250 results on '"Twistor space"'
Search Results
2. The Adjunction Inequality for Weyl-Harmonic Maps
- Author
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Ream Robert
- Subjects
almost-complex manifolds ,twistor space ,weyl geometry ,32q60 ,53c28 ,53c43 ,Mathematics ,QA1-939 - Abstract
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
- Published
- 2020
- Full Text
- View/download PDF
3. Generalized almost even-Clifford manifolds and their twistor spaces
- Author
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Luis Fernando Hernández-Moguel and Rafael Herrera
- Subjects
Physics ,Pure mathematics ,010102 general mathematics ,twistor space ,01 natural sciences ,even-clifford structure ,15a66 ,generalized complex structure ,Twistor theory ,53c15 ,Computer Science::Emerging Technologies ,32l25 ,53c28 ,53d18 ,0103 physical sciences ,Generalized complex structure ,QA1-939 ,Twistor space ,Mathematics::Differential Geometry ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Mathematics - Abstract
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.
- Published
- 2021
4. Global Torelli theorem for irreducible symplectic orbifolds
- Author
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Grégoire Menet
- Subjects
Pure mathematics ,Mathematics::Complex Variables ,Generalization ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Torelli theorem ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,Twistor space ,010307 mathematical physics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Mathematics ,Symplectic geometry - Abstract
We propose a generalization of Verbitsky's global Torelli theorem in the framework of compact K\"ahler irreducible holomorphically symplectic orbifolds by adapting Huybrechts' proof (arXiv:1106.5573). As intermediate step needed, we also provide a generalization of the twistor space and the projectivity criterion based on works of Campana (arXiv:math/0402243) and Huybrechts (arXiv:math/0106014) respectively., Comment: 23 pages, final version to appear in Journal de Math\'ematiques pures et appliqu\'ees
- Published
- 2020
5. The Adjunction Inequality for Weyl-Harmonic Maps
- Author
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Robert K. Ream
- Subjects
Mathematics - Differential Geometry ,Physics ,Pure mathematics ,Minimal surface ,almost-complex manifolds ,Harmonic map ,Sigma ,Conformal map ,53c43 ,Mathematics::Spectral Theory ,Type (model theory) ,twistor space ,Adjunction ,weyl geometry ,53c28 ,Differential Geometry (math.DG) ,32q60 ,QA1-939 ,FOS: Mathematics ,Twistor space ,Mathematics::Differential Geometry ,Geometry and Topology ,Connection (algebraic framework) ,Mathematics::Representation Theory ,Mathematics - Abstract
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\chi \left( {{T_f}\sum } \right) + \chi \left( {{N_f}\sum } \right) \le \pm {c_1}\left( {f*{T^{\left( {1,0} \right)}}M} \right).The ±J-holomorphic curves are automatically Weyl-minimal and satisfy the corresponding equality. These results generalize results of Eells-Salamon and Webster for minimal surfaces in Kähler 4-manifolds as well as their extension to almost-Kähler 4-manifolds by Chen-Tian, Ville, and Ma.
- Published
- 2020
6. Geodesic rigidity of conformal connections on surfaces
- Author
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Thomas Mettler
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Geodesic ,General Mathematics ,Conformal map ,Twistor space ,Projective structures ,Mathematics - Algebraic Geometry ,symbols.namesake ,Geodesic rigidity ,Euler characteristic ,Conformal connections ,FOS: Mathematics ,Primary 53A20 ,Secondary 53C24 ,53C28 ,Algebraic Geometry (math.AG) ,Mathematics ,Quantitative Biology::Biomolecules ,Surface (topology) ,Manifold ,Differential Geometry (math.DG) ,Metric (mathematics) ,symbols ,Conformal connection ,Mathematics::Differential Geometry - Abstract
We show that a conformal connection on a closed oriented surface $\Sigma$ of negative Euler characteristic preserves precisely one conformal structure and is furthermore uniquely determined by its unparametrised geodesics. As a corollary it follows that the unparametrised geodesics of a Riemannian metric on $\Sigma$ determine the metric up to constant rescaling. It is also shown that every conformal connection on the $2$-sphere lies in a complex $5$-manifold of conformal connections, all of which share the same unparametrised geodesics., Comment: 16 pages, exposition improved, references added
- Published
- 2021
7. Integrable Complex Structures on Twistor Spaces
- Author
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Steven Gindi
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Integrable system ,General Mathematics ,010102 general mathematics ,Fibered knot ,01 natural sciences ,Twistor theory ,Differential Geometry (math.DG) ,0103 physical sciences ,Metric (mathematics) ,FOS: Mathematics ,Twistor space ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
We introduce integrable complex structures on twistor spaces fibered over complex manifolds. We then show, in particular, that the twistor spaces associated with generalized Kahler, SKT and strong HKT manifolds all naturally admit complex structures. Moreover, in the strong HKT case we construct a metric and three compatible complex structures on the twistor space that have equal torsions., Comment: New results on Hermitian properties of twistor spaces
- Published
- 2019
8. Example of a Stable but Fiberwise Nonstable Bundle on the Twistor Space of a Hyper-Kähler Manifold
- Author
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A. Yu. Tomberg
- Subjects
Pure mathematics ,General Mathematics ,Bundle ,Twistor space ,Kähler manifold ,Mathematics - Published
- 2019
9. Twistor space of a generalized quaternionic manifold
- Author
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Guillaume Deschamps
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Hypercomplex number ,Mathematics - Complex Variables ,General Mathematics ,Bismut connection ,Manifold ,Connection (mathematics) ,Twistor theory ,Differential Geometry (math.DG) ,53D18, 53C28, 51P05, 32Q15, 32Q60 ,Generalized complex structure ,FOS: Mathematics ,Twistor space ,Mathematics::Differential Geometry ,Complex Variables (math.CV) ,Quaternion ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
We first make a little survey of the twistor theory for hypercomplex, generalized hypercomplex, quaternionic or generalized quaternionic manifolds. This last theory was iniated by Pantilie, who shows that any generalized almost quaternionic manifold equipped with an appropriate connection admit a twistor space with an almost generalized complex structure. The aim of this article is to give an integrability criterion for this generalized almost complex structure and to give some examples especially in the case of generalized hyperk\"ahler manifolds using the generalized Bismut connection, introduced by Gualtieri., Comment: 19 pages
- Published
- 2021
10. Twistor spaces on foliated manifolds
- Author
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Robert Wolak and Rouzbeh Mohseni
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Mathematics::Dynamical Systems ,General Mathematics ,Twistor theory ,High Energy Physics::Theory ,Differential Geometry (math.DG) ,Normal bundle ,FOS: Mathematics ,Foliation (geology) ,Twistor space ,Mathematics::Differential Geometry ,53C12, 53C28 ,Mathematics::Symplectic Geometry ,Orbifold ,Mathematics - Abstract
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.
