Deforming $G_2$ conifolds.

Spiro Karigiannis, University of Waterloo
Fine Hall 314

A theorem of Dominic Joyce says that the moduli space of compact $G_2$ manifolds is smooth of dimension equal to the third Betti number of the manifold. We study the moduli space question for noncompact $G_2$ manifolds with one end, asymptotic to a metric cone of $G_2$ holonomy. This includes the explicit Bryant--Salamon manifolds as examples. We prove that this moduli space is smooth and unobstructed when the rate of convergence to the cone at infinity lies within a certain range. The dimension of this moduli space includes a component which is topological and a component which is analytic, arising from the existence of certain solutions to an eigenvalue equation on the link of the asymptotic cone. We also consider the moduli space question for compact $G_2$ manifolds with isolated conical singularities. In this case there are always analytic obstructions, and we describe these. This is joint work with Jason Lotay of University College London.