L’Enseignement Mathématique

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Volume 56, Issue 3/4, 2010, pp. 287–313
DOI: 10.4171/LEM/56-3-3

Deformations along subsheaves

Stefan Kebekus[1], Stavros Kousidis[2] and Daniel Lohmann[3]

(1) Universität Freiburg, Germany
(2) Universität zu Köln, Germany
(3) Universität Freiburg, Germany

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.

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Kebekus Stefan, Kousidis Stavros, Lohmann Daniel: Deformations along subsheaves. Enseign. Math. 56 (2010), 287-313. doi: 10.4171/LEM/56-3-3