- Published
- 2021
11. Complex multiplication in twistor spaces
- Author
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Daniel Huybrechts
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Complex multiplication ,0102 computer and information sciences ,01 natural sciences ,K3 surface ,Twistor theory ,Mathematics - Algebraic Geometry ,High Energy Physics::Theory ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,010201 computation theory & mathematics ,FOS: Mathematics ,Twistor space ,Transcendental number ,Mathematics::Differential Geometry ,0101 mathematics ,Algebraic number ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
Despite the transcendental nature of the twistor construction, the algebraic fibres of the twistor space of a K3 surface share certain arithmetic properties. We prove that for a polarized K3 surface with complex multiplication, all algebraic fibres of its twistor space away from the equator have complex multiplication as well., Comment: 21 pages. Revision takes into account various insightful comments of two anonymous referees. In particular the equation of the CM extension for the twistor fibre (Corollary 3.10) has been corrected and from Section 3.2 the assumption (\ell'.\ell')>0 has been added. To appear in IMRN
- Published
- 2021
- Full Text
- View/download PDF
12. Energy of sections of the Deligne–Hitchin twistor space
- Author
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Markus Roeser, Florian Beck, and Sebastian Heller
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Twistor methods in differential geometry ,General Mathematics ,Holomorphic function ,Computer Science::Digital Libraries ,01 natural sciences ,Twistor theory ,Mathematics::Algebraic Geometry ,Line bundle ,0103 physical sciences ,FOS: Mathematics ,Compact Riemann surface ,0101 mathematics ,ddc:510 ,Relationships between algebraic curves and integrable systems ,Mathematics::Symplectic Geometry ,Hyper-Kähler and quaternionic Kähler geometry ,Mathematics ,Energy functional ,Meromorphic function ,Mathematics::Complex Variables ,Vector bundles on curves and their moduli ,010102 general mathematics ,Differential geometric aspects of harmonic maps ,Dewey Decimal Classification::500 | Naturwissenschaften::510 | Mathematik ,Moduli space ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Differential Geometry (math.DG) ,Computer Science::Mathematical Software ,Twistor space ,010307 mathematical physics ,Mathematics::Differential Geometry - Abstract
We study a natural functional on the space of holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface, generalizing the energy of equivariant harmonic maps corresponding to twistor lines. We give a link to a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by Hitchin. Moreover, we prove that for a certain class of real holomorphic sections of the Deligne-Hitchin moduli space, the functional is basically given by the Willmore energy of corresponding (equivariant) conformal map to the 3-sphere. As an application we use the functional to distinguish new components of real holomorphic sections of the Deligne-Hitchin moduli space from the space of twistor lines., 33 pages
- Published
- 2021
- Full Text
- View/download PDF
13. On the existence of compact scalar-flat Kähler surfaces
- Author
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M. PONTECORVO
- Subjects
kähler metric ,twistor space ,Mathematics ,QA1-939 - Abstract
A compact compler surface with non-trivial canonical bundle and a Kähler metric ofzero scalar curvature must be a ruled surface. It is also known that not every ruled surface can admit such extremal Kähler metrics. In this paper we review recentjoint work with Kim and LeBrun in which deforma- tion theory of pairs of singular complex spaces it is used to show that any ruled surface (M, J) has blow-ups (M, J) which admit Kähler metrics of zero scalar curvature.
- Published
- 1995
14. Sub-Riemannian Geodesics on Nested Principal Bundles
- Author
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Mauricio Godoy Molina and Irina Markina
- Subjects
Geodesic ,Principal (computer security) ,Structure (category theory) ,Mathematics::Metric Geometry ,Lie group ,Twistor space ,Mathematics::Differential Geometry ,Parametric equation ,Exotic sphere ,Action (physics) ,Mathematical physics ,Mathematics - Abstract
We study the interplay between geodesics on two non-holonomic systems that are related by the action of a Lie group on them. After some geometric preliminaries, we use the Hamiltonian formalism to write the parametric form of geodesics. We present several geometric examples, including a non-holonomic structure on the Gromoll-Meyer exotic sphere and twistor space.
- Published
- 2020
15. Deligne pairings and families of rank one local systems on algebraic curves
- Author
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Gerard Freixas i Montplet, Richard Wentworth, Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), and Department of Mathematics, University of Maryland
- Subjects
Pure mathematics ,Holomorphic function ,58J52 ,01 natural sciences ,symbols.namesake ,Mathematics::Algebraic Geometry ,Line bundle ,Mathematics::K-Theory and Homology ,Analytic torsion ,0101 mathematics ,Connection (algebraic framework) ,[MATH]Mathematics [math] ,Mathematics ,Meromorphic function ,Algebra and Number Theory ,Mathematics::Complex Variables ,Riemann surface ,010102 general mathematics ,14C40 ,16. Peace & justice ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,symbols ,Twistor space ,Geometry and Topology ,Algebraic curve ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,Analysis - Abstract
For smooth families $\mathcal{X} \to S$ of projective algebraic curves and holomorphic line bundles $\mathcal{L, M} \to X$ equipped with flat relative connections, we prove the existence of a canonical and functorial “intersection” connection on the Deligne pairing $\langle \mathcal{L, M} \rangle \to S$. This generalizes the construction of Deligne in the case of Chern connections of hermitian structures on $\mathcal{L}$ and $\mathcal{M}$. A relationship is found with the holomorphic extension of analytic torsion, and in the case of trivial fibrations we show that the Deligne isomorphism is flat with respect to the connections we construct. Finally, we give an application to the construction of a meromorphic connection on the hyperholomorphic line bundle over the twistor space of rank one flat connections on a Riemann surface.
- Published
- 2020
16. Quaternion-Kähler manifolds near maximal fixed point sets of $S^{1}$-symmetries
- Author
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Aleksandra Borówka
- Subjects
Pure mathematics ,Applied Mathematics ,010102 general mathematics ,Connection (principal bundle) ,Kähler manifold ,Fixed point ,Submanifold ,01 natural sciences ,Line bundle ,0103 physical sciences ,Twistor space ,010307 mathematical physics ,0101 mathematics ,Hyperkähler manifold ,Distribution (differential geometry) ,Mathematics - Abstract
Using quaternionic Feix–Kaledin construction, we provide a local classification of quaternion-Kähler metrics with a rotating $$S^1$$S1-symmetry with the fixed point set submanifold S of maximal possible dimension. For any real-analytic Kähler manifold S equipped with a line bundle with a real-analytic unitary connection with curvature proportional to the Kähler form, we explicitly construct a holomorphic contact distribution on the twistor space obtained by the quaternionic Feix–Kaledin construction from these data. Conversely, we show that quaternion-Kähler metrics with a rotating $$S^1$$S1-symmetry induce on the fixed point set of maximal dimension a Kähler metric together with a unitary connection on a holomorphic line bundle with curvature proportional to the Kähler form and the two constructions are inverse to each other. Moreover, we study the case when S is compact, showing that in this case the quaternion-Kähler geometry is determined by the Kähler metric on the fixed point set (of maximal possible dimension) and by the contact line bundle along the corresponding submanifold on the twistor space. Finally, we relate the results to the c-map construction showing that the family of quaternion-Kähler manifolds obtained from a fixed Kähler metric on S by varying the line bundle and the hyperkähler manifold obtained by hyperkähler Feix–Kaledin construction from S are related by hyperkähler/quaternion-Kähler correspondence.
