Strongly homotopy Lie algebras and deformations of calibrated submanifolds

For an element Ψ in the graded vector space Ω∗(M, T M ) of tangent bundle valued forms on a smooth manifold M , a Ψ-submanifold is defined as a submanifold N of M such that Ψ|N ∈ Ω∗(N, T N ). The class of Ψ-submanifolds encompasses calibrated submanifolds, complex sub- manifolds and all Lie subgroups in compact Lie groups. The graded vector space Ω∗(M, T M ) carries a natural graded Lie algebra structure, given by the Fr ̈olicher-Nijenhuis bracket [−, −]F N . When Ψ is an odd degree element with [Ψ, Ψ]F N = 0, we associate to a Ψ-submanifold N a strongly homotopy Lie algebra, which governs the formal and (under certain assumptions) smooth deformations of N as a Ψ-submanifold, and we show that under certain assumptions these deformations form an ana- lytic variety. As an application we revisit formal and smooth deformation theory of complex closed submanifolds and of φ-calibrated closed submanifolds, where φ is a parallel form in a real analytic Riemannian manifold.