Search

Let X = X¯ − D be a smooth quasi-projective curve. We previously constructed a Deligne–Hitchin moduli space with Hecke gauge groupoid for connections of rank 2. We extend this construction to the case of any rank r, although still keeping a genericity hypothesis. The formal neighborhood of a preferred section corresponding to a tame harmonic bundle is governed by a mixed twistor structure. [ABSTRACT FROM AUTHOR]

In this paper, we provide new rigidity results for four-dimensional Riemannian manifolds and their twistor spaces. In particular, using the moving frame method, we prove that C P 3 is the only twistor space whose Bochner tensor is parallel; moreover, we classify Hermitian Ricci-parallel and locally symmetric twistor spaces and we show the nonexistence of conformally flat twistor spaces. We also generalize a result due to Atiyah, Hitchin and Singer concerning the self-duality of a Riemannian four-manifold. [ABSTRACT FROM AUTHOR]

Let X be a quasi-projective curve, compactified to (Y, D) with X = Y - D . We construct a Deligne–Hitchin twistor space out of moduli spaces of framed λ -connections of rank 2 over Y with logarithmic singularities and quasi-parabolic structure along D. To do this, one should divide by a Hecke-gauge groupoid. Tame harmonic bundles on X give preferred sections, and the relative tangent bundle along a preferred section has a mixed twistor structure with weights 0, 1, 2. The weight 2 piece corresponds to the deformations of the KMS structure including parabolic weights and the residues of the λ -connection. [ABSTRACT FROM AUTHOR]

A study is made of algebraic curves and surfaces in the flag manifold F = S U (3) / T 2 , and their configuration relative to the twistor projection π from F to the complex projective plane P 2 , defined with the help of an anti-holomorphic involution j . This is motivated by analogous studies of algebraic surfaces of low degree in the twistor space P 3 of the 4-dimensional sphere S 4 . Deformations of twistor fibers project to real surfaces in P 2 , whose metric geometry is investigated. Attention is then focussed on toric del Pezzo surfaces that are the simplest type of surfaces in F of bidegree (1 , 1) . These surfaces define orthogonal complex structures on specified dense open subsets of P 2 relative to its Fubini-Study metric. The discriminant loci of various surfaces of bidegree (1 , 1) are determined, and bounds given on the number of twistor fibers that are contained in more general algebraic surfaces in F . [ABSTRACT FROM AUTHOR]

In this paper, we generalize the construction of Deligne–Hitchin twistor space by gluing two certain Hodge moduli spaces. We investigate some properties of such generalized Deligne–Hitchin twistor space as a complex analytic manifold. More precisely, we show it admits holomorphic sections whose normal bundle contains a semistable subbundle with positive degree and whose energy is semi-negative, and it carries a balanced metric. Moreover, we also study the automorphism groups of the Hodge moduli spaces and the generalized Deligne–Hitchin twistor spaces. [ABSTRACT FROM AUTHOR]

Motivated by the recent interest in even-Clifford structures and in generalized complex and quaternionic geometries, we introduce the notion of generalized almost even-Clifford structure. We generalize the Arizmendi-Hadfield twistor space construction on even-Clifford manifolds to this setting and show that such a twistor space admits a generalized complex structure under certain conditions.

A hyperkähler manifold M has a family of induced complex structures indexed by a two-dimensional sphere S 2 ≅ CP 1 . The twistor space of M is a complex manifold Tw (M) together with a natural holomorphic projection Tw (M) → CP 1 , whose fiber over each point of CP 1 is a copy of M with the corresponding induced complex structure. We remove one point from this sphere (corresponding to one fiber in the twistor space), and for the case of M = R 4 n , n ∈ N , equipped with the standard hyperkähler structure, we construct one quantization that replaces the family of Berezin–Toeplitz quantizations parametrized by S 2 - { p t } . We provide semiclassical asymptotics for this quantization. [ABSTRACT FROM AUTHOR]