- Published
- 2020
17. Jumps, folds and singularities of Kodaira moduli spaces
- Author
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Paul Tod, James Gundry, and Maciej Dunajski
- Subjects
Pure mathematics ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,Holomorphic function ,Space (mathematics) ,01 natural sciences ,Manifold ,Moduli space ,Twistor theory ,Normal bundle ,Cone (topology) ,0103 physical sciences ,Twistor space ,Mathematics::Differential Geometry ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
For any integer k we construct an explicit example of a twistor space which contains a one-parameter family of jumping rational curves, where the normal bundle changes from O(1) � O(1) to O(k) � O(2 k). For k > 3 the resulting anti-self-dual Ricci-flat manifold is a Zariski cone in the space of holomorphic sections of O(k). In the case k = 2 we recover the canonical example of Hitchin's folded hyper-Kahler manifold, where the jumping lines form a three-parameter family. We show that in this case there exist normalisable solutions to the Schrodinger equation which extend through the fold.
- Published
- 2018
18. Generalized metrics and generalized twistor spaces
- Author
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Johann Davidov
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,53D18, 53C28 ,General Mathematics ,010102 general mathematics ,Space (mathematics) ,01 natural sciences ,Manifold ,Twistor theory ,Differential Geometry (math.DG) ,Bundle ,0103 physical sciences ,Metric (mathematics) ,FOS: Mathematics ,Tangent space ,Twistor space ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
The twistor construction for Riemannian manifolds is extended to the case of manifolds endowed with generalized metrics (in the sense of generalized geometry \`a la Hitchin). The generalized twistor space associated to such a manifold is defined as the bundle of generalized complex structures on the tangent spaces of the manifold compatible with the given generalized metric. This space admits natural generalized almost complex structures whose integrability conditions are found in the paper. An interesting feature of the generalized twistor spaces discussed in it is the existence of intrinsic isomorphisms., Comment: typos corrected, minor changes, to appear in Math. Z
- Published
- 2018
19. A note on Lagrangian submanifolds of twistor spaces and their relation to superminimal surfaces
- Author
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Reinier Storm
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Mathematics::Complex Variables ,Circle bundle ,010102 general mathematics ,Structure (category theory) ,Fibration ,Surface (topology) ,Submanifold ,01 natural sciences ,Hermitian matrix ,Twistor theory ,Computational Theory and Mathematics ,Differential Geometry (math.DG) ,0103 physical sciences ,FOS: Mathematics ,Twistor space ,010307 mathematical physics ,Geometry and Topology ,Mathematics::Differential Geometry ,0101 mathematics ,53C28, 53C42 ,Mathematics::Symplectic Geometry ,Analysis ,Mathematics - Abstract
In this paper a bijective correspondence between superminimal surfaces of an oriented Riemannian 4-manifold and particular Lagrangian submanifolds of the twistor space over the 4-manifold is proven. More explicitly, for every superminimal surface a submanifold of the twistor space is constructed which is Lagrangian for all the natural almost Hermitian structures on the twistor space. The twistor fibration restricted to the constructed Lagrangian gives a circle bundle over the superminimal surface. Conversely, if a submanifold of the twistor space is Lagrangian for all the natural almost Hermitian structures, then the Lagrangian projects to a superminimal surface and is contained in the Lagrangian constructed from this surface. In particular this produces many Lagrangian submanifolds of the twistor spaces C P 3 and F 1 , 2 ( C 3 ) with respect to both the Kahler structure as well as the nearly Kahler structure. Moreover, it is shown that these Lagrangian submanifolds are minimal submanifolds.
- Published
- 2019
20. Almost complex structures that are harmonic maps
- Author
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Absar Ul Haq, Johann Davidov, and Oleg Mushkarov
- Subjects
Pure mathematics ,010308 nuclear & particles physics ,010102 general mathematics ,Harmonic map ,Structure (category theory) ,General Physics and Astronomy ,Mathematics::Geometric Topology ,01 natural sciences ,law.invention ,law ,0103 physical sciences ,Hermitian manifold ,Twistor space ,Mathematics::Differential Geometry ,Geometry and Topology ,0101 mathematics ,Mathematics::Symplectic Geometry ,Manifold (fluid mechanics) ,Mathematical Physics ,Mathematics - Abstract
We find geometric conditions on a four-dimensional almost Hermitian manifold under which the almost complex structure is a harmonic map or a minimal isometric imbedding of the manifold into its twistor space.
- Published
- 2018
21. Twistor interpretation of slice regular functions
- Author
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Amedeo Altavilla
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Mathematics - Complex Variables ,010102 general mathematics ,53C28, 30G35, 53C55, 14J26 ,General Physics and Astronomy ,01 natural sciences ,Twistor theory ,Lift (mathematics) ,Differential Geometry (math.DG) ,0103 physical sciences ,Algebraic surface ,FOS: Mathematics ,Twistor space ,010307 mathematical physics ,Geometry and Topology ,Complex Variables (math.CV) ,0101 mathematics ,Arrangement of lines ,Mathematical Physics ,Mathematics - Abstract
Given a slice regular function $f:\Omega\subset\mathbb{H}\to \mathbb{H}$, with $\Omega\cap\mathbb{R}\neq \emptyset$, it is possible to lift it to a surface in the twistor space $\mathbb{CP}^{3}$ of $\mathbb{S}^4\simeq \mathbb{H}\cup \{\infty\}$ (see~\cite{gensalsto}). In this paper we show that the same result is true if one removes the hypothesis $\Omega\cap\mathbb{R}\neq \emptyset$ on the domain of the function $f$. Moreover we find that if a surface $\mathcal{S}\subset\mathbb{CP}^{3}$ contains the image of the twistor lift of a slice regular function, then $\mathcal{S}$ has to be ruled by lines. Starting from these results we find all the projective classes of algebraic surfaces up to degree 3 in $\mathbb{CP}^{3}$ that contain the lift of a slice regular function. In addition we extend and further explore the so-called twistor transform, that is a curve in $\mathbb{G}r_2(\mathbb{C}^4)$ which, given a slice regular function, returns the arrangement of lines whose lift carries on. With the explicit expression of the twistor lift and of the twistor transform of a slice regular function we exhibit the set of slice regular functions whose twistor transform describes a rational line inside $\mathbb{G}r_2(\mathbb{C}^4)$, showing the role of slice regular functions not defined on $\mathbb{R}$. At the end we study the twistor lift of a particular slice regular function not defined over the reals. This example shows the effectiveness of our approach and opens some questions., Comment: 29 pages
- Published
- 2018
22. An inclusive immersion into a quaternionic manifold and its invariants
- Author
-
Kazuyuki Hasegawa
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Holomorphic function ,01 natural sciences ,Twistor theory ,53C26 ,53C28 ,Quaternionic representation ,0103 physical sciences ,Immersion (mathematics) ,Twistor space ,Mathematics::Differential Geometry ,010307 mathematical physics ,Algebraic curve ,0101 mathematics ,Invariant (mathematics) ,Quaternionic projective space ,Mathematics - Abstract
金沢大学人間社会研究域学校教育系 / Institute of Human and Social science, Teacher Education, We introduce a quaternionic invariant for an inclusive immersion into a quaternionic manifold, which is a quaternionic object corresponding to the Willmore functional. The lower bound of this invariant is given by topological invariant and the equality case can be characterized in terms of the natural twistor lift. When the ambient manifold is the quaternionic projective space and the natural twistor lift is holomorphic, we obtain a relation between the quaternionic invariant and the degree of the image of the natural twistor lift as an algebraic curve. Moreover the first variation formula for the invariant is obtained. As an application of the formula, if the natural twistor lift is a harmonic section, then the surface is a stationary point under any variations such that the induced complex structures do not vary. © 2017, Springer-Verlag Berlin Heidelberg., Embargo Period 12 months
- Published
- 2017
23. SASAKIAN TWISTOR SPINORS AND THE FIRST DIRAC EIGENVALUE
- Author
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Eui Chul Kim
- Subjects
Spinor ,General Mathematics ,010102 general mathematics ,Dirac (software) ,Dirac algebra ,Dirac operator ,01 natural sciences ,Twistor theory ,symbols.namesake ,0103 physical sciences ,symbols ,Twistor space ,010307 mathematical physics ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematical physics ,Mathematics - Published
- 2016
24. Non-Abelian Hodge Theory and Related Topics
- Author
-
Pengfei Huang, Département de Mathématiques [Nice], Université Nice Sophia Antipolis (... - 2019) (UNS), Université Côte d'Azur (UCA)-Université Côte d'Azur (UCA), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Université Côte d'Azur (UCA)
- Subjects
Pure mathematics ,[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] ,01 natural sciences ,Stratification (mathematics) ,oper ,Mathematics - Algebraic Geometry ,stratification ,Mathematics::Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,conformal limit ,0101 mathematics ,Abelian group ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Mathematics ,Hodge theory ,010102 general mathematics ,Hitchin section ,Moduli space ,twistor space ,non-Abelian Hodge theory ,14D20, 14D21, 32G20, 53C07, 57N80 ,Higgs boson ,moduli space ,$\lambda$-connection ,Twistor space ,010307 mathematical physics ,Geometry and Topology ,Analysis - Abstract
International audience; This paper is a survey aimed on the introduction of non-Abelian Hodge theory that gives the correspondence between flat bundles and Higgs bundles. We will also introduce some topics arising from this theory, especially some recent developments on the study of the relevant moduli spaces together with some interesting open problems.
