Stability of coassociative conical singularities (Q1940483)

Coassociative 4-folds are calibrated submanifolds of 7-manifolds with \(G_2\) structure. The author studies the stability of coassociative 4-folds with conical singularities under perturbations of the ambient \(G_2\) structure by defining an invariant of a coassociative cone which is called the stability index. He calculates explicitly the stability index for cones on group orbits, and he also describes the stability index for cones fibred by 2-planes over algebraic curves using the degree and genus of the curve and the spectrum of the Laplacian on the link. Finally, the author applies his results to construct the first known example of coassociative 4-folds with conical singularities in compact manifolds with \(G_2\) holonomy. The main results are the following: Theorem 1.1. If a coassociative integral current has a multiplicity-one Jacobi integrable tangent cone with isolated singularity at an interior point \(p\), then it has a conical singularity at \(p\) (see \S3). Theorem 1.2. The only stable homogeneous coassociative cones are coassociative 4-planes and the \(\mathrm{Sp}(1)\)-invariant coassociative cone given in Example 4.2 (see \S5). Theorem 1.3. Given a pair of maximal deformation families of Fano 3-folds, one can construct a one-parameter family of compact manifolds with \(G_2\) holonomy, which contain coassociative K3 surfaces with conical singularities (see \S7).