SMOOTH MODULI SPACES OF ASSOCIATIVE SUBMANIFOLDS

Let M-7 be a smooth manifold equipped with a G(2)-structure I center dot, and Y-3 be a closed compact I center dot-associative submanifold. McLean [Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (1998), 705-747] proved that the moduli space a"(3)(Y, I center dot) of the I center dot-associative deformations of Y has vanishing virtual dimension. In this paper, we perturb I center dot into a G(2)-structure psi in order to ensure the smoothness of a"(3)(Y, psi) near Y. If Y is allowed to have a boundary moving in a fixed coassociative submanifold X, it was proved in Gayet and Witt [Deformations of associative submanifolds with boundary, Adv. Math. 226 (2011), 2351-2370] that the moduli space a"(3)(Y, X) of the associative deformations of Y with boundary in X has finite virtual dimension. We show here that a generic perturbation of the boundary condition X into X' gives the smoothness of a"(3)(Y, X'). In another direction, we use Bochner's technique to prove a vanishing theorem that forces a"(3)(Y) or a"(3)(Y, X) to be smooth near Y. For every case, some explicit families of examples will be given.