Basic intersection cohomology of conical fibrations

We introduce notions of singular fibration and singular Seifert fibration. These notions naturally generalize that o locally trivial fibration to the category of stratified pseudomanifolds. For singular foliations determined by such fibrations, we prove the de Rham theorem for basic intersection cohomology recently introduced by the present authors. One of the main examples of such a structure is the natural projection to the space of fibers of a singular Riemannian foliation determined by a Lie group action on a compact smooth manifold.     

Fuzzy Fractional Monodromy and the Section–Hyperboloid

A theorem about the matrix of fractional monodromy will be formulated. The monodromy corresponds to going around a fiber with a singular point of oscillator type with arbitrary resonance. The reason of fractional monodromy and fuzziness of such a monodromy is explained. Some ideas for the proof of the theorem are given. A few remarks about the semi-global structure of singular lagrangian fibration are made. Lecture held in the Seminario Matematico e Fisico di Milano on November 8, 2004 Received: June 2007     

A Geometric Construction of the Hyperbolic Fibrations Associated with a Flock, q Even

Baker et al. [2] show, via algebraic methods, that a regular hyperbolic fibration of PG(3, q) with constant back half gives rise to a flock of a quadratic cone in PG(3, q), and conversely. In this paper a geometric construction for q even of the flock from the hyperbolic fibration, and conversely, will be described. A proof will be given that this geometric construction indeed corresponds to the known algebraic one. Deirdre Luyckx - The author is Postdoctoral Fellow of the Fund for Scientific Research—Flanders (Belgium) (F.W.O.—Vlaanderen).     

Iitaka's fibrations via multiplier ideals

We give a new characterization of Iitaka's fibration of algebraic varieties associated to line bundles. Introducing an ``intersection number' of line bundles and curves by using the notion of multiplier ideal sheaves, Iitaka's fibration can be regarded as a ``numerically trivial fibration' in terms of this intersection theory.

     

An example of special lagrangian fibration

On the total space of the line bundle π:π1*T*P1(?)π2*T*P1→P1×P1,a complete Ricci-flat Kaehler metric and a smooth special Lagrangian fibration are given. This special Lagrangian fibration is smoothly built up of 4 Harvey-Lawson's models in 4 directions.     

Lightlike Hypersurfaces on Manifolds Endowed with a Conformal Structure of Lorentzian Signature

The authors study the geometry of lightlike hypersurfaces on manifolds (M, c) endowed with a pseudoconformal structure c = CO(n – 1, 1) of Lorentzian signature. Such hypersurfaces are of interest in general relativity since they can be models of different types of physical horizons. On a lightlike hypersurface, the authors consider the fibration of isotropic geodesics and investigate their singular points and singular submanifolds. They construct a conformally invariant normalization of a lightlike hypersurface intrinsically connected with its geometry and investigate affine connections induced by this normalization. The authors also consider special classes of lightlike hypersurfaces. In particular, they investigate lightlike hypersurfaces for which the elements of the constructed normalization are integrable.     

Schreier theory for singular extensions of categorical groups and homotopy classification∗ ∗

If G is a categorical group, a G-module is defined to be a braided categorical group (A c) together with an action of G on (A,c). In this work we define the notions of singular extension of G by the G-module (A,c) and of 1-cocycle of G with coefficients in (A,c) and we obtain, first, a bijection between the set of equivalence classes of singular extensions of G by (Ac) and the set of equivalence classes of 1-cocycles. Next, we associate to any G-module (Ac) a Kan fibration of simplicial sets ?: Ner(GAc)) → Ner(G)and then we show that there is a bijection between the set of equivalence classes of singular extensions of G by (A,c) and Γ[Ner(G,A,c/Ner(G)]the set of fibre homotopy classes of cross-sections of the fibration ?.     

Linear systems on rational elliptic surfaces and elliptic fibrations on K3 surfaces

We consider K3 surfaces which are double covers of rational elliptic surfaces. The former are endowed with a natural elliptic fibration, which is induced by the latter. There are also other elliptic fibrations on such K3 surfaces, which are necessarily induced by special linear systems on the rational elliptic surfaces. We describe these linear systems. In particular, we observe that every conic bundle on the rational surface induces a genus 1 fibration on the K3 surface and we classify the singular fibers of the genus 1 fibration on the K3 surface it terms of singular fibers and special curves on the conic bundle on the rational surface.     

