Volume 12 (2012), Number 1. Abstracts

M. Borovoi and T. Schlank. A Cohomological Obstruction to Weak Approximation for Homogeneous Spaces [PDF]

Let X be a homogeneous space, X = G/H, where G is a connected linear algebraic group over a number field k and HG is a k-subgroup (not necessarily connected). Let S be a finite set of places of k. We compute a Brauer–Manin obstruction to weak approximation for X in S in terms of Galois cohomology.

Keywords. Brauer–Manin obstruction, weak approximation, homogeneous spaces, linear algebraic groups, Brauer group, Galois.

2000 Mathematics Subject Classification. Primary: 14M17; Secondary: 14G05, 20G10, 20G30.

A. Buryak. The Classes of the Quasihomogeneous Hilbert Schemes of Points on the Plane [PDF]

In this paper we give a formula for the classes (in the Grothendieck ring of complex quasiprojective varieties) of irreducible components of (1,k)-quasihomogeneous Hilbert schemes of points on the plane. We find a new simple geometric interpretation of the (q,t)-Catalan numbers. Finally, we investigate a connection between (1,k)-quasihomogeneous Hilbert schemes and homogeneous nested Hilbert schemes.

Keywords. Hilbert scheme, torus action, (q,t)-Catalan numbers.

2010 Mathematics Subject Classification. 14C05, 05A17.

F. Colombo, M. Luna-Elizarrarás, I. Sabadini, M. Shapiro, and D. Struppa. A Quaternionic Treatment of the Inhomogeneous div-rot System [PDF]

In this paper we study the inhomogeneous div-rot system (div \vec{f} = g0, rot \vec{f} = \vec{g}) where the datum (g0,\vec{g}) consists of a continuous scalar and a continuous vector field, respectively. We embed the system in its appropriate quaternionic setting, and by using the right inverse of the Moisil–Teodorescu operator, we provide a necessary and sufficient condition for the solvability of the system and we describe its general solution. As a byproduct we obtain an explicit integral expression for the right inverse for the operators div and rot. Finally, we show how the same problem could have been studied using algebraic analysis, and we use this different approach to obtain some additional results.

Keywords. div-rot system, right inverse operator, algebraic analysis, cohomology vanishing.

2000 Mathematics Subject Classification. 47F05, 47G10, 35F05.

W. Ebeling and S. Gusein-Zade. Orbifold Euler Characteristics for Dual Invertible Polynomials [PDF]

To construct mirror symmetric Landau–Ginzburg models, P. Berglund, T. Hübsch and M. Henningson considered a pair (f,G) consisting of an invertible polynomial f and an abelian group G of its symmetries together with a dual pair (f̃,G̃). Here we study the reduced orbifold Euler characteristics of the Milnor fibers of f and f̃ with the actions of the groups G and G̃ respectively and show that they coincide up to a sign.

Keywords. Invertible polynomials, group actions, orbifold Euler characteristic

2000 Mathematics Subject Classification. 14J33, 32S55, 57R18.

A. Esterov. Tropical Varieties with Polynomial Weights and Corner Loci of Piecewise Polynomials [PDF]

We find a relation between mixed volumes of several polytopes and the convex hull of their union, deducing it from the following fact: the mixed volume of a collection of polytopes only depends on the product of their support functions (rather than on the individual support functions). For integer polytopes, this dependence is a certain specialization of the isomorphism between two well-known combinatorial models for the cohomology of toric varieties, however, this construction has not been extended to arbitrary polytopes so far (partially due to the lack of combinatorial tools capable of substituting for toric geometry when vertices are not rational). We provide such an extension, which leads to an explicit formula for the mixed volume in terms of the product of support functions, and may also be interesting because of the combinatorial tools (tropical varieties with polynomial weights and their corner loci) that appear in our construction. As an example of another possible application of these new objects, we notice that every tropical subvariety in a tropical manifold M can be locally represented as the intersection of M with another tropical variety (possibly with negative weights), and conjecture certain generalizations of this fact to singular M. The above fact about subvarieties of a tropical manifold may be of independent interest, because it implies that the intersection theory on a tropical manifold, which was recently constructed by Allerman, Francois, Rau and Shaw, is locally induced from the ambient vector space.

