Applications of balanced pairs

Let (X, Y) be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to (X, Y), and then give some equivalent characterizations when a relative cotorsion pair is hereditary or perfect. We prove that if the X-resolution dimension of Y (resp. Y-coresolution dimension of X) is finite, then the bounded homotopy category of Y (resp. X) is contained in that of X (resp. Y). As a consequence, we get that the right X-singularity category coincides with the left Y-singularity category if the X-resolution dimension of Y and the Y-coresolution dimension of X are finite.     

Free groups and ends of graphs

Let X be a connected graph. An automorphism of X is said to be parabolic if it leaves no finite subset of vertices in X invariant and fixes precisely one end of X and hyperbolic if it leaves no finite subset of vertices in X invariant and fixes precisely two ends of X. Various questions concerning dynamics of parabolic and hyperbolic automorphisms are discussed.The set of ends which are fixed by some hyperbolic element of a group G acting on X is denoted by ?(G). If G contains a hyperbolic automorphism of X and G fixes no end of X, then G contains a free subgroup F such that ?(F) is dense in ?(G) with respect to the natural topology on the ends of X.As an application we obtain the following: A group which acts transitively on a connected graph and fixes no end has a free subgroup whose directions are dense in the end boundary.     

Chow points of $${\mathbb{C}}$$-orbits

We consider free algebraic actions of the additive group of complex numbers on a complex vector space X embedded in the complex projective space. We find an explicit formula for the map p that assigns to a generic point x ? X the Chow point of the closure of the orbit through x. The properties Hausdorff quotient topology and proper action are equivalently characterized by the closure of the image of p in the closed Chow variety.     

Hardy and Carleson Measure Spaces Associated with Operators on Spaces of Homogeneous Type

Let (X, d, μ) be a metric measure space with doubling property. The Hardy spaces associated with operators L were introduced and studied by many authors. All these spaces, however, were first defined by L ²(X) functions and finally the Hardy spaces were formally defined by the closure of these subspaces of L ²(X) with respect to Hardy spaces norms. A natural and interesting question in this context is to characterize the closure. The purpose of this paper is to answer this question. More precisely, we will introduce \({CMO}_{L}^{p}(X)\), the Carleson measure spaces associated with operators L, and characterize the Hardy spaces associated with operators L via \(({CMO}_{L}^{p}(X))'\), the distributions of \({CMO}_{L}^{p}(X)\).     

Classification of finite groups satisfying a minimal condition

If H is a subgroup of a finite group G then we denote the normal closure of H in G by H ^G . We call G a PE-group if every minimal subgroup X of G satisfies N ^G (X) ∩ X ^G = X. The authors classify the finite non-PE-groups whose maximal subgroups of even order are PE-groups.     

On the stability of tangent bundle on double coverings

Let Y be a smooth projective surface defined over an algebraically closed field k with char k≠2, and let π : X →Y be a double covering branched along a smooth divisor. We show that y_X is stable with respect to π~*H if the tangent bundle y_Y is semi-stable with respect to some ample line bundle H on Y.     

Morse inequalities for Fourier components of Kohn–Rossi cohomology of CR manifolds with $$S^1$$-action

Let X be a compact connected CR manifold of dimension \(2n-1, n\ge 2\) with a transversal CR \(S^1\)-action on X. We study the Fourier components of the Kohn–Rossi cohomology with respect to the \(S^1\)-action. By studying the Szegö kernel of the Fourier components we establish the Morse inequalities on X. Using the Morse inequalities we have established on X we prove that there are abundant CR functions on X when X is weakly pseudoconvex and strongly pseudoconvex at a point.     

Grauert’s line bundle convexity,reduction and Riemann domains

We consider a convexity notion for complex spaces X with respect to a holomorphic line bundle L over X. This definition has been introduced by Grauert and, when L is analytically trivial, we recover the standard holomorphic convexity. In this circle of ideas, we prove the counterpart of the classical Remmert’s reduction result for holomorphically convex spaces. In the same vein, we show that if H⁰(X,L) separates each point of X, then X can be realized as a Riemann domain over the complex projective space Pⁿ, where n is the complex dimension of X and L is the pull-back of O(1).     

Variations on a theorem of Arhangel’skii and Pytkeev

We identify continuous real-valued functions on a Tychonoff space X with their (closed) graphs thus allowing for C(X) to naturally inherit the lower Vietoris topology from the ambient hyperspace. We then calculate a bitopological version of tightness using the weak Lindelöf numbers of finite powers of X. We also characterize bitopological versions of countable fan and strong fan tightness of the point-open topology with respect to the lower Vietoris topology on C(X) in terms of suitable covering properties of the powers X ⁿ formulated using the language of S ₁ and S _fin selection principles.     

On a Variety Related to the Commuting Variety of a Reductive Lie Algebra

For a reductive Lie algbera over an algbraically closed field of charasteristic zero, we consider a Borel subgroup B of its adjoint group, a Cartan subalgebra contained in the Lie algebra of B and the closure X of its orbit under B in the Grassmannian. The variety X plays an important role in the study of the commuting variety. In this note, we prove that X is Gorenstein with rational singularities.     

Determinant Bundle Over the Universal Moduli Space of Principal Bundles Over the Teichmüller Space

For a Riemann surface X and the moduli of regularly stable G-bundles M, there is a naturally occuring “adjoint” vector bundle over X × M. One can take the determinant of this vector bundle with respect to the projection map onto M. Our aim here is to study the curvature of the determinant bundle as the conformal structure on X varies over the Teichmüller space.     