- Published
- 2019
25. A family of integrable perturbed Kepler systems
- Author
-
Anatol Odzijewicz, E. Wawreniuk, and Aneta Sliżewska
- Subjects
Hamiltonian mechanics ,Integrable system ,010102 general mathematics ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,53D20, 53D05, 70F15, 70F05, 70E20 ,Poisson distribution ,01 natural sciences ,Kepler ,Jacobi elliptic functions ,symbols.namesake ,0103 physical sciences ,symbols ,Twistor space ,010307 mathematical physics ,0101 mathematics ,Mathematical Physics ,Mathematics ,Mathematical physics - Abstract
In the framework of the Poisson geometry of twistor space we consider a family of perturbed 3-dimensional Kepler systems. We show that Hamilton equations of this systems are integrated by quadratures. Their solutions for some subcases are given explicitly in terms of Jacobi elliptic functions., 26 pages
- Published
- 2019
26. Projective geometry and the quaternionic Feix-Kaledin construction
- Author
-
David M. J. Calderbank and Aleksandra Borówka
- Subjects
Mathematics - Differential Geometry ,Connection (fibred manifold) ,Pure mathematics ,Mathematics(all) ,General Mathematics ,Curvature ,01 natural sciences ,symbols.namesake ,FOS: Mathematics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics ,Complex line ,Riemann surface ,Applied Mathematics ,010102 general mathematics ,Submanifold ,Manifold ,Differential Geometry (math.DG) ,symbols ,Twistor space ,Mathematics::Differential Geometry ,Complex manifold ,53A20, 53B10, 53C26, 53C28, 32L25 - Abstract
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
- Published
- 2019
27. Real Holomorphic Sections of the Deligne–Hitchin Twistor Space
- Author
-
Markus Röser, Indranil Biswas, Sebastian Heller, and Ecole Internationale des Sciences du Traitement de l'Information (EISTI)
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] ,Complex system ,Holomorphic function ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,Compact Riemann surface ,0101 mathematics ,Invariant (mathematics) ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Mathematics ,53C26, 53C28, 14H60 ,Mathematics::Complex Variables ,010102 general mathematics ,Harmonic map ,Statistical and Nonlinear Physics ,Moduli space ,Differential Geometry (math.DG) ,Twistor space ,010307 mathematical physics - Abstract
We study the holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface that are invariant under the natural anti-holomorphic involutions of the moduli space. Their relationships with the harmonic maps are established. As a bi-product, a question of Simpson on such sections, posed in \cite{Si2}, is answered., Comment: Final version; to appear in Communications in Mathematical Physics
- Published
- 2019
28. Non-existence of orthogonal complex structures on the round 6-sphere
- Author
-
Ana Cristina Ferreira and Universidade do Minho
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Science & Technology ,Primary 53C15, Secondary 53C55, 32L25 ,010102 general mathematics ,Structure (category theory) ,6-sphere ,01 natural sciences ,Twistor space ,Complex structure ,Computational Theory and Mathematics ,Differential Geometry (math.DG) ,Round metric ,0103 physical sciences ,Metric (mathematics) ,FOS: Mathematics ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,10. No inequality ,Analysis ,Mathematics ,Matemáticas [Ciências Naturais] ,Ciências Naturais::Matemáticas - Abstract
"Available online 14 November 2017", In this short note, we review the well-known result that there is no orthogonal complex structure on the 6-sphere with respect to the round metric., info:eu-repo/semantics/publishedVersion
- Published
- 2019
- Full Text
- View/download PDF
29. General Schlesinger Systems and Their Symmetry from the View Point of Twistor theory
- Author
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Damiran Tseveenamijil and Hironobu Kimura
- Subjects
Weyl group ,Pure mathematics ,Group (mathematics) ,010102 general mathematics ,Statistical and Nonlinear Physics ,01 natural sciences ,Twistor theory ,Algebra ,symbols.namesake ,0103 physical sciences ,symbols ,Twistor space ,010307 mathematical physics ,Isomonodromic deformation ,0101 mathematics ,Symmetry (geometry) ,Hypergeometric function ,Abelian group ,Mathematical Physics ,Mathematics - Abstract
Isomonodromic deformation of linear differential equations on ℙ1 with regular and irregular singular points is considered from the view point of twistor theory. We give explicit form of isomonodromic deformation using the maximal abelian subgroup H of G = GLN+1(ℂ) which appeared in the theory of general hypergeometric functions on a Grassmannian manifold. This formulation enables us to obtain a group of symmetry for the nonlinear system which is an Weyl group analogue NG (H)/H.
- Published
- 2021
30. Totally complex submanifolds of a complex Grassmann manifold of 2-planes
- Author
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Kazumi Tsukada
- Subjects
Pure mathematics ,Mathematics::Complex Variables ,Complex projective space ,010102 general mathematics ,Mathematical analysis ,Structure (category theory) ,Kähler manifold ,01 natural sciences ,Linear subspace ,Twistor theory ,Computational Theory and Mathematics ,Grassmannian ,0103 physical sciences ,Cotangent bundle ,Twistor space ,Mathematics::Differential Geometry ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Mathematics::Symplectic Geometry ,Analysis ,Mathematics - Abstract
A complex Grassmann manifold G 2 ( C m + 2 ) of all 2-dimensional complex subspaces in C m + 2 has two nice geometric structures – the Kahler structure and the quaternionic Kahler structure. We study totally complex submanifolds of G 2 ( C m + 2 ) with respect to the quaternionic Kahler structure. We show that the projective cotangent bundle P ( T ⁎ C P m + 1 ) of a complex projective space C P m + 1 is a twistor space of the quaternionic Kahler manifold G 2 ( C m + 2 ) . Applying the twistor theory, we construct maximal totally complex submanifolds of G 2 ( C m + 2 ) from complex submanifolds of C P m + 1 . Then we obtain many interesting examples. In particular we classify maximal homogeneous totally complex submanifolds. We show the relationship between the geometry of complex submanifolds of C P m + 1 and that of totally complex submanifolds of G 2 ( C m + 2 ) .
