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The twistor space of a Riemannian 4-manifold carries two almost complex structures, $J_+$ and $J_-$, and a natural closed 2-form $\omega$. This article studies limits of manifolds for which $\omega$ tames either $J_+$ or $J_-$. This amounts to a curvature inequality involving self-dual Weyl curvature and Ricci curvature, and which is satisfied, for example, by all anti-self-dual Einstein manifolds with non-zero scalar curvature. We prove that if a sequence of manifolds satisfying the curvature inequality converges to a hyperk\"ahler limit X (in the $C^2$ pointed topology) then X cannot contain a holomorphic 2-sphere (for any of its hyperk\"ahler complex structures). In particular, this rules out the formation of bubbles modelled on ALE gravitational instantons in such families of metrics.

Starting from a complex manifold S with a real-analytic c-projective structure whose curvature has type (1,1), and a complex line bundle L with a connection whose curvature has type (1,1), we construct the twistor space Z of a quaternionic manifold M with a quaternionic circle action which contains S as a totally complex submanifold fixed by the action. This extends a construction of hypercomplex manifolds, including hyperkaehler metrics on cotangent bundles, obtained independently by B. Feix and D. Kaledin. When S is a Riemann surface, M is a self-dual conformal 4-manifold, and the quotient of M by the circle action is an Einstein-Weyl manifold with an asymptotically hyperbolic end, and our construction coincides with a construction presented by the first author in a previous paper. The extension also applies to quaternionic Kaehler manifolds with circle actions, as studied by A. Haydys and N. Hitchin., 28 pages, (v2) added material on Swann bundles, quaternionic Kaehler metrics and the Haydys-Hitchin correspondence, (v3) refereed version, restructured content, to appear in TAMS

The space H of asymptotically (left-) shear-free cuts of the J + (good cuts) of an asymptotically flat space-time M is defined. The connection between this space and the asymptotic projective twistor space PJ of M is discussed, and this relation is used to prove that H is four-complex-dimensional for sufficiently ‘calm’ gravitational radiation in M . The metric on H -space is defined by a simple contour integral expression and is found to be complex Riemannian. The good cut equation governing H -space is solved to three orders by a Taylor series and the solution is used to demonstrate that the curvature of H -space is always a self dual (left flat) solution of the Einstein vacuum equations.

Let $M$ be a hyperkaehler manifold, and $F$ a torsion-free and reflexive coherent sheaf on $M$. Assume that $F$ (outside of its singularities) admits a connection with a curvature which is invariant under the standard SU(2)-action on 2-forms. If the curvature is square-integrable, then $F$ is stable and its singularities are hyperkaehler subvarieties in $M$. Such sheaves (called hyperholomorphic sheaves) are well understood. In the present paper, we study sheaves admitting a connection with SU(2)-invariant curvature which is not necessarily square-integrable. This situation arises often, for instance, when one deals with higher direct images of holomorphic bundles. We show that such sheaves are stable., Comment: 37 pages, version 11, reference updated, corrected many minor errors and typos found by the referee

The notions of a twistor space of a contact manifold and a contact connection on such a manifold have been introduced by L. Vezzoni as extensions of the corresponding notions in the case of a symplectic manifold. Given a contact connection on a contact manifold one can define an almost $CR$-structure on its twistor space and Vezzoni has found the integrability condition for this structure. In the present paper it is observed that the $CR$-structure is induced by an almost contact metric structure. The main goal of the paper is to obtain necessary and sufficient conditions for normality of this structure in terms of the curvature of the given contact connection. Illustrating examples are discussed at the end of the paper., Comment: typos corrected, minor changes

The CR twistor space is defined over a contact metric 5-manifold M. Like the 4-dim twistor theory, the integrability of the almost CR twistor structure is discussed in terms of the Weyl conformal curvature and also the scalar curvature of M.

In this paper we provide twistorial examples of compact Hermitian manifolds with positive holomorphic bisectional curvature. We also observe that the so-called “squashed” metric on CP 3 , the twistor space of the sphere S 4

We introduce some canonical 2-form in the twistor bundles of any Riemannian manifoldM. This form is always closed and turns out to be nondegenerate in the following cases: 1. The curvature ofM is pinched. 2. M is an Einstein four-dimensional manifold of positive or negative curvature. 3. M is self-dual and the Ricci curvature is pinched.

In this paper we generalize the definition of symplectic connection to the contact case. It turns out that any odd-dimensional manifold equipped with a contact form admits contact connections and that any Sasakian structure induces a canonical contact connection. Furthermore (as in the symplectic case), any contact connection induces an almost CR structure on the contact twistor space which is integrable if and only if the curvature of the connection is of Ricci-type.

Vasiliev equations facilitate globally defined formulations of higher-spin gravity in various correspondence spaces associated with different phases of the theory. In the four-dimensional case this induces a map from a generally covariant formulation in spacetime with higher-derivative interactions to a formulation in terms of a deformed symplectic structure on a noncommutative doubled twistor space, sending spacetime boundary conditions to various sectors of an associative star-product algebra. We look at observables given by integrals over twistor space defining composite zero-forms in spacetime that do not break any local symmetries and that are closed on shell. They can be evaluated locally in spacetime and interpreted as building blocks for dual amplitudes. To regularize potential twistor-space divergencies arising in their curvature expansion, we propose a closed-contour prescription that respects associativity and hence higher-spin gauge symmetry. As a sample calculation, we examine next-to-leading corrections to quasi-amplitudes for twistor-space plane waves, and find cancellations that we interpret using transgression properties in twistor space., Comment: 53 pages, enlarged version with clarifications added, changes in presentation, typos corrected and references added

We present a calculus whereby the curvature of a geometry arising from any generalized twistor correspondence is related to an obstruction-theoretic classification of the infinitesimal neighborhoods of submanifolds of its twistor space. The crux of the argument involves a relation between Kodaira—Spencer maps and the Penrose transform.

