The proposals of Joyce (2018) and Doan and Walpuski (2019) on counting closed associative submanifolds
of
-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
,
establishing transversality results in both fixed and one-parameter families of coclosed
-structures. We prove that for
a generic coclosed
-structure
(or a generic path thereof) there are no CS associative submanifolds
having singularities modeled on cones with stability index greater than
(or
, respectively).
We establish that associative cones whose links are null-torsion holomorphic curves in
have stability index
greater than
, and all
special Lagrangian cones in
have stability index greater than or equal to
with equality only for
the Harvey–Lawson
-cone
and a transverse pair of planes. Next, we study the desingularizations
of CS associative submanifolds in a one-parameter family of coclosed
-structures.
Consequently, we derive desingularization results relating the above
transitions for CS associative submanifolds with a Harvey–Lawson
-cone
singularity and for associative submanifolds with a transverse self-intersection.
