Hessian of the natural Hermitian form on twistor spaces

TL;DR

This paper analyzes the Hessian of the Hermitian form on various twistor spaces, revealing convexity properties and excluding certain non-K"ahler structures, advancing understanding of geometric structures in complex differential geometry.

Contribution

It computes the Hessian of the Hermitian form on twistor spaces of hyperk"ahler, anti-self-dual, and quaternionic K"ahler manifolds, establishing convexity properties and ruling out non-K"ahler strong KT twistor spaces.

Findings

Strong convexity of cycle space on hyperk"ahler twistor space

Convexity of 1-cycle space on anti-self-dual twistor space

No non-K"ahler strong KT twistor space found

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.