scholarly journals Complex Multiplication in Twistor Spaces

2020 ◽  
Author(s):  
D Huybrechts
Keyword(s):  
Twistor Space ◽  
Complex Multiplication ◽  
K3 Surface ◽  
Twistor Spaces ◽  
Arithmetic Properties ◽  
Twistor Construction

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 polarised K3 surface with complex multiplication, all algebraic fibres of its twistor space away from the equator have complex multiplication as well.

scholarly journals Geometry of some twistor spaces of algebraic dimension one

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Nobuhiro Honda
Keyword(s):  
Twistor Space ◽  
Projective Planes ◽  
Canonical System ◽  
K3 Surface ◽  
Twistor Spaces ◽  
Algebraic Dimension ◽  
General Fiber ◽  
Algebraic Reduction ◽  
Dimension One ◽  
Complex Projective

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

scholarly journals Generalized almost even-Clifford manifolds and their twistor spaces

2021 ◽  
Vol 8 (1) ◽  
pp. 96-124
Author(s):  
Luis Fernando Hernández-Moguel ◽  
Rafael Herrera
Keyword(s):  
Twistor Space ◽  
Complex Structure ◽  
Twistor Spaces ◽  
Recent Interest ◽  
Generalized Complex Structure ◽  
Generalized Complex ◽  
Space Construction ◽  
Clifford Structures

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.

scholarly journals Integrable Complex Structures on Twistor Spaces

2019 ◽  
Vol 70 (3) ◽  
pp. 937-963
Author(s):  
Steven Gindi
Keyword(s):  
Twistor Space ◽  
Complex Structures ◽  
Complex Manifolds ◽  
Twistor Spaces

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.

scholarly journals Energy of sections of the Deligne–Hitchin twistor space

2020 ◽  
Author(s):  
Florian Beck ◽  
Sebastian Heller ◽  
Markus Röser
Keyword(s):  
Moduli Space ◽  
Compact Riemann Surface ◽  
Twistor Space ◽  
Harmonic Maps ◽  
Energy Functional ◽  
Conformal Map ◽  
Twistor Spaces ◽  
Holomorphic Sections ◽  
Meromorphic Connection ◽  
Willmore Energy

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 show that the energy is the residue of the pull-back along the section of a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by Hitchin. As a byproduct, we show the existence of a hyper-Kähler potentials for new components of real holomorphic sections of twistor spaces of hyper-Kähler manifolds with rotating $$S^1$$ S 1 -action. Additionally, we prove that for a certain class of real holomorphic sections of the Deligne–Hitchin moduli space, the energy 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.

scholarly journals On the Supersingular Reduction of K3 Surfaces with Complex Multiplication

2018 ◽  
Vol 2020 (20) ◽  
pp. 7306-7346
Author(s):  
Kazuhiro Ito
Keyword(s):  
Comparison Theorem ◽  
Complex Multiplication ◽  
K3 Surfaces ◽  
Explicit Description ◽  
Brauer Group ◽  
K3 Surface ◽  
Good Reduction ◽  
Picard Number ◽  
Reduction Modulo ◽  
Modulo P

Abstract We study the good reduction modulo $p$ of $K3$ surfaces with complex multiplication. If a $K3$ surface with complex multiplication has good reduction, we calculate the Picard number and the height of the formal Brauer group of the reduction. Moreover, if the reduction is supersingular, we calculate its Artin invariant under some assumptions. Our results generalize some results of Shimada for $K3$ surfaces with Picard number $20$. Our methods rely on the main theorem of complex multiplication for $K3$ surfaces by Rizov, an explicit description of the Breuil–Kisin modules associated with Lubin–Tate characters due to Andreatta, Goren, Howard, and Madapusi Pera, and the integral comparison theorem recently established by Bhatt, Morrow, and Scholze.

