# Deformation along subsheaves, II

### Clemens Jörder

Universität Freiburg, Germany### Stefan Kebekus

Universität Freiburg, Germany

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## Abstract

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

In case where $X$ is complex-symplectic and $F$ 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.