On the Classification of ALE Kähler Manifolds

Hein, Hans-Joachim; Răsdeaconu, Rareş; Suvaina, Ioana

doi:10.1093/imrn/rnz376

Cited by 4 publications

(4 citation statements)

References 45 publications

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“…Our proofs of Theorems B-C crucially rely on the fact that K M is trivial, but for ALE spaces as in Theorem C(3) it turns out that no conditions on K M are required at all [54].…”

Section: Resultsmentioning

confidence: 99%

Quasi-asymptotically conical Calabi–Yau manifolds

2019

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We construct new examples of quasi-asymptotically conical (QAC) Calabi-Yau manifolds that are not quasi-asymptotically locally Euclidean (QALE). We do so by first providing a natural compactification of QAC-spaces by manifolds with fibred corners and by giving a definition of QAC-metrics in terms of an associated Lie algebra of smooth vector fields on this compactification. Thanks to this compactification and the Fredholm theory for elliptic operators on QAC-spaces developed by the second author and Mazzeo, we can in many instances obtain Kähler QAC-metrics having Ricci potential decaying sufficiently fast at infinity. This allows us to obtain QAC Calabi-Yau metrics in the Kähler classes of these metrics by solving a corresponding complex Monge-Ampère equation.

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“…Our proofs of Theorems B-C crucially rely on the fact that K M is trivial, but for ALE spaces as in Theorem C(3) it turns out that no conditions on K M are required at all [54].…”

Section: Resultsmentioning

confidence: 99%

Quasi-asymptotically conical Calabi–Yau manifolds

2019

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“…Another question that arises is that of higher dimensions. However, in this case, it has been proven by Hein, Radeasconu and Suvaina in [19] that an ALE model asymptotic to a singularity C m /G has to be isomorphic to a deformation of a resolution of the quotient singularity C m /G. However, by Schlessinger's rigidity theorem [40], such singularities are actually rigid; as a consequence, in complex dimension greater than 3, the only available ALE model, up to biholomorphism, is the resolution of the singularity.…”

Section: Examples and Perspectivesmentioning

confidence: 78%

“…On the "ALE" side, the scalar curvature is zero where ρ ≤ r ε , and is given by second derivatives of g Rε in {r ε ≤ ρ ≤ 2r ε }. Thus, using (19) and factoring in the rescaling, we obtain…”

Section: Estimation Of the Hermitian Scalar Curvature Ofĵ εmentioning

confidence: 99%

Almost-Kähler smoothings of compact complex surfaces with $A_1$ singularities

Vernier

2020

Journal of Symplectic Geometry

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This paper is concerned with the existence of metrics of constant Hermitian scalar curvature on almost-Kähler manifolds obtained as smoothings of a constant scalar curvature Kähler orbifold, with A1 singularities. More precisely, given such an orbifold that does not admit nontrivial holomorphic vector fields, we show that an almost-Kähler smoothing (Mε, ωε) admits an almost-Kähler structure (Ĵε,ĝε) of constant Hermitian curvature. Moreover, we show that for ε > 0 small enough, the (Mε, ωε) are all symplectically equivalent to a fixed symplectic manifold (M ,ω) in which there is a surface S homologous to a 2-sphere, such that [S] is a vanishing cycle that admits a representant that is Hamiltonian stationary forĝε.

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“…With extra structures, partial classifications of AC manifolds are possible. For example, in the Kähler setting, there is a more rigidity to AC manifolds; indeed, it is shown in [27], that all Kähler ALE manifolds are obtained as resolutions of deformations of an orbifold singularity C {G. Also all Clalabi-Yau AC manifolds polynomially asymptotic (up to diffeomorphism) to the hyperquadric in C n`1 are classified [15].…”

Section: Introductionmentioning

confidence: 99%

Cultural Heritage Lost: Case Study of Isfahan, Iran

Asadi¹

2023

Multiculturalism and Interculturalism - Managing Diversity in Cross-Cultural Environment

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This research examines the natural and man-made hazards such as the case of urban development, nature, and legislation in the city of Isfahan in Iran. The goals are to show how cultural heritage is at risk of extinction, what the causes are, and how to find and offer solutions. Similar cases and previously studied have been conducted but although these studies all concluded how to use CH as a source of income from attracting tourism, and therefore, there were few details on all kinds of threats to our cultural heritage at once in recent years. On the other hand, the unsustainable progress alongside economic breakdown caused by international sanctions created a budget crisis which leads that has caused the Cultural Heritage Organization to not receive adequate funding. The study was designed as a single-case, qualitative method in which data were gathered through in-depth interviews and discussions with randomly chosen local residents, active heritage experts, and managers in the field of heritage preservation and urban planning. Isfahan has been chosen as a representative example of a metropolis with a variety of heritage participation projects. Although solving budget problems requires experts in this field so that we can maintain our CH sites, learning from the traditional art of the past and rethinking the concept of urban development management is essential to end the crisis of destruction of cultural heritage.

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On the Classification of ALE Kähler Manifolds

Abstract: The underlying complex structure of an ALE Kähler manifold is exhibited as a resolution of a deformation of an isolated quotient singularity. As a consequence, there exist only finitely many diffeomorphism types of minimal ALE Kähler surfaces with a given group at infinity.

Cited by 4 publications

References 45 publications

Quasi-asymptotically conical Calabi–Yau manifolds

Quasi-asymptotically conical Calabi–Yau manifolds

Almost-Kähler smoothings of compact complex surfaces with $A_1$ singularities

Cultural Heritage Lost: Case Study of Isfahan, Iran

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