- Published
- 2016
31. Twistor spaces of hypercomplex manifolds are balanced
- Author
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Artour Tomberg
- Subjects
Twistor theory ,Pure mathematics ,Hypercomplex number ,General Mathematics ,Metric (mathematics) ,Structure (category theory) ,Twistor space ,Mathematics::Differential Geometry ,Mathematics::Symplectic Geometry ,Hermitian matrix ,Hyperkähler manifold ,Manifold ,Mathematics - Abstract
A hypercomplex structure on a differentiable manifold consists of three integrable almost complex structures that satisfy quaternionic relations. If, in addition, there exists a metric on the manifold which is Hermitian with respect to the three structures, and such that the corresponding Hermitian forms are closed, the manifold is said to be hyperkahler. In the paper “Non-Hermitian Yang–Mills connections” [13] , Kaledin and Verbitsky proved that the twistor space of a hyperkahler manifold admits a balanced metric; these were first studied in the article “On the existence of special metrics in complex geometry” [17] by Michelsohn. In the present article, we review the proof of this result and then generalize it and show that twistor spaces of general compact hypercomplex manifolds are balanced.
- Published
- 2015
32. On deformations of the dispersionless Hirota equation
- Author
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Wojciech Kryński
- Subjects
Mathematics - Differential Geometry ,Class (set theory) ,Integrable system ,010308 nuclear & particles physics ,010102 general mathematics ,General Physics and Astronomy ,01 natural sciences ,Dispersionless equation ,Twistor theory ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Differential Geometry (math.DG) ,Simple (abstract algebra) ,0103 physical sciences ,FOS: Mathematics ,Twistor space ,Geometry and Topology ,0101 mathematics ,Mathematical Physics ,Mathematical physics ,Mathematics - Abstract
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.
- Published
- 2017
33. Pure Connection Formulation, Twistors and the Chase for a Twistor Action for General Relativity
- Author
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Yannick Herfray, Laboratoire de Physique de l'ENS Lyon ( Phys-ENS ), École normale supérieure - Lyon ( ENS Lyon ) -Université Claude Bernard Lyon 1 ( UCBL ), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Physique de l'ENS Lyon (Phys-ENS), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), and École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL)
- Subjects
High Energy Physics - Theory ,effective Lagrangian: chiral ,General relativity ,FOS: Physical sciences ,integrability ,01 natural sciences ,[ PHYS.HTHE ] Physics [physics]/High Energy Physics - Theory [hep-th] ,Twistor theory ,High Energy Physics::Theory ,twistor ,SU(2) theory ,0103 physical sciences ,Euclidean geometry ,general relativity ,010306 general physics ,Mathematical Physics ,Mathematics ,Mathematical physics ,[PHYS.HTHE]Physics [physics]/High Energy Physics - Theory [hep-th] ,010308 nuclear & particles physics ,Complex line ,Graviton ,Statistical and Nonlinear Physics ,Hermitian matrix ,Connection (mathematics) ,High Energy Physics - Theory (hep-th) ,SU(2) ,gravitation ,Twistor space ,Mathematics::Differential Geometry - Abstract
This paper establishes the relation between traditional results from (euclidean) twistor theory and chiral formulations of General Relativity (GR), especially the pure connection formulation. Starting from a $SU(2)$-connection only we show how to construct natural complex data on twistor space, mainly an almost Hermitian structure and a connection on some complex line bundle. Only when this almost Hermitian structure is integrable is the connection related to an anti-self-dual-Einstein metric and makes contact with the usual results. This leads to a new proof of the non-linear-graviton theorem. Finally we discuss what new strategies this "connection approach" to twistors suggests for constructing a twistor action for gravity. In appendix we also review all known chiral Lagrangians for GR., This is the version published in J.Math.Phys. As compare to the previous version, some paragraph were rewritten to make the text easier to read and some typos corrected
- Published
- 2017
34. Twistor Spaces and Compact Manifolds Admitting Both Kähler and Non-Kähler Structures
- Author
-
Ljudmila Kamenova
- Subjects
Twistor theory ,Pure mathematics ,Simply connected space ,Differentiable manifold ,Twistor space ,Geometry and Topology ,Quaternion ,Surface (topology) ,Mathematical Physics ,Hyperkähler manifold ,Manifold ,Mathematics - Abstract
In this expository paper we review some twistor techniques and recall the problem of finding compact differentiable manifolds that can carry both Kahler and non-Kahler complex structures. Such examples were constructed independently by Atiyah, Blanchard and Calabi in the 1950’s. In the 1980’s Tsanov gave an example of a simply connected manifold that admits both Kahler and non-Kahler complex structures - the twistor space of a $K3$ surface. Here we show that the quaternion twistor space of a hyperkahler manifold has the same property.
