The Relationship of Partial Metric Varieties and Commuting Powers Varieties

Holland et al. (Algebra Univers 67:1–18, 2012) considered varieties ${\mathcal E}_n$ of lattice-ordered groups defined by partial metrics, and showed for all n that ${\mathcal E}_n$ is contained within the variety ${\mathcal L}_n$ defined by x ⁿ y ⁿ ?=?y ⁿ x ⁿ . They also showed that if n were prime, then ${\mathcal E}_n = {\mathcal L}_n$ . Letting ${\mathcal A}^2$ denote the metabelian variety (defined at the beginning of Section 2), this article continues their work, showing that for all n, ${\mathcal L}_n \cap {\mathcal A}^2 \subseteq {\mathcal E}_n$ while showing that if n is not prime, ${\mathcal L}_n \not\subseteq {\mathcal E}_n$ .     

Real Commuting Differential Operators Connected with Two-Dimensional Abelian Varieties

We indicate smooth real commuting differential operators whose eigenvalues and eigenfunctions are parametrized by principally polarized abelian varieties.     

Real Commuting Differential Operators Connected with Two-Dimensional Abelian Varieties

We indicate smooth real commuting differential operators whose eigenvalues and eigenfunctions are parametrized by principally polarized abelian varieties.     

Commuting Varieties of Lie Algebras over Fields of Prime Characteristic

Let K be an algebraically closed field of positive characteristic and let G be a reductive group over K with Lie algebra . This paper will show that under certain mild assumptions on G, the commuting variety ( ) is an irreducible algebraic variety.     

Roitman's Theorem for Singular Projective Varieties

We construct an Abel–Jacobi mapping on the Chow group of 0-cycles of degree 0, and prove a Roitman theorem, for projective varieties over C with arbitrary singularities. Along the way, we obtain a new version of the Lefschetz Hyperplane theorem for singular varieties.     

On the Regularity of Codimension Two Surfaces and Threefolds with Singular Loci

ABSTRACT

For projective codimension two surfaces and threefolds whose singular locus is one dimensional, we get the sharp Castelnuovo–Mumford regularity bound in terms of degrees of defining equations and give the classification of nearly extremal cases. This is a generalization of the result of Bertram et al.     

Euler Obstruction and Defects of Functions on Singular Varieties

Several authors have proved Lefschetz type formulas for thelocal Euler obstruction. In particular, a result of this typehas been proved that turns out to be equivalent to saying thatthe local Euler obstruction, as a constructible function, satisfiesthe local Euler condition (in bivariant theory) with respectto general linear forms. The purpose of the paper is to determinewhat prevents the local Euler obstruction from satisfying thelocal Euler condition with respect to functions which are singularat the considered point. This is measured by an invariant (or‘defect’) of such functions. An interpretation ofthis defect is given in terms of vanishing cycles, which allowsit to be calculated algebraically. When the function has anisolated singularity, the invariant can be defined geometrically,via obstruction theory. This invariant unifies the usual conceptsof the Milnor number of a function and the local Euler obstructionof an analytic set.     

Algebraic Versus Homological Equivalence for Singular Varieties

Let Y ? ?^N be a possibly singular projective variety, defined over the field of complex numbers. Let X be the intersection of Y with h general hypersurfaces of sufficiently large degrees. Let d > 0 be an integer, and assume that dimY = n + h and dimY_sing ≤ min {d + h ? 1, n ? 1}. Let Z be an algebraic cycle on Y of dimension d + h, whose homology class in H_2(d+h)(Y; ?) is nonzero. In the present article, we prove that the restriction of Z to X is not algebraically equivalent to zero. This is a generalization to the singular case of a result due to Nori in the case Y is smooth. As an application we provide explicit examples of singular varieties for which homological equivalence is different from the algebraic one.     

On Singular Varieties Having an Extremal Secant Line

ABSTRACT

Let X be a nondegenerate subvariety of degree d and codimension e in the projective space ? ⁿ . If X is smooth, any multisecant line to X cuts X along a 0-dimensional scheme of length at most d ? e + 1. Moreover, smooth varieties X having a (d ? e + 1)-secant line (an extremal secant line) have been completely classified, extending del Pezzo and Bertini classification of varieties of minimal degree. In this article, we almost completely classify possibly singular varieties having an extremal secant line, without any assumptions on the singularities of X. First, we show that, if e ≠ 2, a multisecant line to X meets X along a 0-dimensional scheme of length at most d ? e + 1. Then, we completely classify singular varieties having a (d ? e + 1)-secant line for e ≠ 3. A partial result is provided in case e = 3.     

