Volume 2006, Issue 600

We study the exterior product on 0-cycles modulo rational equivalence. The main tools used are higher cycle- and Abel-Jacobi-classes developed in [ J. Lewis , A filtration on the Chow groups of a complex projective variety, Compos. Math. 128 (2001), no. 3, 299–322.] and [ M. Kerr , Higher Abel-Jacobi maps for 0-cycles, preprint, math.AG/0504086.]. The theorem of [ A. Rosenschon and M. Saito , Cycle map for strictly decomposable cycles, Amer. J. Math. 125 (2003), no. 4, 773–790.] (applied to 0-cycles) appears as a special case of our results.

The goal of this paper is to study point distributions in the multi-dimensional unit cube which possess the structure of finite abelian groups with respect to certain p -ary arithmetic operations. Such distributions can be thought of as finite subgroups in a compact totally disconnected group of the Cantor type. We apply the methods of L q harmonic analysis to estimate very precisely the L q -discrepancies for such distributions. Following this approach, we explicitly construct point distributions with the minimal order of the L q -discrepancy for each q , 1 < q < ∞.

Let k be a characteristic zero field, C a k -algebra and M a square zero two sided ideal of C . We obtain a new mixed complex, simpler than the canonical one, giving the Hochschild and cyclic homologies of C relative to M . This complex resembles the canonical reduced mixed complex of an augmented algebra. We begin the study of our complex showing that it has a harmonic decomposition like to the one considered by Cuntz and Quillen for the normalized mixed complex of an algebra. We also give new proofs of two theorems of Goodwillie, obtaining a light improvement of one of them.

We study birational transformations into elliptic fibrations and birational automorphisms of quasismooth anticanonically embedded weighted Fano 3-fold hypersurfaces with terminal singularities classified by A. R. Iano-Fletcher, J. Johnson, J. Kollár, and M. Reid.

We show that strictly convex surfaces expanding by the inverse Gauß curvature flow converge to infinity in finite time. After appropriate rescaling, they converge to spheres. We describe the algorithm to find our main test function.

Introduction In symplectic geometry, it is often useful to consider the so-called Poisson bracket on the algebra of functions on a C ∞ symplectic manifold M . The bracket determines, and is determined by, the symplectic form; however, many of the features of symplectic geometry are more conveniently described in terms of the Poisson bracket. When one turns to the study of symplectic manifolds in the holomorphic or algebro-geometric setting, one expects the Poisson bracket to be even more useful because of the following observation: the bracket is a purely algebraic structure, and it generalizes immediately to singular algebraic varieties and complex-analytic spaces. The appropriate notion of singularities for symplectic algebraic varieties has been introduced recently by A. Beauville [ A. Beauville , Symplectic singularities, Invent. Math. 139 (2000), 541–549.] and studied by Y. Namikawa [ Y. Namikawa , Deformation theory of singular symplectic n -folds, Math. Ann. 319 (2001), 597–623.], [ Y. Namikawa , Extension of 2-forms and symplectic varieties, J. reine angew. Math. 539 (2001), 123–147.]. However, the theory of singular symplectic algebraic varieties is in its starting stages; in particular, to the best of our knowledge, the Poisson methods have not been used yet. This is the goal of the present paper.

In the Grothendieck cohomology of a flag variety G/H there are two canonical additive bases, namely, the Demazure basis [ M. Demazure , Désingularisation des variétés de Schubert généralisées, Ann. Sci. Éc. Norm. Sup. (4) 7 (1974), 53–88.] and the Grothendieck basis [ B. Kostant and S. Kumar , T -equivariant K -theory of generalized flag varieties, J. Diff. Geom. 32 (1990), no. 2, 549–603.]. We present explicit formulae that reduce the multiplication of these basis elements to the Cartan numbers of G .

We describe a set of defining relations of the tame automorphism group TA 3 ( F ) of the polynomial algebra F [x 1 , x 2 , x 3 ] in variables x 1 , x 2 , x 3 over an arbitrary field F of characteristic 0.