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## Blow-ups of locally conformally Kahler manifolds

A locally conformally Khler (LCK) manifold is a complex manifold which admits a covering endowed with a Kähler metric with respect to which the covering group acts through homotheties. We show that the blow-up of a compact LCK manifold along a complex submanifold admits an LCK structure if and only if this submanifold is globally conformally Kähler. We also prove that a twistor space (of a compact four-manifold, a quaternion-Kähler manifold, or a Riemannian manifold) cannot admit an LCK metric, unless it is Kähler.

An LCK manifold with potential is a compact quotient of a Kähler manifold X equipped with a positive Kähler potential f, such that the monodromy group acts on X by holomorphic homotheties and multiplies f by a character. The LCK rank is the rank of the image of this character, considered as a function from the monodromy group to real numbers. We prove that an LCK manifold with potential can have any rank between 1 and b1(M). Moreover, LCK manifolds with proper potential (ones with rank 1) are dense. Two errata to our previous work are given in the last Section.

A manifold M is locally conformally Kähler (LCK) if it admits a Kähler covering M˜ with monodromy acting by holomorphic homotheties. Let M be an LCK manifold admitting a holomorphic conformal flow of diffeomorphisms, lifted to a non-isometric homothetic flow on M˜ . We show that M admits an automorphic potential, and the monodromy group of its conformal weight bundle is Z.

A locally conformally Kähler (LCK) manifold is a complex manifold M admitting a Kähler covering \tilde{M}, such that its monodromy acts on this covering by homotheties. A compact LCK manifold is called LCK with potential if its covering admits an automorphic Kähler potential. It is known that in this case \tilde{M} is an algebraic cone, that is, the set of all non-zero vectors in the total space of an anti-ample line bundle over a projective orbifold. We start with an algebraic cone C, and show that the set of Kähler metrics with potential which could arise from an LCK structure is in bijective correspondence with the set of pseudoconvex shells, that is, pseudoconvex hypersurfaces in C meeting each orbit of the associated \mathbb{R}^{>0}-action exactly once and transversally. This is used to produce explicit LCK and Vaisman metrics on Hopf manifolds, generalizing earlier work by Gauduchon-Ornea, Belgun and Kamishima-Ornea.

We present an overview of recent results in locally conformally K¨ahler geometry, with focus on the topological properties which obstruct the existence of such structures on compact manifolds.

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.