scholarly journals Calabi flow on toric varieties with bounded Sobolev constant, I

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Hongnian Huang
Keyword(s):  
Ricci Curvature ◽  
Toric Variety ◽  
Toric Varieties ◽  
Kähler Manifold ◽  
Kähler Metric ◽  
Calabi Flow ◽  
Kahler Manifold ◽  
Sobolev Constant ◽  
Kahler Metric ◽  
Uniformly Bounded

AbstractLet (X, P) be a toric variety. In this note, we show that the C0-norm of the Calabi flow φ(t) on X is uniformly bounded in [0, T) if the Sobolev constant of φ(t) is uniformly bounded in [0, T). We also show that if (X, P) is uniform K-stable, then the modified Calabi flow converges exponentially fast to an extremal Kähler metric if the Ricci curvature and the Sobolev constant are uniformly bounded. At last, we discuss an extension of our results to a quasi-proper Kähler manifold.

scholarly journals TWISTING OF N=1 SUSY GAUGE THEORIES AND HETEROTIC TOPOLOGICAL THEORIES

1995 ◽  
Vol 10 (30) ◽  
pp. 4325-4357 ◽  
Author(s):  
A. JOHANSEN
Keyword(s):  
Complex Structure ◽  
Kähler Manifold ◽  
Dolbeault Cohomology ◽  
Kähler Metric ◽  
Yang Mills ◽  
Kahler Manifold ◽  
Brst Charge ◽  
Kahler Metric ◽  
Compact Kähler Manifold ◽  
Topological Theories

It is shown that D=4N=1 SUSY Yang-Mills theory with an appropriate supermultiplet of matter can be twisted on a compact Kähler manifold. The conditions for cancellation of anomalies of BRST charge are found. The twisted theory has an appropriate BRST charge. We find a nontrivial set of physical operators defined as classes of the cohomology of this BRST operator. We prove that the physical correlators are independent of the external Kähler metric up to a power of a ratio of two Ray-Singer torsions for the Dolbeault cohomology complex on a Kähler manifold. The correlators of local physical operators turn out to be independent of antiholomorphic coordinates defined with a complex structure on the Kähler manifold. However, a dependence of the correlators on holomorphic coordinates can still remain. For a hyper-Kähler metric the physical correlators turn out to be independent of all coordinates of insertions of local physical operators.

Conformally Kähler manifolds

1954 ◽  
Vol 50 (1) ◽  
pp. 16-19 ◽  
Author(s):  
W. J. Westlake
Keyword(s):  
Conformal Geometry ◽  
Kähler Manifold ◽  
Hermitian Manifold ◽  
Necessary And Sufficient Condition ◽  
Kähler Metric ◽  
Kahler Manifold ◽  
Hermitian Space ◽  
Necessary And Sufficient ◽  
Simply Connected ◽  
Kahler Metric

Introduction. The present paper is concerned with the conformal geometry of Hermitian spaces. In the first part we find a necessary and sufficient condition for a Hermitian space to be conformally Kähler, that is, conformal to some Kähler space. The condition is that a certain conformal tensor, , vanishes identically. Then, defining a Hermitian manifold as in Hodge (3), we consider such a manifold where the restriction is made that at every point the tensor is zero. This will be called a conformally Kähler manifold, and conditions under which it may be given a Kähler metric are obtained. It is found that any conformally Kähler manifold may be given a Kähler metric provided it is simply-connected or that its fundamental group is of finite order.

THE TIAN–YAU–ZELDITCH ASYMPTOTIC EXPANSION FOR REAL ANALYTIC KÄHLER METRICS

2004 ◽  
Vol 01 (03) ◽  
pp. 253-263 ◽  
Author(s):  
ANDREA LOI
Keyword(s):  
Asymptotic Expansion ◽  
Kähler Manifold ◽  
Distortion Function ◽  
Kähler Metrics ◽  
Kähler Metric ◽  
Kahler Metrics ◽  
Kahler Manifold ◽  
Real Analytic ◽  
Kahler Metric ◽  
Compact Kähler Manifold

Let M be a compact Kähler manifold endowed with a real analytic and polarized Kähler metric g and let Tmω(x) be the corresponding Kempf's distortion function. In this paper we compute the coefficients of Tian–Yau–Zelditch asymptotic expansion of Tmω(x) using quantization techniques alternative to Lu's computations in [10].

