Cohomological characterisation of monads

Let X be an n‐dimensional smooth projective variety with an n‐block collection , with , of coherent sheaves on X that generate the bounded derived category . We give a cohomological characterisation of torsion‐free sheaves on X that are the cohomology of monads of the form where . We apply the result to get a cohomological characterisation when X is the projective space, the smooth hyperquadric or the Fano threefold V₅. We construct a family of monads on a Segre variety and apply our main result to this family.     

On the extremal Betti numbers of the binomial edge ideal of closed graphs

We study the equality of the extremal Betti numbers of the binomial edge ideal and those of its initial ideal for a closed graph G. We prove that in some cases there is a unique extremal Betti number for and as a consequence there is a unique extremal Betti number for and these extremal Betti numbers are equal.     

Boundaries for algebras of analytic functions on function module Banach spaces

We consider the uniform algebra of continuous and bounded functions that are analytic on the interior of the closed unit ball of a complex Banach function module X. We focus on norming subsets of , i.e., boundaries, for such algebra. In particular, if X is a dual complex Banach space whose centralizer is infinite‐dimensional, then the intersection of all closed boundaries is empty. This also holds in case that X is an ‐sum of infinitely many Banach spaces and further, the torus is a boundary.     

On admissible topologies in JB*‐triples

Given a complex JB*‐triple X, we define and study admissible topologies on X, i.e., locally convex topologies τ on X coarser than the norm topology, invariant under the group of surjective linear isometries of X, and such that the triple product is jointly ‐continuous on bounded subsets of X. As a consequence of the joint ‐continuity of the triple product, all holomorphic automorphisms of the open unit ball are homeomorphisms of and the natural action is jointly ‐continuous on .     

A Mehta–Ramanathan theorem for linear systems with basepoints

Let be a normal complex projective polarized variety and an H‐semistable sheaf on X. We prove that the restriction to a sufficiently positive general complete intersection curve passing through a prescribed finite set of points remains semistable, provided that at each , the variety X is smooth and the factors of a Jordan–Hölder filtration of are locally free. As an application, we obtain a generalization of Miyaoka's generic semipositivity theorem.     

Reduction theorems for Sobolev embeddings into the spaces of Hölder,Morrey and Campanato type

Let X be a rearrangement‐invariant Banach function space on Q where Q is a cube in and let be the Sobolev space of real‐valued weakly differentiable functions f satisfying . We establish a reduction theorem for an embedding of the Sobolev space into spaces of Campanato, Morrey and Hölder type. As a result we obtain a new characterization of such embeddings in terms of boundedness of a certain one‐dimensional integral operator on representation spaces.     

Holonomy group scheme of an integral curve

Let Y be a projective variety over a field k (of arbitrary characteristic). Assume that the normalization X of Y is such that is normal, being the algebraic closure of k. We define a notion of strong semistability for vector bundles on Y. We show that a vector bundle on Y is strongly semistable if and only if its pull back to X is strongly semistable and hence it is a tensor category. In case , we show that strongly semistable vector bundles on Y form a neutral Tannakian category. We define the holonomy group scheme of Y to be the Tannakian group scheme for this category. For a strongly semistable principal G‐bundle , we construct a holonomy group scheme. We show that if Y is an integral complex nodal curve, then the holonomy group of a strongly semistable vector bundle on Y is the Zariski closure of the (topological) fundamental group of Y.     

Two‐sided estimates and positivity conditions for solutions of linear operator equations in a Banach lattice

Let A and be bounded linear operators in a Banach lattice B, and M be a positive operator in B. The paper deals with the equation where X should be found and are real numbers. Two‐sided estimates and positivity conditions for a solution of that equation are established. The illustrative examples are also presented.     

Dissipative operators and additive perturbations in locally convex spaces

Let be a densely defined operator on a Banach space X. Characterizations of when generates a C₀‐semigroup on X are known. The famous result of Lumer and Phillips states that it is so if and only if is dissipative and is dense in X for some . There exists also a rich amount of Banach space results concerning perturbations of dissipative operators. In a recent paper Tyran–Kamińska provides perturbation criteria of dissipative operators in terms of ergodic properties. These results, and others, are shown to remain valid in the setting of general non–normable locally convex spaces. Applications of the results to concrete examples of operators on function spaces are also presented.     

On the existence of RUC systems in rearrangement invariant spaces

We characterize rearrangement invariant spaces X on [0, 1] with the property that each orthonormal system in X which is uniformly bounded in some Marcinkiewicz space , for equivalent to , , is a system of Random Unconditional Convergence (RUC system).     

