Compactification équivariante d’un tore (d’après Brylinski et Künnemann)

We give a combinatorial, self-contained proof of the existence of a smooth equivariant compactification for an algebraic torus defined over an arbitrary field.     

Necessary and Sufficient Conditions for Mcshane’s Identity and Variations

Greg McShane introduced a remarkable identity for lengths of simple closed geodesics on the once punctured torus with a complete, finite volume hyperbolic structure. Bowditch later generalized this and gave sufficient conditions for the identity to hold for general type-preserving representations of a free group on two generators Γ to SL(2,C), this was further generalized by the authors to obtain sufficient conditions for a generalized McShane’s identity to hold for arbitrary (not necessarily type-preserving) non-reducible representations in Tan et al. (Submitted). Here we extend the above by giving necessary and sufficient conditions for the generalized McShane identity to hold (Akiyoshi, Miyachi and Sakuma had proved it for type-preserving representations). We also give a version of Bowditch’s variation of McShane’s identity to once-punctured torus bundles, in the case where the monodromy is generated by a reducible element, and provide necessary and sufficient conditions for the variations to hold.     

Classification of two-orbit varieties

We obtain the classification of two-orbit varieties, i.e. the normal complete complex algebraic varieties on which a reductive complex algebraic group acts with two orbits. We prove also Luna's conjecture saying that these varieties are spherical, i.e. admit a dense orbit of a Borel subgroup. Received: September 1, 2000     

Residues in toric varieties

We study residues on a complete toric variety X, which are defined in terms of the homogeneous coordinate ring of X.We first prove a global transformation law for toric residues. When the fan of the toric variety has a simplicial cone of maximal dimension, we can produce an element with toric residue equal to 1. We also show that in certain situations, the toric residue is an isomorphism on an appropriate graded piece of the quotient ring. When X is simplicial, we prove that the toric residue is a sum of local residues. In the case of equal degrees, we also show how to represent X as a quotient (Y\{0})/C* such that the toric residue becomes the local residue at 0 in Y.     

Log Structures on Generalized Semi-Stable Varieties

In this paper, a class of morphisms which have a kind of singularity weaker than normal crossing is considered. We construct the obstruction such that the so-called semi-stable log structures exists if and only if the obstruction vanishes. In the case of no power, if the obstruction vanishes, then the semi-stable log structure is unique up to a unique isomorphism. So we obtain a kind of canonical structure on this family of morphisms.     

左分式半群的几个性质

讨论了若干半群类在取左分式半群下的封闭性，给出了关于左发式半群的一个同构定理。     

Canonical models and the law of requisite variety

The law of requisite variety from cybernetics is shown to be related to the reachability and observability properties of a dynamical control system. In particular, it is established that the transmission of all input variety to the system output is possible if and only if the system is canonical, i.e., completely reachable and completely observable.Dedicated to G. LeitmannThis work was partially supported by the National Science Foundation under Grants Nos. CEE-81-00491 and CEE-81-10778.     

k-Arcs,Hyperovals, Partial Flocks and Flocks

Some recent results on k-arcs and hyperovals of PG(2,q),on partial flocks and flocks of quadratic cones of PG(3,q),and on line spreads in PG(3,q) are surveyed. Also,there is an appendix on how to use Veronese varieties as toolsin proving theorems.     

Strictly pseudoconvex domains and algebraic varieties

In the paper, we consider applications of strictly pseudoconvex domains to the problems of algebraicity and rationality. We give a new proof of the Kodaira theorem on the algebraicity of a surface and we also prove a multidimensional version of this theorem. Theorems analogous to the Hodge index theorem and the Lefschetz theorem about (1, 1)-classes are obtained for strictly pseudoconvex domains. Conjectures on the geometry of strictly pseudoconvex domains on algebraic surfaces are formulated. Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 414–422, September, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00225 and by the International Science Foundation under grant No. 508.