scholarly journals Ricci-flat Kähler metrics on canonical bundles

2002 ◽  
Vol 132 (3) ◽  
pp. 471-479 ◽  
Author(s):  
ROGER BIELAWSKI
Keyword(s):  
Kähler Manifold ◽  
Zero Section ◽  
Canonical Bundle ◽  
Kähler Metrics ◽  
Kähler Metric ◽  
Kahler Metrics ◽  
Ricci Flat ◽  
Kahler Manifold ◽  
Real Analytic ◽  
Kahler Metric

We prove the existence of a (unique) S1-invariant Ricci-flat Kähler metric on a neighbourhood of the zero section in the canonical bundle of a real-analytic Kähler manifold X, extending the metric on X.

Quaternionic Transformations of a Non-Positive Quaternionic Kähler Manifold

1997 ◽  
Vol 08 (03) ◽  
pp. 301-316 ◽  
Author(s):  
D. V. Alekseevsky ◽  
S. Marchiafava
Keyword(s):  
Kähler Manifold ◽  
Parameter Group ◽  
Kähler Metric ◽  
Quaternionic Structure ◽  
Kahler Manifold ◽  
Quaternionic Kähler Manifold ◽  
De Rham ◽  
Simply Connected ◽  
Kahler Metric ◽  
Quaternionic Kähler

Let (M,g,Q) be a simply connected, complete, quaternionic Kähler manifold without flat de Rham factor. Then any 1-parameter group of transformations of M which preserve the quaternionic structure Q preserves also the metric g. Moreover, if (M,g) is irreducible then the quaternionic Kähler metric g on (M,Q) is unique up to a homothety.

scholarly journals The Calabi Flow with Rough Initial Data

2019 ◽  
Author(s):  
Weiyong He ◽  
Yu Zeng
Keyword(s):  
Initial Data ◽  
Heat Equation ◽  
Riemannian Manifolds ◽  
Kähler Metric ◽  
Calabi Flow ◽  
Existence Time ◽  
Rough Initial Data ◽  
Short Time ◽  
Kahler Metric ◽  
Time Existence

Abstract In this paper, we prove that there exists a dimensional constant $\delta> 0$ such that given any background Kähler metric $\omega $, the Calabi flow with initial data $u_0$ satisfying \begin{equation*} \partial \bar \partial u_0 \in L^\infty (M)\ \textrm{and}\ (1- \delta )\omega < \omega_{u_0} < (1+\delta )\omega, \end{equation*}admits a unique short-time solution, and it becomes smooth immediately, where $\omega _{u_0}: = \omega +\sqrt{-1}\partial \bar \partial u_0$. The existence time depends on initial data $u_0$ and the metric $\omega $. As a corollary, we get that the Calabi flow has short-time existence for any initial data satisfying \begin{equation*} \partial \bar \partial u_0 \in C^0(M)\ \textrm{and}\ \omega_{u_0}> 0, \end{equation*}which should be interpreted as a “continuous Kähler metric”. A main technical ingredient is a new Schauder-type estimates for biharmonic heat equation on Riemannian manifolds with time-weighted Hölder norms.

scholarly journals BOUNDING SECTIONAL CURVATURE ALONG THE KÄHLER–RICCI FLOW

2009 ◽  
Vol 11 (06) ◽  
pp. 1067-1077 ◽  
Author(s):  
WEI-DONG RUAN ◽  
YUGUANG ZHANG ◽  
ZHENLEI ZHANG
Keyword(s):  
Sectional Curvature ◽  
Chern Class ◽  
Ricci Flow ◽  
Kähler Manifold ◽  
Ricci Soliton ◽  
Curvature Operator ◽  
Kahler Manifold ◽  
First Chern Class ◽  
Compact Kähler Manifold ◽  
Uniformly Bounded

If a normalized Kähler–Ricci flow g(t), t ∈ [0,∞), on a compact Kähler manifold M, dim ℂ M = n ≥ 3, with positive first Chern class satisfies g(t) ∈ 2πc1(M) and has curvature operator uniformly bounded in Ln-norm, the curvature operator will also be uniformly bounded along the flow. Consequently, the flow will converge along a subsequence to a Kähler–Ricci soliton.

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.

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