Algebraic Varieties Research Articles

Bohr Almost Periodic Sets of Toral Type

A locally finite multiset \((\Lambda ,c),\) \(\Lambda \subset {\mathbb {R}}^n, c : \Lambda \mapsto \{1,...,b\}\) defines a Radon measure \(\mu := \sum _{\lambda \in \Lambda } c(\lambda )\, \delta _\lambda \) that is Bohr almost periodic in the sense of Favorov if the convolution \(\mu *f\) is Bohr almost periodic for every \(f \in C_c({\mathbb {R}}^n).\) If it is of toral type: the Fourier transform \({\mathfrak {F}} \mu \) equals zero outside of a rank \(m < \infty \) subgroup, then there exists a compactification \(\psi : {\mathbb {R}}^n \mapsto {\mathbb {T}}^m\) of \(\mathbb R^n,\) a foliation of \({\mathbb {T}}^m,\) and a pair \((K,\kappa )\) where \(K := \overline{\psi (\Lambda )}\) and \(\kappa \) is a measure supported on K such that \({\mathfrak {F}} \kappa = ({\mathfrak {F}} \mu ) \circ \widehat{\psi }\) where \({\widehat{\psi }} : \widehat{{\mathbb {T}}^m} \mapsto \widehat{{\mathbb {R}}^n}\) is the Pontryagin dual of \(\psi .\) For \((\Lambda ,c)\) uniformly discrete, we prove that every connected component of K is homeomorphic to \({\mathbb {T}}^{m-n}\) embedded transverse to the foliation and the homotopy of its embedding is a rank \(m-n\) subgroup S of \({\mathbb {Z}}^m,\) and we compute its density as a function of S and \(\psi .\) For \(n = 1\) and K, a nonsingular real algebraic variety, this construction gives all Fourier quasicrystals recently characterized by Olevskii and Ulanovskii.

Hodge theory on Alexander invariants – A survey

We survey recent developments in the study of Hodge theoretic aspects of Alexander-type invariants associated with smooth complex algebraic varieties.

Nilpotent varieties and metabelian varieties

We deal with varieties of nonassociative algebras having polynomial growth of codimensions. We describe some results obtained in recent years in the class of left nilpotent algebras of index two. Recently the authors established a correspondence between the growth rates for left nilpotent algebras of index two and the growth rates for commutative or anticommutative metabelian algebras that allows to transfer the results concerning varieties of left nilpotent algebras of index two to varieties of commutative or anticommutative metabelian algebras.

Ranks of homotopy and cohomology groups for rationally elliptic spaces and algebraic varieties

We discuss inequalities between the values of \emph{homotopical and cohomological Poincare polynomials} of the self-products of rationally elliptic spaces. For rationally elliptic quasi-projective varieties, we prove inequalities between the values of generating functions for the ranks of the graded pieces of the weight and Hodge filtrations of the canonical mixed Hodge structures on homotopy and cohomology groups. Several examples of such mixed Hodge polynomials and related inequalities for rationally elliptic quasi-projective algebraic varieties are presented. One of the consequences is that the homotopical (resp. cohomological) mixed Hodge polynomial of a rationally elliptic toric manifold is a sum (resp. a product) of polynomials of projective spaces. We introduce an invariant called \emph{stabilization threshold} $\frak{pp} (X;\varepsilon)$ for a simply connected rationally elliptic space $X$ and a positive real number $\varepsilon$, and we show that the Hilali conjecture implies that $\frak{pp} (X;1) \le 3$.

Kernel L‐Ideals and L‐Congruence on a Subclass of Ockham Algebras

In this paper, we study L‐congruences and their kernel in a subclass of the variety of Ockham algebras A. We prove that the class of kernel L‐ideals of an Ockham algebra forms a complete Heyting algebra. Moreover, for a given kernel L‐ideal ξ on A, we obtain the least and the largest L‐congruences on A having ξ as its kernel.

Equivariant completions of affine spaces

We survey recent results on open embeddings of the affine space $\mathbb{C}^n$ into a complete algebraic variety $X$ such that the action of the vector group $\mathbb{G}_a^n$ on $\mathbb{C}^n$ by translations extends to an action of $\mathbb{G}_a^n$ on $X$. We begin with the Hassett-Tschinkel correspondence describing equivariant embeddings of $\mathbb{C}^n$ into projective spaces and present its generalization for embeddings into projective hypersurfaces. Further sections deal with embeddings into flag varieties and their degenerations, complete toric varieties, and Fano varieties of certain types. Bibliography: 109 titles.

