The Absolute of a Topological Space and its Application to Abelian l-groups

We give a axiomatic treatment of the absolute in the category of all topological spaces and characterize it as a cover with respect to the full subcategory of extremally disconnected spaces. As an application, we obtain the strongly projectable hull of an abelian l-group as a unique lifting of the absolute of its spectrum. Dedicated to B. V. M.     

Characterization of standard embeddings between complex Grassmannians by means of varieties of minimal rational tangents

In 1993,Tsal proved that a proper holomorphic mapping f:Ω→Ω' from an irreducible bounded symmetric domainΩof rank≥2 into a bounded symmetric domainΩ' is necessarily totally geodesic provided that r':=rank(Ω')≤rank(Ω):= r,proving a conjecture of the author's motivated by Hermitian metric rigidity.As a first step in the proof,Tsai showed that df preserves almost everywhere the set of tangent vectors of rank 1.Identifying bounded symmetric domains as open subsets of their compact duals by means of the Borel embedding,this means that the germ of f at a general point preserves the varieties of minimal rational tangents(VMRTs). In another completely different direction Hwang-Mok established with very few exceptions the Cartan- Fubini extension priniciple for germs of local biholomorphisms between Fano manifolds of Picard num- ber 1,showing that the germ of map extends to a global biholomorphism provided that it preserves VMRTs.We propose to isolate the problem of characterization of special holomorphic embeddings between Fano manifolds of Picard number 1,especially in the case of classical manifolds such as ratio- nal homogeneous spaces of Picard number 1,by a non-equidimensional analogue of the Cartan-Fubini extension principle.As an illustration we show along this line that standard embeddings between com- plex Grassmann manifolds of rank≤2 can be characterized by the VMRT-preserving property and a non-degeneracy condition,giving a new proof of a result of Neretin's which on the one hand paves the way for far-reaching generalizations to the context of rational homogeneous spaces and more generally Fano manifolds of Picard number 1,on the other hand should be applicable to the study of proper holomorphic mappings between bounded domains carrying some form of geometric structures.     

关于有限阿贝尔群单因子个数的均值(Ⅱ)

令ｔ（Ｇ）为有限阿贝尔群Ｇ的单因子个数．本文得到了∑｜Ｇ｜≤ｘｔ（Ｇ）带有更好余项的渐近公式．     

Deformation Quantization of Algebraic Varieties

The paper is devoted to peculiarities of the deformation quantization in the algebro-geometric context. A direct application of the formality theorem to an algebraic Poisson manifold gives a canonical sheaf of categories deforming coherent sheaves. The global category is very degenerate in general. Thus, we introduce a new notion of a semiformal deformation, a replacement in algebraic geometry of an actual deformation (versus a formal one). Deformed algebras obtained by semiformal deformations are Noetherian and have polynomial growth. We propose constructions of semiformal quantizations of projective and affine algebraic Poisson manifolds satisfying certain natural geometric conditions. Projective symplectic manifolds (e.g. K3 surfaces and Abelian varieties) do not satisfy our conditions, but projective spaces with quadratic Poisson brackets and Poisson–Lie groups can be semiformally quantized.     

On a theorem of Tate

We study applications of divisibility properties of recurrence sequences to Tate’s theory of abelian varieties over finite fields.      

Determination of phenolic compounds in modern and old varieties of durum wheat using liquid chromatography coupled with time-of-flight mass spectrometry

An evaluation of the grain functional components of Italian durum wheat cultivars was conducted. The raw material was obtained from the field trial performed in 2006–2007 at the Experimental Farm of the University of Bologna, (Bologna, Italy). The aim of this study was to define the phytochemical profile of ten varieties, comprised of old and modern durum wheat genotypes, including quantitative and qualitative phenolic and flavonoid content (free and bound forms). The results showed that mean values of total phenolic compound and total flavonoid content in old wheat varieties (878.2 ± 19.0 μmol gallic acid equivalent/100 g of grain and 122.6 ± 25.4 μmol catechin equivalent/100 g of grain, respectively) did not differ significantly from those detected in modern genotypes (865.9 ± 128.9 μmol gallic acid equivalent/100 g and 123.5 ± 20.6 μmol catechin equivalent/100 g, respectively). However, the HPLC–ESI-TOF-MS analysis highlighted remarkable differences between modern and old cultivars. The interpretation of the mass spectra allowed the identification of 70 phenolic compounds, including coumarins, phenolic acids, anthocyanins, flavones, isoflavones, proanthocyanidins, stilbenes and lignans. The free extracts of ancient wheat varieties showed the presence of a mean number of phenolic compounds and isomer forms (8.7 ± 2.5 and 7.7 ± 4.7 respectively) significantly higher than in modern genotypes (4.4 ± 2.9 and 2.0 ± 2.4, respectively). A similar trend was observed also for the bound phenolic fraction. Moreover, the phytochemical profiles showed the presence of unique phenolic compounds in both free and bound fractions of some of the investigated wheat genotypes. Results highlighted that investigated old wheat cultivars may offer unique nutraceutical values for their peculiar contents in bioactive phytochemicals, suggesting their uses into a wide range of regular and specialty products naturally enriched with health-promoting compounds.     

Cohomology of the Orlik–Solomon Algebras and Local Systems

The paper provides a combinatorial method to decide when the space of local systems with nonvanishing first cohomology on the complement to an arrangement of lines in a complex projective plane has as an irreducible component a subgroup of positive dimension. Partial classification of arrangements having such a component of positive dimension and a comparison theorem for cohomology of Orlik–Solomon algebra and cohomology of local systems are given. The methods are based on Vinberg–Kac classification of generalized Cartan matrices and study of pencils of algebraic curves defined by mentioned positive dimensional components.     

On multiplication groups of relatively free quasigroups isotopic to Abelian groups

If Q is a quasigroup that is free in the class of all quasigroups which are isotopic to an Abelian group, then its multiplication group Mlt Q is a Frobenius group. Conversely, if Mlt Q is a Frobenius group, Q a quasigroup, then Q has to be isotopic to an Abelian group. If Q is, in addition, finite, then it must be a central quasigroup (a T-quasigroup).This work was supported by institutional grant MSM 113200007, and by Grant Agency of Czech Republic, grant number 201/99/0263.     

Gribov ambiguity in gauge theories

Gribov ambiguity in gauge field theories is discussed and it is shown that such an ambiguity exists even for Abelian theories in covariant gauge at finite temperature. Both geometric and algebraic proofs are presented. In view of the importance of non-perturbative methods, some special gauges are given in which such ambiguities do not exist or are not relevant. The significance of these in the study of confinement in QCD is pointed out.     

Length 3 complexes of abelian sheaves and Picard 2-stacks

We define a tricategory T[−2,0] of length 3 complexes of abelian sheaves, whose hom-bigroupoids consist of weak morphisms of such complexes. We also define a 3-category 2Pic(S) of Picard 2-stacks, whose hom-2-groupoids consist of additive 2-functors. We prove that these categories are triequivalent as tricategories. As a consequence we obtain a generalization of Deligne's analogous result about Picard stacks in SGA4, Exposé XVIII (Deligne (1973) [11]).