Metric foliations and curvature

We prove a rigidity theorem for Riemannian fibrations of flat spaces over compact bases and give a metric classification of compact four-dimensional manifolds of nonnegative curvature that admit totally geodesic Riemannian foliations.     

Groupe de Chow de dimension zéro des fibrations en variétés de Severi–Brauer

We study here the O-dimension Chow group of some particular smooth varieties which are fibrations over curves (i.e. are equipped with a proper, surjective morphism over a smooth curve). We obtain finiteness results for fibrations, whose generic fibres are Severi-Brauer varieties of squarefree index, when the group field is a number field (or in some particular cases a finitely generated field over Q).     

A rigidity theorem for group extensions

We consider the class of discrete groups which arise as fundamental groups of iterated surface fibrations; that is, of complexes obtained from a sequence of fibrations in which all bases and the initial fibre are hyperbolic surfaces. Group theoretically, this corresponds to studying the class of iterated extensions of hyperbolic surface groups. In [4], for the case of a single extension we conjectured and partially established that no group can arise from more than a finite number of such extensions. Here we show that the result holds in complete generality. As remarked in [4], the result has a strong affinity with the rigidity theorems of Parshin [7] and Arakelov [1] for fibred (complex) algebraic surfaces.     

On almost holomorphic Lagrangian fibrations

We prove that the pull back of an ample line bundle by an almost holomorphic Lagrangian fibration is nef. As an application, we show birational semi rigidity of Lagrangian fibrations.     

Relatively Stable Bundles over Elliptic Fibrations

We consider a relative Fourier‐Mukai transform de.ned on elliptic fibrations over an arbitrary base scheme. This is used to construct relative Atiyah sheaves and generalize Atiyah and Tu's results about semistable sheaves over elliptic curves to the case of elliptic fibrations. Moreover we show that this transform preserves relative (semi)stability of sheaves of positive relative degree.     

Classification of Lagrangian fibrations over a Klein bottle

This paper completes the classification of regular Lagrangian fibrations over compact surfaces. Mishachev (Diff Geom Appl 6:301–320, 1996) classifies regular Lagrangian fibrations over \mathbbT²{\mathbb{T}^2}. The main theorem in Fried et al. (Comment Math Helv 56(4):487–523, 1981) is used to in order to classify integral affine structures on the Klein bottle K ² and, hence, regular Lagrangian fibrations over this space.     

Fibrations spinorielles et twisteurs généralisés

In the first part my purpose is to find existence conditions for a spin structure on a manifold in a purely differential geometric scope, without the machinery of the algebraic topology and avoiding systematically any matricial formalism. A spin structure can be considered as aG-structure (using a classical terminology). Groups, called by myself “groups of spinoriality” play an essential part in this problem. We study them with a few details in the second part of chapter I. The second part is devoted to introduce the notion of twistors, generalizing studies already made in particular cases. Twistorial fibrations naturally go with spin fibrations. Broadly, twistors of order 2 are for the conformal group of signature (p, q), that are spinors for the group of isometry of signature (p+1,q+1).     

Elliptic fibrations on cubic surfaces

We classify elliptic fibrations birational to a nonsingular, minimal cubic surface over a field of characteristic zero. Our proof is adapted to provide computational techniques for the analysis of such fibrations, and we describe an implementation of this analysis in computer algebra.     

On Lifting Fibrations of Genus One

We investigate the problem of lifting fibrations of genus one on algebraic surfaces of Kodaira dimension zero. We prove that fibrations on the following surfaces lift: Enriques surfaces, K3 surfaces covering Enriques surfaces, certain hyperelliptic, and quasi-hyperelliptic surfaces.     

On Nonrational Fibers of del Pezzo Fibrations over Curves

We consider threefold del Pezzo fibrations over a curve germ whose central fiber is nonrational. Under the additional assumption that the singularities of the total space are at worst ordinary double points, we apply a suitable base change and show that there is a one-to-one correspondence between such fibrations and certain nonsingular del Pezzo fibrations equipped with a cyclic group action.

     

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.     

Filtered model of a fibration and rational obstruction theory

For a Serre fibration a filtered model is defined and obstructions to the homotopy equivalence of two fibrations (maps), to the section problem and to the extension problem in rational homotopy theory are given.     

