a given Kähler class. The Yau-Tian-Donaldson conjecture predicts that this is equivalent to an algebraic notion of stability. In this talk, I will discuss a necessary and sufficient condition for the projectivisation of a slope semistable vector bundle to admit cscK metrics in adiabatic classes, when the base admits a cscK metric. In particular, this shows that the existence of cscK metrics is equivalent to K-stability in this setting. Moreover, our construction reduces K-stability to a finite dimensional criterion in terms of intersection numbers associated to the vector bundle. This is joint work with Annamaria Ortu.

Global algebraic functions on the moduli space of parabolic connections

In the framework of Simpson, there is a correspondence between three moduli spaces: the de Rham moduli space, the Dolbeault moduli space and the Betti moduli space. The de Rham moduli space is the moduli space of integrable connections, the Dolbeault moduli space is the moduli space of Higgs bundles and the Betti moduli space is the moduli space of representations of a fundamental group. All these moduli spaces are constructed as algebraic varieties, but their structures seem to be different. In the logarithmic case, we consider the de Rham moduli space as the moduli space of parabolic connections with a fixed determinant line bundle. In that case, we can prove that the transcendence degree of the ring of global algebraic functions on the de Rham moduli space is less than or equal to half the dimension of the moduli space. As a corollary, we can see that the Riemann--Hilbert morphism from the de Rham moduli space to the Betti moduli space is not algebraic. The content of this talk is based on the joint work with Biswas, Komyo and Saito.

Japanese page only

Japanese page only

Non-Hausdorff Differential Geometry

It is standard practice to impose the Hausdorff property within the definition of a manifold. This is for good reason: Hausdorffness gives us access to partitions of unity subordinate to any open cover, and these in turn can be used to construct various objects of geometric interest. In this talk we will boldly take the opposite approach and see what happens when we relax the Hausdorff property in the definition of a manifold. A priori, it may seem that such spaces are too difficult to study without arbitrarily-existent partitions of unity. However, through a series of tricks it is possible to avoid this problem and construct various things like smooth structures, bundles, differential forms and integrals. We will keep the talk pedagogical and start with the basics: firstly we will review the topological properties of non-Hausdorff manifolds, and then we will slowly add more and more structure of interest, until we build up to a description of cohomology. As a final goal, we will describe de Rham cohomology and prove a non-Hausdorff version of de Rham’s theorem, all without appealing to partitions of unity. Finally, we finish with some ongoing ideas regarding the Cech-de Rham equivalence for non-Hausdorff manifolds.

Subriemannian geometry and spectral analysis

A regular subriemannian manifold M carries a geometric hypoelliptic operator, the intrinsic sublaplacian. Due to a degeneracy of its symbol, geometric and analytic e ects can be observed in the study of this operator, which have no counterpart in Riemannian geometry. During the last decades inverse spectral problems in subriemannian geometry have been studied by various authors. Typical approaches are based on the analysis of the induced subriemannian heat or wave equation. In this talk we survey some results in subriemannian geometry. In particular, we address the spectral theory of the sublaplacian in the case of certain compact nilmanifolds or, more generally, for H-type foliations. This talk is based on joint work with K. Furutani, C. Iwasaki, A. Laaroussi, I. Markina and S. Vega-Molino.

Homogeneous Structures on Three- and Four-dimensional Lie groups

In this talk, we will introduce the notion of homogeneous pseudo-Riemannian structures and demonstrate how to establish homogeneity and natural reductiveness of 3- and 4-dimensional Lie groups through a tensor satisfying certain geometric partial differential equations involving the metric and the curvature of a given manifold. These equations are known as Ambrose-Singer equations. We will begin by examining homogeneous structures on three-dimensional unimodular and non-unimodular Lie groups, proving the existence of homogeneous Lorentzian structures that differ from the canonical ones without being naturally reductive, a phenomenon with no Riemannian counterpart. Using these homogeneous structures, we will show how to classify naturally reductive 3-dimensional Lorentzian manifolds. Our focus will then shift to the geometric properties of four-dimensional nilpotent Lie groups endowed with a family of non-flat left-invariant Lorentzian metrics. We will conduct a comprehensive classification of homogeneous structures for each metric and meticulously examine the distinctive properties characterizing each structure. Additionally, we will provide a specific example demonstrating the presence of a naturally reductive, non-flat, left-invariant Lorentzian metric on the 2-nilpotent Lie group, where the center exhibits degeneracy. Furthermore, we will establish the existence of a non-canonical homogeneous structure. As an application, we will demonstrate the existence of naturally reductive left-invariant Lorentzian metrics on the four-dimensional 3-nilpotent Lie group.

Topological recursion and twisted Higgs bundles

Prior works relating meromorphic Higgs bundles to topological recursion have considered non-singular models that allow the recursion to be carried out on a smooth Riemann surface. I will discuss some recent work where we define a "twisted topological recursion" on the spectral curve of a twisted Higgs bundle, and show that the g=0 components of the recursion compute the Taylor expansion of the period matrix of the spectral curve, mirroring a result of for ordinary Higgs bundles and topological recursion. I will also discuss some current work relating topological recursion to a new viewpoint of quantization of Higgs bundles.

Homology of metric spaces