K-stability of constant scalar curvature polarization

By T. Mabuchi. 2008

61 citations recorded by the source corpus.

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 arguments are based on a forthcoming paper "A stronger concept of K-stability." )

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Citations in the corpus

On blowing up extremal Kähler manifolds (2010, background): “pic are Thomas [32] and PhongSturm [26]. Note that it is essentially known that if an extremal metric exists then the manifold is relatively K-polystable (see Donaldson [10], Stoppa [28, 30], Mabuchi [21]), although this depends on the precise deﬁnition of stability being used. A natural problem is ”
The Calabi conjecture and K-stability (2010, result): “ty (cf. e. g. [RT07, section 3]). So far, one direction of Conjecture 1.2, i. e. , the claim that the existence of cscK implies Kpolystability is proved, due to the works of [Don05], [CT08], [Stp09], [Mab08b] and [Mab09], though the converse is only proved for some special cases at present. If c1(X)”
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”
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].”
On the moduli of Kähler-Einstein Fano manifolds (2012, methodology): “n02]. As originally expected, the existence of Ka¨hler-Einstein metrics on a Fano manifold is proved to be equivalent to the K-stability. The proof of stability of KE Fano manifolds is established by [Mab08], [Mab09], [Ber12] and for the converse direction, please consult [CDS12a], [CDS12b], [CDS13]”
Alpha invariant and K-stability of Q-Fano varieties (2010, background): “The case where Aut(X,L) is not discrete is studied in [17] and [18].”
Regularity of weak minimizers of the K-energy and applications to properness and K-stability (2016, methodology): “As a consequence of Theorem 1.5 and the techniques of [Ti2, Be3] we obtain a result on K-polystability, originally proved by Mabuchi ([Ma3, Main Theorem] see also [Ma2]), using a completely diﬀerent argument.”
Compact moduli space of Kahler-Einstein Fano varieties (2014, methodology): “tion In this short paper, we construct compactiﬁed moduli algebraic space of Fano manifolds which have Ka¨hler-Einstein metrics or equivalently (thanks to the achievements [CDS], combined with [Ber], [Mab1],[Mab2]) are K-polystable, following the (precise) conjecture in [OSS] formulated with C. Spot”
Extremal K\"ahler metrics (2014, background): “y. In Section 2 we will discuss two such notions of stability: K-stability, and a slight reﬁnement of it which we call Kb-stability. As a consequence of work of Tian [54], Donaldson [18, 20], Mabuchi [34], Stoppa [44], Stoppa-Sz´ekelyhidi [45], Paul [39], Berman [7], and others, there are now many s”