On the Irreducibility of Commuting Varieties Associated with Involutions of Simple Lie Algebras

Let be a reductive Lie algebra over an algebraically closed field of characteristic zero and an arbitrary -grading. We consider the variety , which is called the commuting variety associated with the -grading. Earlier it was proved by the author that is irreducible, if the -grading is of maximal rank. Now we show that is irreducible for and (E₆,F₄). In the case of symmetric pairs of rank one, we show that the number of irreducible components of is equal to that of nonzero non--regular nilpotent G ₀-orbits in . We also discuss a general problem of the irreducibility of commuting varieties.     

Logarithmic Vanishing Theorems and Arakelov–Parshin Boundedness for Singular Varieties

This article can be divided into two loosely connected parts. The first part is devoted to proving a singular version of the logarithmic Kodaira–Akizuki–Nakano vanishing theorem of Esnault and Viehweg in the style of Navarro-Aznar et al. This in turn is used to prove other vanishing theorems. In the second part, these vanishing theorems are used to prove an Arakelov–Parshin type boundedness result for families of canonically polarized varieties with rational Gorenstein singularities.     

A Remark on Characterizations of Projective Spaces among Singular Varieties

Let X be a projective variety of dimension n ≥ 2 with at worst log-terminal singularities and let be an ample vector bundle of rank r. By partially extending previous results due to Andreatta and Wiśniewski in the smooth case, we prove that if r = n then , while if r = n − 1 and X has only isolated singularities, then either or n = 2 and X is the quadric cone Q ₂. Received: April 20, 2006. Revised: April 5, 2007.     

The Theorem of Mather on Generic Projections for Singular Varieties

The theorem of Mather on generic projections of smooth algebraic varieties is also proved for the singular ones.     

On Rational Varieties Smooth Except at a Seminormal Singular Point

We construct a class of projective rational varieties X of any dimension m ≥ 1, which are smooth except at a point O, with the projective space ? ^m as normalization, having smooth branches, and reduced projectivized tangent cone in O. The Hilbert function of X is considered and is explicitly computed when the point O is seminormal. Indeed, we study seminormality, obtaining necessary and sufficient conditions for O to be seminormal and show that in such case the tangent cone is reduced and seminormal.     

Shift-Type Properties of Commuting,Completely Non Doubly Commuting Pairs of Isometries

Pairs (V, V′) of commuting, completely non doubly commuting isometries are studied. We show, that the space of the minimal unitary extension of V (denoted by U) is a closed linear span of subspaces reducing U to bilateral shifts. Moreover, the restriction of V′ to the maximal subspace reducing V to a unitary operator is a unilateral shift. We also get a new hyperreducing decomposition of a single isometry with respect to its wandering vectors which strongly corresponds with Lebesgue decomposition.     

Commuting algebraic correspondences and groups

In this paper, the naturalness of the appearance of the group approach in the study of problems of algebraic commutation is considered. New, previously unknown solutions are obtained. Special attention is paid to an example of commutation obtained from the icosahedral equation. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 662–670, May, 1997. Translated by A. I. Shtern     

Commuting extensions of operators

The maximal commuting proper extensions of a closed Hermitian operator and a dual pair of continuous operators in a Hilbert space are described; the criteria of their existence are established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 6, pp. 744–752, June, 1993.     

Commuting extensions and cubature formulae

Based on a novel point of view on 1-dimensional Gaussian quadrature, we present a new approach to d-dimensional cubature formulae. It is well known that the nodes of 1-dimensional Gaussian quadrature can be computed as eigenvalues of the so-called Jacobi matrix. The d-dimensional analog is that cubature nodes can be obtained from the eigenvalues of certain mutually commuting matrices. These are obtained by extending (adding rows and columns to) certain noncommuting matrices A₁,...,A_d, related to the coordinate operators x₁,...,x_d, in R^d. We prove a correspondence between cubature formulae and “commuting extensions” of A₁,...,A_d, satisfying a compatibility condition which, in appropriate coordinates, constrains certain blocks in the extended matrices to be zero. Thus, the problem of finding cubature formulae can be transformed to the problem of computing (and then simultaneously diagonalizing) commuting extensions. We give a general discussion of existence and of the expected size of commuting extensions and briefly describe our attempts at computing them.