On K-Polystability of cscK Manifolds with Transcendental Cohomology Class

2018 ◽  
Vol 2020 (9) ◽  
pp. 2769-2817 ◽  
Author(s):  
Zakarias Sjöström Dyrefelt
Keyword(s):  
Scalar Curvature ◽  
Cohomology Class ◽  
Vector Fields ◽  
Constant Scalar Curvature ◽  
Energy Functional ◽  
Kahler Manifolds ◽  
Kähler Metric ◽  
Holomorphic Vector Fields ◽  
Singularity Type ◽  
Kahler Metric

Abstract In this paper we study K-polystability of arbitrary (possibly non-projective) compact Kähler manifolds admitting holomorphic vector fields. As a main result we show that existence of a constant scalar curvature Kähler (cscK) metric implies geodesic K-polystability, in a sense that is expected to be equivalent to K-polystability in general. In particular, in the spirit of an expectation of Chen–Tang [28] we show that geodesic K-polystability implies algebraic K-polystability for polarized manifolds, so our main result recovers a possibly stronger version of results of Berman–Darvas–Lu [10] in this case. As a key part of the proof we also study subgeodesic rays with singularity type prescribed by singular test configurations and prove a result on asymptotics of the K-energy functional along such rays. In an appendix by R. Dervan it is moreover deduced that geodesic K-polystability implies equivariant K-polystability. This improves upon the results of [39] and proves that existence of a cscK (or extremal) Kähler metric implies equivariant K-polystability (resp. relative K-stability).

Partially regular and cscK metrics

2020 ◽  
Vol 31 (10) ◽  
pp. 2050079
Author(s):  
Andrea Loi ◽  
Fabio Zuddas
Keyword(s):  
Scalar Curvature ◽  
Complex Manifold ◽  
Positive Constant ◽  
Bergman Kernel ◽  
Constant Scalar Curvature ◽  
Kähler Metric ◽  
Kähler Form ◽  
Kahler Metric

A Kähler metric [Formula: see text] with integral Kähler form is said to be partially regular if the partial Bergman kernel associated to [Formula: see text] is a positive constant for all integer [Formula: see text] sufficiently large. The aim of this paper is to prove that for all [Formula: see text] there exists an [Formula: see text]-dimensional complex manifold equipped with strictly partially regular and cscK metric [Formula: see text]. Further, for [Formula: see text], the (constant) scalar curvature of [Formula: see text] can be chosen to be zero, positive or negative.

scholarly journals Remarks on extremal Kähler metrics on ruled manifolds

1992 ◽  
Vol 126 ◽  
pp. 89-101 ◽  
Author(s):  
Akira Fujiki
Keyword(s):  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Einstein Metric ◽  
Kähler Metrics ◽  
Algebraic Subgroup ◽  
Kähler Metric ◽  
Kahler Metrics ◽  
Extremal Kähler Metrics ◽  
Kahler Metric ◽  
Compact Kähler Manifold

Let X be a compact Kähler manifold and γ Kähler class. For a Kàhler metric g on X we denote by Rg the scalar curvature on X According to Calabi [3][4], consider the functional defined on the set of all the Kähler metrics g whose Kähler forms belong to γ, where dvg is the volume form associated to g. Such a Kähler metric is called extremal if it gives a critical point of Ф. In particular, if Rg is constant, g is extremal. The converse is also true if dim L(X) = 0, where L(X) is the maximal connected linear algebraic subgroup of AutoX (cf. [5]). Note also that any Kähler-Einstein metric is of constant scalar curvature.

