Related papers: K-stability for K\"ahler Manifolds

We show that a polarised manifold with a constant scalar curvature K\"ahler metric and discrete automorphisms is K-stable. This refines the K-semistability proved by S. K. Donaldson.

We study the existence of extremal K\"ahler metrics on K\"ahler manifolds. After introducing a notion of relative K-stability for K\"ahler manifolds, we prove that K\"ahler manifolds admitting extremal K\"ahler metrics are relatively…

In this paper, we shall show that a polarized algebraic manifold is K-stable if the polarization class admits a Kaehler metric of constant scalar curvature. This generalizes the results of Chen-Tian, Donaldson and Stoppa. (Parts of the…

We prove continuity results for new stability thresholds related to uniform K-stability and deduce that uniform K-stability is an open condition in the K\"ahler cone of any compact K\"ahler manifold, thus establishing an algebro-geometric…

We prove the Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds, that is, for projective manifolds equipped with a holomorphic action of a compact Lie group with at least one real hypersurface orbit. Contrary to what seems to be…

Consider a compact K\"ahler manifold which either admits an extremal K\"ahler metric, or is a small deformation of such a manifold. We show that the blowup of the manifold at a point admits an extremal K\"ahler metric in K\"ahler classes…

We introduce a norm on the space of test configurations, which we call the minimum norm. We conjecture that uniform K-stability with respect to this norm is equivalent to the existence of a constant scalar curvature K\"ahler metric. This…

We prove that constant scalar curvature K\"ahler (cscK) manifolds with transcendental cohomology class are K-semistable, naturally generalising the situation for polarised manifolds. Relying on a very recent result by R. Berman, T. Darvas…

We show that if a Fano manifold $M$ is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then $M$ admits a K\"ahler-Einstein metric. This is a strengthening of the solution of the…

Consider a polarized complex manifold (X,L) and a ray of positive metrics on L defined by a positive metric on a test configuration for (X,L). For most of the common functionals in K\"ahler geometry, we prove that the slope at infinity…

We study the stability of compact pseudo-K\"ahler manifolds, i.e. compact complex manifolds $X$ endowed with a symplectic form compatible with the complex structure of $X$. When the corresponding metric is positive-definite, $X$ is K\"ahler…

Using dynamical stability of symplectic curvature flow, we show that on a compact Calabi-Yau manifold, any small symplectic deformation of a K\"ahler form remains K\"ahler.

We survey some recent developments in the direction of the Yau-Tian-Donaldson conjecture, which relates the existence of constant scalar curvature K\"ahler metrics to the algebro-geometric notion of K-stability. The emphasis is put on the…

We embed polarised orbifolds with cyclic stabiliser groups into weighted projective space via a weighted form of Kodaira embedding. Dividing by the (non-reductive) automorphisms of weighted projective space then formally gives a moduli…

If $M$ is a projective manifold in $P^N$, then one can associate to each one parameter subgroup $H$ of $SL(N+1)$ the Mumford $\mu$ invariant. The manifold $M$ is Chow-Mumford stable if $\mu$ is positive for all $H$. Tian has defined the…

In this paper, we study the degeneration and stability of K\"ahler structures on Calabi--Yau manifolds, namely compact K\"ahler manifolds with trivial canonical bundles, from the viewpoint of deformation theory and Hodge theory. Using the…

In this paper we study K-polystability of arbitrary (possibly non-projective) compact K\"ahler manifolds admitting holomorphic vector fields. As a main result, we show that existence of a constant scalar curvature K\"ahler (cscK) metric…

We prove various results involving arcs - which generalise test configurations - within the theory of K-stability. Our main result characterises coercivity of the Mabuchi functional on spaces of Fubini-Study metrics in terms of uniform…

In this paper, assuming that a polarized algebraic manifold $(X,L)$ is strongly K-stable, we shall show that the polarization class $c_1(L)$ admits a constant scalar curvature Kaehler metric.

For a small polarised deformation of a constant scalar curvature K\"ahler manifold, under some cohomological vanishing conditions, we prove that K-polystability along nearby polarisations implies the existence of a constant scalar curvature…