Manifolds with locally standard half-dimensional torus actions represent a large and important class of spaces. Cohomology rings of such manifolds are known in particular cases but in general even Betti numbers are difficult to compute. Our approach to this problem is the following: we consider the orbit type filtration on a manifold with locally standard action and study the induced spectral sequence in homology. It collapses at a second page only in the case when the orbit space is homologically trivial. The cohomology ring in this case was already computed. Nevertheless, we can completely describe the spectral sequence under more general assumptions, namely when all proper faces of the orbit space are acyclic. The theory of sheaves and cosheaves on finite partially ordered sets is used in the computation. The second page of the spectral sequence can be described as the cohomology of a certain sheaf on the dual simplicial poset, whose value on a simplex is the homology of the corresponding toric orbit. We study this and related sheaves and establish the generalizations of the Poincare duality and the Zeeman-McCrory spectral sequence for sheaves of ideals of exterior algebras.
We enumerate the local Petrovskii lacunas (that is, the domains of local regularity of the principal fundamental solutions of strictly hyperbolic PDE's with constant coefficients in R^N) at the parabolic singular points of their wavefronts (i.e., at the points of types P_8^1, P_8^2, +X_9, -X_9, X_9^1, X_9^2, J_10^1, J_10^3). These points form the next difficult class of the natural classification of singular points after the so-called simple singularities A_k, D_k, E_6, E_7, E_8, studied previously. Also we promote a computer program counting for topologically different Morsifications of critical points of smooth functions, and hence also for local components of the complement of a generic wavefront at its singular points.
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 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.
It is shown that every m-by-n matrix that consists of zeros and ones and has Boolean rank equal to n has a column in which at least sqrt(n)/2-1 entries are zero. It is proved that the bound obtained is asymptotically best possible. As an application of the results obtained we show that matrices of full Boolean rank can not have arbitrarily large size provided that those matrices have their tropical or determinantal ranks bounded.