PRECOMPACTNESS OF THE CALABI ENERGY

1996 ◽  
Vol 07 (02) ◽  
pp. 245-254 ◽  
Author(s):  
SANTIAGO R. SIMANCA
Keyword(s):  
Continuous Function ◽  
Scalar Curvature ◽  
Complex Manifold ◽  
Vector Fields ◽  
Convergent Sequence ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Holomorphic Vector Fields ◽  
Extremal Kähler Metrics ◽  
Kahler Metric

For any complex manifold of Kähler type, the L2-norm of the scalar curvature of an extremal Kähler metric is a continuous function of the Kähler class. In particular, if a convergent sequence of Kähler classes are represented by extremal Kähler metrics, the corresponding sequence of L2-norms of the scalar curvatures is convergent. Similarly, the sequence of holomorphic vector fields associated with a sequence of extremal Kähler metrics with converging Kähler classes is convergent.

scholarly journals Stable Polarized del Pezzo Surfaces

2018 ◽  
Vol 2020 (18) ◽  
pp. 5477-5505 ◽  
Author(s):  
Ivan Cheltsov ◽  
Jesus Martinez-Garcia
Keyword(s):  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Del Pezzo Surfaces ◽  
Sufficient Condition ◽  
Kähler Metric ◽  
Simple Sufficient Condition ◽  
Del Pezzo ◽  
Kahler Metric

Abstract We give a simple sufficient condition for $K$-stability of polarized del Pezzo surfaces and for the existence of a constant scalar curvature Kähler metric in the Kähler class corresponding to the polarization.

UNIQUENESS OF EXTREMAL KÄHLER METRICS FOR AN INTEGRAL KÄHLER CLASS

2004 ◽  
Vol 15 (06) ◽  
pp. 531-546 ◽  
Author(s):  
TOSHIKI MABUCHI
Keyword(s):  
Scalar Curvature ◽  
Recent Result ◽  
Complex Manifold ◽  
Constant Scalar Curvature ◽  
Kähler Metrics ◽  
Kähler Metric ◽  
Kahler Metrics ◽  
Extremal Kähler Metrics ◽  
Kahler Metric

For an integral Kähler class on a compact connected complex manifold, an extremal Kähler metric, if any, in the class is unique up to the action of Aut 0(M). This generalizes a recent result of Donaldson (see [4] for cases of metrics of constant scalar curvature) and that of Chen [3] for c1(M)≤0.

scholarly journals K-SEMISTABILITY OF OPTIMAL DEGENERATIONS

2020 ◽  
Vol 71 (3) ◽  
pp. 989-995
Author(s):  
Ruadhaí Dervan
Keyword(s):  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Kähler Metric ◽  
Kahler Metric

Abstract K-polystability of a polarized variety is an algebro-geometric notion conjecturally equivalent to the existence of a constant scalar curvature Kähler metric. When a variety is K-unstable, it is expected to admit a ‘most destabilizing’ degeneration. In this note we show that if such a degeneration exists, then the limiting scheme is itself relatively K-semistable.

scholarly journals Quaternionic contact 4n + 3-manifolds and their 4n-quotients

2021 ◽  
Author(s):  
Yoshinobu Kamishima
Keyword(s):  
Scalar Curvature ◽  
Lie Group ◽  
Contact Structure ◽  
Einstein Manifolds ◽  
Hermitian Metrics ◽  
Kähler Metric ◽  
Formula One ◽  
Quaternion Space ◽  
Kahler Metric ◽  
Do So

AbstractWe study some types of qc-Einstein manifolds with zero qc-scalar curvature introduced by S. Ivanov and D. Vassilev. Secondly, we shall construct a family of quaternionic Hermitian metrics $$(g_a,\{J_\alpha \}_{\alpha =1}^3)$$ ( g a , { J α } α = 1 3 ) on the domain Y of the standard quaternion space $${\mathbb {H}}^n$$ H n one of which, say $$(g_a,J_1)$$ ( g a , J 1 ) is a Bochner flat Kähler metric. To do so, we deform conformally the standard quaternionic contact structure on the domain X of the quaternionic Heisenberg Lie group$${{\mathcal {M}}}$$ M to obtain quaternionic Hermitian metrics on the quotient Y of X by $${\mathbb {R}}^3$$ R 3 .

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