Narrow Results

In this short note we discuss some properties of multiplier ideal sheaves on singular varieties, for example, in how far they are compatible with fibre products.

In this paper, we show that a smooth toric variety $X$ of Picard number $r\le 3$ always admits a nef primitive collection supported on a hyperplane admitting non-trivial intersection with the cone $\mathrm{\text{Nef}}(X)$ of numerically effective divisors and cutting a facet of the pseudo-effective cone $\mathrm{\text{Eff}}(X)$, that is $\mathrm{\text{Nef}}(X)\cap \partial \overline{\mathrm{\text{Eff}}}(X)\ne \{0\}$. In particular, this means that $X$ admits non-trivial and non-big numerically effective divisors. Geometrically, this guarantees the existence of a fiber type contraction morphism over a smooth toric variety of dimension and Picard number lower than those of $X$, so giving rise to a classification of smooth and complete toric varieties with $r\le 3$. Moreover, we revise and improve results of Oda–Miyake by exhibiting an extension of the above result to projective, toric, varieties of dimension $n=3$ and Picard number $r=4$, allowing us to classifying all these threefolds. We then improve results of Fujino–Sato, by presenting sharp (counter)examples of smooth, projective, toric varieties of any dimension $n\ge 4$ and Picard number $r=4$ whose non-trivial nef divisors are big, that is $\mathrm{\text{Nef}}(X)\cap \partial \overline{\mathrm{\text{Eff}}}(X)=\{0\}$. Producing those examples represents an important goal of computational techniques in definitely setting an open geometric problem. In particular, for $n=4$, the given example turns out to be a weak Fano toric fourfold of Picard number 4.

Stationary discs of fibrations over the unit circle ∂D are considered. It is shown that if all fibers of a fibration Σ⊆∂D×Cⁿ over the unit circle ∂D are strongly pseudoconvex hypersurfaces in Cⁿ, then for every stationary disc f of the fibration Σ one can define the partial indices of f. In the case all fibers of Σ are strictly convex, it is proved that all partial indices of a stationary disc f are 0. It is also proved that in the case a stationary disc f of the fibration Σ is non-degenerate, the only possible partial indices of f are 0, 1 and –1. In particular, these results give information on the polynomial hull of Σ and new proofs of results related to the smoothness of the Kobayashi metric on some strongly pseudoconvex domains in Cⁿ.

As it is well-known, all Vassiliev invariants of degree one of a knot K ⊂ ℝ³ are trivial. There are nontrivial Vassiliev invariants of degree one, when the ambient space is not ℝ³. Recently, T. Fiedler introduced such invariants of a knot in an ℝ¹-fibration over a surface F. They take values in the free ℤ-module generated by all the free homotopy classes of loops in F. Here, we generalize them to the most refined Vassiliev invariant of degree one. The ranges of values of all these invariants are explicitly described.

We also construct a similar invariant of a two-component link in an ℝ¹-fibration. It generalizes the linking number.

We prove the generalized Margulis lemma with a uniform index bound on an Alexandrov $n$-space $X$ with curvature bounded below, i.e. small loops at $p\in X$ generate a subgroup of the fundamental group of the unit ball $B_1(p)$ that contains a nilpotent subgroup of index $\le w(n)$, where $w(n)$ is a constant depending only on the dimension $n$. The proof is based on the main ideas of V. Kapovitch, A. Petrunin and W. Tuschmann, and the following results:

(1) We prove that any regular almost Lipschitz submersion constructed by Yamaguchi on a collapsed Alexandrov space with curvature bounded below is a Hurewicz fibration. We also prove that such fibration is uniquely determined up to a homotopy equivalence.

(2) We give a detailed proof on the gradient push, improving the universal pushing time bound given by V. Kapovitch, A. Petrunin and W. Tuschmann, and justifying in a specific way that the gradient push between regular points can always keep away from extremal subsets.

This paper describes the Leray spectral sequence associated to a differential fibration. The differential fibration is described by base and total differential graded algebras. The cohomology used is noncommutative differential sheaf cohomology. For this purpose, a sheaf over an algebra is a left module with zero curvature covariant derivative. As a special case, we can recover the Serre spectral sequence for a noncommutative fibration.

Type I collagen fibrils have circular cross sections with radii mostly distributed in between 50 and 100 nm and are characterized by an axial banding pattern with a period of 67 nm. The constituent long molecules of those fibrils, the so-called triple helices, are densely packed but their nature is such that their assembly must conciliate two conflicting requirements. One is a double-twist around the axis of the fibril induced by their chirality and the other is a periodic layered organization, corresponding to the axial banding, built by specific lateral interactions. We examine here how such a conflict could contribute to the control of the radius of a fibril. We develop our analysis with the help of two geometrical archetypes: the Hopf fibration and the algorithm of phyllotaxis. The first one provides an ideal template for a twisted bundle of fibres and the second ensures the best homogeneity and local isotropy possible for a twisted dense packing with circular symmetry. This approach shows that, as the radius of a fibril with constant double-twist increases, the periodic layered organization can not be preserved without moving from planar to helicoidal configurations. Such changes of configurations are indeed made possible by the edge dislocations naturally present in the phyllotactic pattern. The distribution of those defects is such that the lateral growth of a fibril should stay limited in the observed range. Because of our limited knowledge about the elastic constants involved, this purely geometrical development stays at a quite conjectural level.

Suppose a sequence $M_j$ of Alexandrov spaces collapses to a space $X$ with only weak singularities. Yamaguchi constructed a map $f_j:M_j\to X$ called an almost Lipschitz submersion for large $j$. We prove that if $M_j$ has a uniform positive lower bound for the volumes of spaces of directions, which is sufficiently large compared to the weakness of singularities of $X$, then $f_j$ is a locally trivial fibration. Moreover, we show some properties on the intrinsic metric and the volume of the fibers of $f_j$.