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.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 235–257.Original Russian Text Copyright © 2005 by M. Saralegi-Aranguren, R. Wolak.This revised version was published online in April 2005 with a corrected issue number.     

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.     

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 ?.     

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.     

On base manifolds of Lagrangian fibrations

We consider base spaces of Lagrangian fibrations from singular symplectic varieties.After defining cohomologically irreducible symplectic varieties,we construct an example of Lagrangian fibration whose base space is isomorphic to a quotient of the projective space.We also prove that the base space of Lagrangian fibration from a cohomologically symplectic variety is isomorphic to the projective space provided that the base space is smooth.     

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     

Singular Fibers in Elliptic Fibrations on the Rational Elliptic Surface

We determine the combinations of singular fibers a locally holomorphic elliptic fibration on the rational elliptic surface can admit. This problem has been answered for globally holomorphic elliptic fibrations by Persson and Miranda [12], [9]; we compare our methods and results to theirs. In particular, we find combinations of singular fibers which can be realized by locally holomorphic fibrations but not by globally holomorphic ones.     

Fibrations of Simplicial Sets

There are infinitely many variants of the notion of Kan fibration that, together with suitable choices of cofibrations and the usual notion of weak equivalence of simplicial sets, satisfy Quillen’s axioms for a homotopy model category. The combinatorics underlying these fibrations is purely finitary and seems interesting both for its own sake and for its interaction with homotopy types. To show that these notions of fibration are indeed distinct, one needs to understand how iterates of Kan’s Ex functor act on graphs and on nerves of small categories.     

Two remarks on sextic double solids

We study the potential density of rational points on double solids ramified along singular reduced sextic surfaces. Also, we investigate elliptic fibration structures on nonsingular sextic double solids defined over a perfect field of characteristic 5.     

On families of curves over 1 with small number of singular fibres

Let f : X → C be a surface together with a fibration over a curve. We recall the Ogg-Shafarevich-Grothendieck formula, study the number of singular fibres, and give examples where this number is minimal.     

Fibration of classifying spaces in the cobordism theory of singular maps

In the cobordism theory of singular smooth maps there exist classifying spaces (analogues of Thom spectra) depending on the set of allowed singularity types. The so-called “key fibration” introduced by A. Szűcs connects these classifying spaces for different sets of allowed singularities. Here we prove the existence of such a fibration using a new, more simple and general argument than that of Szűcs. This makes it possible to extend the range of applications to some negative codimension maps.     

Multiple fibers of del Pezzo fibrations

We prove that a terminal three-dimensional del Pezzo fibration has no fibers of multiplicity >6. We also obtain a rough classification of possible configurations of singular points on multiple fibers and give some examples. To the memory of Professor Vasily Alekseevich Iskovskikh     

Poisson cohomology of broken Lefschetz fibrations

We compute the formal Poisson cohomology of a broken Lefschetz fibration by calculating it at fold and Lefschetz singularities. Near a fold singularity the computation reduces to that for a point singularity in 3 dimensions. For the Poisson cohomology around singular points we adapt techniques developed for the Sklyanin algebra. As a side result, we give compact formulas for the Poisson coboundary operator of an arbitrary Jacobian Poisson structure in 4 dimensions.     

Elliptic K3 surfaces as dynamical models and their Hamiltonian monodromy

This note deals with Lagrangian fibrations of elliptic K3 surfaces and the associated Hamiltonian monodromy. The fibration is constructed through the Weierstraß normal form of elliptic surfaces. There is given an example of K3 dynamical models with the identity monodromy matrix around 12 elementary singular loci.     

A Characterization of Counterexamples to the Kodaira-Ramanujam Vanishing Theorem on Surfaces in Positive Characteristic

The author gives a characterization of counterexamples to the Kodaira-Ramanujam vanishing theorem on smooth projective surfaces in positive characteristic. More precisely, it is reproved that if there is a counterexample to the Kodaira-Ramanujam vanishing theorem on a smooth projective surface X in positive characteristic, then X is either a quasi-elliptic surface of Kodaira dimension 1 or a surface of general type. Furthermore, it is proved that up to blow-ups, X admits a fibration to a smooth projective curve, such that each fiber is a singular curve.     

Real analytic elliptic surfaces

Here, we give an upper bound for the number of connected components of the real locus of several smooth complex compact elliptic surfaces defined over R in terms of the type of the singular fibers of their elliptic fibration.     

On a Relative Fourier–Mukai Transform on Genus One Fibrations

We study relative Fourier–Mukai transforms on genus one fibrations with section, allowing explicitly the total space of the fibration to be singular and non-projective. Grothendieck duality is used to prove a skew–commutativity relation between this equivalence of categories and certain duality functors. We use our results to explicitly construct examples of semi-stable sheaves on degenerating families of elliptic curves.     

Complex projective towers and their cohomological rigidity up to dimension six

A complex projective tower, or simply a ?P-tower, is an iterated complex projective fibration starting from a point. In this paper we classify all six-dimensional ?P-towers up to diffeomorphism, and as a consequence we show that all such manifolds are cohomologically rigid, i.e., they are completely determined up to diffeomorphism by their cohomology rings.