Volume 2010, Issue 642

We consider Frobenius algebras and their bimodules in certain abelian monoidal categories. In particular we study the Picard group of the category of bimodules over a Frobenius algebra, i.e. the group of isomorphism classes of invertible bimodules. The Rosenberg-Zelinsky sequence describes a homomorphism from the group of algebra automorphisms to the Picard group, which however is typically not surjective. We investigate under which conditions there exists a Morita equivalent Frobenius algebra for which the corresponding homomorphism is surjective. One motivation for our considerations is the orbifold construction in conformal field theory.

The purpose of the present article is to initiate Arakelov theory of noncommutative arithmetic surfaces. Roughly speaking, a noncommutative arithmetic surface is a noncommutative projective scheme of cohomological dimension 1 of finite type over Spec ℤ. An important example is the category of coherent right 𝒪-modules, where 𝒪 is a coherent sheaf of 𝒪 X -algebras and 𝒪 X is the structure sheaf of a commutative arithmetic surface X . Since smooth hermitian metrics are not available in our noncommutative setting, we have to adapt the definition of arithmetic vector bundles on noncommutative arithmetic surfaces. Namely, we consider pairs (ℰ, β ) consisting of a coherent sheaf ℰ and an automorphism β of the real sheaf ℰ ℝ induced by ℰ. We define the intersection of two such objects using the determinant of the cohomology and prove a Riemann-Roch theorem for arithmetic line bundles on noncommutative arithmetic surfaces.

A stability study of the correlations of multiplicative arithmetic functions yields necessary and sufficient conditions that the frequency distributions naturally attached to sums of additive arithmetic functions ƒ 1 ( n ) + ƒ 2 ( N – n ) on the integers not exceeding N possess a limiting distribution as N traverses the positive integers, or the positive primes. Moreover, the functions ƒ j may be allowed a wide class of unbounded renormalizations.

We deal with homogeneous toric bundles M over generalized flag manifolds G C / P , where G is a compact semisimple Lie group and P a parabolic subgroup. Using symplectic data, we provide a simple characterization of the homogeneous toric bundles M which are Fano; we then show that a homogeneous toric bundle M admits a Kähler-Ricci solitonic metric if and only if it is Fano.

We give a number of new characterizations of the Jiang–Su algebra 𝒵, both intrinsic and extrinsic, in terms of C *-algebraic, dynamical, topological and K -theoretic conditions. Along the way we study divisibility properties of C *-algebras, we give a precise characterization of those unital C *-algebras of stable rank one that admit a unital embedding of the dimension-drop C *-algebra Z n, n +1 , and we prove a cancellation theorem for the Cuntz semigroup of C *-algebras of stable rank one.

To a definable subset of (or to a scheme of finite type over ℤ p ) one can associate a tree in a natural way. It is known that the corresponding Poincaré series ∑ N λ Z λ ∈ ℤ[[ Z ]] is rational, where N λ is the number of nodes of the tree at depth λ. This suggests that the trees themselves are far from arbitrary. We state a conjectural, purely combinatorial description of the class of possible trees and provide some evidence for it. We verify that any tree in our class indeed arises from a definable set, and we prove that the tree of a definable set (or of a scheme) lies in our class in three special cases: under weak smoothness assumptions, for definable subsets of , and for one-dimensional sets.

In recent papers [Honda, J. Diff. Geom 82: 411–444, 2009], [Honda, Invent. Math 174, 463–504, 2008], we gave explicit description of some new Moishezon twistor spaces. In this paper, developing the method in the papers much further, we explicitly give projective models of a number of new Moishezon twistor spaces, as conic bundles over some rational surfaces (called minitwistor spaces). These include the twistor spaces studied in the papers as very special cases.