In the theory of affine SL(2)-embeddings, which was constructed in 1973 by Popov, a locally transitive action of the group SL(2) on a normal affine three-dimensional variety X is determined by a pair (p/q, r), where 0 < p/q ≤ 1 is a rational number written as an irreducible fraction and called the height of the action, while r is a positive integer that is the order of the stabilizer of a generic point. In the present paper it is shown that the variety X is toric, that is, it admits a locally transitive action of an algebraic torus if and only if the number r is divisible by q - p. For that, the following criterion for an affine G/H-embedding to be toric is proved. Let X be a normal affine variety, G a simply connected semisimple group acting regularly on X, and H ⊂ G a closed subgroup such that the character group X(H) of the group H is finite. If an open equivariant embedding G/H → X is defined, then X is toric if and only if there exist a quasitorus T̂ and a (G × T̂)-module V such that X ≅G V//T̂. In the substantiation of this result a key role is played by Cox's construction in toric geometry.

We prove that the bound from the theorem on 'economic' maps is best possible. Namely, for m > n + d we construct a map from an n-dimensional simplex to an m-dimensional Euclidean space for which (and for any close map) there exists a d-dimensional plane whose preimage has cardinality not less than the upper bound \(dn + n + 1)/(m - n - d)] + d from the theorem on 'economic' maps. Bibliography: 16 titles.

We study the limit as α → 0+ of the long-time dynamics for various approximate α-models of a viscous incompressible fluid and their connection with the trajectory attractor of the exact 3D Navier-Stokes system. The α-models under consideration are divided into two classes depending on the orthogonality properties of the nonlinear terms of the equations generating every particular α-model. We show that the attractors of α-models of class I have stronger properties of attraction for their trajectories than the attractors of α-models of class II. We prove that for both classes the bounded families of trajectories of the α-models considered here converge in the corresponding weak topology to the trajectory attractor 0 of the exact 3D Navier-Stokes system as time t tends to infinity. Furthermore, we establish that the trajectory attractor α of every α-model converges in the same topology to the attractor 0 as α → 0+. We construct the minimal limits min ⊆ 0 of the trajectory attractors α for all α-models as α → 0+. We prove that every such set min is a compact connected component of the trajectory attractor 0, and all the min are strictly invariant under the action of the translation semigroup. © 2016 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

We prove that any G-del Pezzo threefold of degree 4, except for a one-parameter family and four distinguished cases, can be equivariantly reconstructed to the projective space ℙ3, a quadric Q ⊂ ℙ4 , a G-conic bundle or a del Pezzo fibration. We also show that one of these four distinguished varieties is birationally rigid with respect to an index 2 subgroup of its automorphism group. © 2016 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

This paper is on the classical Knotting Problem: for a given manifold N and a number m describe the set of isotopy classes of embeddings N->S^m. We study the specific case of knotted tori, i. e. the embeddings S^p x S^q -> S^m. The classification of knotted tori up to isotopy in the metastable dimension range m>p+3q/2+3/2, p<q+1, was given by A. Haefliger, E. Zeeman and A. Skopenkov. We consider the dimensions below the metastable range, and give an explicit criterion for the finiteness of this set of isotopy classes in the 2-metastable dimension:

Theorem. Assume that p+4q/3+2<m<p+3q/2+2 and m>2p+q+2. Then the set of smooth embeddings S^p x S^q -> S^m up to isotopy is infinite if and only if either q+1 or p+q+1 is divisible by 4.

Our approach to the classification is based on an analogue of the Koschorke exact sequence from the theory of link maps. This sequence involves a new beta-invariant of knotted tori. The exactness is proved using embedded surgery and the Habegger-Kaiser techniques of studying the complement.

