Orbifold fibrations of Eschenburg spaces

Most of the few known examples of compact Riemannian manifolds with positive sectional curvature are the total space of a Riemannian submersion. In this article we show that this is true for all known examples, if we enlarge the category to orbifold fibrations. For this purpose we study all almost free isometric circle actions on positively curved Eschenburg spaces, which give rise to principle orbifold bundle structures, and we examine in detail their geometric properties. In particular, we obtain a new family of 6-dimensional orbifolds with positive sectional curvature whose singular locus consists of just two points.      

Invariant fibrations for some birational maps of ℂ2

Abstract

In this article, we extract and study the zero entropy subfamilies of a certain family of birational maps of the plane. We find these zero entropy mappings and give the invariant fibrations associated to them.     

Sur l'' application de réciprocité pour une surface fibrée en coniques définie sur un corps local

For a proper smooth variety X defined over a local field k, unramified class field theory investigates the reciprocity map _X: SK₁(X) ^ab ₁(X) as introduced by S. Saito. We study this map in the case when X is a surface admitting a proper surjection onto a smooth geometrically connected curve C with a smooth conic as generic fibre. Without any assumption on the reduction of C, we prove that _X is injective modulo n for all n invertible in k and its cokernel is the same as that of _C.     

Minimal number of singular fibers in a Lefschetz fibration

There exists a (relatively minimal) genus Lefschetz fibration with only one singular fiber over a closed (Riemann) surface of genus iff and . The singular fiber can be chosen to be reducible or irreducible. Other results are that every Dehn twist on a closed surface of genus at least three is a product of two commutators and no Dehn twist on any closed surface is equal to a single commutator.

     

Calabi-Yau 4-folds and toric fibrations

We present a general scheme for identifying fibrations in the framework of toric geometry and provide a large list of weights for Calabi-Yau 4-folds. We find 914 164 weights with degree d ≤ 150 whose maximal Newton polyhedra are reflexive and 525 572 weights with degree d ≤ 4000 that give rise to weighted projective spaces such that the polynomial defining a hypersurface of trivial canonical class is transversal. We compute all Hodge numbers, using Batyrev's formulas (derived by toric methods) for the first and Vafa's formulas (obtained by counting of Ramond ground states in N = 2 LG models) for the latter class, checking their consistency for the 109 308 weights in the overlap. Fibrations of k-folds, including the elliptic case, manifest themselves in the N lattice in the following simple way: The polyhedron corresponding to the fiber is a subpolyhedron of that corresponding to the k-fold, whereas the fan determining the base is a linear projection of the fan corresponding to the k-fold.     

Simplicial Homotopical Algebra and Satellites

We study here a notion of simplicial satellites, as a first step towards a characterisation of simplicial derived functors, a problem unsolved since the latter were introduced.The problem comes from the fact that, in contrast with the abelian case, simplicial derived functors do not produce by themselves an exact sequence. Our solution consists in extending them to commutative k-cubes, for all k, forming thus an exact system of functors universal within the connected ones; or, in other words, a system of simplicial satellites. The tool we develop here for this extension is the homotopy kernel of a commutative k-dimensional cubic diagram, generalising the homotopy kernel of a map; its 2-dimensional version has already been proved essential in other homotopical topics.     

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.     

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.     

Compact open topology and CW homotopy type

The class of spaces having the homotopy type of a CW complex is not closed under formation of function spaces. In 1959, Milnor proved the fundamental theorem that, given a space and a compact Hausdorff space X, the space Y^X of continuous functions X→Y, endowed with the compact open topology, belongs to . P.J. Kahn extended this in 1982, showing that if X has finite n-skeleton and π_k(Y)=0, k>n.

Using a different approach, we obtain a further generalization and give interesting examples of function spaces where is not homotopy equivalent to a finite complex, and has infinitely many nontrivial homotopy groups. We also obtain information about some topological properties that are intimately related to CW homotopy type.

As an application we solve a related problem concerning towers of fibrations between spaces of CW homotopy type.