We show that any symmetric, Baire measurable function from the complement of E0 to a finite set is constant on an E0-nonsmooth square. A simultaneous generalization of Galvin’s theorem that Baire measurable colorings admit perfect homogeneous sets and the Kanovei-Zapletal theorem canonizing Borel equivalence relations on E0-nonsmooth sets, this result is proved by relating E0-nonsmooth sets to ...

The study of moduli spaces by means of the period mapping has found its greatest success for moduli spaces of varieties with trivial canonical bundle, or more generally, varieties with Kodaira dimension zero. Now these moduli spaces play a pivotal rôle in the classification theory of algebraic varieties, since varieties with nonnegative Kodaira dimension which are not of general type admit bira...

In this paper we study composition series and em-beddings of Verma modules induced from "nonstandard" Borel subalgebras. This article can be viewed as a generalization of Futorny's work on imaginary Verma modules for Λ\ ' where the center of the Kac-Moody algebra acts nontrivially. Introduction. Let A be an indecomposable symmetrizable generalized Cartan matrix, fl(A) = n_ © f) 0 n+ the triangu...

We show that for any infinite countable group G and for any free Borel action G y X there exists an equivariant class-bijective Borel map from X to the free part Free(2G) of the 2-shift G y 2G. This implies that any Borel structurability which holds for the equivalence relation generated by Gy Free(2G) must hold a fortiori for all equivalence relations coming from free Borel actions of G. A rel...

The singular part of Borel transform of a QCD amplitude near the infrared renormalon can be expanded in terms of higher order Wilson coefficients of the operators associated with the renormalon. In this paper we observe that this expansion gives nontrivial constraints on the Borel amplitude that can be used to improve the accuracy of the ordinary perturbative expansion of the Borel amplitude. I...

Suppose that X is a standard Borel space, E is a countable Borel equivalence relation on X, and μ is an E-invariant Borel probability measure on X. We consider the circumstances under which for every countable non-abelian free group Γ, there is a Borel sequence (·r)r∈R of free actions of Γ on X, generating subequivalence relations Er of E with respect to which μ is ergodic, with the further pro...

We use the notion of Borel generators to give alternative methods for computing standard invariants, such as associated primes, Hilbert series, and Betti numbers, of Borel ideals. Because there are generally few Borel generators relative to ordinary generators, this enables one to do manual computations much more easily. Moreover, this perspective allows us to find new connections to combinator...

We establish the generic inexistence of stationary Borel probability measures for aperiodic Borel actions of countable groups on Polish spaces. Using this, we show that every aperiodic continuous action of a countable group on a compact Polish space has an invariant Borel set on which it has no σ-compact realization.

There are continuum many ≤B-incomparable equivalence relations induced by a free, Borel action of a countable non-abelian free group – and hence, there are 20 many treeable countable Borel equivalence relations which are incomparable in the ordering of Borel reducibility