### A hyperbolic twistor space.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

∗Research supported in part by NSF grant INT-9903302.In previous work a hyperbolic twistor space over a paraquaternionic Kähler manifold was defined, the fibre being the hyperboloid model of the hyperbolic plane with constant curvature −1. Two almost complex structures were defined on this twistor space and their properties studied. In the present paper we consider a twistor space over a paraquaternionic Kähler manifold with fibre given by the hyperboloid of 1-sheet, the anti-de-Sitter plane...

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 $Z$.

For a Fedosov manifold (symplectic manifold equipped with a symplectic torsion-free affine connection) admitting a metaplectic structure, we shall investigate two sequences of first order differential operators acting on sections of certain infinite rank vector bundles defined over this manifold. The differential operators are symplectic analogues of the twistor operators known from Riemannian or Lorentzian spin geometry. It is known that the mentioned sequences form complexes if the symplectic...

On s’intéresse à l’espace de twisteurs réduit d’une variété presque hermitienne, en relisant un article de N.R.O’Brian et J.H.Rawnsley (Ann. Global Anal. Geom., 1985). On traite la question laissée ouverte de la dimension 6. Cet espace est muni d’une structure presque complexe $\mathcal{J}$ en utilisant la distribution horizontale de la connexion hermitienne canonique. On montre qu’une condition nécessaire d’intégrabilité de $\mathcal{J}$ est que la variété soit de type ${W}_{1}\oplus {W}_{4}$ dans la classification de Gray et Hervella. Dans...

It is shown that there exists a twistor space on the n-fold connected sum of complex projective planes nCP2, whose algebraic dimension is one and whose general fiber of the algebraic reduction is birational to an elliptic ruled surface or a K3 surface. The former kind of twistor spaces are constructed over nCP2 for any n ≥ 5, while the latter kind of example is constructed over 5CP2. Both of these seem to be the first such example on nCP2. The algebraic reduction in these examples is induced by...

We define, on smooth manifolds, the notions of almost twistorial structure and twistorial map, thus providing a unified framework for all known examples of twistor spaces. The condition of being a harmonic morphism naturally appears among the geometric properties of submersive twistorial maps between low-dimensional Weyl spaces endowed with a nonintegrable almost twistorial structure due to Eells and Salamon. This leads to the twistorial characterisation of harmonic morphisms between Weyl spaces...

Let M be a hyperkähler manifold, and F a reflexive sheaf on M. Assume that F (away from its singularities) admits a connection ▿ with a curvature Θ which is invariant under the standard SU(2)-action on 2-forms. If Θ is square-integrable, such sheaf is called hyperholomorphic. Hyperholomorphic sheaves were studied at great length in [21]. Such sheaves are stable and their singular sets are hyperkähler subvarieties in M. In the present paper, we study sheaves admitting a connection with SU(2)-invariant...

The Penrose transform gives an isomorphism between the kernel of the $2$-Dirac operator over an affine subset and the third sheaf cohomology group on the twistor space. In the paper we give an integral formula which realizes the isomorphism and decompose the kernel as a module of the Levi factor of the parabolic subgroup. This gives a new insight into the structure of the kernel of the operator.

It is shown that operators occurring in the classical Penrose transform are differential. These operators are identified depending on line bundles over the twistor space.

The conformal infinity of a quaternionic-Kähler metric on a $4n$-manifold with boundary is a codimension $3$ distribution on the boundary called quaternionic contact. In dimensions $4n-1$ greater than $7$, a quaternionic contact structure is always the conformal infinity of a quaternionic-Kähler metric. On the contrary, in dimension $7$, we prove a criterion for quaternionic contact structures to be the conformal infinity of a quaternionic-Kähler metric. This allows us to find the quaternionic-contact structures...

We study the Jones and Tod correspondence between selfdual conformal $4$-manifolds with a conformal vector field and abelian monopoles on Einstein-Weyl $3$-manifolds, and prove that invariant complex structures correspond to shear-free geodesic congruences. Such congruences exist in abundance and so provide a tool for constructing interesting selfdual geometries with symmetry, unifying the theories of scalar-flat Kähler metrics and hypercomplex structures with symmetry. We also show that in the presence...