Volume 51, Nº 4 (2017)

In the article we introduce the notion of logarithmic differential forms with poles along a Cartier divisor given on a variety with singularities, discuss some properties of such forms, and describe highly efficient methods for computing the Poincaré series and generators of modules of logarithmic differential forms in various situations. We also examine several concrete examples by applying these methods to the study of divisors on varieties with singularities of many types, including quasi-homogeneous complete intersections, normal, determinantal, and rigid varieties, and so on.

We consider a parametric family of continuous maps of a compact metric space which continuously depend on a parameter ranging over a metric space. The topological pressure of maps in any such family is studied as a function of the parameter from the viewpoint of the Baire classification of functions.

The existence of unconditional bases of reproducing kernels in the Fock-type spaces F_φ with radial weights φ is studied. It is shown that there exist functions φ(r) of arbitrarily slow growth for which ln r = o(φ(r)) as r → ∞ and there are no unconditional bases of reproducing kernels in the space F_φ. Thus, a criterion for the existence of unconditional bases cannot be given only in terms of the growth of the weight function.

Various weight functions for the model of a 2×n square lattice are defined so that the graded Euler characteristic for the complex corresponding to this model remain equal to the usual one. Differentials preserving these gradings and determining one-dimensional cohomology are also specified.

Systems of equations f₁ = ··· = f_n−1 = 0 in ℝⁿ = {x} having the solution x = 0 are considered under the assumption that the quasi-homogeneous truncations of the smooth functions f₁,..., f_n−1 are independent at x ≠ 0. It is shown that, for n ≠ 2 and n ≠ 4, such a system has a smooth solution which passes through x = 0 and has nonzero Maclaurin series.

The possibility of factoring a product of nuclear operators through operators in the von Neumann–Schatten class is considered. In particular, generally, the product of two nuclear operators can be factored only through a Hilbert–Schmidt operator.