Let f : Y -> X be a morphism of complex projective manifolds, and let F be a subsheaf of the tangent bundle which is closed under the Lie bracket, but not necessarily a foliation. This short paper contains an elementary and very geometric argument to show that all obstructions to deforming the morphism f along the sheaf F lie in the first cohomology group H^1(Y, F_Y) of the sheaf F_Y, which is the image of f^*(F) in f^*(T_X) under the pull-back of the inclusion map. Special cases of this result include the theory of deformation along a (possibly singular) foliation, logarithmic deformation theory and deformations with fixed points.
- Pub Date:
- May 2009
- Mathematics - Algebraic Geometry;
- Mathematics - Complex Variables;
- Mathematics - Differential Geometry;
- Removed an unnecessary projectivity assumption and implemented several smaller changes, suggested to us by the referee. To appear in L'Enseignement Mathematique.