Differential Geometry Seminar (2025)

Extremal Kähler metrics and destabilizers for relative K-polystability of toric varieties

It was conjectured by Székelyhidi that a polarized manifold admits an extremal Kähler metric in the class of polarization if and only if it is relatively K-polystable. Furthermore, the folklore conjecture states that every toric Fano manifold admits an extremal Kähler metric in its first Chern class. For a given toric Fano manifold X, we provide a destabilizing convex function on the corresponding moment polytope P to clarify the relative K-unstability of X. Applying this criteria into a certain toric Fano manifold, we prove that there exists a toric Fano manifold of dimension 10 that does not admit an extremal Kähler metric. This talk is based on joint work with B. Zhou, and another recent work with D. Hwang and H. Sato.

On s-commutative sets in real Grassmannian manifolds and representations of Clifford algebras

An s-commutative set in a quandle or a symmetric space is a set in which, for any two points, the point symmetries at those points are commutative. This notion is a generalization of the antipodal set, which was introduced by Hiroshi Tamaru et al. Moreover we expect that they have geometric information about the quandles or symmetric spaces. Although, for the s-commutative sets, several concrete examples are known, the details of them are not clear. In this talk, I will present a construction of s-commutative sets in real Grassmannian manifolds derived from representations of Clifford algebras. In addition, I will also discuss the relationship between our results and the classification of maximal antipodal sets in real Grassmannian manifolds given by Makiko Tanaka and Hiroyuki Tasaki.

A characterization of Oeljeklaus-Toma manifolds in LCK geometry

Oeljeklaus–Toma (OT) manifolds are known as examples of complex manifolds that do not admit a Kähler metric, and they are regarded as higher-dimensional analogues of Inoue surfaces. OT manifolds are solvmanifolds constructed from number-theoretic data, and some of them admit locally conformally Kähler (LCK) metrics. In this way, a large number of examples of solvmanifolds equipped with LCK metrics have been obtained, and OT manifolds have been actively studied as important examples in LCK geometry. Although their construction may seem intricate, aside from some simple examples, OT manifolds are the only known solvmanifolds admitting LCK metrics. In this talk, I will show that if a certain class of solvmanifolds admits an LCK metric, then it is essentially an OT manifold. Since number theory naturally arises from geometric constraints in our setting, this result suggests that number-theoretic arguments are indispensable in the construction of certain classes of solvmanifolds. This talk is based on the preprint arXiv:2502.12500.