Fibration Structure Research Articles

Density of rational points on elliptic K3 surfaces

Let $X$ be a K3 surface defined over a number field $K$. Assume that $X$ admits a structure of an elliptic fibration or an infinite group of automorphisms. Then there exists a finite extension $K'/K$ such that the set of $K'$-rational points is Zariski dense in $X$.

Certain examples of birationally rigid varieties pencil of double quadrics

We prove that double covers ofP1×Q, where\(Q \subset P^{k + 1}\) is a nondegenerate quadric, and the branch divisor cut out by a hypersurface of the type (2λ, 2k−2) are birationally superrigid. In particular, they have only one structure of a Fano fibration. Therefore, they are nonrational.

Vers une neurogéométrie. Fibrations corticales, structures de contact et contours subjectifs modaux

This work presents some variational models for the cortical algorithms processing Kanizsa modal subjective contours . These models are based on the geometric concepts of fibration and contact structure. The retinoptic structure of the orientation hypercolumns in the visual area V1 is a functionnal architecture which can be mathematically idealized by the fibration having the retinian plane M as base and the projective line P1 as fiber F. The total space E of Pi p is isomorphic to the direct product M x F. The cortico-cortical horizontal connections implement what is called the local triviality of this fibration, and also a Cartan connection defining a parallel transport between neighboring fibers. Then the paper focuses on the geometrical interpretation of the results of Field, Hayes and Hess concerning the association field. It shows that the latter implements what is called the contact structure of the fibration. The association field expresses an integrability condition for the skew curves in E : they have to be a lifting of their projection on the retinian plane M. This model of fibration endowed with a contact structure is then applied to the modal subjective contours and provides a variant of the elastica model developped by B.K.P. Horn and D. Mumford. The key idea is that the lifting of subjective contours satisfy a "geodesic" condition in the cortical fibration E : they have to be of minimal lenght (for an appropriate metrics) among the class of curves satisfying the integrability condition. These "geodesic" models are then reformulated, according to R. Bryant and P. Griffiths, in the more fondamental geometric framework of Lie groups and Cartan's "repère mobile" (Vielbein). Finally, some experimental possibilities are suggested.

K3 fibrations and softly broken N = 4 supersymmetric gauge theories

Global geometry of K3 fibration Calabi-Yau threefolds, with Hodge number h 2,1 = r + 1, is used to define N = 4 softly broken SU( r + 1) gauge theories, with the bare coupling constant given by the dual heterotic dilaton, and the mass of the adjoint hypermultiplet given by the heterotic string tension. The U( r + 1) Donagi-Witten integrable model is also derived from the K3 fibration structure, with the extra U(1) associated to the heterotic dilaton. The case of the SU(2) gauge group is analyzed in detail. String physics beyond the heterotic point particle limit is partially described by the N = 4 softly broken theory.

The Hurwitz transformation and generalized Cayley–Klein parameterization

The fiber bundles and global rotations arising in the Kustaanheimo–Stiefel and Hurwitz transformations are studied. Generalized Cayley–Klein parameterization is introduced in a functional form and is used for the analysis of fibration in Hurwitz transformations. Essential differences in fibration structure between these transformations similar in Cartesian description are established in the angular map.

On a generalization of the hopf fibration, III (Subvarieties in theC-spaces)

Analytic subvarieties inC-spaces are discussed. First, a certain kind of closed 2-formdω is constructed. Then, the subvarieties are studied by means of this 2-form.dω may be considered as the curvature form of a connection whenC-spaces are considered as the toral bundle space over an algebraic variety. This 2-form is horizontal in its nature with respect to the bundle structure and indicates, in general, how different the bundle is from the trivial bundle. Because of this twist in the bundle space, subvarieties in theC-spaces tend to inherit the same structure. In this paper, the inherited fibration structure is studied. The most concrete results are obtained when the fiber torus has complex dimension 2.

Some reasons why algebraic topology is important in neuropsychology: perceptual and cognitive systems as fibrations

The topological structure of fibration (a total space coupled with a base space by a projection mapping) appears to be found throughout the CNS. Neuropsychological structure and functioning are analysed in terms of such fibrations, and application made to perceptual and cognitive systems. The key feature of “mechanistic” cognition appears to be a CNS embodiment of the category of simplicial sets. Specific structures for problem-solving and memory are determined in terms of this category.

The twistor programme

The formalism of twistors provides a new approach to the description of basic physics. The points of Minkowski space-time are represented by 2-dimensional linear subspaces of a complex 4-dimensional vector space (flat twistor space) on which a Hermitian form of signature ++-- is defined. Free massless fields can be represented in terms of the sheaf cohomology of portions of this space. Twistor space (or a suitable part of it) can be expressed in two different ways as a complex fibration. If one or the other fibration structure is deformed, the resulting space represents not empty Minkowski space but, in one case, the general “right-flat” solution of Einstein's vacuum equations and, in the other, the general (left-handed) solution of Maxwell's equations. These provide the most primitive types of interaction (gravitational or electromagnetic) which may generalize to other fields in a comprehensive twistor scheme for the description of elementary particles.

On the structure of fibrations

On the structure of fibrations

F-theory with quantized fluxes

We present evidence that the CHL string in eight dimensions is dual to F-theory compactified on elliptic K3 with a 0(2) monodromy group. The monodromy group 0(2) allows one to turn on the flux of an antisymmetric two form along the base. The Bµ� flux is quantized and therefore the moduli space of the CHL string is disconnected from the moduli space of F-theory/Heterotic strings (as expected). The non-zero Bµ� flux obstructs certain deformations restricting the moduli of elliptic K3 to a 10 dimensional moduli space. We also discuss how one can reconstruct the gauge groups from the elliptic fibration structure.

Boundedness of threefolds of Fano type with Mori fibration structures

We show boundedness of $3$-folds of $\epsilon$-Fano type with Mori fibration structures. The proof is based on the birational boundedness result in our previous work arXiv:1509.08722 combining with arguments in Kawamata \cite{K} and Koll\'ar--Miyaoka--Mori--Takagi \cite{KMMT}.

Affine Kac-Moody algebras, CHL strings and the classification of tops

Candelas and Font introduced the notion of a `top' as half of a three dimensional reflexive polytope and noticed that Dynkin diagrams of enhanced gauge groups in string theory can be read off from them. We classify all tops satisfying a generalized definition as a lattice polytope with one facet containing the origin and the other facets at distance one from the origin. These objects torically encode the local geometry of a degeneration of an elliptic fibration. We give a prescription for assigning an affine, possibly twisted Kac-Moody algebra to any such top (and more generally to any elliptic fibration structure) in a precise way that involves the lengths of simple roots and the coefficients of null roots. Tops related to twisted Kac-Moody algebras can be used to construct string compactifications with reduced rank of the gauge group.