Pure Dimension Research Articles

Dimension two holomorphic distributions on four-dimensional projective space

We study two-dimensional holomorphic distributions on P4. We classify dimension two distributions, of degree at most 2, with either locally free tangent sheaf or locally free conormal sheaf and whose singular scheme has pure dimension one. We show that the corresponding sheaves are split. Next, we investigate the geometry of such distributions, studying from maximally non–integrable to integrable distributions. In the maximally non–integrable case, we show that the distribution is either of Lorentzian type or a push-forward by a rational map of the Cartan prolongation of a singular contact structure on a weighted projective 3-fold. We study distributions of dimension two in P4 whose conormal sheaves are the Horrocks–Mumford sheaves, describing the numerical invariants of their singular schemes which are smooth and connected. Such distributions are maximally non–integrable, uniquely determined by their singular schemes and invariant by a group H5⋊SL(2,Z5)⊂Sp(4,Q), where H5 is the Heisenberg group of level 5. We prove that the moduli spaces of Horrocks–Mumford distributions are irreducible quasi-projective varieties and we determine their dimensions. Finally, we observe that the space of codimension one distributions, of degree d≥6, on P4 has a family of degenerate flat holomorphic Riemannian metrics. Moreover, the degeneracy divisors of such metrics consist of codimension one distributions invariant by H5⋊SL(2,Z5) and singular along a degenerate abelian surface with (1,5)-polarization and level-5-structure.

On Finite Parts of Divergent Complex Geometric Integrals and Their Dependence on a Choice of Hermitian Metric

Let X be a reduced complex space of pure dimension. We consider divergent integrals of certain forms on X that are singular along a subvariety defined by the zero set of a holomorphic section of some holomorphic vector bundle E→X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E \\rightarrow X$$\\end{document}. Given a choice of Hermitian metric on E we define a finite part of the divergent integral. Our main result is an explicit formula for the dependence on the choice of metric of the finite part.

Dimensional Structure of MAPS-15: Validation of the Multidimensional Academic Procrastination Scale.

Academic procrastination is a complex behavior that hampers the cyclical process of self-regulation in learning, impeding the flow of actions necessary to achieve the goals and sub-goals that students have set out to attain. It has a high frequency of occurrence and has been linked to lessened student performance and a decrease in psychological and physical well-being. The objective of this study is to analyze the psychometric characteristics of a new academic procrastination scale MAPS-15 (Multidimensional Academic Procrastination Scale) applicable in self-regulated learning environments through a cross-validation study (exploratory factor analysis and confirmatory factor analysis). The sample consisted of 1289 students from a distance/online university, with a wide age range and sociocultural variability. The students completed self-reported online questionnaires on two dates: during the university access and adaptation phase and before the first period of compulsory exams. One-, two- and three-factor structures were tested as well as a second-order structure. The results support a three-dimensional structure of MAPS-15: core procrastination, a pure dimension of procrastinating behavior and difficulty in carrying out the action; poor time management, a dimension related to time organization and perceived control over time; and work disconnection, a dimension conceptually related to lack of persistence, and work interruptions.

NORMAL AND IRREDUCIBLE ADIC SPACES, THE OPENNESS OF FINITE MORPHISMS, AND A STEIN FACTORIZATION

AbstractWe transfer several elementary geometric properties of rigid-analytic spaces to the world of adic spaces, more precisely to the category of adic spaces which are locally of (weakly) finite type over a non-archimedean field. This includes normality, irreducibility (in particular, irreducible components), and a Stein factorization theorem. Most notably, we show that a finite morphism in our category of adic spaces is automatically open if the target is normal and both source and target are of the same pure dimension. Moreover, our version of the Stein factorization theorem includes a statement about the geometric connectedness of fibers which we have not found in the literature of rigid-analytic or Berkovich spaces.

Cycle class maps for Chow groups of zero-cycles with modulus

For a smooth scheme X of pure dimension d over a field k and an effective Cartier divisor D⊂X whose support is a simple normal crossing divisor, we construct a cycle class mapcycX|D:CH0(X|D)→HNisd(X,KdM(OX,ID)) from the Chow group of zero-cycles with modulus to the cohomology of the relative Milnor K-sheaf.

Chern currents of coherent sheaves

Given a finite locally free resolution of a coherent analytic sheaf $\mathcal F$, equipped with Hermitian metrics and connections, we construct an explicit current, obtained as the limit of certain smooth Chern forms of $\mathcal F$, that represents the Chern class of $\mathcal F$ and has support on the support of $\mathcal F$. If the connections are $(1,0)$-connections and $\mathcal F$ has pure dimension, then the first nontrivial component of this Chern current coincides with (a constant times) the fundamental cycle of $\mathcal F$. The proof of this goes through a generalized Poincar\'e-Lelong formula, previously obtained by the authors, and a result that relates the Chern current to the residue current associated with the locally free resolution.