- Published
- 2017
35. Quantum Riemannian geometry of phase space and nonassociativity
- Author
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Shahn Majid and Edwin J. Beggs
- Subjects
Differential form ,General Mathematics ,58B32 ,FOS: Physical sciences ,General Relativity and Quantum Cosmology (gr-qc) ,Riemannian geometry ,Quantum mechanics ,01 natural sciences ,General Relativity and Quantum Cosmology ,symbols.namesake ,Poisson bracket ,Mathematics - Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Quantum Algebra (math.QA) ,0101 mathematics ,Connection (algebraic framework) ,Mathematical physics ,Mathematics ,lcsh:Mathematics ,010102 general mathematics ,Quantum gravity ,Noncommutative geometry ,Order (ring theory) ,lcsh:QA1-939 ,Poisson geometry ,81R50 ,Differential geometry ,symbols ,Twistor space ,83C57 ,010307 mathematical physics - Abstract
Noncommutative or `quantum' differential geometry has emerged in recent years as a process for quantizing not only a classical space into a noncommutative algebra (as familiar in quantum mechanics) but also differential forms, bundles and Riemannian structures at this level. The data for the algebra quantisation is a classical Poisson bracket, the data for the quantum differential forms is a Poisson-compatible connection it was recently shown that after this, classical data such as classical bundles, metrics etc. all become quantised in a canonical `functorial' way at least to 1st order in deformation theory. There are, however, fresh compatibility conditions between the classical Riemannian and the Poisson structures as well as new physics such as nonassociativity at 2nd order. We give an introduction to this theory and some details for the case of CP${}^n$ where the commutation relations have the canonical form $[w^i,\bar w^j]=\mathrm{i}\lambda\delta_{ij}$ similar to the proposal of Penrose for quantum twistor space. Our work provides a canonical but ultimately nonassociative differential calculus on this algebra and quantises the metric and Levi-Civita connection at lowest order in $\lambda$., Comment: 14 pages latex
- Published
- 2017
36. Hessian of the natural Hermitian form on twistor spaces
- Author
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Noël Le Du, Christophe Mourougane, Guillaume Deschamps, Laboratoire de mathématiques de Brest ( LM ), Université de Brest ( UBO ) -Institut Brestois du Numérique et des Mathématiques ( IBNM ), Université de Brest ( UBO ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de mathématiques de Brest (LM), Université de Brest (UBO)-Institut Brestois du Numérique et des Mathématiques (IBNM), Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,[ MATH.MATH-CV ] Mathematics [math]/Complex Variables [math.CV] ,Closed manifold ,53C28 ,53C26 ,32Q45 ,General Mathematics ,Invariant manifold ,twistor space ,01 natural sciences ,Pseudo-Riemannian manifold ,Twistor theory ,symbols.namesake ,Mathematics - Algebraic Geometry ,High Energy Physics::Theory ,FOS: Mathematics ,Hermitian manifold ,0101 mathematics ,Complex Variables (math.CV) ,hyperkähler manifold ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,4-dimensional Riemannian manifold ,Ricci curvature ,strong KT manifolds ,Mathematics ,quaternionic Kähler manifold ,Mathematics - Complex Variables ,010102 general mathematics ,Mathematical analysis ,Holonomy ,[MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV] ,Mathematics::Geometric Topology ,[ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG] ,[ MATH.MATH-DG ] Mathematics [math]/Differential Geometry [math.DG] ,Differential Geometry (math.DG) ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,symbols ,Twistor space ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,Mathematics::Differential Geometry - Abstract
International audience; We compute the hessian of the natural Hermitian form successively on the Calabi family of a hyperkähler manifold, on the twistor space of a 4-dimensional anti-self-dual Riemannian manifold and on the twistor space of a quaternionic Kähler manifold. We show a strong convexity property of the cycle space of twistor lines on the Calabi family of a hyperkähler 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 Kähler manifold of non-positive scalar curvature. We check that no non-Kähler strong KT manifold occurs as such a twistor space.
- Published
- 2017
37. Harmonic Almost Hermitian Structures
- Author
-
Johann Davidov
- Subjects
Harmonic coordinates ,Pure mathematics ,010308 nuclear & particles physics ,010102 general mathematics ,Mathematical analysis ,Harmonic map ,Riemannian manifold ,01 natural sciences ,Manifold ,Twistor theory ,0103 physical sciences ,Hermitian manifold ,Twistor space ,Mathematics::Differential Geometry ,0101 mathematics ,Complex manifold ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
This is a survey of old and new results on the problem when a compatible almost complex structure on a Riemannian manifold is a harmonic section or a harmonic map from the manifold into its twistor space. In this context, special attention is paid to the Atiyah-Hitchin-Singer and Eells-Salamon almost complex structures on the twistor space of an oriented Riemannian four-manifold.
- Published
- 2017
38. On the Automorphisms of a Rank One Deligne-Hitchin Moduli Space
- Author
-
Sebastian Heller and Indranil Biswas
- Subjects
Modular equation ,Dimension (graph theory) ,Hodge moduli space ,Holomorphic function ,01 natural sciences ,Combinatorics ,Mathematics - Algebraic Geometry ,Mathematics::Group Theory ,Mathematics::Algebraic Geometry ,0103 physical sciences ,λ-connections ,FOS: Mathematics ,14D20, 14J50, 14H60 ,0101 mathematics ,ddc:510 ,Algebraic Geometry (math.AG) ,Mathematical Physics ,Mathematics ,010102 general mathematics ,Automorphism ,Cohomology ,Dewey Decimal Classification::500 | Naturwissenschaften::510 | Mathematik ,Moduli space ,Moduli of algebraic curves ,Algebra ,Moishezon twistor space ,Deligne-Hitchin moduli space ,Twistor space ,010307 mathematical physics ,Geometry and Topology ,Analysis - Abstract
Let $X$ be a compact connected Riemann surface of genus $g \geq 2$, and let ${\mathcal M}_{\rm DH}$ be the rank one Deligne-Hitchin moduli space associated to $X$. It is known that ${\mathcal M}_{\rm DH}$ is the twistor space for the hyper-K\"ahler structure on the moduli space of rank one holomorphic connections on $X$. We investigate the group $\operatorname{Aut}({\mathcal M}_{\rm DH})$ of all holomorphic automorphisms of ${\mathcal M}_{\rm DH}$. The connected component of $\operatorname{Aut}({\mathcal M}_{\rm DH})$ containing the identity automorphism is computed. There is a natural element of $H^2({\mathcal M}_{\rm DH}, {\mathbb Z})$. We also compute the subgroup of $\operatorname{Aut}({\mathcal M}_{\rm DH})$ that fixes this second cohomology class. Since ${\mathcal M}_{\rm DH}$ admits an ample rational curve, the notion of algebraic dimension extends to it by a theorem of Verbitsky. We prove that ${\mathcal M}_{\rm DH}$ is Moishezon.
- Published
- 2017
- Full Text
- View/download PDF
39. Twistor construction of asymptotically hyperbolic Einstein–Weyl spaces
- Author
-
Aleksandra Borówka
- Subjects
Mathematics - Differential Geometry ,conformal Cartan connection ,Pure mathematics ,Holomorphic function ,Conformal map ,Space (mathematics) ,Surface (topology) ,53C28, 32L25, 53A30, 53C25 ,Twistor theory ,minitwistor space ,Differential Geometry (math.DG) ,Computational Theory and Mathematics ,Jones–Tod correspondence ,Cartan connection ,FOS: Mathematics ,asymptotically hyperbolic Einstein–Weyl manifold ,Twistor space ,Geometry and Topology ,Analysis ,Quotient ,Mathematics - Abstract
Starting from a real analytic conformal Cartan connection on a real analytic surface $S$, we construct a complex surface $T$ containing a family of pairs of projective lines. Using the structure on $S$ we also construct a complex $3$-space $Z$, such that $Z$ is a twistor space of a self-dual conformal $4$-fold and $T$ is a quotient of $Z$ by a holomorphic local $\mathbb{C}^*$ action. We prove that $T$ is a minitwistor space of an asymptotically hyperbolic Einstein-Weyl space with $S$ as an asymptotic boundary.
- Published
- 2014
40. Hopf hypersurfaces in complex projective space and half-dimensional totally complex submanifolds in complex 2-plane Grassmannian I
- Author
-
Makoto Kimura
- Subjects
Pure mathematics ,Mathematics::Complex Variables ,Plane (geometry) ,Complex projective space ,Mathematical analysis ,Mathematics::Algebraic Geometry ,Computational Theory and Mathematics ,Grassmannian ,Twistor space ,Mathematics::Differential Geometry ,Geometry and Topology ,Quaternionic projective space ,Mathematics::Symplectic Geometry ,Analysis ,Mathematics - Abstract
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.
