Twisted connected sums and special Riemannian holonomy

By A. Kovalev. 2000

282 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f5a8cdc83f7fb4159504f204d936e18b83c4efb4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Asymptotically cylindrical Calabi–Yau 3–folds from weak Fano 3–folds (2012, methodology): “Kovalev [57] used Fano 3–folds to construct ACyl Calabi–Yau 3–folds with ends asymptotic to C∗ × S where S is a smooth K3 surface and suggested that other constructions of suitable ACyl Calabi–Yau 3–folds might be possible [57, page 148]; we prove that starting only with a weak Fano 3–fold (satisfyi”
Nilpotent structures and collapsing Ricci-flat metrics on the K3 surface (2018, background): “s at each endpoint. Gluing of asymptotically cylindrical geometric structures is a very familiar construction in geometry, see for example [Flo91, KS01] for anti-self-dual metrics in dimension 4, and [Kov03] for holonomy G2 metrics in dimension 7. 1.2. Main results. The main result of this paper giv”
G2-manifolds and associative submanifolds via semi-Fano 3-folds (2012, methodology): “Kovalev [44, Theorem 5.34] uses this to prove that for T sufficiently large there are nearby torsion-free G2–structures (one could also apply more general results of Joyce, see Remark 2.36).”
New aspects of the ddc-lemma (2005, unknown): “In [55], Kovalev produces a series of examples of 1-connected compact G2manifolds from pairs of Fano 3-folds via twisted connected sums.”
G2-instantons on generalised Kummer constructions (2011, background): “One is Kovalev’s twisted connected sum construction [20], which produces G2 –manifolds with “long necks” from certain pairs of Calabi–Yau 3–folds with asymptotically cylindrical ends.”
A surgery for generalized complex structures on 4-manifolds (2006, background): “1 along one of the T 2 fibers, we obtain a typechanging generalized complex structure on X(3) × S(1), where X(3) is the twisted connected sum of the S(1)-bundles Σ × S(1) and S(2) × S(1), in the language of [5].”
G2-instantons over twisted connected sums (2013, methodology): “g/ D G2 is the twisted connected sum construction, suggested by Donaldson, pioneered by Kovalev [16] and later extended and improved by Kovalev and Lee [17] and Corti, Haskins, Nordström and Pacini [2].”
Laplacian flow for closed G2 structures: Shi-type estimates, uniqueness and compactness (2015, background): “This theory has formed the cornerstone of the programme for constructing compact holonomy G2 manifolds, of which there are now many examples (see [13,29]).”
New invariants of G2–structures (2012, background): “Now we sketch the basics of the twisted connected sum construction, ignoring many details that are required to justify that the resulting G2-structures are torsion-free (see [23, 10]).”
The Srní lectures on non-integrable geometries with torsion (2006, methodology): “], [Joy96b] [Joy96c] and the book [Joy00], which also contains a proof of the Calabi Conjecture). Later, compact exceptional holonomy manifolds have also been constructed by other methods by Kovalev ([Kov03]). As we will explain later, the General Holonomy Principle relates manifolds with Hol(M) = S”
M-Theory Dynamics On A Manifold Of G_2 Holonomy (2001, background): “What is needed is some kind of gluing technique, perhaps on the lines of [50].”
New aspects of the dd c -lemma (2004, background): “In [55], Kovalev produces a series of examples of 1-connected compact G2manifolds from pairs of Fano 3-folds via twisted connected sums.”
A new construction of compact torsion-free $G_2$-manifolds by gluing families of Eguchi–Hanson spaces (2017, methodology): “More recently, Corti– Haskins–Nordström–Pacini [6, 7] generalized the Kovalev construction, producing tens of thousands of topologically distinct compact G2 holonomy manifolds.”
Dirichlet Branes and Mirror Symmetry (2009, background): “If one is looking for compact examples, there are again only existence statements available for the metrics (Chapters 11-15 of [274, 314]); to give explicit examples of such metrics, one needs to look for non-compact examples.”
Gauge Theory in higher dimensions, II (2009, background): “This kind of neck-stretching sequence, and the converse “gluing theory” for manifolds with tubular ends, is central in the work of Kovalev [21], and many interesting new examples have been found recently by Kovalev and Nordstrom [22].”
Asymptotically cylindrical Calabi-Yau manifolds (2012, methodology): “Projective manifolds M satisfying the hypotheses of Theorem D were first constructed by Kovalev [24] as blow-ups of Fano 3-folds; this construction yields around one hundred families of ACyl CalabiYau 3-folds with split cross-section S1 ×D.”
Notes on the octonions (2010, methodology): “There are however a number of construction techniques (all based on gluing methods): Joyce’s generalized Kummer construction for G2– and Spin(7)–manifolds [20, 21, 18, 19] based on resolving orbifolds of the form T7/Γ and T8/Γ; a method of Joyce’s for constructing Spin(7)–manifolds from real singula”
M theory and singularities of exceptional holonomy manifolds (2004, unknown): “Kovalev [22], where a smooth, compact 7-manifold X with G2 holonomy is obtained by gluing ‘back-to-back’ two asymptotically cylindrical Calabi-Yau (2)K3 is the only compact 4-manifold admitting metrics with SU(2) holonomy.”