Strongly homotopy Lie algebras and deformations of calibrated submanifolds

Abstract

For an element $\Psi$ in the graded vector space ${\Omega }^{\ast }(M,TM)$ of tangent bundle valued forms on a smooth manifold $M$, a $\Psi$-submanifold is defined as a submanifold $N$ of $M$ such that ${\Psi }_{|N}\in {\Omega }^{\ast }(N,TN)$. The class of $\Psi$-submanifolds encompasses calibrated submanifolds, complex submanifolds and all Lie subgroups in compact Lie groups. The graded vector space ${\Omega }^{\ast }(M,TM)$ carries a natural graded Lie algebra structure, given by the Frölicher-Nijenhuis bracket $[-,-{]}^{FN}$. When $\Psi$ is an odd degree element with $[\Psi ,\Psi {]}^{FN}=0$, we associate to a $\Psi$-submanifold $N$ a strongly homotopy Lie algebra, which governs the formal and (under certain assumptions) smooth deformations of $N$ as a $\Psi$-submanifold, and we show that under certain assumptions these deformations form an analytic variety. As an application we revisit formal and smooth deformation theory of complex closed submanifolds and of $\phi$-calibrated closed submanifolds, where $\phi$ is a parallel form in a real analytic Riemannian manifold.

arXiv commentVer3: 32 p, final version, accepted to Asian Journal of Math. Version 2: 31 p, the multisymplecticity assumption in Theorem 6.4 is replaced by the analyticity assumption, comments welcome. Version 1: 30 pages, comments are welcome!