Deformations and desingularizations of conically singular associative submanifolds

Abstract

The proposals of Joyce [Joy18], and Doan and Walpuski [DW19] on counting closed associative submanifolds of $G_2$-manifolds depend on various conjectural transitions. This article contributes to the study of transitions arising from the degenerations of associative submanifolds into conically singular (CS) associative submanifolds. First, we study the moduli space of CS associative submanifolds with isolated singularities modeled on associative cones in $\mathbb R^7$, establishing transversality results in both fixed and one-parameter family of co-closed $G_2 $-structures. We prove that for a generic co-closed $G_2$-structure (or a generic path thereof) there are no CS associative submanifolds having singularities modeled on cones with stability-index greater than $0$ (or $1$, respectively). We establish that associative cones whose links are null-torsion holomorphic curves in $S^6$ have stability-index greater than $4$, and all special Lagrangian cones in $\mathbb C^3$ have stability-index greater than or equal to $1$ with equality only for the Harvey-Lawson $T^2$-cone and a transverse pair of planes. Next, we study the desingularizations of CS associative submanifolds in a one-parameter family of co-closed $G_2$-structures. Consequently, we derive desingularization results relating the above transitions for CS associative submanifolds with a Harvey-Lawson $T^2$-cone singularity and for associative submanifolds with a transverse self-intersection.