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The theory of twistors on foliated manifolds is developed. We construct the twistor space of the normal bundle of a foliation. It is demonstrated that the classical constructions of the twistor theory lead to foliated objects and permit to formulate and prove foliated versions of some well-known results on holomorphic mappings. Since any orbifold can be understood as the leaf space of a suitably defined Riemannian foliation we obtain orbifold versions of the classical results as a simple consequence of the results on foliated mappings. [ABSTRACT FROM AUTHOR]

Several methods use the Fourier transform from momentum space to twistor space to analyze scattering amplitudes in Yang–Mills theory. However, the transform has not been defined as a concrete complex integral when the twistor space is a three-dimensional complex projective space. To the best of our knowledge, this is the first study to define it as well as its inverse in terms of a concrete complex integral. In addition, our study is the first to show that the Fourier transform is an isomorphism from the zeroth Čech cohomology group to the first one. Moreover, the well-known twistor operator representations in twistor theory literature are shown to be valid for the Fourier transform and its inverse transform. Finally, we identify functions over which the application of the operators is closed. [ABSTRACT FROM AUTHOR]

The class of hyper-CR Einstein–Weyl structures on R 3 can be described in terms of the solutions to the dispersionless Hirota equation. In the present paper we show that simple geometric constructions on the associated twistor space lead to deformations of the Hirota equation that have been introduced recently by B. Kruglikov and A. Panasyuk. Our method produces also the hyper-CR equation and can be applied to other geometric structures related to different twistor constructions. [ABSTRACT FROM AUTHOR]

We show that Hopf hypersurfaces in complex projective space are constructed from half-dimensional totally complex submanifolds in complex 2-plane Grassmannian and Legendrian submanifolds in the twistor space. [ABSTRACT FROM AUTHOR]

Let M be a hyperkähler manifold, and η a closed, positive (1, 1)-form with rk η < dim M . We associate to η a family of complex structures on M , called a degenerate twistor family, and parametrized by a complex line. When η is a pullback of a Kähler form under a Lagrangian fibration L , all the fibers of degenerate twistor family also admit a Lagrangian fibration, with the fibers isomorphic to that of L . Degenerate twistor families can be obtained by taking limits of twistor families, as one of the Kähler forms in the hyperkähler triple goes to η . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: In this note, we prove that the CR manifold induced from the canonical parabolic geometry of a quaternionic contact (qc) manifold via a Fefferman-type construction is equivalent to the CR twistor space of the qc manifold defined by O. Biquard. [Copyright &y& Elsevier]

In this article, we show that the space of nodal rational curves, which is so called a Severi variety (of rational curves), on any non-singular projective surface is always equipped with a natural Einstein-Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein-Weyl structure on the space of smooth rational curves on a complex surface, given by Hitchin. As geometric objects naturally associated to Einstein-Weyl structure, we investigate null surfaces and geodesics on the Severi varieties. Also, we see that if the projective surface has an appropriate real structure, then the real locus of the Severi variety becomes a positive definite Einstein-Weyl manifold. Moreover, we construct various explicit examples of rational surfaces having 3-dimensional Severi varieties of rational curves. [ABSTRACT FROM AUTHOR]

Abstract: Let M be a -manifold. We consider an almost CR-structure on the sphere bundle of unit tangent vectors on M, called the CR twistor space. This CR-structure is integrable if and only if M is a holonomy -manifold. We interpret -instanton bundles as CR-holomorphic bundles on its twistor space. [Copyright &y& Elsevier]

We describe a new construction of anti-self-dual metrics on four-manifolds. These metrics are characterized by the property that their twistor spaces project as affine line bundles over surfaces. To any affine bundle with the appropriate sheaf of local translations, we associate a solution of a second-order partial differential equations system D2 V = 0 on a five-dimensional manifold $${\mathbf{Y}}$$. The solution V and its differential completely determine an anti-self-dual conformal structure on an open set in { V = 0}. We show how our construction applies in the specific case of conformal structures for which the twistor space $${\mathcal{Z}}$$ has $${ \dim\left|-\frac{1}{2}K_\mathcal{Z}\right|\geq 2}$$, projecting thus over $${\mathbb C\mathbb P_2}$$ with twistor lines mapping onto plane conics. [ABSTRACT FROM AUTHOR]

Abstract: We investigate the behavior of derivatives of the fundamental solution of a parabolic equation for the square of Dirac operator on a twistor space when the metric is blown up in the base space direction. Such a blowing up operation is expected to be an effective method for extracting some intrinsic values from various geometric invariants, most of whose cores consist of some derivatives of the fundamental solution. [Copyright &y& Elsevier]

We shall describe the twistor space of a hyperka¨hler 4n-manifold with an isometric S1-action which is holomorphic for one of the complex structures, scales the corresponding holomorphic symplectic form and whose fixed point set has complex dimension n.We deduce that any hyperka¨hler metric on the cotangent bundle of a real-analytic Ka¨hler manifold which is compatible with the canonical holomorphic symplectic structure, extends the given Ka¨hler metric and for which the S1-action by scalar multiplication in the fibres is isometric is unique in a neighbourhood of the zero section. These metrics have been constructed independently by the author and Kaledin. [Copyright &y& Elsevier]

Abstract: In this note, we consider surfaces in self-dual Einstein manifolds whose twistor lifts are harmonic sections. In particular, we state the stability of the twistor lifts as harmonic sections and determine such surfaces of genus zero. This paper is a short survey of our previous results in Hasegawa (2007, 2009, 2011) . [Copyright &y& Elsevier]