Volume 11 (2011), Number 2. Abstracts
F. Bogomolov and Yu. Tschinkel. Reconstruction of Higher-Dimensional Function Fields [PDF]
We determine the function fields of varieties of dimension ≥2 defined over the algebraic closure of 𝔽p, modulo purely inseparable extensions, from the quotient by the second term in the lower central series of their pro-ℓ Galois groups.
Keywords. Galois groups, function fields.
2000 Mathematics Subject Classification. 12F10, 14E08, 19C20, 19C30.
A. Caine. Toric Poisson Structures [PDF]
Let Tℂ be a complex algebraic torus and let X(Σ) be a complete nonsingular toric variety for Tℂ. In this paper, a real Tℂ-invariant Poisson structure ΠΣ is constructed on the complex manifold X(Σ), the symplectic leaves of which are the Tℂ-orbits in X(Σ). It is shown that each leaf admits an effective Hamiltonian action by a sub-torus of the compact torus T⊂Tℂ. However, the global action of Tℂ on (X(Σ),ΠΣ) is Poisson but not Hamiltonian. The main result of the paper is a lower bound for the first Poisson cohomology of these structures. For the simplest case, X(Σ)=ℂP1, the Poisson cohomology is computed using a Mayer–Vietoris argument and known results on planar quadratic Poisson structures. In this example, the bound is optimal. The paper concludes by studying the interaction of ΠΣ with the symplectic structure on ℂPn, where the modular vector field with respect to a particular Delzant Liouville form admits a curious formula in terms of Delzant moment data. This formula enables one to compute the zero locus of this modular vector field and relate it to the Euclidean geometry of the moment simplex.
Keywords. Poisson cohomology, modular class, momentum map, toric variety.
2000 Mathematics Subject Classification. 53D17, 14M25, 37J15.
A. Felikson and S. Natanzon. Double Pants Decompositions of 2-Surfaces [PDF]
We consider the union of two pants decompositions of the same orientable 2-dimensional surface of any genus g. Each pants decomposition corresponds to a handlebody bounded by this surface, so two pants decompositions correspond to a Heegaard splitting of a 3-manifold. We introduce a groupoid acting on double pants decompositions. This groupoid is generated by two simple transformations (called flips and handle twists), each transformation changing only one curve of the double pants decomposition. We prove that the groupoid acts transitively on all double pants decompositions corresponding to Heegaard splittings of a 3-dimensional sphere. As a corollary, we prove that the mapping class group of the surface is contained in the groupoid.
Keywords. Pants decomposition, Heegaard splitting, Mapping class group.
2000 Mathematics Subject Classification. 57M50.
Yu. Ilyashenko. Weak Total Rigidity for Polynomial Vector Fields of Arbitrary Degree [PDF]
We prove that in the space of the polynomial vector fields of arbitrary degree n with n+1 different singular points at infinity the set of vector fields that are orbitally topologically equivelent to a generic vector field (modulo affine equivalence) is no more than countable.
This is the second one of two closely related papers. It was started after the first one, “Total rigidity of generic quadratic vector fields”, was completed. The present paper is motivated by the problem stated at the end of the first paper. The problem remains open. A slightly weaker problem is solved below.
This paper is independent on the first one. For the sake of convinience, it is published first.
Keywords. Foliations, topological equivalence, rigidity.
2000 Mathematics Subject Classification. 37F75.
K. Kaveh and A. Khovanskii. Newton Polytopes for Horospherical Spaces [PDF]
A subgroup H of a reductive group G is horospherical if it contains a maximal unipotent subgroup. We describe the Grothendieck semigroup of invariant subspaces of regular functions on G/H as a semigroup of convex polytopes. From this we obtain a formula for the number of solutions of a system of equations f1(x) = ⋯ = fn(x) = 0 on G/H, where n = dim(G/H) and each fi is a generic element from an invariant subspace Li of regular functions on G/H. The answer is in terms of the mixed volume of polytopes associated to the Li. This generalizes the Bernstein–Kushnirenko theorem from toric geometry. We also obtain similar results for the intersection numbers of invariant linear systems on G/H.
Keywords. Reductive group, moment polytope, Newton polytope, horospherical variety, Bernstein–Kushnirenko theorem, Grothendieck group.
2000 Mathematics Subject Classification. 14M17, 14M25.
J. Lebl and H. Peters. Polynomials Constant on a Hyperplane and CR Maps of Hyperquadrics [PDF]
We prove a sharp degree bound for polynomials constant on a hyperplane with a fixed number of distinct monomials for dimensions 2 and 3. We study the connection with monomial CR maps of hyperquadrics and prove similar bounds in this setup with emphasis on the case of spheres. The results support generalizing a conjecture on the degree bounds to the more general case of hyperquadrics.
Keywords. Polynomials constant on a hyperplane, CR mappings of spheres and hyperquadrics, monomial mappings, degree estimates, Newton diagram.
2000 Mathematics Subject Classification. 14P99, 05A20, 32H35, 11C08.
L. Positselski. Mixed Artin–Tate Motives with Finite Coefficients [PDF]
The goal of this paper is to give an explicit description of the triangulated categories of Tate and Artin–Tate motives with finite coefficients ℤ/m over a field K containing a primitive m-root of unity as the derived categories of exact categories of filtered modules over the absolute Galois group of K with certain restrictions on the successive quotients. This description is conditional upon (and its validity is equivalent to) certain Koszulity hypotheses about the Milnor K-theory/Galois cohomology of K. This paper also purports to explain what it means for an arbitrary nonnegatively graded ring to be Koszul. Tate motives with integral coefficients are discussed in the “Conclusions” section.
Keywords. Tate motives, Artin–Tate motives, motives with finite coefficients, Milnor–Bloch–Kato conjecture, Beilinson–Lichtenbaum conjecture, K(π,1) conjecture, Koszulity conjecture, Koszul algebras, nonflat Koszul rings, silly filtrations.
2010 Mathematics Subject Classification. 12G05, 19D45, 16S37.
Moscow Mathematical Journal