Algebraic cycles and EPW cubes

Let X be a hyperkähler variety with an anti‐symplectic involution ι. According to Beauville's conjectural “splitting property”, the Chow groups of X should split in a finite number of pieces such that the Chow ring has a bigrading. The Bloch–Beilinson conjectures predict how ι should act on certain of these pieces of the Chow groups. We verify part of this conjecture for a 19‐dimensional family of hyperkähler sixfolds that are “double EPW cubes” (in the sense of Iliev–Kapustka–Kapustka–Ranestad). This has interesting consequences for the Chow ring of the quotient , which is an “EPW cube” (in the sense of Iliev–Kapustka–Kapustka–Ranestad).     

The Radon‐Nikodym property,generalized bounded variation and the average range

A generalized bounded variation characterization of Banach spaces possessing the Radon‐Nikodym property is given in terms of the average range. We prove that a Banach space X has the Radon‐Nikodym property if and only if for each function of generalized bounded variation on [0, 1], the average range is a nonempty set at almost all .     

Stability and integrability of horizontally conformal maps and harmonic morphisms

In this article, we study stability of minimal fibers and integrability of horizontal distribution for horizontally conformal maps and harmonic morphisms. Let be a horizontally conformal submersion. We prove that if the horizontal distribution is integrable, then any minimal fiber of φ is volume‐stable. This result is an improved version of the main theorem in [15]. As a corollary, we obtain if φ is a submersive harmonic morphism whose fibers are totally geodesic, and the horizontal distribution is integrable, then any fiber of φ is volume‐stable and so such a map φ is energy‐stable if M is compact. We also show that if is a horizontally conformal map from a compact Riemannian manifold M into an orientable Riemannian manifold N which is horizontally homothetic, and if the pull‐back of the volume form of N is harmonic, then the horizontal distribution is integrable and φ is a harmonic morphism.     

Multiple summability properties for Cohen ‐summing operators

We prove that Cohen p‐summing operators satisfy multiple summability properties. Some of these multiple summability properties are new even in the linear case. For example, we prove that the multilinear functional associated to a Cohen p‐summing n‐linear operator is multiple ‐summing.     

Microlocal analysis on wonderful varieties. Regularized traces and global characters

Let be a connected reductive complex algebraic group with split real form . Consider a strict wonderful ‐variety X equipped with its σ‐equivariant real structure, and let X be the corresponding real locus. Further, let E be a real differentiable G‐vector bundle over X . In this paper, we introduce a distribution character for the regular representation of G on the space of smooth sections of E given in terms of the spherical roots of , and show that on a certain open subset of G of transversal elements it is locally integrable and given by a sum over fixed points.     

Arithmetic toric varieties

We study toric varieties over a field k that split in a Galois extension using Galois cohomology with coefficients in the toric automorphism group. Part of this Galois cohomology fits into an exact sequence induced by the presentation of the class group of the toric variety. This perspective helps to compute the Galois cohomology, particularly for cyclic Galois groups. We use Galois cohomology to classify k‐forms of projective spaces when is cyclic, and we also study k‐forms of surfaces.     

Sections of the regular simplex – Volume formulas and estimates

We state a general formula to compute the volume of the intersection of the regular n‐simplex with some k‐dimensional subspace. It is known that for central hyperplanes the one through the centroid containing vertices gives the maximal volume. We show that, for fixed small distances of a hyperplane to the centroid, the hyperplane containing vertices is still volume maximizing. The proof also yields a new and short argument for the result on central sections. With the same technique we give a partial result for the minimal central hyperplane section. Finally, we obtain a bound for k‐dimensional sections.     

Level‐ limit linear series

We consider all one‐parameter families of smooth curves degenerating to a singular curve X and describe limits of linear series along such families. We treat here only the simplest case where X is the union of two smooth components meeting transversely at a point P. We introduce the notion of level‐δ limit linear series on X to describe these limits, where δ is the singularity degree of the total space of the degeneration at P. If the total space is regular, that is, , we recover the limit linear series introduced by Osserman in 11 . So we extend his treatment to a more general setup. In particular, we construct a projective moduli space parameterizing level‐δ limit linear series of rank r and degree d on X, and show that it is a new compactification, for each δ, of the moduli space of Osserman exact limit linear series. Finally, we generalize 7 by associating with each exact level‐δ limit linear series on X a closed subscheme of the dth symmetric product of X, and showing that, if is a limit of linear series on the smooth curves degenerating to X, then is the limit of the corresponding spaces of divisors. In short, we describe completely limits of divisors along degenerations to such a curve X.     

Well‐posedness of fractional degenerate differential equations with finite delay on vector‐valued functional spaces

We study the well‐posedness of the fractional degenerate differential equations with finite delay on Lebesgue–Bochner spaces , periodic Besov spaces and periodic Triebel–Lizorkin spaces , where A and M are closed linear operators on a Banach space X satisfying , F is a bounded linear operator from (resp. and ) into X, where is given by when and . Using known operator‐valued Fourier multiplier theorems, we give necessary or sufficient conditions for the well‐posedness of in the above three function spaces.