Details
| Original language | English |
|---|---|
| Article number | 105 |
| Journal | Journal of Geometric Analysis |
| Volume | 33 |
| Issue number | 3 |
| Early online date | 30 Jan 2023 |
| Publication status | Published - Mar 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.
Keywords
- Asymptotically conical manifolds, Einstein manifolds, Special holonomy
ASJC Scopus subject areas
- Mathematics(all)
- Geometry and Topology
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In: Journal of Geometric Analysis, Vol. 33, No. 3, 105, 03.2023.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Topology of Asymptotically Conical Calabi–Yau and G2 Manifolds and Desingularization of Nearly Kähler and Nearly G2 Conifolds
AU - Schiemanowski, Lothar
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.
PY - 2023/3
Y1 - 2023/3
N2 - 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.
AB - 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.
KW - Asymptotically conical manifolds
KW - Einstein manifolds
KW - Special holonomy
UR - http://www.scopus.com/inward/record.url?scp=85147125013&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2205.15922
DO - 10.48550/arXiv.2205.15922
M3 - Article
AN - SCOPUS:85147125013
VL - 33
JO - Journal of Geometric Analysis
JF - Journal of Geometric Analysis
SN - 1050-6926
IS - 3
M1 - 105
ER -