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

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.


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
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