Volume 3 (2003), Number 1. Abstracts

A. Bondal, M. Van den Bergh. Generators and Representability of Functors in Commutative and Noncommutative Geometry [PDF]

We give a sufficient condition for an Ext-finite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in the existence of a strong generator. We prove that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, and are hence saturated. In contrast, the similar category for a smooth compact analytic surface with no curves is not saturated.

Keywords. Saturation, generators, representability, triangulated categories.

2000 Mathematics Subject Classification. Primary 18E30.

P. Etingof, S. Gelaki. The Classification of Finite-Dimensional Triangular Hopf Algebras over an Algebraically Closed Field of Characteristic 0 [PDF]

We explain that a new theorem of Deligne on symmetric tensor categories [De2] implies, in a straightforward manner, that any finite dimensional triangular Hopf algebra over an algebraically closed field of characteristic zero has the Chevalley property, and in particular the list of finite dimensional triangular Hopf algebras over such a field, given in [AEG], [EG3], is complete. We also use Deligne's theorem to settle a number of questions about triangular Hopf algebras, raised in our previous publications, and generalize Deligne's result to nondegenerate semisimple categories in positive characteristic p, by using the lifting methods developed in [ENO].

Keywords. Triangular Hopf algebras, finite supergroups.

2000 Mathematics Subject Classification. 16W30.

B. Helffer, T. Hoffmann-Ostenhof, N. Nadirashvili. Periodic Schrödinger Operators and Aharonov—Bohm Hamiltonians [PDF]

Let H=−Δ+V be a two-dimensional Schröodinger operator defined on a domain Ω⊂R² with Dirichlet boundary conditions. Suppose that H and Ω are invariant with respect to translations in the x₁ direction, so that V(x₁,x₂)=V(x₁+1, x₂); suppose in addition that V(x₁, x₂)=V(−x₁, x₂) and that (x₁,x₂)∈Ω implies (x₁+1,x₂)∈Ω and (−x₁,x₂)∈Ω. We investigate the associated Floquet operator H^(q), 0≤q<1. In particular, we show that the lowest eigenvalue λ_q is simple for q≠1/2 and strictly increasing in q for 0<q<1/2 and that the associated complex-valued eigenfunction u_q has empty zero set.

For the Dirichlet realization of the Aharonov—Bohm Hamiltonian in an annulus-like domain with an axis of symmetry,

Keywords. Schröodinger operator, magnetic field, eigenvalues.

2000 Mathematics Subject Classification. 35B05.

G. Laptev. Non-Existence of Global Solutions for Higher-Order Evolution Inequalities in Unbounded Cone-Like Domains [PDF]

We use the test function method developed by Mitidieri and Pohozaev to get a priori estimates and non-existence results for semi-linear ``higher-order evolution inequalities'' in unbounded cone-like domains. As a model we consider the problem in a cone K with the positive initial-boundary conditions

Keywords. Blow-up, partial differential inequalities, non-existence cone, cone-like domain.

2000 Mathematics Subject Classification. Primary 35G25; Secondary 35R45, 35K55, 35L70.

M. Larsen, V. Lunts. Motivic Measures and Stable Birational Geometry [PDF]

We study the motivic Grothendieck group of algebraic varieties from the point of view of stable birational geometry. In particular, we obtain a counterexample to a conjecture of M. Kapranov on the rationality of motivic zeta-functions.

Keywords. Grothendieck group, motivic zeta-function, stable birational equivalence.

2000 Mathematics Subject Classification. Primary: 14F42, 14E05.

A. Odesskii. Set-Theoretical Solutions to the Yang—Baxter Relation from Factorization of Matrix Polynomials and θ-Functions [PDF]

New set-theoretical solutions to the Yang—Baxter Relation are constructed. These solutions arise from the decompositions ``in different order'' of matrix polynomials and θ-functions. We also construct a ``local action of the symmetric group'' in these cases, generalizations of the action of the symmetric group S_N given by the set-theoretical solution.

Keywords. Yang—Baxter relation, set-theoretical solution, local action of the symmetric group, matrix polynomials, matrix θ-functions.

2000 Mathematics Subject Classification. Primary: 81R50.

A. Polishchuk. Triple Massey Products on Curves, Fay's Trisecant Identity and Tangents to the Canonical Embedding [PDF]

We show that Fay's trisecant identity follows from the A_∞-constraint satisfied by certain triple Massey products in the derived category of coherent sheaves on a curve. We also deduce the matrix analogue of this identity that can be conveniently formulated using quasideterminants of matrices with noncommuting entries. On the other hand, looking at more special Massey products, we derive a formula for the tangent line to a canonically embedded curve at a given point.

Keywords. Massey products, theta functions, quasideterminant.

