Riemannian metrics with holonomy G_2 on 7-manifolds can be defined in terms of a harmonic 3-form. 3-dimensional submanifolds calibrated by the 3-form are called associative. Like all calibrated submanifolds, they minimise volume in their homology class. The deformations of an associative may be obstructed, but if they are not then the associative is rigid, i.e. the moduli space is discrete.

Kovalev constructed examples of compact G_2-manifolds as "twisted connected sums" of asymptotically cylindrical Calabi-Yau 3-folds, in turns constructed from Fano 3-folds. I will describe work with Corti, Haskins and Pacini to construct compact G_2-manifolds by the same method but starting from weak Fano 3-folds, which are more plentiful. Moreover, rigid complex curves in the weak Fano give rise to associative submanifolds, and these are the first examples of associatives in compact G_2-manifolds that are known to be rigid.
In at least some special cases, all associatives in a homology class arise this way, allowing them to be counted.