Journal Articles Quarterly Journal of Mathematics Year : 2014

Smooth moduli spaces of associative submanifolds

Damien Gayet

Abstract

Let $M^7$ be a smooth manifold equipped with a $G_2$-structure $\phi$, and $Y^3$ be an closed compact $\phi$-associative submanifold. In \cite{McL}, R. McLean proved that the moduli space $\bm_{Y,\phi}$ of the $\phi$-associative deformations of $Y$ has vanishing virtual dimension. In this paper, we perturb $\phi$ into a $G_2$-structure $\psi$ in order to ensure the smoothness of $\bm_{Y,\psi}$ near $Y$. If $Y$ is allowed to have a boundary moving in a fixed coassociative submanifold $X$, it was proved in \cite{GaWi} that the moduli space $\bm_{Y,X}$ 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 $\bm_{Y,X'}$. In another direction, we use the Bochner technique to prove a vanishing theorem that forces $\bm_Y$ or $\bm_{Y,X}$ to be smooth near $Y$. For every case, some explicit families of examples will be given.

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Dates and versions

hal-00532891 , version 1 (04-11-2010)
hal-00532891 , version 2 (08-09-2012)
hal-00532891 , version 3 (13-08-2013)

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Damien Gayet. Smooth moduli spaces of associative submanifolds. Quarterly Journal of Mathematics, 2014, pp.10.1093/qmath/hat042. ⟨10.1093/qmath/hat042⟩. ⟨hal-00532891v3⟩
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