### Article

## Complex rotation numbers

We investigate the notion of complex rotation number which was introduced by V.I.Arnold in 1978. Let f: R/Z -> R/Z be a (real) analytic orientation preserving circle diffeomorphism and let omega in C/Z be a parameter with positive imaginary part. Construct a complex torus by glueing the two boundary components of the annulus { z in C/Z : 0< Im(z) < Im(omega)} via the map f+omega. This complex torus is isomorphic to C/(Z+ tau Z) for some appropriate tau in C/Z. According to V.Moldavskis, if the ordinary rotation number rot(f+omega0) is Diophantine and if omega tends to omega0 non tangentially to the real axis, then tau tends to rot(f+omega0). We show that the Diophatine and non tangential assumptions are unnecessary: if rot(f+omega0) is irrational then tau tends to rot(f+omega0) as omega tends to omega0. This, together with results of N. Goncharuk [4], motivates us to introduce a new fractal set (``bubbles'') given by the limit values of tau as omega tends to the real axis. For the rational values of rot (f+omega0), these limits do not necessarily coincide with rot(f+omega0) and form a countable number of analytic loops in the upper half-plane.

We summarize some of the recent works, devoted to the study of one-dimensional (pseudo)group actions and codimension one foliations. We state a conjectural alternative for such actions (generalizing the already obtained results) and describe the properties in both alternative cases. We also discuss the generalizations for holomorphic one-dimensional actions. Finally, we state some open questions that seem to be already within the reach.

A three-parametrical family of ODEs on a torus arises from a model of Josephson effect in a resistive case when a Josephson junction is biased by a sinusoidal microwave current. We study asymptotics of Arnold tongues of this family on the parametric plane (the third parameter is fixed) and prove that the boundaries of the tongues are asymptotically close to Bessel functions.

We prove that any compact Kahler 3-dimensional manifold which has no nontrivial complex subvarieties is a torus. This is a very special case of a general conjecture on the structure of so-called simple manifolds, central in the bimeromorphic classication of compact Kahler manifolds. The proof follows from the Brunella pseudo-eectivity theorem, combined with fundamental results of Siu and of the second author on the Le- long numbers of closed positive (1;1)-currents, and with a version of the hard Lefschetz theorem for pseudo-eective line bundles, due to Takegoshi and Demailly-Peternell- Schneider. In a similar vein, we show that a normal compact and Kahler 3-dimensional analytic space with terminal singularities and nef canonical bundle is a cyclic quotient of a simple nonprojective torus if it carries no eective divisor. This is a crucial step towards completing the bimeromorphic classication of compact Kahler threefolds.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.