The twistor space \Z of an oriented Riemannian 4-manifold M admits a natural 1-parameter family of Riemannian metrics h_t compatible with the almost complex structures J_1 and J_2 introduced, respectively, by Atiyah, Hitchin and Singer, and Eells and Salamon. In this paper we compute the first Chern form of the almost Hermitian manifold (\Z,h_t,J_n), n=1,2 and find the geometric conditions on M under which the curvature of its Chern connection D^n is of type (1,1). We also describe the twistor spaces of constant holomorphic sectional curvature with respect to D^n and show that the Nijenhuis tensor of J_2 is D^2-parallel provided the base manifold M is Einstein and self-dual., 14 pages, to appear in Rocky Mountain J. Math

International audience; Our purpose here is to present a fairly large class of algebraic dimension zero threefolds, namely the twistor spaces of certain self-dual 4-manifolds.

What it means for (relative) sheaf cohomology classes to have a pole of a given order on a surface S in twistor space will be defined and that they can be described in terms of some formal neighborhood sheaves will be shown. In space‐time, S corresponds to a foliation by α‐surfaces and the filtration of cohomology gives a filtration on the fields that extends the idea of being algebraically special along the foliation. We use this idea also for the case of the ‘‘double‐valued’’ congruence associated with a world line, in which case the filtration applied to sourced fields is essentially a multipole expansion. In the case of curved space‐times, it will be shown that if a certain curvature condition holds, then the space of leaves of a foliation by α surfaces has an ambient twistor space defined to first order, and we relate this to an extended version of Robinson’s theorem.

We give a simple interpretation of the adapted complex structure of Lempert-Szoke and Guillemin-Stenzel: it is given by a polar decomposition of the complexified manifold. We then give a twistorial construction of an SO(3)-invariant hypercomplex structure on a neighbourhood of $X$ in $TTX$, where $X$ is a real-analytic manifold equipped with a linear connection. We show that the Nahm equations arise naturally in this context: for a connection with zero curvature and arbitrary torsion, the real sections of the twistor space can be obtained by solving Nahm's equations in the Lie algebra of certain vector fields. Finally, we show that, if we start with a metric connection, then our construction yields an SO(3)-invariant hyperk\"ahler metric., Comment: some corrections, a reference added, to appear in International J. of Mathematics

[Davidov Johann; Давидов Йохан]; [Mushkarov Oleg; Muškarov Oleg; Мушкаров Олег]; [Grantcharov G.; Грънчаров Г.]

In this paper we obtain several curvature properties of the twistor and reflector spaces of a paraquaternionic Kähler manifold and prove the existence of both positive and negative mixed 3-Sasakian structures in a principal SO (2, 1)-bundle over a paraquaternionic Kähler manifold.

Let (M, g) be a 2n-dimensional oriented Riemannian manifold, let P(M) = P(M, SO(2n)) be the principal SO(2n)-bundle of oriented orthonormal frames over M and let Z(M) = P(M)/U(n) be the Twistor Space of M.

We show that asymptotic projective twistor space PT+ is an Einstein–Kahler manifold of positive curvature. We then use the Chern–Moser theory of hypersurfaces in complex manifolds to show that the Kahler curvature of PT+ closely related to the CR curvature of its boundary. We also give a proof that the Kahler potential function defining the boundary satisfies the complex Monge–Ampere equations.

Asymptotic twistor space T is a 4‐complex‐dimensional Kahler manifold (of signature ++−−) which can be constructed from an asymptotically flat space–time containing gravitational radiation. The properties of this Kahler structure are investigated, the Kahler metric being of a particular type, arising from a scalar Σ with special homogeneity properties. The components of the Kahler curvature Kαβγδ are found explicitly in terms of the asymptotic Weyl curvature of the space–time. When gravitational radiation is present, Kαβγδ ≠0, whereas for a stationary field Kαβγδ=0. The ’’Ricci‐flat’’ condition Kαβαγ=0 is found always to hold.

A holomorphic connection on (1, 0)-vector fields which is intrinsically defined on any curved twistor space is described. Although it is a local operator, it is given in terms of the nonlocal geometry of the twistor space corresponding to the local geometry of the spacetime. The connection is represented in local coordinates by a system ofnonlinear first-order partial differential operators. It has torsion but no curvature. A parallelism is given explicitly, and an example is computed.

We show that asymptotic twistor space PJ+ is an Einstein Kahler manifold of positive curvature. We relate the curvature of PJ+ to the CR-curvature of its boundary and we show that the function defining the boundary satisfies the complex Monge-Ampere equations.

The non-linear graviton construction of Penrose (9) and the ℋ-space construction of Newman (6) are two complementary techniques for constructing complex four dimensional space-times with quadratic metric and anti-self-dual curvature tensor.In the former, the space-time is the space of holomorphic sections of a complex fibre space obtained by deforming part of flat twistor space. In the latter the space-time is the space of regular solutions of a differential equation, the good cut equation.Pathologies arise in the non-linear graviton construction when the normal bundle of a holomorphic section changes. This is reflected in the ℋ-space construction by a change in the character of the solutions of the linearized good cut equation, the Newman equation.