Extremal metrics and K‐stability

By G'abor Sz'ekelyhidi. 2004

199 citations recorded by the source corpus.

We propose an algebraic geometric stability criterion for a polarised variety to admit an extremal Kähler metric. This generalises conjectures by Yau, Tian and Donaldson, which relate to the case of Kähler–Einstein and constant scalar curvature metrics. We give a result in geometric invariant theory that motivates this conjecture, and an example computation that supports it.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d9bcc136cd2b195e4293d12bc9e732683bda13a0

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Extremal metrics on blowups (2006, background): “[37, 26, 35, 13]) it is now clear that the existence problem is related to a suitable stability condition (in the sense of GIT [29]) for (M,J, c).”
Filtrations and test-configurations (2011, background): “This picture can be extended to bundles of toric varieties, in particular to ruled surfaces, following [27].”
Hamiltonian 2-forms in Kähler geometry, III extremal metrics and stability (2005, methodology): “Now, following Székelyhidi [50], we note that the weight wx of the linear action of the isotropy algebra gx on L∗x is given by wx = 〈βx, ·〉 : gx → R for some βx ∈ gx , which is the orthogonal projection of μ(x) onto gx .”
On blowing up extremal Kähler manifolds (2010, methodology): “The relationship of this definition using moment maps to geometric invariant theory is a version of the Kempf-Ness theorem [16] and is worked out in [31] (see also Kirwan [17]).”
Extremal Kähler metrics on projective bundles over a curve (2009, unknown): “Székelyhidi introduced [66, 67] the notion of relative K-(poly)stability with respect to a maximal torus of the automorphism group of the pair (M,L) — which is the same as the reduced automorphism group Ãut0(M,J) — and the similar implication ‘extremal ⇒ relatively K-polystable’ was recently establi”
Lower bounds on the Calabi functional (2005, unknown): “This would be a variant of the conjectures in [23], [21], [8 ]r elating K-stability to the existence of constant scalar curvature metrics (and the extension of that discussion to extremal metrics in [ 19 ]).”
Kähler metrics with constant weighted scalar curvature and weighted K‐stability (2018, methodology): “With the above in mind, the main goal of this paper is to introduce a suitable notion of (v,w)K-stability associated to (X,α,T,P, v,w) as above, extending the corresponding notion in the cscK and the extremal cases, introduced by Tian [83, 84], Donaldson [34], and Székelyhidi [77], and extensively ”
Existence of approximate Hermitian-Einstein structures on semi-stable bundles (2010, methodology): “The analogue of the Donaldson functional is in this case the Mabuchi K-energy [19], and several analogues of Mumford-Takemoto stability have also been introduced, including Chow-Mumford stability, K-stability (Tian [28], Donaldson [11]), uniform K-stability (Szekelyhidi [27]), slope-stability (Ross-”
Lectures on Stability and Constant Scalar Curvature (2008, methodology): “A slightly stronger version of K-stability has been proposed by Szekelyhidi [135].”
Extremal metrics and K-stability (PhD thesis) (2006, background): “Such a criterion was proposed by the author in [31] (see also Mabuchi [22] for a different definition).”
RELATIVE K-STABILITY AND MODIFIED K-ENERGY ON TORIC MANIFOLDS (2006, methodology): “In this section, we recall the notation of relative K-stability introduced in [Sz] and compute the modified Futaki invariant for a test configuration in the sense of Donaldson ([D2]).”