Deformations along subsheaves
Abstract
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 pullback 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2009
 arXiv:
 arXiv:0905.2749
 Bibcode:
 2009arXiv0905.2749K
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Complex Variables;
 Mathematics  Differential Geometry;
 14D15;
 13D10;
 32G10
 EPrint:
 Removed an unnecessary projectivity assumption and implemented several smaller changes, suggested to us by the referee. To appear in L'Enseignement Mathematique.