Deformations of asymptotically cylindrical G2-manifolds
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We prove that for a 7-dimensional manifold M with cylindrical ends the moduli space of exponentially asymptotically cylindrical torsion-free G2-structures is a smooth manifold (if non-empty), and study some of its local properties. We also show that the holonomy of the induced metric of an exponentially asymptotically cylindrical G2-manifold is exactly G2 if and only if the fundamental group π 1(M) is finite and neither M nor any double cover of M is homeomorphic to a cylinder.
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The restriction map H^2(X) -> H^2(M) has a 1-dimensional kernel spanned by the Poincaré dual of the fibers, i.e. c1(X), and maps onto the 9-dimensional subspace H^{1,1}(M) ⊂ H^2(M) of classes containing a closed form which is (1,1) with respect to the complex structure of X.
“More precisely, the restriction map H 2 (X) → H 2 (M ) has a 1-dimensional kernel (spanned by the Poincaré dual of the fibers, i.e. c 1 (X)) and maps onto the 9-dimensional subspace H 1,1 (M ) ⊂ H 2 (M ) of classes containing a closed form which is (1, 1) with respect to the complex structure of X.”
From Gravitational instantons from rational elliptic surfaces (2011) · cites this paper as [47]
Split into 2 atoms — 0 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 1 unverifiable. Expand the claim to see per-atom verdicts.
Experimental — accuracy not yet measuredFor any ALH Riemannian metric g on M, the classes in im(H^2_c(M) -> H^2(M)) are in bijection with the L^2-harmonic 2-forms with respect to g.
“If g is any ALH Riemannian metric on M , then these classes are in bijection with the L 2 -harmonic 2-forms with respect to g.”
From Gravitational instantons from rational elliptic surfaces (2011) · cites this paper as [47]
Theorem 5.10 (p. 15) gives an isomorphism between L²-harmonic forms and the compactly supported classes in de Rham cohomology, which includes degree two. It states this for oriented exponentially asymptotically cylindrical (EAC) manifolds; our copy does not establish the claim for any ALH Riemannian metric. (LLM, conf=0.84)
Experimental — accuracy not yet measuredOne way of proving the unobstructedness in Theorem 1.10 is by taking products with T^3 and appealing to Nordström's work on G_2-structures, and the parameter count is consistent with Nordström's Proposition 3.5.
“One way of proving the unobstructedness in Theorem 1.10 is by taking products with T 3 and appealing to Nordström's work [47] on G 2 -structures.Our parameter count is consistent with [47, Proposition 3.5], though our proof provides a specific bit of extra information concerning Lie derivatives; see also Remark 4.6.”
From Gravitational instantons from rational elliptic surfaces (2011) · cites this paper as [47]
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Besse, A. L. (1987) · doi:10.1007/978-3-540-74311-8 · Cited by 2,164 · see citing works
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Gravitational instantons from rational elliptic surfaces
Hein, H.-J. (2011) · Journal of the American Mathematical Society · doi:10.1090/s0894-0347-2011-00723-6 · Cited by 87 · see citing works
- Authors
- NORDSTRÖM, J.
- Affiliations
- University of Cambridge (GB)
- Corresponding author
- NORDSTRÖM, JOHANNES
- Venue
- Mathematical Proceedings of the Cambridge Philosophical Society
- Publisher
- Cambridge University Press (CUP)
- Published
- 2008-09
- Volume
- 145
- Issue
- 2
- Pages
- 311-348
- ISSN
- 0305-0041 (ISSN-L)
- 1469-8064
- OpenAlex
- W2155057600
- Cited by
- 21
- References
- 1
- Citations by year
- 2024: 1 · 2021: 2 · 2020: 1 · 2019: 1 · 2018: 1 · 2017: 2 · 2016: 3 · 2015: 2 · 2013: 2
- FWCI
- 2.62
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Mathematical Proceedings of the Cambridge Philosophical Society (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- 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
- G2 manifolds, moduli spaces, smoothness of moduli spaces, holonomy, finite fundamental group, 7-dimensional manifolds
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