scholarly journals Uniqueness of optimal symplectic connections

2021 ◽  
Vol 9 ◽  
Author(s):  
Ruadhaí Dervan ◽  
Lars Martin Sektnan
Keyword(s):  
Automorphism Group ◽  
Holomorphic Vector Bundle ◽  
Compact Subgroup ◽  
Constant Scalar Curvature ◽  
Absolute Minimum ◽  
Kähler Metric ◽  
Symplectic Connections ◽  
Kahler Metric ◽  
Symplectic Connection ◽  
Geometric Partial Differential Equation

Abstract Consider a holomorphic submersion between compact Kähler manifolds, such that each fibre admits a constantscalar curvature Kähler metric. When the fibres admit continuous automorphisms, a choice of fibrewise constant scalarcurvature Kähler metric is not unique. An optimal symplectic connection is a choice of fibrewise constant scalar curvature Kähler metric satisfying a geometric partial differential equation. The condition generalises the Hermite-Einstein condition for a holomorphic vector bundle through the induced fibrewise Fubini-Study metric on the associated projectivisation. We prove various foundational analytic results concerning optimal symplectic connections. Our main result proves that optimal symplectic connections are unique, up to the action of the automorphism group of the submersion, when they exist. Thus optimal symplectic connections are canonical relatively Kähler metrics when they exist. In addition, we show that the existence of an optimal symplectic connection forces the automorphism group of the submersion to be reductive and that an optimal symplectic connection is automatically invariant under a maximal compact subgroup of this automorphism group. We also prove that when a submersion admits an optimal symplectic connection, it achieves the absolute minimum of a natural log norm functional, which we define.

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.

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.

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 Kobayashi—Hitchin corrispondence for twisted vector bundles

2021 ◽  
Vol 8 (1) ◽  
pp. 1-95
Author(s):  
Arvid Perego
Keyword(s):  
Vector Bundle ◽  
Compact Manifold ◽  
Vector Bundles ◽  
Holomorphic Vector Bundle ◽  
Kähler Manifold ◽  
Kahler Manifolds ◽  
Kähler Metric ◽  
Holomorphic Vector Bundles ◽  
Kahler Metric ◽  
Compact Kähler Manifold

Abstract We prove the Kobayashi—Hitchin correspondence and the approximate Kobayashi—Hitchin correspondence for twisted holomorphic vector bundles on compact Kähler manifolds. More precisely, if X is a compact manifold and g is a Gauduchon metric on X, a twisted holomorphic vector bundle on X is g−polystable if and only if it is g−Hermite-Einstein, and if X is a compact Kähler manifold and g is a Kähler metric on X, then a twisted holomorphic vector bundle on X is g−semistable if and only if it is approximate g−Hermite-Einstein.

scholarly journals Hierarchical structure of physical Yukawa couplings from matter field Kähler metric

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Keiya Ishiguro ◽  
Tatsuo Kobayashi ◽  
Hajime Otsuka
Keyword(s):  
String Theory ◽  
Hierarchical Structure ◽  
Matter Field ◽  
Heterotic String ◽  
Heterotic String Theory ◽  
Kähler Metric ◽  
Yukawa Couplings ◽  
String Compactifications ◽  
Kahler Metric ◽  
Matter Fields

Abstract We study the impacts of matter field Kähler metric on physical Yukawa couplings in string compactifications. Since the Kähler metric is non-trivial in general, the kinetic mixing of matter fields opens a new avenue for realizing a hierarchical structure of physical Yukawa couplings, even when holomorphic Yukawa couplings have the trivial structure. The hierarchical Yukawa couplings are demonstrated by couplings of pure untwisted modes on toroidal orbifolds and their resolutions in the context of heterotic string theory with standard embedding. Also, we study the hierarchical couplings among untwisted and twisted modes on resolved orbifolds.

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