Hessian of the metric form on twistor spaces

Guillaume Deschamps (1) , Noël Le Du (2) , Christophe Mourougane (2)

1 LM - Laboratoire de mathématiques de Brest
2 IRMAR - Institut de Recherche Mathématique de Rennes

We compute the hessian of the natural metric form on the twistor space of a 4-dimensional anti-self dual Riemannian manifold. We then adapt the computations to the case of the twistor space of a hyperkähler manifold. In this case, we show a strong positivity property of the hessian of the metric form and prove, as an application, a convexity property of the component of the twistor lines in the cycle space.

Mots clés: twistor space; 4-dimensional Riemannian manifold; hyperkähler manifold

Domaines: Géométrie différentielle [math.DG]; Variables complexes [math.CV]; Géométrie algébrique [math.AG]

Guillaume Deschamps, Noël Le Du, Christophe Mourougane. Hessian of the metric form on twistor spaces. 2012. ⟨hal-00665165v1⟩