G 2 -manifolds and associative submanifolds via semi-Fano 3 -folds

Abstract

We construct many new topological types of compact ${\mathrm{G}}_2$-manifolds, that is, Riemannian $7$-manifolds with holonomy group ${\mathrm{G}}_2$. 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 realized as twisted connected sums in a variety of ways, raising questions about the global structure of the moduli space of ${\mathrm{G}}_2$-metrics. Many of the ${\mathrm{G}}_2$-manifolds we construct 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 ${\mathrm{G}}_2$-metrics. By varying the semi-Fanos used to build different ${\mathrm{G}}_2$-metrics on the same $7$-manifold we can change the number of rigid associative $3$-folds we produce.