Oriented Chow groups, Hermitian K-theory and the Gersten conjecture

We show that the oriented Chow groups of Barge–Morel appear in the E ₂-term of the coniveau spectral sequence for Hermitian K-theory. This includes a localization theorem and the Gersten conjecture (over infinite base fields) for Hermitian K-theory. We also discuss the conjectural relationship between oriented and higher oriented Chow groups and Levine’s homotopy coniveau spectral sequence when applied to Hermitian K-theory.     

Remarks on the critical graph conjecture

The vertex-critical graph conjecture (critical graph conjecture respectively) states that every vertex-critical (critical) graph has an odd number of vertices. In this note we prove that if G is a critical graph of even order, then G has at least three vertices of less-than-maximum valency. In addition, if G is a 3-critical multigraph of smallest even order, then G is simple and has no triangles.     

Chow groups and Borel-Moore schemes

Summary In this paper we study the Chow groups of schemes for which the class map to Borel-Moore homology is an isomorphism. Then we determine the Chow groups of the scheme Cop^k P ⁿ parametrizing finite coplanary subschemes of lenght k ofP ⁿ and of the variety of «complete S-tuples» of Le Barz.The authors were partially supported by the DGICYT.     

On topological filtrations of groups

The (weak) geometric simple connectivity and the quasi-simple filtration are topological notions of manifolds, which may be defined for discrete groups too. It turns out that they are equivalent for finitely presented groups, but the main problem is the absence of examples of groups which do not satisfy them. In this note we study some algebraic classes of groups with respect to these properties.     

Remarks on the Lax conjecture for hyperbolic polynomials

A conjecture of Lax [P. Lax, Differential equations, difference equations and matrix theory, Commun. Pure Appl. Math. 11 (1958) 175–194] says that every hyperbolic polynomial in two space variables is the determinant of a symmetric hyperbolic matrix. The conjecture has recently been proved by Lewis–Parillo–Ramana, based on previous work of Dubrovin and Helton–Vinnikov. In this note we prove related results for polynomials in several space variables which have rotational symmetries.     

Zero cycles on conic fibrations and a conjecture of Bloch

Let X be a smooth projective surface over a number field k. Let (CH₀(X)) denote the Chow group of zero-cyles modulo rational equivalence on X. Let CH₀(X) be the subgroup of CH ₀(X) consisting of classes which vanish when going over to an arbitrary completion of k. Bloch put forward a conjecture asserting that this group is isomorphic to the Tate-Shafarevich group of a certain Galois module atttached to X. In this paper, we disprove this general conjecture. We produce a conic bundle X over an elliptic curve, for which the group (CH₀(X) is not zero, but the Galois-theoretic Tate-Shafarevich group vanishes.     

Remarks on a conjecture on certain integer sequences

Summary The periodicity of sequences of integers <InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>(a_{n})_{n\in\mathbb Z}$ satisfying the inequalities <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation> 0 \le a_{n-1}+\lambda a_n +a_{n+1} < 1 \ (n \in {\mathbb Z}) $$ is studied for real $ \lambda $ with $|\lambda|< 2$. Periodicity is proved in case $ \lambda $ is the golden ratio; for other values of $ \lambda $ statements on possible period lengths are given. Further interesting results on the morphology of periods are illustrated. The problem is connected to the investigation of shift radix systems and of Salem numbers.     

Robinson's conjecture on Abelian groups

Let x₁,…,x_n be elements of a finite abelian group G, having respective orders k₁,…,k_n such that (x₁−1)(x₂−1)⋯(x_n−1)=0 in $\text{Z}\text{(G)}$, where n⩽1. We prove that min k_i≤n−1 with equalityossible if only if n−1 is prime. If all k_i are equal, and not divisible by the cube of a prime, we prove $\text{n⩾k}{}_1\text{(1+}\text{1}\text{r)}$ where r is the least prime dividing k₁. We also establish an inequality concerning coverings of a set by subsets.     