2000 Mathematics Subject Classification. Primary 14H42; Secondary 15A15.

J. Rebelo, R. Silva. The Multiple Ergodicity of Nondiscrete Subgroups of Diff^ω (S¹) [PDF]

We deal with nondiscrete subgroups of Diff^ω (S¹), the group of orientation-preserving analytic diffeomorphisms of the circle. If Γ is such a group, we consider its natural diagonal action ˜Γ on the n-dimensional torus Tⁿ. A complete characterization of those groups Γ whose corresponding ˜Γ-action on Tⁿ is not piecewise ergodic (see Introduction) for all n∈N is obtained (see Theorem A). Theorem A can also be interpreted as an extension of Lie's classification of Lie algebras on S¹ to general nondiscrete subgroups of S¹.

Keywords. Diagonal action, ergodicity, vector fields.

2000 Mathematics Subject Classification. 58F11, 22E65.

S. Ryom-Hansen. A q-Analogue of Kempf's Vanishing Theorem [PDF]

We use deep properties of Kashiwara's crystal basis to show that the induction functor H_k⁰(−) introduced by Andersen, Polo and Wen satisfies an analogon of Kempf's vanishing theorem for k a field.

Keywords. Kempf Vanishing, Quantum groups, crystal basis, H_k⁰(−), Demazure modules.

2000 Mathematics Subject Classification. 17B37, 20G42.

B. Saussol, S. Troubetzkoy, S. Vaienti. Recurrence and Lyapunov Exponents [PDF]

We prove two inequalities between the Lyapunov exponents of a diffeomorphism and some characteristics of its local recurrence properties. We give examples of linear hyperbolic maps of the torus showing that each of the inequalities is optimal.

Keywords. Return time, Lyapunov exponents.

2000 Mathematics Subject Classification. Primary: 37B20; Secondary: 37C45.

V. Vologodsky. Hodge Structure on the Fundamental Group and its Application to p-Adic Integration [PDF]

We study the unipotent completion Π^dR_un(x₀, x₁, X_K) of the de Rham fundamental groupoid of a smooth algebraic variety over a local non-Archimedean field K of characteristic 0. We show that the vector space Π^dR_un(x₀, x₁, X_K) carries a certain additional structure. That is a Q^ur_p-space Π_un(x₀, x₁, X_K) equipped with a σ-semi-linear operator φ, a linear operator N satisfying the relation Nφ = pφN, and a weight filtration W_• together with a canonical isomorphism Π^dR_un(x₀, x₁, X_K)⊗ _K \overline K ∼ Π_un(x₀, x₁, X_K) ⊗ _Q^urp \overline K. We prove that an analogue of the monodromy conjecture holds for Π_un(x₀, x₁, X_K).

As an application, we show that the vector space Π^dR_un(x₀, x₁, X_K) possesses a distinguished element. In other words, given a vector bundle E on X_K$ together with a unipotent integrable connection, we have a canonical isomorphism E_x0 ∼ E_x1 between the fibres. This construction is a generalisation of Colmez's p-adic integration (rk E=2) and Coleman's p-adic iterated integrals (X_K is a curve with good reduction).

In the second part, we prove that, for a smooth variety X_K0 over an unramified extension of Q_p with good reduction and r ≤ (p−1)/2, there is a canonical isomorphism Π^dR_r(x₀, x₁, X_K0) ⊗ B_dR ∼ Π^et_r(x₀, x₁, X_{\overline K0}) ⊗ B_dR compatible with the action of the Galois group (Π^dR_r(x₀, x₁, X_K0) stands for the level r quotient of Π^dR_un(x₀, x₁, X_K)). In particular, this implies the crystalline conjecture for the fundamental group (for r ≤ (p−1)/2).

Keywords. Crystalline cohomology, Hodge structure, p-adic integration.

2000 Mathematics Subject Classification. Primary 14D10, 11G25; Secondary 14D07.

R. Yuncken. Regular Tessellations of the Hyperbolic Plane by Fundamental Domains of a Fuchsian Group [PDF]

For positive integers p and q with 1/p+1/q<1/2, a tessellation of type {p,q} is a tessellation of the hyperbolic plane by regular p-gons with q p-gons meeting at each vertex. In this paper, a necessary and sufficient condition on the integers p and q is established to determine when a tessellation of type {p,q} can be realized as a tessellation of the hyperbolic plane by fundamental domains of some Fuchsian group. Specifically, a tessellation of type {p,q} is a tessellation by fundamental domains if and only if q has a prime divisor less than or equal to p.

Keywords. Fuchsian group, regular tessellation, hyperbolic plane, fundamental domain.

2000 Mathematics Subject Classification. 20H10.

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