Deformation along subsheaves, II

TL;DR

This paper provides criteria for when infinitesimal deformations of a subspace within a complex manifold, constrained by a subsheaf of the tangent bundle, lead to positive-dimensional deformation families, with applications to complex-symplectic geometry.

Contribution

It offers elementary, geometric criteria for deformations along subsheaves, including cases related to Hamiltonian vector fields and Lagrangian submanifolds.

Findings

Criteria for positive-dimensional deformation families

Partially reproduces unobstructedness results for Lagrangian submanifolds

Elementary geometric proof using flow maps

Abstract

Let Y be a compact reduced subspace of a complex manifold X, and let F be a subsheaf of the tangent bundle T_X which is closed under the Lie bracket. This paper discusses criteria to guarantee that infinitesimal deformations of the inclusion morphism Y -> X give rise to positive-dimensional deformation families, deforming the inclusion map "along the sheaf F". In case where X is complex-symplectic and F is the sheaf of Hamiltonian vector fields, this partially reproduces known results on unobstructedness of deformations of Lagrangian submanifolds. Written for the IMPANGA Lecture Notes series, this paper aims at simplicity and clarity of argument. It does not strive to present the shortest proofs or most general results available. 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.