Kähler-Einstein metrics on Fano manifolds
Lead Research Organisation:
University of Nottingham
Department Name: Sch of Mathematical Sciences
Abstract
The fabric of modern geometry is designed around questions that lead to the existence of canonical metrics on manifolds. A classical example is the Riemannian metrics with constant Gauss curvature on Riemann surfaces. The higher dimensional analogue sparks the hope of finding an "Einstein metric" on a given manifold. When the manifold in question is Kähler, then the desired metric is called Kähler-Einstein.
Manifolds can be simplified to have positive or negative curvature, or be flat. The existence of a Kähler-Einstein metric when the curvature is negative or flat is known, thanks to the celebrated work of Aubin and Yau. The existence of such metric is obstructed in the positive curvature case. Due to the pioneering work of Donaldson et al, the existence of a Kähler-Einstein metric in this case is determined by an algebraic stability condition on the underlying Fano variety. However, it is difficult to verify such stability condition for a given Fano variety.
Based on some recent developments in the field, we aim to produce and fine-tune a new method to check whether a given Fano variety is K-(semi)stable or not. The plan is to apply this new method to various challenging examples.
Manifolds can be simplified to have positive or negative curvature, or be flat. The existence of a Kähler-Einstein metric when the curvature is negative or flat is known, thanks to the celebrated work of Aubin and Yau. The existence of such metric is obstructed in the positive curvature case. Due to the pioneering work of Donaldson et al, the existence of a Kähler-Einstein metric in this case is determined by an algebraic stability condition on the underlying Fano variety. However, it is difficult to verify such stability condition for a given Fano variety.
Based on some recent developments in the field, we aim to produce and fine-tune a new method to check whether a given Fano variety is K-(semi)stable or not. The plan is to apply this new method to various challenging examples.
Organisations
Publications
Abban H
(2022)
Stability of fibrations over one-dimensional bases
in Duke Mathematical Journal
Abban H
(2023)
Seshadri constants and K-stability of Fano manifolds
in Duke Mathematical Journal
Abban H
(2022)
K-stability of Fano varieties via admissible flags
in Forum of Mathematics, Pi
Duarte Guerreiro T
(2022)
Sarkisov links from toric weighted blowups of $\mathbb{P}^4$ at a point
in Portugaliae Mathematica
Kim I
(2022)
K-stability of log del Pezzo hypersurfaces with index 2
in International Journal of Mathematics