Deformations along subsheaves

TL;DR

This paper provides a geometric argument showing that obstructions to deforming a morphism along a subsheaf of the tangent bundle are contained in a specific first cohomology group, extending deformation theory to non-foliation sheaves.

Contribution

It introduces an elementary geometric proof that obstructions to deformation along certain subsheaves lie in a specific cohomology group, broadening deformation theory beyond foliations.

Findings

Obstructions to deformation are in H^1(Y, F_Y).

Applicable to deformation along singular foliations.

Includes cases like logarithmic deformation and fixed points.

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 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.