Topology of Asymptotically Conical Calabi–Yau and G2 Manifolds and Desingularization of Nearly Kähler and Nearly G2 Conifolds

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  • Lothar Schiemanowski

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OriginalspracheEnglisch
Aufsatznummer105
FachzeitschriftJournal of Geometric Analysis
Jahrgang33
Ausgabenummer3
Frühes Online-Datum30 Jan. 2023
PublikationsstatusVeröffentlicht - März 2023

Abstract

A natural approach to the construction of nearly G2 manifolds lies in resolving nearly G2 spaces with isolated conical singularities by gluing in asymptotically conical G2 manifolds modelled on the same cone. If such a resolution exits, one expects there to be a family of nearly G2 manifolds, whose endpoint is the original nearly G2 conifold and whose parameter is the scale of the glued in asymptotically conical G2 manifold. We show that in many cases such a curve does not exist. The non-existence result is based on a topological result for asymptotically conical G2 manifolds: if the rate of the metric is below - 3 , then the G2 4-form is exact if and only if the manifold is Euclidean R7. A similar construction is possible in the nearly Kähler case, which we investigate in the same manner with similar results. In this case, the non-existence results is based on a topological result for asymptotically conical Calabi–Yau 6-manifolds: if the rate of the metric is below - 3 , then the square of the Kähler form and the complex volume form can only be simultaneously exact, if the manifold is Euclidean R6.

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Topology of Asymptotically Conical Calabi–Yau and G2 Manifolds and Desingularization of Nearly Kähler and Nearly G2 Conifolds. / Schiemanowski, Lothar.
in: Journal of Geometric Analysis, Jahrgang 33, Nr. 3, 105, 03.2023.

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Schiemanowski L. Topology of Asymptotically Conical Calabi–Yau and G2 Manifolds and Desingularization of Nearly Kähler and Nearly G2 Conifolds. Journal of Geometric Analysis. 2023 Mär;33(3):105. Epub 2023 Jan 30. doi: 10.48550/arXiv.2205.15922, 10.1007/s12220-022-01143-3
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abstract = "A natural approach to the construction of nearly G2 manifolds lies in resolving nearly G2 spaces with isolated conical singularities by gluing in asymptotically conical G2 manifolds modelled on the same cone. If such a resolution exits, one expects there to be a family of nearly G2 manifolds, whose endpoint is the original nearly G2 conifold and whose parameter is the scale of the glued in asymptotically conical G2 manifold. We show that in many cases such a curve does not exist. The non-existence result is based on a topological result for asymptotically conical G2 manifolds: if the rate of the metric is below - 3 , then the G2 4-form is exact if and only if the manifold is Euclidean R7. A similar construction is possible in the nearly K{\"a}hler case, which we investigate in the same manner with similar results. In this case, the non-existence results is based on a topological result for asymptotically conical Calabi–Yau 6-manifolds: if the rate of the metric is below - 3 , then the square of the K{\"a}hler form and the complex volume form can only be simultaneously exact, if the manifold is Euclidean R6.",
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Download

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N1 - Funding Information: The author wishes to thank M. Freibert and H. Weiß for many inspiring discussions, which led the author to consider the problems addressed in this article. The author also wishes to thank the anonymous referee, whose suggestions improved the exposition and sharpened several results.

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