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.

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 Resolution à la Kronheimer of $$\mathbb {C}^3/\Gamma $$ singularities and the Monge–Ampère equation for Ricci-flat Kähler metrics in view of D3-brane solutions of supergravity

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Massimo Bianchi ◽  
Ugo Bruzzo ◽  
Pietro Fré ◽  
Dario Martelli
Keyword(s):  
Exceptional Divisor ◽  
The Other ◽  
Canonical Bundle ◽  
Kähler Metrics ◽  
Kähler Metric ◽  
Kahler Metrics ◽  
Ricci Flat ◽  
Flat Metric ◽  
Kahler Metric ◽  
Monge Ampere

AbstractIn this paper, we analyze the relevance of the generalized Kronheimer construction for the gauge/gravity correspondence. We begin with the general structure of D3-brane solutions of type IIB supergravity on smooth manifolds $$Y^\Gamma $$ Y Γ that are supposed to be the crepant resolution of quotient singularities $$\mathbb {C}^3/\Gamma $$ C 3 / Γ with $$\Gamma $$ Γ a finite subgroup of SU(3). We emphasize that nontrivial 3-form fluxes require the existence of imaginary self-dual harmonic forms $$\omega ^{2,1}$$ ω 2 , 1 . Although excluded in the classical Kronheimer construction, they may be reintroduced by means of mass deformations. Next we concentrate on the other essential item for the D3-brane construction, namely, the existence of a Ricci-flat metric on $$Y^\Gamma $$ Y Γ . We study the issue of Ricci-flat Kähler metrics on such resolutions $$Y^\Gamma $$ Y Γ , with particular attention to the case $$\Gamma =\mathbb {Z}_4$$ Γ = Z 4 . We advance the conjecture that on the exceptional divisor of $$Y^\Gamma $$ Y Γ the Kronheimer Kähler metric and the Ricci-flat one, that is locally flat at infinity, coincide. The conjecture is shown to be true in the case of the Ricci-flat metric on $$\mathrm{tot} K_{{\mathbb {W}P}[112]}$$ tot K W P [ 112 ] that we construct, i.e., the total space of the canonical bundle of the weighted projective space $${\mathbb {W}P}[112]$$ W P [ 112 ] , which is a partial resolution of $$\mathbb {C}^3/\mathbb {Z}_4$$ C 3 / Z 4 . For the full resolution, we have $$Y^{\mathbb {Z}_4}={\text {tot}} K_{\mathbb {F}_{2}}$$ Y Z 4 = tot K F 2 , where $$\mathbb {F}_2$$ F 2 is the second Hirzebruch surface. We try to extend the proof of the conjecture to this case using the one-parameter Kähler metric on $$\mathbb {F}_2$$ F 2 produced by the Kronheimer construction as initial datum in a Monge–Ampère (MA) equation. We exhibit three formulations of this MA equation, one in terms of the Kähler potential, the other two in terms of the symplectic potential but with two different choices of the variables. In both cases, one can establish a series solution in powers of the variable along the fibers of the canonical bundle. The main property of the MA equation is that it does not impose any condition on the initial geometry of the exceptional divisor, rather it uniquely determines all the subsequent terms as local functionals of this initial datum. Although a formal proof is still missing, numerical and analytical results support the conjecture. As a by-product of our investigation, we have identified some new properties of this type of MA equations that we believe to be so far unknown.

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.

Type II compact almost homogeneous manifolds of cohomogeneity one-II

2019 ◽  
Vol 30 (13) ◽  
pp. 1940002
Author(s):  
Daniel Guan
Keyword(s):  
Moduli Space ◽  
Kähler Manifold ◽  
Type Ii ◽  
Homogeneous Manifolds ◽  
Kähler Metrics ◽  
Cohomogeneity One ◽  
Kahler Metrics ◽  
Kahler Manifold ◽  
Blowing Up ◽  
Compact Kähler Manifold

In this paper, we start the program of the existence of the smooth equivariant geodesics in the equivariant Mabuchi moduli space of Kähler metrics on type II cohomogeneity one compact Kähler manifold. In this paper, we deal with the manifolds [Formula: see text] obtained by blowing up the diagonal of the product of two copies of a [Formula: see text].

scholarly journals Curvature and the c-projective mobility of Kähler metrics with hamiltonian 2-forms

2016 ◽  
Vol 152 (8) ◽  
pp. 1555-1575 ◽  
Author(s):  
David M. J. Calderbank ◽  
Vladimir S. Matveev ◽  
Stefan Rosemann
Keyword(s):  
Necessary Conditions ◽  
Kähler Metrics ◽  
Kähler Metric ◽  
Kahler Metrics ◽  
Local Classification ◽  
Complex Cone ◽  
Kahler Metric ◽  
Cone Metric

The mobility of a Kähler metric is the dimension of the space of metrics with which it is c-projectively equivalent. The mobility is at least two if and only if the Kähler metric admits a nontrivial hamiltonian 2-form. After summarizing this relationship, we present necessary conditions for a Kähler metric to have mobility at least three: its curvature must have nontrivial nullity at every point. Using the local classification of Kähler metrics with hamiltonian 2-forms, we describe explicitly the Kähler metrics with mobility at least three and hence show that the nullity condition on the curvature is also sufficient, up to some degenerate exceptions. In an appendix, we explain how the classification may be related, generically, to the holonomy of a complex cone metric.

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.

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 Locally conformally Kähler metrics on Kato surfaces

2011 ◽  
Vol 202 ◽  
pp. 77-81 ◽  
Author(s):  
Marco Brunella
Keyword(s):  
Kähler Metrics ◽  
Kähler Metric ◽  
Kahler Metrics ◽  
Locally Conformally Kähler ◽  
Kahler Metric ◽  
Kato Surfaces

AbstractWe show that every Kato surface admits a locally conformally Kähler metric.

Bergman metrics and geodesics in the space of Kähler metrics on principally polarized abelian varieties

2011 ◽  
Vol 11 (1) ◽  
pp. 1-25 ◽  
Author(s):  
Renjie Feng
Keyword(s):  
Abelian Varieties ◽  
Kähler Manifold ◽  
Invariant Metrics ◽  
Kahler Geometry ◽  
Kähler Metrics ◽  
Complete Asymptotic Expansion ◽  
Kahler Metrics ◽  
Infinite Dimensional ◽  
Kahler Manifold ◽  
Bergman Metrics

AbstractIt is well known in Kähler geometry that the infinite-dimensional symmetric space $\mathcal{H}$ of smooth Kähler metrics in a fixed Kähler class on a polarized Kähler manifold is well approximated by finite-dimensional submanifolds $\mathcal{B}_k\subset\mathcal{H}$ of Bergman metrics of height k. Then it is natural to ask whether geodesics in $\mathcal{H}$ can be approximated by Bergman geodesics in $\mathcal{B}_k$. For any polarized Kähler manifold, the approximation is in the C0 topology. For some special varieties, one expects better convergence: Song and Zelditch proved the C2 convergence for the torus-invariant metrics over toric varieties. In this article, we show that some C∞ approximation exists as well as a complete asymptotic expansion for principally polarized abelian varieties.

scholarly journals Locally conformally Kähler metrics on Kato surfaces

2011 ◽  
Vol 202 ◽  
pp. 77-81 ◽  
Author(s):  
Marco Brunella
Keyword(s):  
Kähler Metrics ◽  
Kähler Metric ◽  
Kahler Metrics ◽  
Locally Conformally Kähler ◽  
Kahler Metric ◽  
Kato Surfaces

AbstractWe show that every Kato surface admits a locally conformally Kähler metric.

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