Deformation along subsheaves, II

Abstract

Let $f \colon Y \to X$ be the inclusion map of a compact reduced subspace of a complex manifold, and let $\mathcal{F} \subseteq T_X$ be a subsheaf of the tangent bundle which is closed under the Lie bracket, but not necessarily a sheaf of $\mathcal{O}_X$-algebras. This paper discusses criteria to guarantee that infinitesimal deformations of $f$ which are induced by $\mathcal{F}$ lift to positive-dimensional deformations of $f$, where $f$ is deformed “along the sheaf $\mathcal{F}$”.

In case where $X$ is complex-symplectic and $\mathcal{F}$ the sheaf of locally Hamiltonian vector fields, this partially reproduces known results on unobstructedness of deformations of Lagrangian submanifolds. The proof is rather elementary and geometric, constructing higher-order liftings of a given infinitesimal deformation using flow maps of carefully crafted time-dependent vector fields.