Classifying smooth lattice polytopes via toric fibrations

We show that any smooth Q-normal lattice polytope P of dimension n and degree d is a strict Cayley polytope if n?2d+1. This gives a sharp answer, for this class of polytopes, to a question raised by V.V. Batyrev and B. Nill.     

On the geometry of the space of smooth fibrations

We study geometrical aspects of the space of smooth fibrations between two given manifolds M and B, from the point of view of Fréchet geometry. As a first result, we show that any connected component of this space is the base space of a Fréchet-smooth principal bundle with the identity component of the group of diffeomorphisms of M as total space. Second, we prove that the space of fibrations is also itself the total space of a smooth Fréchet principal bundle with structure group the group of diffeomorphisms of the base B.     

Noncomplex smooth 4-manifolds with genus-2 Lefschetz fibrations

We construct noncomplex smooth 4-manifolds which admit genus-2 Lefschetz fibrations over . The fibrations are necessarily hyperelliptic, and the resulting 4-manifolds are not even homotopy equivalent to complex surfaces. Furthermore, these examples show that fiber sums of holomorphic Lefschetz fibrations do not necessarily admit complex structures.

     

Del Pezzo surface fibrations obtained by blow-up of a smooth curve in a projective manifold

This Note is devoted to the study of the Fano manifolds X obtained by blow-up along a smooth curve C in a complex projective manifold Y. By the Mori theory, we can ensure the existence of an extremal contraction $\phi :X\to Z$ different from the blow-up $\pi :X\to Y$. Here we give the complete classification of the corresponding pairs $(Y,C)$ in the case where φ is a fiber type contraction of relative dimension 2, i.e. the general fibers of φ are del Pezzo surfaces. In Tsukioka (Thesis, Nancy University 1, 2005), the relative dimension 1 case is also considered. To cite this article: T. Tsukioka, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Diffeomorphisms taking lines to circles, and quaternionic Hopf fibrations

We list all diffeomorphisms between an open subset of the four-dimensional projective space and an open subset of the four-dimensional sphere that take all line segments to arcs of round circles. These diffeomorphisms are restrictions of quaternionic Hopf fibrations and radial projections from hyperplanes to spheres.     

Sources of straight electrostatic flux lines in R 3

It is well known that the point charge, line charge and homogeneously charged plane of infinite extent are sources of electrostatic fields with straight flux lines. It is proved in this note that provided these flux lines occupy all of three-dimensional space, then these charge configurations and appropriate variants of them are the only sources of straight-line electrostatic fields in R ³.     

Topological invariants of stable immersions of oriented 3-manifolds in R4

We show that the $\mathbb{Z}$-module of first order local Vassiliev type invariants of stable immersions of oriented 3-manifolds into ${\mathbb{R}}^4$ is generated by 3 topological invariants: The number of pairs of quadruple points and the positive and negative linking invariants ${?}^{+}$ and ${?}^{?}$ introduced by V. Goryunov (1997) [7]. We obtain the expression for the Euler characteristic of the immersed 3-manifold in terms of these invariants. We also prove that the total number of connected components of the triple points curve is a non-local Vassiliev type invariant.     

Local sections of Serre fibrations with 3-manifold fibers

It was proved by H. Whitney in 1933 that a Serre fibration of compact metric spaces admits a global section provided every fiber is homeomorphic to the unit interval [0,1]. An extension of the Whitney's theorem to the case when all fibers are homeomorphic to some fixed compact two-dimensional manifold was proved by the authors (Brodsky et al. (2008) [2]). The main result of this paper proves the existence of local sections in a Serre fibration with all fibers homeomorphic to some fixed compact three-dimensional manifold.     

Symmetries of Lagrangian fibrations

We construct fiber-preserving anti-symplectic involutions for a large class of symplectic manifolds with Lagrangian torus fibrations. In particular, we treat the K3 surface and the six-dimensional examples constructed by Castaño-Bernard and Matessi (2009) [8], which include a six-dimensional symplectic manifold homeomorphic to the quintic threefold. We interpret our results as corroboration of the view that in homological mirror symmetry, an anti-symplectic involution is the mirror of duality. In the same setting, we construct fiber-preserving symplectomorphisms that can be interpreted as the mirror to twisting by a holomorphic line bundle.     

Indecomposables of multiplicative fibrations

Given a multiplicative fibration we study the module of indecomposables for a prime.

     

三维全空间上Ginzburg Landau方程的整体吸引子

作者在三维全空间中考虑研究复Ginzburg Landau方程(CGL)的解的长时间行为。通过引入权空间，应用内插不等式和在权空间的先验估计，获得复 Ginzburg Landau方程整体解的存在性，进一步建立了整体吸引子的存在性。     

Characterisation of plus-constructive fibrations

For the numerous applications of the plus-construction, a key question concerns when a fibration F → E → B induces another F⁺ → E⁺ → B⁺. A complete solution (with proof) is given, together with a more easily verifiable simplification in special cases.     

Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces

A rational Lagrangian fibration f on an irreducible symplectic variety V is a rational map which is birationally equivalent to a regular surjective morphism with Lagrangian fibers. By analogy with K3 surfaces, it is natural to expect that a rational Lagrangian fibration exists if and only if V has a divisor D with Bogomolov–Beauville square 0. This conjecture is proved in the case when V is the Hilbert scheme of d points on a generic K3 surface S of genus g under the hypothesis that its degree 2g−2 is a square times 2d−2. The construction of f uses a twisted Fourier–Mukai transform which induces a birational isomorphism of V with a certain moduli space of twisted sheaves on another K3 surface M, obtained from S as its Fourier–Mukai partner.