Hamiltonian 2-forms in Kähler geometry, III extremal metrics and stability

By V. Apostolov, D. Calderbank, P. Gauduchon, Christina W. Tønnesen-Friedman. 2005

166 citations recorded by the source corpus.

This paper concerns the existence and explicit construction of extremal Kähler metrics on total spaces of projective bundles, which have been studied in many places. We present a unified approach, motivated by the theory of Hamiltonian 2-forms (as introduced and studied in previous papers in the series) but this paper is largely independent of that theory.We obtain a characterization, on a large family of projective bundles, of the ‘admissible’ Kähler classes (i.e., those compatible with the bundle structure in a way we make precise) which contain an extremal Kähler metric. In many cases every Kähler class is admissible. In particular, our results complete the classification of extremal Kähler metrics on geometrically ruled surfaces, answering several long-standing questions.We also find that our characterization agrees with a notion of K-stability for admissible Kähler classes. Our examples and nonexistence results therefore provide a fertile testing ground for the rapidly developing theory of stability for projective varieties, and we discuss some of the ramifications. In particular we obtain examples of projective varieties which are destabilized by a non-algebraic degeneration.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ad85ed4841f4ec2ec0a144263b2319c665b7e47e

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Extremal metrics and K‐stability (2004, unknown): “For a torus T in G with Lie algebra t, define two subalgebras of g: gT := {α ∈ g | [α, β] = 0 for all β ∈ t}; gT⊥ := {α ∈ gT | 〈α, β〉 = 0 for all β ∈ t} ⊂ gT .”
Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics (2008, background): “ 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 group. DI”
Lectures on Stability and Constant Scalar Curvature (2008, background): “For example, there is a very rich literature on geometric constructions of metrics of constant scalar curvature, which we did not discuss, see e.g. [ 2 , 3, 75, 82, 116, 121].”
Filtrations and test-configurations (2011, background): “Another reason is that Apostolov–Calderbank–Gauduchon–Tønnesen–Friedman [1] have found an example of a manifold that does not admit an extremal metric, but does not appear to be destabilized by a test-configuration.”
Relative K-stability of extremal metrics (2009, background): “Note that by an example in [1] relative K-polystability may not be sufficient to ensure the existence of an extremal metric, so it is likely that Conjectures 1.”
An obstruction to the existence of constant scalar curvature K (2004, background): “ bounded from below [Do4, CT] (resp. proper in the Ka¨hler-Einstein Fano case when aut(X) = 0 [Ti3]); in turn this implies K-semistability [PT] (resp. K-stability). We remark that a recent example in [ACGT] suggests that algebraic K-stability may not be enough to guarantee the existence of a cscK me”
CM Stability and the Generalized Futaki Invariant I (2006, background): “For example, the reader is invited to consu lt , as a rather small sample, the following papers [9, 8, 5, 28, 29, 1, 21, 22, 15].”
On the Calabi Flow (2006, background): “There are many beautiful works in constructing special CscK metric or CextK metric on many K¨ahler manifolds (cf. Lebrun-Simanca [35] [36] [4] and [43] etc and reference therein).”
Stability, birational transformations and the Kahler-Einstein problem (2010, background): “On the other hand an example of Apostolov et al [1] shows that K-stability is probably not the correct criterion for the existence of these metrics.”