Hessian of the natural Hermitian form on twistor spaces
Differential Geometry
2018-05-24 v2 Algebraic Geometry
Complex Variables
Abstract
We compute the hessian of the natural Hermitian form successively on the Calabi family of a hyperk\"ahler manifold, on the twistor space of a 4-dimensional anti-self-dual Riemannian manifold and on the twistor space of a quaternionic K\"ahler manifold. We show a strong convexity property of the cycle space of twistor lines on the Calabi family of a hyperk\"ahler 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 K\"ahler manifold of non-positive scalar curvature. We check that no non-K\"ahler strong KT manifold occurs as such a twistor space.
Keywords
Cite
@article{arxiv.1202.0183,
title = {Hessian of the natural Hermitian form on twistor spaces},
author = {Guillaume Deschamps and Noël Le Du and Christophe Mourougane},
journal= {arXiv preprint arXiv:1202.0183},
year = {2018}
}