Compatible complex structures on twistor space

Deschamps, Guillaume

Annales de l'Institut Fourier, Tome 61 (2011) no. 6, pp. 2219-2248.

Soit $M$ une 4-variété riemannienne. L’espace de twisteur associé est un fibré qui admet une métrique naturelle. Le but de cet article est d’étudier les structures complexes sur $Z$ qui sont compatibles avec la fibration et la métrique. Les résultats obtenu permettent d’exprimer des propriétés métriques sur $M$ (courbure scalaire nulle, Kähler à courbure scalaire nulle...) en termes de propriétés des structures complexes de l’espace de twisteur $Z$.

Let $M$ be a Riemannian 4-manifold. The associated twistor space is a bundle whose total space $Z$ admits a natural metric. The aim of this article is to study properties of complex structures on $Z$ which are compatible with the fibration and the metric. The results obtained enable us to translate some metric properties on $M$ (scalar flat, scalar-flat Kähler...) in terms of complex properties of its twistor space $Z$.

Classification : 53C28, 52C26
Keywords: twistor space, complex structure, scalar-flat, scalar-flat Kähler, locally conformally Kähler, quaternionic Kähler.
Mot clés : espace de twisteur, structure complexe, courbure scalaire nulle, Kähler à courbure scalaire nulle, localement conformément Kähler, quaternionique Kähler.

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