Keywords. Tropical variety, mixed volume, matroid fan, piecewise polynomial, corner locus, intersection theory, cohomology, differential ring, toric variety.

2000 Mathematics Subject Classification. 14T05, 14M25, 52A39.

C. Lambert and C. Rousseau. Complete System of Analytic Invariants for Unfolded Differential Linear Systems with an Irregular Singularity of Poincaré Rank 1 [PDF]

In this paper we give a complete system of analytic invariants for the unfoldings of nonresonant linear differential systems with an irregular singularity of Poincaré rank 1 at the origin over a fixed neighborhood Dr. The unfolding parameter ε is taken in a sector S pointed at the origin of opening larger than 2π in the complex plane, thus covering a whole neighborhood of the origin. For each parameter value ε ∈ S, we cover Dr with two sectors and, over each sector, we construct a well chosen basis of solutions of the unfolded linear differential systems. This basis is used to find the analytic invariants linked to the monodromy of the chosen basis around the singular points. The analytic invariants give a complete geometric interpretation to the well-known Stokes matrices at ε=0: this includes the link (existing at least for the generic cases) between the divergence of the solutions at ε=0 and the presence of logarithmic terms in the solutions for resonance values of the unfolding parameter. Finally, we give a realization theorem for a given complete system of analytic invariants satisfying a necessary and sufficient condition, thus identifying the set of modules.

Keywords. Stokes phenomenon, irregular singularity, unfolding, confluence, divergent series, monodromy, Riccati matrix differential equation, analytic classification, summability, realization.

2000 Mathematics Subject Classification. Primary: 34M35, 34M40, 34M50, 34M03; Secondary: 37G10, 34E10, 37G05.

V. Malyshev. Fixed Points for One-Dimensional Particle System with Strong Interaction [PDF]

We consider hamiltonian N particle system on the finite segment with nearest-neighbor Coulomb interaction and external force F. We study the fixed points of such system and show that the distances between neighbors are asymptotically, for large N, the same for any F.

Keywords. Statistical physics, hamiltonian systems, submicroscale, fixed points.

2000 Mathematics Subject Classification. 82C21.

T. Panov and Yu. Ustinovsky. Complex-Analytic Structures on Moment-Angle Manifolds [PDF]

We show that the moment-angle manifolds corresponding to complete simplicial fans admit non-Kähler complex-analytic structures. This generalises the known construction of complex-analytic structures on polytopal moment-angle manifolds, coming from identifying them as LVM-manifolds. We proceed by describing Dolbeault cohomology and some Hodge numbers of moment-angle manifolds by applying the Borel spectral sequence to holomorphic principal bundles over toric varieties.

Keywords. Moment-angle manifold, simplicial fan, simple polytope, complex structure, Dolbeault cohomology, Hodge numbers.

2000 Mathematics Subject Classification. 32J18, 32L05, 32Q55, 57R19, 14M25.

A. Penskoi. Extremal Spectral Properties of Lawson Tau-Surfaces and the Lamé Equation [PDF]

Extremal spectral properties of Lawson tau-surfaces are investigated. The Lawson tau-surfaces form a two-parametric family of tori or Klein bottles minimally immersed in the standard unitary three-dimensional sphere. A Lawson tau-surface carries an extremal metric for some eigenvalue of the Laplace–Beltrami operator. Using theory of the Lamé equation we find explicitly these extremal eigenvalues.

Keywords. Lawson minimal surfaces, extremal metric, Lamé equation, Magnus–Winkler–Ince equation.

2000 Mathematics Subject Classification. 58E11, 58J50.

A. Vershik. Totally Nonfree Actions and the Infinite Symmetric Group [PDF]

We consider totally nonfree (TNF) actions of groups and the corresponding adjoint invariant (AD) measures on lattices of the subgroups of the given group. The main result is the description of all adjoint-invariant and TNF measures on the lattice of subgroups of the infinite symmetric group S. The problem is closely related to the theory of characters and factor representations of groups.

Keywords. Totally nonfree actions, infinite symmetric group, random subgroups.

2000 Mathematics Subject Classification. 37A15, 20B35, 22D40.

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