Vanishing theorems for associative submanifolds
Résumé
Let M a 7-dimensional manifold with holonomy in G2, and containing a compact associative 3-dimensional submanifold Y . It is known since McLean [9] that the problem of the associative deformations of Y is related to a Dirac-like operator on Y , hence is elliptic and of vanishing index. In this paper we adapt the Bochner method to give a sufficient metric condition which implies that Y is isolated. In the case where Y has boundary in a coassociative 4-dimensional submanifold X, it has been proved in [5] that the associated deformation problem is still elliptic, and moreover its index is in general non vanishing. In this case, we add to the former condition a new one related to the geometry of the boundary. They imply the smoothness of the moduli space of associative deformations of Y . Thanks those general results, we are able to describe explicit smooth moduli spaces for some families of examples in R7 and in Calabi-Yau extensions. In particular, we prove that the moduli space of associative deformations of the product of a special lagrangian by S1 in a Calabi-Yau manifold times S1 is always smooth and of dimension b1(L) + 1.
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