Whitehead groups and the Bass conjecture

This paper will be concerned with proving that certain Whitehead groups of torsion-free elementary amenable groups are torsion groups and related results, and then applying these results to the Bass conjecture. In particular we shall establish the strong Bass conjecture for an arbitrary elementary amenable group. Mathematics Subject Classification (2000): 19A31, 19B28, 16A27, 16E20, 20C07The first author was supported in part by the National Science Foundation     

Remarks on Hard Lefschetz conjectures on Chow groups Dedicated in celebration of SCIENCE CHINA on the occasion of its 60th birthday

We propose two conjectures of Hard Lefschetz type on Chow groups and prove them for some special cases. For abelian varieties, we show that they are equivalent to the well-known conjectures of Beauville and Murre.     

Generic points of quadrics and Chow groups

In this article we study the sufficient conditions for the k̅-defined element of the Chow group of a smooth variety to be k-rational (defined over k). For 0-cycles this question was addressed earlier. Our methods work for cycles of arbitrary dimension. We show that it is sufficient to check this property over the generic point of a quadric of sufficiently large dimension. Among the applications one should mention the uniform construction of fields with all known u-invariants.     

Equivariant Chow groups for torus actions

We study Edidin and Graham's equivariant Chow groups in the case of torus actions. Our main results are: (i) a presentation of equivariant Chow groups in terms of invariant cycles, which shows how to recover usual Chow groups from equivariant ones; (ii) a precise form of the localization theorem for torus actions on projective, nonsingular varieties; (iii) a construction of equivariant multiplicities, as functionals on equivariant Chow groups; (iv) a construction of the action of operators of divided differences on theT-equivariant Chow group of any scheme with an action of a reductive group with maximal torusT. We apply these results to intersection theory on varieties with group actions, especially to Schubert calculus and its generalizations. In particular, we obtain a presentation of the Chow ring of any smooth, projective spherical variety.     

Pure motives with representable Chow groups

Let k be an algebraically closed field. We show using Kahn's and Sujatha's theory of birational motives that a Chow motive over k whose Chow groups are all representable (in the sense of Definition 2.1) belongs to the full and thick subcategory of motives generated by the twisted motives of curves.     

Solvable groups and Bass' conjecture

In this Note, we examine Bass' conjecture on the values of the Hattori-Stallings rank of idempotent matrices with coefficients in group rings, in the special case of solvable groups. We show how one can use a result of Linnell, in order to impose some restrictions on the structure of a (potential) counter-example.     

A conjecture on arithmetic fundamental groups

The conjecture is the following: Over an algebraic variety over a finite field, the geometric monodromy group of every smooth is finite. We indicate how to prove this for rank 2, using results of Drinfeld. We also show that the conjecture implies that certain deformation rings of Galois representations are complete intersection rings. This material is based upon work supported by the National Science Foundation under Grant No. 9970049.     

On the Chow groups of certain cubic fourfolds

This note is about the Chow groups of a certain family of smooth cubic fourfolds. This family is characterized by the property that each cubic fourfold X in the family has an involution such that the induced involution on the Fano variety F of lines in X is symplectic and has a K3 surface S in the fixed locus. The main result establishes a relation between X and S on the level of Chow motives. As a consequence, we can prove finite-dimensionality of the motive of certain members of the family.     

Witt groups and the elementary type conjecture

Let A be an algebra with involution * over a field F of characteristic zero and Id(A, *) the ideal of the free algebra with involution of *-identities of A. By means of the representation theory of the hyperoctahedral group Z ₂wrS _n we give a characterization of Id(A, *) in case the sequence of its *-codimensions is polynomially bounded. We also exhibit an algebra G ₂ with the following distinguished property: the sequence of *-codimensions of Id(G ₂, *) is not polynomially bounded but the *-codimensions of any T-ideal U properly containing Id(G ₂, *) are polynomially bounded.