K3 surfaces with algebraic period ratios have complex multiplication

2015 ◽  
Vol 11 (05) ◽  
pp. 1709-1724
Author(s):  
Paula Tretkoff
Keyword(s):  
Number Theory ◽  
Vector Space ◽  
Hodge Structure ◽  
Complex Multiplication ◽  
K3 Surfaces ◽  
K3 Surface ◽  
Detailed Proof ◽  
Tate Group ◽  
Calabi Yau Manifolds

Let Ω be a non-zero holomorphic 2-form on a K3 surface S. Suppose that S is projective algebraic and is defined over [Formula: see text]. Let [Formula: see text] be the [Formula: see text]-vector space generated by the numbers given by all the periods ∫γ Ω, γ ∈ H2(S, ℤ). We show that, if [Formula: see text], then S has complex multiplication, meaning that the Mumford–Tate group of the rational Hodge structure on H2(S, ℚ) is abelian. This result was announced in [P. Tretkoff, Transcendence and CM on Borcea–Voisin towers of Calabi–Yau manifolds, J. Number Theory 152 (2015) 118–155], without a detailed proof. The converse is already well known.

Twistors and fields with sources on worldlines

1985 ◽  
Vol 397 (1812) ◽  
pp. 143-155 ◽  
Keyword(s):  
Minkowski Space ◽  
Twistor Space ◽  
Rest Mass ◽  
Double Cover ◽  
Ruled Surface ◽  
Hausdorff Spaces ◽  
Twistor Spaces ◽  
Non Hausdorff Spaces ◽  
Massless Fields

It is shown that zero-rest-mass fields with sources on an analytic worldline are naturally defined on a double cover of some region of Minkowski space. Twistor spaces are constructed that correspond to such regions and these turn out to be non-Hausdorff spaces, obtained by identifying two copies of regions in ordinary twistor space, except on a ruled surface that corresponds to the worldline. It is shown that cohomology classes on the twistor space corresponds to sourced fields on Minkowski space, thus extending the twistor descrip­tion of massless fields.

scholarly journals Twistor spaces and the adiabatic limits of Dirac operators

2001 ◽  
Vol 164 ◽  
pp. 53-73 ◽  
Author(s):  
Masayoshi Nagase
Keyword(s):  
Dirac Operator ◽  
Spin Structure ◽  
Twistor Space ◽  
Dirac Operators ◽  
Twistor Spaces ◽  
Adiabatic Limits ◽  
Twisted Dirac Operator

We show that a (Spinq-style) twistor space admits a canonical Spin structure. The adiabatic limits of η-invariants of the associated Dirac operator and of an intrinsically twisted Dirac operator are then investigated.

ON NON(ANTI)COMMUTATIVE SUPER TWISTOR SPACES

2010 ◽  
Vol 07 (04) ◽  
pp. 655-668
Author(s):  
TADASHI TANIGUCHI ◽  
NAOYA MIYAZAKI
Keyword(s):  
Clifford Algebra ◽  
Twistor Space ◽  
Twistor Spaces ◽  
Associative Product

The main purpose of this article is a proposal of non(anti)commutative super twistor space by making the odd coordinates θ not anticommuting, but satisfying Clifford algebra relations. Despite the deformation, we can introduce a deformed associative product which is globally defined on P3|N.

Some arithmetic properties of the Legendre polynomials

1957 ◽  
Vol 53 (2) ◽  
pp. 265-268 ◽  
Author(s):  
L. Carlitz
Keyword(s):  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Complex Multiplication ◽  
Elliptic Functions ◽  
Arithmetic Properties ◽  
Special Values ◽  
Right Hand ◽  
Image Position ◽  
The Right ◽  
Divisibility Properties

1. Good (4) has proved the formulawhere Pn(x) is the Legendre polynomial of degree n and t is any integer greater than n. The form of the right-hand side suggests that (1) may be of use in deriving arithmetic properties of Pn(x).Elsewhere (1) the writer indicated a connexion between divisibility properties of Pm(a) for special values of a and the complex multiplication of elliptic functions. If p = 2m + 1 is an odd prime, put

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