We study the holomorphic vector bundles E over the twistor space Tw (M) of a compact simply connected hyperkähler manifold M. We give a characterization of the semistability condition for E in terms of its restrictions to the holomorphic sections of the holomorphic twistor projection π : Tw (M) ⟶ CP 1 . It is shown that if E admits a holomorphic connection, then E is holomorphically trivial and the holomorphic connection on E is trivial as well. For any irreducible vector bundle E on Tw (M) of prime rank, we prove that its restriction to the generic fibre of π is stable. On the other hand, for a K3 surface M, we construct examples of irreducible vector bundles of any composite rank on Tw (M) whose restriction to every fibre of π is non-stable. We have obtained a new method of constructing irreducible vector bundles on hyperkähler twistor spaces; this method is employed in constructing these examples. [ABSTRACT FROM AUTHOR]

In this paper we study an analog of minimal surfaces called Weyl-minimal surfaces in conformal manifolds with a Weyl connection (M4, c, D). We show that there is an Eells-Salamon type correspondence between nonvertical 𝒥-holomorphic curves in the weightless twistor space and branched Weyl-minimal surfaces. When (M, c, J) is conformally almost-Hermitian, there is a canonical Weyl connection. We show that for the canonical Weyl connection, branched Weyl-minimal surfaces satisfy the adjunction inequality

Let E be an oriented vector bundle of rank 4 over a torus T 2 with a metric h and E ^ the twistor space associated with E. We will study the horizontality in E ^ with respect to the connection ∇ ^ induced by an h-connection ∇ . We will describe the subset of the fiber E ^ a of E ^ at a point a ∈ T 2 given by horizontality along normal polygonal curves in R 2 for an initial value at a in the case where the subset is finite. We will see that if E ^ has a partially horizontal section Ω , then there exists an h-connection ∇ ′ related to ∇ such that h, ∇ ′ and Ω give a Kähler structure of E. We will make analogous discussions for an oriented vector bundle of rank 4 over T 2 with a neutral metric. [ABSTRACT FROM AUTHOR]

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]

Given a smooth complex projective variety M and a smooth closed curve X ⊂ M such that the homomorphism of fundamental groups π 1 (X) ⟶ π 1 (M) is surjective, we study the restriction map of Higgs bundles, namely from the Higgs bundles on M to those on X. In particular, we investigate the interplay between this restriction map and various types of branes contained in the moduli spaces of Higgs bundles on M and X. We also consider the setup where a finite group is acting on M via holomorphic automorphisms or anti-holomorphic involutions, and the curve X is preserved by this action. Branes are studied in this context. [ABSTRACT FROM AUTHOR]

In this paper, we show that if (X, g) is an oriented four-dimensional Einstein manifold which is self-dual or anti-self-dual then superminimal surfaces in X of appropriate spin enjoy the Calabi–Yau property, meaning that every immersed surface of this type from a bordered Riemann surface can be uniformly approximated by complete superminimal surfaces with Jordan boundaries. The proof uses the theory of twistor spaces and the Calabi–Yau property of holomorphic Legendrian curves in complex contact manifolds. [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]

We propose a generalization of Verbitsky's global Torelli theorem in the framework of compact Kähler irreducible holomorphically symplectic orbifolds by adapting Huybrechts' proof [17]. As intermediate step needed, we also provide a generalization of the twistor space and the projectivity criterion based on works of Campana [8] and Huybrechts [21] respectively. [ABSTRACT FROM AUTHOR]

In this paper, we extend the notion of Bekenstein–Hawking entropy for a cover of a site. We deduce a new class of discrete dynamical system on a site and we introduce the Bekenstein–Hawking entropy for each member of it. We present an upper bound for the Bekenstein–Hawking entropy of the iterations of a dynamical system. We define a conjugate relation on the set of dynamical systems on a site and we prove that the Bekenstein–Hawking entropy preserves under this relation. We also prove that the twistor correspondence preserves the Bekenstein–Hawking entropy. [ABSTRACT FROM AUTHOR]

We construct new twistorial examples of non-Kähler almost Hermitian manifolds with Hermitian Ricci tensor by means of a natural almost Hermitian structures on the twistor space of an almost Hermitian four manifold. [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]

In this short note, we review the well-known result that there is no orthogonal complex structure on S 6 with respect to the round metric. [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]

A conformal map from a Riemann surface to the Euclidean four-space is explained in terms of its twistor lift. A local factorization of a differential of a conformal map is obtained. As an application, the factorization of a differential provides an upper bound of the area of a super-conformal map around a branch point. [ABSTRACT FROM AUTHOR]