G2-manifolds and associative submanifolds via semi-Fano 3-folds

By Alessio Corti, M. Haskins, Johannes Nordstrom, T. Pacini. 2012

188 citations recorded by the source corpus.

We construct many new topological types of compact G_2-manifolds, i.e. Riemannian 7-manifolds with holonomy group G_2. To achieve this we extend the twisted connected sum construction first developed by Kovalev and apply it to the large class of asymptotically cylindrical Calabi–Yau 3-folds built from semi-Fano 3-folds constructed previously by the authors. In many cases we determine the diffeomorphism type of the underlying smooth 7-manifolds completely; we find that many 2-connected 7-manifolds can be realised as twisted connected sums in a variety of ways, raising questions about the global structure of the moduli space of G_2-metrics. Many of the G_2-manifolds we construct contain compact rigid associative 3-folds, which play an important role in the the higher-dimensional enumerative geometry (gauge theory/calibrated submanifolds) approach to defining deformation invariants of G_2-metrics. By varying the semi-Fanos used to build different G_2-metrics on the same 7-manifold we can change the number of rigid associative 3-folds we produce.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e1725d4c33f591168b2570f6d5b6fde2ccad6dbd

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

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.”
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”
Asymptotically cylindrical Calabi-Yau manifolds (2012, methodology): “i). The exponential convergence in (i) is important because it is used in an essential way in the socalled twisted connected sum construction of compact Riemannian 7-manifolds with holonomy group G 2 [9, 8, 22], ﬁrst suggested by Donaldson and then pioneered by Kovalev in [22]. At present no complet”