# Deformations along subsheaves

### Stefan Kebekus

Universität Freiburg, Germany### Stavros Kousidis

Universität zu Köln, Germany### Daniel Lohmann

Universität Freiburg, Germany

## Abstract

Let $f \: Y \to X$ 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 $H^1\bigl( Y,\, \mathcal F_Y \bigr)$, where $\mathcal F_Y \subseteq f^*(T_X)$ is the image of $f^*(\mathcal F)$ 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.

## Cite this article

Stefan Kebekus, Stavros Kousidis, Daniel Lohmann, Deformations along subsheaves. Enseign. Math. 56 (2010), no. 3, pp. 287–313

DOI 10.4171/LEM/56-3-3