A Combinatorial Approach to the Number of Solutions of Systems of Homogeneous Polynomial Equations over Finite Fields

We give a complete conjectural formula for the number $e_r(d,m)$ of maximum possible ${\mathbb{F}}q$-rational points on a projective algebraic variety defined by $r$ linearly independent homogeneous polynomial equations of degree $d$ in $m+1$ variables with coefficients in the finite field ${\mathbb{F}}q$ with $q$ elements, when $d<q$. It is shown that this formula holds in the affirmative for several values of $r$. In the general case, we give explicit lower and upper bounds for $e_r(d,m)$ and show that they are sometimes attained. Our approach uses a relatively recent result, called the projective footprint bound, together with results from extremal combinatorics such as the Clements-Lindstrom Theorem and its variants. Applications to the problem of determining the generalized Hamming weights of projective Reed-Muller codes are also included.

Hilbert series of mixed braid monoid $ MB_{2, 2} $

&lt;abstract&gt;&lt;p&gt;Hilbert series is a simplest way to calculate the dimension and the degree of an algebraic variety by an explicit polynomial equation. The mixed braid group $ B_{m, n} $ is a subgroup of the Artin braid group $ B_{m+n} $. In this paper we find the ambiguity-free presentation and the Hilbert series of canonical words of mixed braid monoid $ M\!B_{2, 2} $.&lt;/p&gt;&lt;/abstract&gt;

Cohomology of algebraic varieties over non-archimedean fields

Abstract We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed nontrivially valued nonarchimedean field K based on Hrushovski-Loeser’s stable completion. In parallel, we develop a sheaf cohomology of definable subsets in o-minimal expansions of the tropical semi-group $\Gamma _{\infty }$ , where $\Gamma $ denotes the value group of K. For quasi-projective varieties, both cohomologies are strongly related by a deformation retraction of the stable completion homeomorphic to a definable subset of $\Gamma _{\infty }$ . In both contexts, we show that the corresponding cohomology theory satisfies the Eilenberg-Steenrod axioms, finiteness and invariance, and we provide natural bounds of cohomological dimension in each case. As an application, we show that there are finitely many isomorphism types of cohomology groups in definable families. Moreover, due to the strong relation between the stable completion of an algebraic variety and its analytification in the sense of V. Berkovich, we recover and extend results on the singular cohomology of the analytification of algebraic varieties concerning finiteness and invariance.

The Hartogs extension phenomenon in almost homogeneous algebraic varieties

We study the Hartogs extension phenomenon in noncompact almost homogeneous algebraic varieties, and we prove a cohomological and a weight criterion for the Hartogs phenomenon. In the case of spherical varieties we prove a criterion for the Hartogs phenomenon in terms of coloured fans. Bibliography: 28 titles.

The Hartogs extension phenomenon in almost homogeneous algebraic varieties

Изучается феномен продолжения Гартогса в некомпактных почти однородных алгебраических многообразиях и доказываются когомологический и весовой критерии. В случае сферических многообразий получен критерий феномена Гартогса в терминах цветных вееров. Библиография: 28 названий.

Equivariant completions of affine spaces

Работа содержит обзор недавних результатов об открытых вложениях аффинного пространства $\mathbb{C}^n$ в полные алгебраические многообразия $X$, для которых действие векторной группы $\mathbb{G}_a^n$ на $\mathbb{C}^n$ параллельными переносами продолжается до действия $\mathbb{G}_a^n$ на $X$. В первой части работы мы подробно изучаем соответствие Хассетта-Чинкеля, описывающее эквивариантные вложения $\mathbb{C}^n$ в проективные пространства, и приводим его обобщение на случай вложений в проективные гиперповерхности. Последующие разделы посвящены изучению вложений в многообразия флагов и в их вырождения, в полные торические многообразия и в многообразия Фано определeнных типов. Библиография: 109 названий.

On the Monodromies and the Limit Mixed Hodge Structures of Families of Algebraic Varieties

We study the monodromies and the limit mixed Hodge structures of families of complete intersection varieties over a punctured disk in the complex plane. For this purpose, we express their motivic nearby fibers in terms of the geometric data of some Newton polyhedra. In particular, the limit mixed Hodge numbers and some part of the Jordan normal forms of the monodromies of such a family will be described very explicitly.

