K-stability of constant scalar curvature Kähler manifolds

By J. Stoppa. 2008

182 citations recorded by the source corpus.

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

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/74396fa67de4046485e4d81670b589449efb0dc4

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In Applied Mathematics

In Geometry and Topology

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Citations in the corpus

Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics (2008, background): “imit cycle is a smooth (or normal) scheme. This is the deﬁnition adopted by most researchers in the ﬁeld today . As a small sample the reader should consult the following interesting papers ([3],[36],[38],[39],[2],[22],[52] and [21]) . In 2004 the CM 3In this paper Galways denotes the special linear”
Extremal K\"ahler metrics (2014, background): “As a consequence of work of Tian [54], Donaldson [18, 20], Mabuchi [34], Stoppa [44], Stoppa-Székelyhidi [45], Paul [39], Berman [7], and others, there are now many satisfactory results that show that the existence of an extremal metric implies various notions of stability.”
Stability, birational transformations and the Kahler-Einstein problem (2010, background): “It seems reasonable to hope that the argument of Stoppa in [10] can be extended to show that K − stability =⇒ b− stability, and, assuming this can be done, we get KE=⇒ b-stability.”
The J-flow and stability (2013, methodology): “We now strengthen this to strict positivity, with a perturbation method similar to Stoppa’s work [16] on K-stability.”
Compact Moduli Spaces of Del Pezzo Surfaces and K\"ahler-Einstein metrics (2012, background): “3.2. CM line bundle comparison. In this subsection we study GIT stability of Ka¨hler-Einstein log Del Pezzo surfaces. For smooth Kahler-Einstein manifolds, it is known that they are K-polystable (cf. [73, 68, 44]). This has been generalized to the singular setting in [11], and we state the two dimen”
The Kähler-Ricci flow and K-polystability (2008, methodology): “This was improved by Stoppa [19] who shows that existence of a cscK metric implies K-polystability (using Donaldson’s deﬁnition) when the manifold has no holomorphic vector ﬁelds.”
A generalization of the Ross--Thomas slope theory (2009, background): “Let us recall that a polarized manifold with a constant scalar curvature Kähler metric is K-polystable, due to the works of [5], [2], [22], [12] and [13].”
Uniform stability of twisted constant scalar curvature K\"ahler metrics (2014, background): “This is in contrast Stoppa’s perturbation argument in the untwisted setting [44], which perturbs the manifold itself, even at the level of topological spaces.”
Filtrations and test-configurations (2011, methodology): “This result is a strengthening of Stoppa’s result [26], whose conclusion under the same assumptions is that (X, L) is K-stable, since it implies that the Futaki invariant has to be bounded away from zero uniformly along certain families of test-configurations.”
Alpha invariant and K-stability of Q-Fano varieties (2010, unknown): “2 (in particular, [33], [11], [9], [30]), one direction is proved as follows.”
Extremal Kähler metrics on projective bundles over a curve (2009, result): “-polystable. This conjecture is still open, but the implication ‘CSC ⇒ K-polystable’ in the conjecture is now well-established, thanks to work by S. K. Donaldson [20], X. Chen–G. Tian [15], J. Stoppa [64], and T. Mabuchi [53, 54]. In order to account for extremal Ka¨hler metrics of non-constant scal”
On blowing up extremal Kähler manifolds (2010, background): “In comparison, Stoppa’s result [29] implies that if BlpM is K-polystable for the polarizations π∗L−εE for sufficiently small ε, then p ∈M is semistable with respect to the linearization L.”