Hessian of the natural Hermitian 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 Hermitian form successively on the Calabi family of a hyperkähler manifold, on the twistor space of a 4-dimensional anti-self-dual Riemannian manifold and on the twistor space of a quaternionic Kähler manifold. We show a strong convexity property of the cycle space of twistor lines on the Calabi family of a hyperkähler manifold. We also prove convexity properties of the 1-cycle space of the twistor space of a 4-dimensional anti-self-dual Einstein manifold of non-positive scalar curvature and of the 1-cycle space of the twistor space of a quaternionic Kähler manifold of non-positive scalar curvature. We check that no non-Kähler strong KT manifold occurs as such a twistor space.

Mots clés: twistor space; 4-dimensional Riemannian manifold; hyperkähler manifold; quaternionic Kähler manifold; strong KT manifolds

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

Guillaume Deschamps, Noël Le Du, Christophe Mourougane. Hessian of the natural Hermitian form on twistor spaces. Bulletin de la société mathématique de France, 2017, 145 (1), pp.1-27. ⟨10.24033/bsmf.2729⟩. ⟨hal-00665165v2⟩