Ruadhaí Dervan

K-stability for Kähler manifolds

We formulate a notion of K-stability for Kahler manifolds, and prove one direction of the Yau-Tian-Donaldson conjecture in this setting. More precisely, we prove that the Mabuchi functional being bounded below (resp. coercive) implies K-semistability (resp. uniformly K-stable). In particular this shows that the existence of a constant scalar curvature Kahler metric implies K-semistability, and K-stability if one assumes the automorphism group is discrete. We also show how Stoppas argument holds in the Kahler case, giving a simpler proof of this K-stability statement.

Stable maps in higher dimensions

We formulate a notion of stability for maps between polarised varieties which generalises Kontsevich’s definition when the domain is a curve and Tian-Donaldson’s definition of K-stability when the target is a point. We give some examples, such as Kodaira embeddings and fibrations. We prove the existence of a projective moduli space of canonically polarised stable maps, generalising the Kontsevich-Alexeev moduli space of stable maps in dimensions one and two. We also state an analogue of the Yau–Tian-Donaldson conjecture in this setting, relating stability of maps to the existence of certain canonical Kähler metrics.