Deformations along subsheaves

Abstract

Let $fY\to X$ be a morphism of complex manifolds, and assume that $Y$ is compact. Let $\mathcal{F}\subseteq T_X$ be a subsheaf which is closed under the Lie bracket. The present paper contains an elementary and very geometric argument to show that all obstructions to deforming $f$ along the sheaf $\mathcal{F}$ lie in $H^1(Y,{\mathcal{F}}_Y)$, where ${\mathcal{F}}_Y\subseteq f^{\ast }(T_X)$ is the image of $f^{\ast }(\mathcal{F})$ under the pull-back of the inclusion map. Special cases of this result include Miyaoka's theory of deformation along a foliation, Keel-McKernan's logarithmic deformation theory and deformations with fixed points.