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.

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 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.

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.

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.

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).

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 .

scholarly journals Lagrangian foliations and Anosov symplectomorphisms on Kähler manifolds

2020 ◽  
pp. 1-11
Author(s):  
M. J. D. HAMILTON ◽  
D. KOTSCHICK
Keyword(s):  
The Other ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Kähler Metric ◽  
Other Hand ◽  
Kähler Form ◽  
Lagrangian Foliation ◽  
The One ◽  
Kahler Metric

Abstract We investigate parallel Lagrangian foliations on Kähler manifolds. On the one hand, we show that a Kähler metric admitting a parallel Lagrangian foliation must be flat. On the other hand, we give many examples of parallel Lagrangian foliations on closed flat Kähler manifolds which are not tori. These examples arise from Anosov automorphisms preserving a Kähler form.

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.

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