\(G_2\)-manifolds and associative submanifolds via semi-Fano 3-folds. (English) Zbl 1343.53044

Corti, Alessio; Haskins, Mark; Nordström, Johannes; Pacini, Tommaso

Duke Math. J. 164, No. 10, 1971-2092 (2015).

Compact \(G_2\)-manifolds, that is, Riemannian 7-manifolds whose holonomy group is the compact exceptional Lie group \(G_2\), play a distinguished role in both geometry and theoretical physics.
In the paper under review, the authors construct many new topological types of compact \(G_2\)-manifolds and prove many new results about them.
In order to implement the twisted connected sum construction the authors construct first exponentially ACyl Calabi-Yau structures on suitable quasiprojective 3-folds and solve the matching problem, that is they show how to find pairs of exponentially ACyl Calabi-Yau 3-folds for which there exists a hyper-Kähler rotation. Then they construct many new topological types of compact \(G_2\)-manifolds by applying the twisted connected sum construction to ACyl Calabi-Yau 3-folds of semi-Fano type.
Next, much more precise topological information about twisted connected sum \( G_2\)-manifolds are obtained. One application is the determination for the first time of the diffeomorphism type of many compact \(G_2\)-manifolds. Geometric transitions between \(G_2\)-metrics on different 7-manifolds mimicking flopping behavior among semi-Fano 3-folds and conifold transitions between Fano and semi-Fano 3-folds are described.
Many of the \(G_2\)-manifolds constructed in this paper contain compact rigid associative 3-folds, which play an important role in 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, the number of rigid associative 3-folds produced can be changed.
The paper is very interesting and nicely illustrated with examples.

Reviewer: Andreea Olteanu (Bucureşti)

MSC:

53C29	Issues of holonomy in differential geometry
53C38	Calibrations and calibrated geometries
53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)
14J28	\(K3\) surfaces and Enriques surfaces
14J32	Calabi-Yau manifolds (algebro-geometric aspects)
14J45	Fano varieties

Keywords:

Einstein manifolds; Ricci-flat manifolds; special holonomy; exceptional holonomy; Fano varieties; weak Fano varieties; ACyl Calabi-Yau structures; calibrated submanifolds; associative submanifolds; differential topology