Integral approximation of the characteristic function of an interval and the Jackson inequality in C($$ \mathbb{T} $$)

We give an analog of D.O. Orlov’s theorem on semiorthogonal decompositions of the derived category of projective bundles for the case of equivariant derived categories. Under the condition that the action of a finite group on the projectivization X of a vector bundle E is compatible with the twisted action of the group on the bundle E, we construct a semiorthogonal decomposition of the derived category of equivariant coherent sheaves on X into subcategories equivalent to the derived categories of twisted sheaves on the base scheme.     

On the singularity of Quillen metrics

Let π: X → S be a holomorphic map from a compact Kähler manifold (X,g _X ) to a compact Riemann surface S. Let Σ_π be the critical locus of π and let Δ  =  π(Σ_π) be the discriminant locus. Let (ξ, h _ξ) be a holomorphic Hermitian vector bundle on X. We determine the singularity of the Quillen metric on det Rπ_*ξ near Δ with respect to g _X | _TX/S and h _ξ.     

Continuous reducibility and dimension of metric spaces

If (X, d) is a Polish metric space of dimension 0, then by Wadge’s lemma, no more than two Borel subsets of X are incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space (X, d) of positive dimension, there are uncountably many Borel subsets of (X, d) that are pairwise incomparable with respect to continuous reducibility. In general, the reducibility that is given by the collection of continuous functions on a topological space \((X,\tau )\) is called the Wadge quasi-order for \((X,\tau )\). As an application of the main result, we show that this quasi-order, restricted to the Borel subsets of a Polish space \((X,\tau )\), is a well-quasiorder if and only if \((X,\tau )\) has dimension 0. Moreover, we give further examples of applications of the construction of graph colorings that is used in the proofs.     

Weak rigid monoidal category

We define the right regular dual of an object X in a monoidal category C; and give several results regarding the weak rigid monoidal category. Based on the definition of the right regular dual, we construct a weak Hopf algebra structure of H = End(F) whenever (F; J) is a fiber functor from category C to Vec and every X ∈ C has a right regular dual. To conclude, we give a weak reconstruction theorem for a kind of weak Hopf algebra.     

Symmetric function kernels and sweeping of measures

This is a potential theoretic study of balayage (sweeping) of a positive Radon measure ω on a locally compact (Hausdorff) space X onto a closed, or, more generally, a quasiclosed set A ? X (that is, a set which can be approximated in outer capacity by closed sets). The setting is that of potentials with respect to a suitable symmetric function kernel G: X × X → [0,+∞]. We consider energy capacity, not as a set function, but as a functional, acting on positive numerical functions on X. The finiteness of the upper capacity of the function 1 _A Gω is sufficient for the possibility of the sweeping in question (1 _A denoting the indicator function of A and Gω the G-potential of ω).     

Interpolation of vector measures

Let (Ω, Σ) be a measurable space and m ₀: Σ → X ₀ and m ₁: Σ → X ₁ be positive vector measures with values in the Banach Köthe function spaces X ₀ and X ₁. If 0 < α < 1, we define a new vector measure [m ₀, m ₁] _α with values in the Calderón lattice interpolation space X ₀ ^1?ga X ₁ ^α and we analyze the space of integrable functions with respect to measure [m ₀, m ₁] _α in order to prove suitable extensions of the classical Stein-Weiss formulas that hold for the complex interpolation of L ^p -spaces. Since each p-convex order continuous Köthe function space with weak order unit can be represented as a space of p-integrable functions with respect to a vector measure, we provide in this way a technique to obtain representations of the corresponding complex interpolation spaces. As applications, we provide a Riesz-Thorin theorem for spaces of p-integrable functions with respect to vector measures and a formula for representing the interpolation of the injective tensor product of such spaces.     

Generalized varieties of sums of powers

Let X ? P^N be an irreducible, non-degenerate variety. The generalized variety of sums of powers V S P_H^X(h) of X is the closure in the Hilbert scheme Hilb_h (X) of the locus parametrizing collections of points {x₁,..., x_h} such that the (h -1)-plane >x₁,..., x_h> passes through a fixed general point p ∈ P^N. When X = V_dⁿ is a Veronese variety we recover the classical variety of sums of powers V S P(F, h) parametrizing additive decompositions of a homogeneous polynomial as powers of linear forms. In this paper we study the birational behavior of V S P_H^X(h). In particular, we show how some birational properties, such as rationality, unirationalityand rational connectedness, of V S P_H^X(h) are inherited from the birational geometry of variety X itself.     

A unifying approach to the Margolis–Meakin and Birget–Rhodes group expansion

Let G be a group. We show that the Birget–Rhodes prefix expansion \(G^{Pr}\) and the Margolis–Meakin expansion M(X; f) of G with respect to \(f:X\rightarrow G\) can be regarded as inverse subsemigroups of a common E-unitary inverse semigroup P. We construct P as an inverse subsemigroup of an E-unitary inverse monoid \(U/\zeta \) which is a homomorphic image of the free product U of the free semigroup \(X^+\) on X and G. The semigroup P satisfies a universal property with respect to homomorphisms into the permissible hull C(S) of a suitable E-unitary inverse semigroup S, with \(S/\sigma _S=G\), from which the characterizing universal properties of \(G^{Pr}\) and M(X; f) can be recaptured easily.     

Random difference equations with subexponential innovations

We consider the random difference equations S = _d (X + S)Y and T = _d X + TY, where = _d denotes equality in distribution, X and Y are two nonnegative random variables, and S and T on the right-hand side are independent of (X, Y). Under the assumptions that X follows a subexponential distribution with a nonzero lower Karamata index, that Y takes values in [0, 1] and is not degenerate at 0 or 1, and that (X, Y) fulfills a certain dependence structure via the conditional tail probability of X given Y, we derive some asymptotic formulas for the tail probabilities of the weak solutions S and T to these equations. In doing so we also obtain some by-products which are interesting in their own right.