The Sard problem in step 2 and in filiform Carnot groups,

We study the Sard problem for the endpoint map in some well-known classes of Carnot groups. Our first main result deals with step 2 Carnot groups, where we provide lower bounds (depending only on the algebra of the group) on the codimension of the abnormal set; it turns out that our bound is always at least 3, which improves the result proved in Le Donne et al. [Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016) 1639–1666] and settles a question emerged in Ottazzi and Vittone [ESAIM: COCV 25 (2019) 18]. In our second main result we characterize the abnormal set in filiform groups and show that it is either a horizontal line, or a 3-dimensional algebraic variety.

Infinitely many Lagrangian fillings

We prove that all maximal-tb positive Legendrian torus links $(n,m)$ in the standard contact $3$-sphere, except for $(2,m)$, $(3,3)$, $(3,4)$ and $(3,5)$, admit infinitely many Lagrangian fillings in the standard symplectic 4-ball. This is proven by constructing infinite order Lagrangian concordances that induce faithful actions of the modular group $\mathrm{PSL}(2,\mathbb{Z})$ and the mapping class group $M_{0,4}$ into the coordinate rings of algebraic varieties associated to Legendrian links. In particular, our results imply that there exist Lagrangian concordance monoids with subgroups of exponential-growth, and yield Stein surfaces homotopic to a $2$-sphere with infinitely many distinct exact Lagrangian surfaces of higher-genus. We also show that there exist infinitely many satellite and hyperbolic knots with Legendrian representatives admitting infinitely many exact Lagrangian fillings.

Hodge filtration on local cohomology, Du Bois complex and local cohomological dimension

Abstract We study the Hodge filtration on the local cohomology sheaves of a smooth complex algebraic variety along a closed subscheme Z in terms of log resolutions and derive applications regarding the local cohomological dimension, the Du Bois complex, local vanishing and reflexive differentials associated to Z.

Classification of limit varieties of 𝒥-trivial monoids

A variety of algebras is a limit variety if it is non-finitely based but all its proper subvarieties are finitely based. We present a new pair of limit varieties of monoids and show that together with the five limit varieties of monoids previously discovered by Jackson, Zhang and Luo and the first-named author, there are exactly seven limit varieties of -trivial monoids.

On the symmetric polynomials in the variety of Grassmann algebras

𝐾 karakteristiği sıfır olan bir cisim ve 𝐿, Grassmann cebirleri tarafından üretilen varyetede, 𝐾 cismi üzerinde rankı 2 olan birleşmeli cebir olsun. Bu çalışmada, 𝐿 cebirinin 𝐿𝑆2 simetrik polinomlar alt cebiri incelenmiş ve 𝐿𝑆2 için sonlu bir üreteç kümesi verilmiştir.

On the Axiomatisability of the Dual of Compact Ordered Spaces

AbstractWe prove that the category of Nachbin’s compact ordered spaces and order-preserving continuous maps between them is dually equivalent to a variety of algebras, with operations of at most countable arity. Furthermore, we observe that the countable bound on the arity is the best possible: the category of compact ordered spaces is not dually equivalent to any variety of finitary algebras. Indeed, the following stronger results hold: the category of compact ordered spaces is not dually equivalent to (i) any finitely accessible category, (ii) any first-order definable class of structures, and (iii) any class of finitary algebras closed under products and subalgebras. An explicit equational axiomatisation of the dual of the category of compact ordered spaces is obtained; in fact, we provide a finite one, meaning that our description uses only finitely many function symbols and finitely many equational axioms. In preparation for the latter result, we establish a generalisation of a celebrated theorem by Mundici: our result—whose proof is independent of Mundici’s theorem—asserts that the category of unital commutative distributive lattice-ordered monoids is equivalent to the category of what we call MV-monoidal algebras.Abstract taken directly from the thesis.E-mail: marco.abbadini.uni@gmail.comURL: https://air.unimi.it/retrieve/handle/2434/812809/1698986/phd_unimi_R11882.pdf

Invariants of algebraic varieties over imperfect fields

We introduce four invariants of algebraic varieties over imperfect fields, each of which measures either geometric non-normality or geometric non-reducedness. The first objective of this article is to establish fundamental properties of these invariants. We then apply our results to curves over imperfect fields. In particular, we establish a genus change formula and prove the boundedness of non-smooth regular curves of genus one. We also compute our invariants for some explicit examples.