Special/canonical metrics play a central role in modern complex geometry. A major question is to characterize their existence on given complex manifolds, famous examples including the Kobayashi-Hitchin correspondence and existence of Kähler-Einstein metrics via the Yau-Tian-Donaldson conjecture. The latter is a good prototype, showing that Kähler-Einstein metrics exist if and only if the underlying manifold satisfies an algebro-geometric condition called K-stability. More generally, special metrics occur naturally as solutions to geometric partial differential equations (PDE), whose solvability is however often hard or currently impossible to test in practice. In the SMKG we focus on key PDE's in Kähler geometry, notably Calabi's extremal metrics as well as solutions to Donaldson's J-equation, the deformed Hermitian-Yang-Mills equation, and Z-critical or generalised Monge-Ampère equations. Our main aim is to find effective (geometric) criteria for solvability of these infinite families of geometric PDE. This is motivated by finding finitely many optimal destabilizing subvarieties to numerical criteria (or in the longer perspective, optimal destabilizing test configurations to e.g. K-stability), and also has direct implications for proving existence of wall-chamber type decompositions for geometric PDE.
The group often interacts with the CMCG group led by Cristiano Spotti, as well as G. Bérczi, A. Swann, and A. Otiman.
Villum Young Investigator "Effective Testing in Complex Geometry" (June 1st 2024 – 2029). PI Z. Sjöström Dyrefelt.