- Published
- 2014
41. Einstein–Weyl geometry, dispersionless Hirota equation and Veronese webs
- Author
-
Maciej Dunajski and Wojciech Kryński
- Subjects
Mathematics - Differential Geometry ,High Energy Physics - Theory ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,General Mathematics ,FOS: Physical sciences ,Poisson distribution ,symbols.namesake ,Mathematics::Algebraic Geometry ,Differential Geometry (math.DG) ,High Energy Physics - Theory (hep-th) ,FOS: Mathematics ,symbols ,Heisenberg group ,Twistor space ,Mathematics::Differential Geometry ,Exactly Solvable and Integrable Systems (nlin.SI) ,Einstein ,Pencil (mathematics) ,Mathematics ,Mathematical physics - Abstract
We exploit the correspondence between the three-dimensional Lorentzian Einstein-Weyl geometries of the hyper-CR type, and the Veronese webs to show that the former structures are locally given in terms of solutions to the dispersionless Hirota equation. We also demonstrate how to construct hyper-CR Einstein--Weyl structures by Kodaira deformations of the flat twistor space $T\CP^1$, and how to recover the pencil of Poisson structures in five dimensions illustrating the method by an example of the Veronese web on the Heisenberg group., Comment: 11 pages. Minor changes. Final version, to appear in the Mathematical Proceedings of the Cambridge Philosophical Society
- Published
- 2014
42. Twistor transforms of quaternionic functions and orthogonal complex structures
- Author
-
Simon Salamon, Caterina Stoppato, and Graziano Gentili
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Mathematics - Complex Variables ,Applied Mathematics ,General Mathematics ,Parabola ,53C28, 30G35, 53C55, 14J26 ,Function (mathematics) ,Domain (mathematical analysis) ,Twistor theory ,Mathematics - Algebraic Geometry ,Differential Geometry (math.DG) ,FOS: Mathematics ,Twistor space ,Mathematics::Differential Geometry ,Klein quadric ,Complex Variables (math.CV) ,Quartic surface ,Quaternion ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
The theory of slice regular functions of a quaternion variable is applied to the study of orthogonal complex structures on domains \Omega\ of R^4. When \Omega\ is a symmetric slice domain, the twistor transform of such a function is a holomorphic curve in the Klein quadric. The case in which \Omega\ is the complement of a parabola is studied in detail and described by a rational quartic surface in the twistor space CP^3., Comment: Some explanation added in section 1, other minor amendments and reformatting; to appear in JEMS
- Published
- 2014
43. The first Chern class and conformal area for a twistor holomorphic immersion
- Author
-
Kazuyuki Hasegawa
- Subjects
Unit sphere ,Pure mathematics ,Twistor holomorphic surface ,Chern class ,General Mathematics ,Mathematical analysis ,Holomorphic function ,Conformal area ,Twistor lift ,Physics::History of Physics ,Twistor theory ,Normal bundle ,Immersion (mathematics) ,Twistor space ,Mathematics::Symplectic Geometry ,Euler class ,First Chern class ,Mathematics - Abstract
We obtain an inequality involving the first Chern class of the normal bundle and the conformal area for a twistor holomorphic surface. Using this inequality, we can improve an inequality obtained by T. Friedrich for the Euler class of the normal bundle of a twistor holomorphic surface in the four-dimensional space form. Moreover, as a corollary, we see that the area of a superminimal surface in the unit sphere is an integer multiple of {Mathematical expression}, which is essentially proved by E. Calabi. © 2014 Mathematisches Seminar der Universität Hamburg and Springer-Verlag Berlin Heidelberg., in Press
- Published
- 2014
44. ‐Algebras, the BV Formalism, and Classical Fields
- Author
-
Tommaso Macrelli, Martin Wolf, Christian Sämann, Branislav Jurčo, and Lorenzo Raspollini
- Subjects
0303 health sciences ,Batalin–Vilkovisky formalism ,030302 biochemistry & molecular biology ,General Physics and Astronomy ,Classical field theory ,Twistor theory ,High Energy Physics::Theory ,03 medical and health sciences ,Formalism (philosophy of mathematics) ,Theoretical physics ,Differential geometry ,Fiber bundle ,Twistor space ,Gauge theory ,030304 developmental biology ,Mathematics - Abstract
We summarise some of our recent works on L∞‐algebras and quasi‐groups with regard to higher principal bundles and their applications in twistor theory and gauge theory. In particular, after a lightning review of L∞‐algebras, we discuss their Maurer–Cartan theory and explain that any classical field theory admitting an action can be reformulated in this context with the help of the Batalin–Vilkovisky formalism. As examples, we explore higher Chern–Simons theory and Yang–Mills theory. We also explain how these ideas can be combined with those of twistor theory to formulate maximally superconformal gauge theories in four and six dimensions by means of L∞‐quasi‐isomorphisms, and we propose a twistor space action.
- Published
- 2019
45. Twistor Space for Rolling Bodies
- Author
-
Pawel Nurowski and Daniel An
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Circle bundle ,Statistical and Nonlinear Physics ,Symmetry group ,Twistor theory ,Differential Geometry (math.DG) ,Homogeneous space ,FOS: Mathematics ,Twistor space ,Configuration space ,Symmetry (geometry) ,Mathematical Physics ,Distribution (differential geometry) ,Mathematics - Abstract
On a natural circle bundle T(M) over a 4-dimensional manifold M equipped with a split signature metric g, whose fibers are real totally null selfdual 2-planes, we consider a tautological rank 2 distribution D obtained by lifting each totally null plane horizontally to its point in the fiber. Over the open set where g is not antiselfdual, the distribution D is (2,3,5) in T(M). We show that if M is a Cartesian product of two Riemann surfaces (S1,g1) and (S2,g2), and if g=g1--g2, then the circle bundle T(S1 x S2) is just the configuration space for the physical system of two solid bodies B1 and B2, bounded by the surfaces S1 and S2 and rolling on each other. The condition for the two bodies to roll on each other `without slipping or twisting' identifies the restricted velocity space for such a system with the tautological distribution D on T(S1 x S2). We call T(S1 x S2) the twistor space, and D the twistor distribution for the rolling bodies. Among others we address the following question: "For which pairs of bodies does the restricted velocity distribution (which we identify with the twistor distribution D) have the simple Lie group G2 as its group of symmetries?" Apart from the well known situation when the boundaries S1 and S2 of the two bodies have constant curvatures whose ratio is 1:9, we unexpectedly find three different types of surfaces, which when bounding a body that rolls `without slipping or twisting' on a plane, have D with the symmetry group G2. Although we have found the differential equations for the curvatures of S1 and S2 that gives D with G2 symmetry, we are unable to solve them in full generality so far., An extended version of a talk given on 5th of September 2012 by one of the authors at the conference `The interaction of geometry and representation theory. Exploring new frontiers". The conference was organized by A. Cap, A. L. Carey, A. R. Gover, C. R. Graham and J. Slovak and took place at the Erwin Schrodinger Institute in Vienna, Austria; http://www.mat.univie.ac.at/~cap/esiprog/Nurowski.pdf
- Published
- 2013
46. Ruled austere submanifolds of dimension four
- Author
-
Thomas A. Ivey and Marianty Ionel
