Strongly homotopy Lie algebras and deformations of calibrated submanifolds

For an element $\mathrm{\Psi }$ in the graded vector space ${\mathrm{\Omega }}^{\ast }(M,TM)$ of tangent bundle valued forms on a smooth manifold $M$, a $\mathrm{\Psi }$-submanifold is defined as a submanifold $N$ of $M$ such that ${\mathrm{\Psi }}_{\mathrm{\mid }N}\in {\mathrm{\Omega }}^{\ast }(N,TN)$. The class of $\mathrm{\Psi }$-submanifolds encompasses calibrated submanifolds, complex submanifolds and all Lie subgroups in compact Lie groups. The graded vector space ${\mathrm{\Omega }}^{\ast }(M,TM)$ carries a natural graded Lie algebra structure, given by the Fr"olicher-Nijenhuis bracket $[-,-{]}^{FN}$. When $\mathrm{\Psi }$ is an odd degree element with $[\mathrm{\Psi },\mathrm{\Psi }{]}^{FN}=0$, we associate to a $\mathrm{\Psi }$-submanifold $N$ a strongly homotopy Lie algebra, which governs the formal and (under certain assumptions) smooth deformations of $N$ as a $\mathrm{\Psi }$-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.