A characterisation of elementary fibrations

In the categorical approach to logic proposed by Lawvere, which systematically uses adjoints to describe the logical operations, equality is presented in the form of a left adjoint to reindexing along diagonal arrows in the base. Taking advantage of the modular perspective provided by category theory, one can look at those Grothendieck fibrations which sustain just the structure of equality, the so-called elementary fibrations, aka fibrations with equality.The present paper provides a characterisation of elementary fibrations which is a substantial generalisation of the one already available for faithful fibrations. The characterisation is based on a particular structure in the fibres which may be understood as proof-relevant equality predicates equipped with a principle of indiscernibility of identicals à la Leibniz. We exemplify this structure for several classes of fibrations, in particular, for fibrations used in the semantics of the identity type of Martin-Löf type theory. We close the paper discussing some fibrations related to Hofmann and Streicher's groupoid model of the identity type and showing that one of them is elementary.     

Fibrations on four-folds with trivial canonical bundles

Four-folds with trivial canonical bundles are divided into six classes according to their holonomy group. We consider examples that are fibred by abelian surfaces over the projective plane. We construct such fibrations in five of the six classes, and prove that there is no such fibration in the sixth class. We classify all such fibrations whose generic fibre is the Jacobian of a genus two curve.     

Variétés de Waldhausen et fibrations sur le cercle

Let K be a finite union of Seifert fibres in a Waldhausen manifold. We study the fibrations over the circle of Mwhich equip M with cm open book structure with binding K. In general, these fibrations are transversal to the Waldhausen decomposition, and their monodromies are, up to isotopy, quasi-periodic diffeomorphisms of surfaces. We describe the relations between these monodromies and the topology of (M. K). In particuliar, we prove that there exists, up to conjugation, a finite number (possibly equal to zero) of such monodromies with all their Dehn twists being negative.     

Toric sheaves,stability and fibrations

For an equivariant reflexive sheaf over a polarised toric variety, we study slope stability of its reflexive pullback along a toric fibration. Examples of such fibrations include equivariant blow-ups and toric locally trivial fibrations. We show that stability (resp. unstability) is preserved under such pullbacks for so-called adiabatic polarisations. In the strictly semistable situation, under locally freeness assumptions, we provide a necessary and sufficient condition on the graded object to ensure stability of the pulled back sheaf. As applications, we provide various stable perturbations of semistable tangent sheaves, either by changing the polarisation, or by blowing-up a subvariety. Finally, our results apply uniformly in specific flat families and induce injective maps between the associated moduli spaces.     

Quartic del Pezzo surfaces over function fields of curves

We classify quartic del Pezzo surface fibrations over the projective line via numerical invariants, giving explicit examples for small values of the invariants. For generic such fibrations, we describe explicitly the geometry of spaces of sections to the fibration, and mappings to the intermediate Jacobian of the total space. We exhibit examples where these are birational, which has applications to arithmetic questions, especially over finite fields.     

Über einige formale Eigenschaften von Faserungen und h-Faserungen

The aim of this paper is to study some formal properties of fibrations (=continuous maps which have the covering homotopy property), cofibrations (=maps which have the homotopy extension property), h-fibrations ([2],6.4) and h-cofibrations([2],2.2). We introduce the notion of a generalized homotopy system in a category (2). This notion will be selfdual. To show that fibrations and cofibrations, h-fibrations and h-cofibrations are dual notions, no use will be made of adjoint functors. Our approach admits the transition from a categoryC to other categories, e.g. to the categoryC _B of objects over a given object B ofC.     

Strongly rigid and I-rigid rings

In this paper we consider questions connected with the problem of rigidity of rings for the class of finite-rank torsion-free rings (A ring is called rigid, if it has only trivial endomorphisms). We study strongly rigid and I-rigid rings [3] (A ring R is called strongly rigid, if any ring quasi-isomorphic to R is rigid). The main results of this paper are the characterization of strongly rigid rings and the establishment of relations between strongly rigid and I-rigid rings. It is shown that the verification of strong rigidity of the ring reduces to testing the rigidity of some field of algebraic numbers over its subfield. This gives rise to examples of non-trivial strongly rigid rings.     

The Geography Problem of 4-Manifolds with Various Structures

We review results about the geography problem of complex, symplectic and Einstein 4-manifolds. Finally we discuss the same problem for Lefschetz fibrations. We show that in the geography of symplectic 4-manifolds and Lefschetz fibrations the slopes = c₁ ²/c₂ do not admit gaps — following similar results of Sommese in the complex case.