On the birational rigidity of a series of multidimensional fano varieties

We prove the birational rigidity of algebraic varieties admitting a fibration into a pencil of hypersurfaces of degreeM in ℙ ^M having degree 2 over the base. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 455–460, September, 2000.     

Nielsen numbers of n-valued fiber maps

The Nielsen number for n-valued multimaps, defined by Schirmer, has been calculated only for the circle. A concept of n-valued fiber map on the total space of a fibration is introduced. A formula for the Nielsen numbers of n-valued fiber maps of fibrations over the circle reduces the calculation to the computation of Nielsen numbers of single-valued maps. If the fibration is orientable, the product formula for single-valued fiber maps fails to generalize, but a “semi-product formula" is obtained. In this way, the class of n-valued multimaps for which the Nielsen number can be computed is substantially enlarged. Dedicated, with gratitude, to Felix Browder who, long ago, encouraged and supported a young topologist’s interest in fixed point theory     

Convergence Conditions for Interpolation Fractions at the Nodes Distinct from the Singular Points of a Function

We consider an interpolation process for the class of functions with finitely many singular points by means of the rational functions whose poles coincide with the singular points of the function under interpolation. The interpolation nodes constitute a triangular matrix and are distinct from the singular points of the function. We find a necessary and sufficient condition for uniform convergence of sequences of interpolation fractions to the function under interpolation on every compact set disjoint from the singular points of the function and other conditions for convergence.Original Russian Text Copyright © 2005 Lipchinskii A. G.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 822–833, July–August, 2005.     

Decompositions into codimension two spheres and approximate fibrations

Let M denote an orientable (n+2)-manifold (n?1) and G an upper semicontinuous decomposition of M into compacta having the shape of the n-sphere. In this context it is shown that the decomposition space is a 2-manifold. Moreover, it is established that the decomposition map is an approximate fibration for n>1, while for n=1 the map is an approximate fibration over the complement of a locally finite set.     

Taut foliations in punctured surface bundles, I

Given a fibred, compact, orientable 3-manifold with single boundarycomponent, we show that a fibration with fiber surface of negativeEuler characteristic can be perturbed to yield taut foliationsrealizing an open interval of boundary slopes about the boundaryslope of the fibration. These taut foliations extend to tautfoliations in the corresponding surgery manifolds. 2000 MathematicsSubject Classification: primary 57M25; secondary 57R30.     

An Example of a Fiber in Fibrations Whose Serre Spectral Sequences Collapse

We give an example of a space X with the property that every orientable fibration with the fiber X is rationally totally non-cohomologous to zero, while there exists a nontrivial derivation of the rational cohomology of X of negative degree.     

M-纤维式纤维化的特征与诱导M-纤维式纤维化

本文研究了M-纤维式纤维化的问题,利用M-纤维式升腾函数获得了M-纤维式纤维化的特征,以及证明了M-纤维式上纤维化诱导的映射空间之间的M-纤维式映射在一定条件下也是M-纤维式纤维化.     

On bifurcation braid monodromy of elliptic fibrations

We propose to study a new kind of monodromy homomorphism for families of regular elliptic fibrations of a given differentiable fibration type to get a hold on topological properties of moduli stacks of elliptic surfaces.In specific cases, including the most significant one, when all singular fibres are nodal irreducible rational curves, we compute the corresponding monodromy group, a subgroup of the mapping class group of the fibration base punctured at the singular values of the fibration.We study a tentative algebraic characterisation and give implications for the group of diffeomorphisms compatible with the fibration.     

Multifibrations. A Class of Shape Fibrations with the Path Lifting Property

In this paper we introduce a class of maps possessing a multivalued homotopy lifting property with respect to every topological space. We call these maps multifibrations and they represent a formally stronger concept than that of shape fibration. Multifibrations have the interesting property of being characterized in a completely intrinsic way by a path lifting property involving only the total and the base space of the fibration. We also show that multifibrations (and also, with some restrictions, shape fibrations) have a lifting property for homotopies of fine multivalued maps. This implies, when the spaces considered are metric compacta, that the possibility of lifting a fine multivalued map is a property of the corresponding strong shape morphism and not of the particular map considered.