Deformations and desingularizations of conically singular associative submanifolds

TL;DR

This paper investigates the moduli space and desingularization of conically singular associative submanifolds in $G_2$-manifolds, establishing transversality and stability results crucial for understanding transitions in these geometric structures.

Contribution

It provides new transversality results for moduli spaces of CS associative submanifolds and analyzes stability indices of various associative cones, advancing the understanding of their degenerations and desingularizations.

Findings

No CS associative submanifolds with high stability index for generic structures.

Associative cones with null-torsion holomorphic links have stability index > 4.

Desingularization results connect transitions involving Harvey-Lawson cones and self-intersections.

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.