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Gauss map ,Ruled submanifolds ,Euclidean space ,Holomorphic function ,Primary 53B25, Secondary 53B35, 53C38, 58A15 ,Type (model theory) ,Austere submanifolds ,Generalized helicoid ,Connection (mathematics) ,Differential Geometry (math.DG) ,Computational Theory and Mathematics ,Grassmannian ,FOS: Mathematics ,Twistor space ,Exterior differential systems ,Mathematics::Differential Geometry ,Geometry and Topology ,Analysis ,Mathematics - Abstract
We classify 4-dimensional austere submanifolds in Euclidean space ruled by 2-planes. The algebraic possibilities for second fundamental forms of an austere 4-fold M were classified by Bryant, falling into three types which we label A, B, and C. We show that if M is 2-ruled of Type A, then the ruling map from M into the Grassmannian of 2-planes in R^n is holomorphic, and we give a construction for M starting with a holomorphic curve in an appropriate twistor space. If M is 2-ruled of Type B, then M is either a generalized helicoid in R^6 or the product of two classical helicoids in R^3. If M is 2-ruled of Type C, then M is either a one of the above, or a generalized helicoid in R^7. We also construct examples of 2-ruled austere hypersurfaces in R^5 with degenerate Gauss map., Comment: 20 pages
- Published
- 2012
47. Generalized quaternionic manifolds
- Author
-
Radu Pantilie
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Applied Mathematics ,Mathematical analysis ,Space (mathematics) ,Differential Geometry (math.DG) ,Quaternionic representation ,Generalized complex structure ,FOS: Mathematics ,Twistor space ,53D18, 53C26, 53C28 ,Mathematics::Differential Geometry ,Complex manifold ,Quaternionic projective space ,Mathematics::Symplectic Geometry ,Symplectic manifold ,Vector space ,Mathematics - Abstract
We initiate the study of the generalized quaternionic manifolds by classifying the generalized quaternionic vector spaces, and by giving two classes of nonclassical examples of such manifolds. Thus, we show that any complex symplectic manifold is endowed with a natural (nonclassical) generalized quaternionic structure, and the same applies to the heaven space of any three-dimensional Einstein-Weyl space. In particular, on the product $Z$ of any complex symplectic manifold $M$ and the sphere there exists a natural generalized complex structure, with respect to which $Z$ is the twistor space of $M$., Comment: 10 pages, improved version
- Published
- 2012
48. Homogeneity for a Class of Riemannian Quotient Manifolds
- Author
-
Joseph A. Wolf
- Subjects
Mathematics - Differential Geometry ,53C20, 53C26, 53C35, 22F30 ,General Mathematics ,Fibered knot ,Rank (differential topology) ,53C20 ,01 natural sciences ,Combinatorics ,0103 physical sciences ,FOS: Mathematics ,Order (group theory) ,0101 mathematics ,Mathematics ,010102 general mathematics ,Pure Mathematics ,53C35 ,53C26 ,math.DG ,Computational Theory and Mathematics ,Differential Geometry (math.DG) ,Symmetric space ,Homogeneous space ,Isometry ,Twistor space ,010307 mathematical physics ,Geometry and Topology ,Mathematics::Differential Geometry ,Isometry group ,22F30 ,Analysis - Abstract
We study Riemannian coverings φ : M ˜ → Γ \ M ˜ where M ˜ is a normal homogeneous space G / K 1 fibered over another normal homogeneous space M = G / K and K is locally isomorphic to a nontrivial product K 1 × K 2 . The most familiar such fibrations π : M ˜ → M are the natural fibrations of Stiefel manifolds S O ( n 1 + n 2 ) / S O ( n 1 ) over Grassmann manifolds S O ( n 1 + n 2 ) / [ S O ( n 1 ) × S O ( n 2 ) ] and the twistor space bundles over quaternionic symmetric spaces (= quaternion-Kaehler symmetric spaces = Wolf spaces). The most familiar of these coverings φ : M ˜ → Γ \ M ˜ are the universal Riemannian coverings of spherical space forms. When M = G / K is reasonably well understood, in particular when G / K is a Riemannian symmetric space or when K is a connected subgroup of maximal rank in G, we show that the Homogeneity Conjecture holds for M ˜ . In other words we show that Γ \ M ˜ is homogeneous if and only if every γ ∈ Γ is an isometry of constant displacement. In order to find all the isometries of constant displacement on M ˜ we work out the full isometry group of M ˜ , extending Elie Cartan's determination of the full group of isometries of a Riemannian symmetric space. We also discuss some pseudo-Riemannian extensions of our results.
- Published
- 2016
49. Gravity in Twistor Space and its Grassmannian Formulation
- Author
-
David Skinner, Lionel Mason, and Freddy Cachazo
- Subjects
High Energy Physics - Theory ,Gravity (chemistry) ,010308 nuclear & particles physics ,Mathematical analysis ,FOS: Physical sciences ,General Relativity and Quantum Cosmology (gr-qc) ,01 natural sciences ,Methods of contour integration ,BCFW recursion ,General Relativity and Quantum Cosmology ,Twistor theory ,Scattering amplitude ,High Energy Physics - Theory (hep-th) ,Simple (abstract algebra) ,Grassmannian ,0103 physical sciences ,Twistor space ,Geometry and Topology ,010306 general physics ,Mathematical Physics ,Analysis ,Mathematics ,Mathematical physics - Abstract
We prove the formula for the complete tree-level $S$-matrix of $\mathcal{N}=8$ supergravity recently conjectured by two of the authors. The proof proceeds by showing that the new formula satisfies the same BCFW recursion relations that physical amplitudes are known to satisfy, with the same initial conditions. As part of the proof, the behavior of the new formula under large BCFW deformations is studied. An unexpected bonus of the analysis is a very straightforward proof of the enigmatic $1/z^2$ behavior of gravity. In addition, we provide a description of gravity amplitudes as a multidimensional contour integral over a Grassmannian. The Grassmannian formulation has a very simple structure; in the N$^{k-2}$MHV sector the integrand is essentially the product of that of an MHV and an $\overline{{\rm MHV}}$ amplitude, with $k+1$ and $n-k-1$ particles respectively.
- Published
- 2016
50. TWISTOR SPACES FOR HYPERKAHLER IMPLOSIONS
- Author
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Andrew Swann, Andrew Dancer, and Frances Kirwan
- Subjects
Pure mathematics ,Algebra and Number Theory ,Holomorphic function ,Twistor theory ,Complex vector bundle ,Irreducible representation ,Lie algebra ,Equivariant map ,Maximal torus ,Twistor space ,Geometry and Topology ,Mathematics::Symplectic Geometry ,Analysis ,Mathematics - Abstract
We study the geometry of the twistor space of the universal hyperkahler implosion $Q$ for $SU(n)$. Using the description of $Q$ as a hyperkahler quiver variety, we construct a holomorphic map from the twistor space $\mathcal{Z}_Q$ of $Q$ to a complex vector bundle over $\mathbb{P}^1$, and an associated map of $Q$ to the affine space $\mathcal{R}$ of the bundle’s holomorphic sections. The map from $Q$ to $\mathcal{R}$ is shown to be injective and equivariant for the action of $SU(n) \times T^{n-1} \times SU(2)$. Both maps, from $Q$ and from $\mathcal{Z}_Q$, are described in detail for $n = 2$ and $n = 3$. We explain how the maps are built from the fundamental irreducible representations of $SU(n)$ and the hypertoric variety associated to the hyperplane arrangement given by the root planes in the Lie algebra of the maximal torus. This indicates that the constructions might extend to universal hyperkahler implosions for other compact groups.
- Published
- 2016
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