JournalslemVol. 56, No. 3/4pp. 287–313

Deformations along subsheaves

  • Stefan Kebekus

    Universität Freiburg, Germany
  • Stavros Kousidis

    Universität zu Köln, Germany
  • Daniel Lohmann

    Universität Freiburg, Germany
Deformations along subsheaves cover
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Abstract

Let fYXf \: Y \to X be a morphism of complex manifolds, and assume that YY is compact. Let FTX\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 ff along the sheaf F\mathcal F lie in H1(Y,FY)H^1\bigl( Y,\, \mathcal F_Y \bigr), where FYf(TX)\mathcal F_Y \subseteq f^*(T_X) is the image of f(F)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