Deformation along subsheaves, II

  • Clemens Jörder

    Universität Freiburg, Germany
  • Stefan Kebekus

    Universität Freiburg, Germany
Deformation along subsheaves, II cover
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Let f ⁣:YXf \colon Y \to X be the inclusion map of a compact reduced subspace of a complex manifold, and let FTX\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 OX\mathcal{O}_X-algebras. This paper discusses criteria to guarantee that infinitesimal deformations of ff which are induced by F\mathcal{F} lift to positive-dimensional deformations of ff, where ff is deformed “along the sheaf F\mathcal{F}”.

In case where XX is complex-symplectic and F\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.