BIMSA > Differential Geometry Seminar Toric Fano manifolds that do not admit extremal Kähler metrics
Toric Fano manifolds that do not admit extremal Kähler metrics
Organizers
Kenji Fukaya , Lynn Heller , Sebastian Heller , Kotaro Kawai
Speaker
Naoto Yotsutani
Time
Tuesday, September 16, 2025 2:40 PM - 5:30 PM
Venue
A7-201
Abstract
It was conjectured by Székelyhidi that a polarized manifold admits an extremal Kähler metric in the polarization class if and only if it is relatively K-polystable. In addition, a well-known folklore conjecture asserts that every toric Fano manifold admits an extremal Kähler metric in its first Chern class. For a given toric Fano manifold X, we construct a destabilizing convex function on the associated moment polytope, thereby demonstrating the relative K-unstability of X. Applying relative K-unstability criterion to a specific toric Fano manifold, we show that there exists a 10-dimensional toric Fano manifold that does not admit an extremal Kähler metric. This talk is based on joint work with D. S. Hwang and H. Sato.