Continuous Morse-Smale flows on closed manifolds whose nonwandering set consists of three equilibrium positions are considered. Necessary and sufficient conditions for topological equivalence of such flows are obtained and the topological structure of the underlying manifolds is described. Bibliography: 36 titles. © 2016 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

We study the Cox realization of an affine variety, that is, a canonical representation of a normal affine variety with finitely generated divisor class group as a quotient of a factorially graded affine variety by an action of the Neron-Severi quasitorus. The realization is described explicitly for the quotient space of a linear action of a finite group. A universal property of this realization is proved, and some results in the divisor theory of an abstract semigroup emerging in this context are given. We show that each automorphism of an affine variety can be lifted to an automorphism of the Cox ring normalizing the grading. It follows that the automorphism group of an affine toric variety of dimension ≥ 2 without nonconstant invertible regular functions has infinite dimension. We obtain a wild automorphism of the three-dimensional quadratic cone that rises to the Anick automorphism of the polynomial algebra in four variables.

We put forward a method for constructing semiorthogonal decompositions of the derived category of G-equivariant sheaves on a variety X under the assumption that the derived category of sheaves on X admits a semiorthogonal decomposition with components preserved by the action of the group G on X. This method is used to obtain semiorthogonal decompositions of equivariant derived categories for projective bundles and blow-ups with a smooth centre as well as for varieties with a full exceptional collection preserved by the group action. Our main technical tool is descent theory for derived categories.

We classify $\mathbb Q$-Fano threefolds of Fano index > 2 and sufficiently big degree.

We say that a group G acts infinitely transitively on a set X if for every m ε N the induced diagonal action of G is transitive on the cartesian mth power X m\δ with the diagonals removed. We describe three classes of affine algebraic varieties such that their automorphism groups act infinitely transitively on their smooth loci. The first class consists of normal affine cones over flag varieties, the second of nondegenerate affine toric varieties, and the third of iterated suspensions over affine varieties with infinitely transitive automorphism groups. Bibliography: 42 titles.

We prove that every complete foliation (M, F) of codimension q > 1 is either Riemannian or a (Conf (S^q), S^q)-foliation. We further prove that if (M, F) is not Riemannian, it has a global attractor which is either a nontrivial minimal set or a closed leaf or a union of two closed leaves. In particular, every proper conformal non-Riemannian foliation (M, F) has a global attractor which is either a closed leaf or a union of two closed leaves, and the space of all non-closed leaves is a connected q-dimensional orbifold. We show that every countable group of conformal transformations of the sphere S^q can be realized as a global holonomy group of complete conformal foliation. Examples of complete conformal foliations with exceptional and exotic minimal sets as global attractors are constructed.

It is demonstrated that every (0, 1)-matrix of size n×m having Boolean rank n contains a column with at least √n/2 − 1 zero entries. This bound is shown to be asymptotically optimal. As a corollary, it is established that the size of a full-rank Boolean matrix is bounded from above by a function of its tropical and determinantal ranks.

The Barth-Van de Ven-Tyurin-Sato Theorem states that any finite-rank vector bundle on the complex projective ind-space P∞ is isomorphic to a direct sum of line bundles. We establish sufficient conditions on a locally complete linear ind-variety X which ensure that the same result holds on X. We then exhibit natural classes of locally complete linear ind-varieties which satisfy these sufficient conditions. © 2015 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

An algebraic technique is presented that does not use results of model theory and makes it possible to construct a general Galois theory of arbitrary nonlinear systems of partial differential equations. The algebraic technique is based on the search for prime differential ideals of special form in tensor products of differential rings. The main results demonstrating the work of the technique obtained are the theorem on the constructedness of the differential closure and the general theorem on the Galois correspondence for normal extensions.

In this paper a six-valued two-dimensional formal group with ring of coefficients **Λ**2, lying in **Ω***U*[1/2], is constructed. It is proved that the ring **Λ**2[1/2] coincides with the image of the ring **Ω***SU*[1/2] in the ring **Ω***U*[1/2].

In this work we explicitly calculate the syzygies of the quadratic Veronese embedding ℙ (V) ⊂ ℙ (Sym2V) as representations of the group GL(V ). Resolutions of the sheaves Oℙ (V)(i) are also constructed in the category D(ℙ (Sym2 V)). © 2017 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.