Orbifolds, Sheaves and Groupoids

We characterize orbifolds in terms of their sheaves, and show that orbifolds correspond exactly to a specific class of smooth groupoids. As an application, we construct fibered products of orbifolds and prove a change-of-base formula for sheaf cohomology.     

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.      

Intersecting brane models and F-theory in six dimensions

We analyze six-dimensional supergravity theories coming from intersecting brane models on the toroidal orbifold T⁴/_Z2$T^4/{\mathbb{Z}}_2$. We use recently developed tools for mapping general 6D supergravity theories to F-theory to identify F-theory constructions dual to the intersecting brane models. The F-theory picture illuminates several aspects of these models. In particular, we have some new insight into the matter spectrum on intersecting branes, and analyze gauge group enhancement as branes approach orbifold points. These novel features of intersecting brane models are also relevant in four dimensions, and are confirmed in 6D using more standard Chan–Paton methods.     

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.

     

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.     

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.     

Convergence of Calabi–Yau manifolds

In this paper, we study the convergence of Calabi–Yau manifolds under Kähler degeneration to orbifold singularities and complex degeneration to canonical singularities (including the conifold singularities), and the collapsing of a family of Calabi–Yau manifolds.     

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.