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Cited by 192Geometry and complex manifoldsGeometric Analysis and Curvature FlowsAdvanced Differential Geometry Research

Twisted connected sums and special Riemannian holonomy

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Kovalev, Alexei·2003·Journal für die reine und angewandte Mathematik (Crelles Journal)
DOI 10.1515/crll.2003.097
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Abstract

We give a new, connected-sum-like construction of Riemannian metrics with special holonomy G_2 on compact 7-manifolds. The construction is based on a gluing theorem for appropriate elliptic partial differential equations. As a prerequisite, we also obtain asymptotically cylindrical Riemannian manifolds with holonomy SU(3) building up on the work of Tian and Yau. Examples of new topological types of compact 7-manifolds with holonomy G_2 are constructed using Fano 3-folds.

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  • The appropriate dimension count for gluing ALH spaces by their ends to construct hyper-Kähler metrics on K3 is 2 • 24 + 5 + (3 + 1) = 57.

    “The appropriate dimension count is 2 • 24 + 5 + (3 + 1) = 57.”

    From Gravitational instantons from rational elliptic surfaces (2011) · cites this paper as [34]

    The introduction (p. 2) describes gluing asymptotically cylindrical seven-manifolds to construct G₂ manifolds, with hyper-Kähler K3 surfaces in the cross-sections. Section 6 (p. 24) gives a different dimension count for moduli of polarized K3 surfaces. Searches for ALH-space gluing and the terms of the claimed calculation found no passage supporting the dimension count 2 × 24 + 5 + (3 + 1) = 57. (LLM, conf=0.77)

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Research works printed but not linked: 33

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Authors
Kovalev, A.
Corresponding author
Kovalev, Alexei
Venue
Journal für die reine und angewandte Mathematik (Crelles Journal)
Publisher
Walter de Gruyter GmbH
Published
2003-01-02
Volume
2003
Issue
565
ISSN
  • 0075-4102 (ISSN-L)
  • 1435-5345
DOI
10.1515/crll.2003.097
OpenAlex
W2963001356
Cited by
192
References
1
Citations by year
2026: 2 · 2025: 5 · 2024: 8 · 2023: 9 · 2022: 7 · 2021: 11 · 2020: 18 · 2019: 11 · 2018: 21 · 2017: 9 · 2016: 6 · 2015: 10 · 2014: 1 · 2013: 6 · 2012: 4
FWCI
12.77
Type
journal article
Language
de
Open access
Locations
  • Journal für die reine und angewandte Mathematik (Crelles Journal) (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Funders
Engineering and Physical Sciences Research Council
Topics
  • Physical Sciences › Mathematics › Geometry and Topology › Geometry and complex manifolds
  • Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Advanced Differential Geometry Research
Keywords
twisted connected sum, special holonomy, G2 holonomy, SU(3) holonomy, gluing theorem, elliptic equations, Fano threefolds

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