A Dual Multi-Head Contextual Attention Network for Hyperspectral Image Classification

To learn discriminative features, hyperspectral image (HSI), containing 3-D cube data, is a preferable means of capturing multi-head self-attention from both spatial and spectral domains if the burden in model optimization and computation is low. In this paper, we design a dual multi-head contextual self-attention (DMuCA) network for HSI classification with the fewest possible parameters and lower computation costs. To effectively capture rich contextual dependencies from both domains, we decouple the spatial and spectral contextual attention into two sub-blocks, SaMCA and SeMCA, where depth-wise convolution is employed to contextualize the input keys in the pure dimension. Thereafter, multi-head local attentions are implemented as group processing when the keys are alternately concatenated with the queries. In particular, in the SeMCA block, we group the spatial pixels by evenly sampling and create multi-head channel attention on each sampling set, to reduce the number of the training parameters and avoid the storage increase. In addition, the static contextual keys are fused with the dynamic attentional features in each block to strengthen the capacity of the model in data representation. Finally, the decoupled sub-blocks are weighted and summed together for 3-D attention perception of HSI. The DMuCA module is then plugged into a ResNet to perform HSI classification. Extensive experiments demonstrate that our proposed DMuCA achieves excellent results over several state-of-the-art attention mechanisms with the same backbone.

A pointwise norm on a non-reduced analytic space

Let X be a possibly non-reduced space of pure dimension. We introduce a pointwise Hermitian norm on smooth (0,q)-forms, in particular on holomorphic functions, on X. The norm is canonical, up to equivalence, where the underlying reduced space is a manifold. We prove that the space of holomorphic functions is complete with respect to the natural topology induced by this norm.

The direct image of generalized divisors and the Norm map between compactified Jacobians

The Norm map between Jacobians and the Prym scheme are widely used in the description of spectral data of G-Higgs pairs, for G varying among classical Lie group. The aim of this paper is to generalize the Norm map to compactified Jacobians and provide a compactification for the Prym scheme. Given a finite, flat morphism between curves (e.g. embeddable noetherian schemes of pure dimension 1), we first define the notion of direct and inverse image for generalized divisors and generalized line bundles. In the case when we deal with (possibly reducible, non-reduced) projective curves over a field and the codomain curve is smooth, we introduce the compactified Jacobians parametrizing torsion-free rank-1 sheaves and then we study the Norm and inverse image maps between compactified Jacobians. Finally, we introduce and study the Prym stack defined as the kernel of the Norm map.

Comparison of Poisson structures on moduli spaces

Let X be a complex irreducible smooth projective curve, and let {{mathbb {L}}} be an algebraic line bundle on X with a nonzero section sigma _0. Let {mathcal {M}} denote the moduli space of stable Hitchin pairs (E,, theta ), where E is an algebraic vector bundle on X of fixed rank r and degree delta , and theta , in , H^0(X,, {mathcal {E}nd}(E)otimes K_Xotimes {{mathbb {L}}}). Associating to every stable Hitchin pair its spectral data, an isomorphism of {mathcal {M}} with a moduli space {mathcal {P}} of stable sheaves of pure dimension one on the total space of K_Xotimes {{mathbb {L}}} is obtained. Both the moduli spaces {mathcal {P}} and {mathcal {M}} are equipped with algebraic Poisson structures, which are constructed using sigma _0. Here we prove that the above isomorphism between {mathcal {P}} and {mathcal {M}} preserves the Poisson structures.

On the Briançon-Skoda theorem for analytic local rings with singularities

In this note, we study validity of the original Briançon-Skoda theorem for analytic local rings. By introducing a new concept of multiplier ideal sheaves associated to plurisubharmonic functions on any (reduced) complex space of pure dimension, we establish the original Briançon-Skoda theorem for normal weakly rational singularities (not essentially of finite type over C and Cohen-Macaulay). Moreover, we also characterize two dimensional rational singularities by the original Briançon-Skoda theorem.

Fibres non r\xe9duites d\u2019un sch\xe9ma arithm\xe9tique

(Non-reduced fibers of an arithmetic scheme) For a reduced projective scheme over the ring of integers of a number field, the set of places over which the fibres of the scheme are not reduced is a finite set. We give an explicit upper bound for the product of the norms of places in this set. For this purpose, we introduce a generalization of the notion of height over the adelic ring. We reduce the general case of a scheme of pure dimension to the case of a hypersurface by using the theory of Chow varieties. The case of a hypersurface is then treated with the help of the resultant of the equation of the hypersurface with some partial derivatives of the equation.

Pure Goldie Dimensions for Exactly Definable Categories

It is shown that if A is an object in an exactly definable category $${\mathcal {C}}$$ such that A has finite pure Goldie dimension and that every pure monomorphism $$A\rightarrow A$$ is an isomorphism, then its endomorphism ring $$\hbox {End}_{{\mathcal {C}}}(A)$$ is semilocal. Also, it is proved that every subobject of a pure quotient finite dimensional pure injective object of an exactly definable category has a semilocal endomorphism ring.

The analytic class number formula for 1‐dimensional affine schemes

We derive an analytic class number formula valid for an order in a product of S-integers in global fields, or equivalently for reduced finite-type affine schemes of pure dimension 1 over Z.

Defining integer-valued functions in rings of continuous definable functions over a topological field

Let [Formula: see text] be an expansion of either an ordered field [Formula: see text], or a valued field [Formula: see text]. Given a definable set [Formula: see text] let [Formula: see text] be the ring of continuous definable functions from [Formula: see text] to [Formula: see text]. Under very mild assumptions on the geometry of [Formula: see text] and on the structure [Formula: see text], in particular when [Formula: see text] is [Formula: see text]-minimal or [Formula: see text]-minimal, or an expansion of a local field, we prove that the ring of integers [Formula: see text] is interpretable in [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] is definably connected of pure dimension [Formula: see text], then [Formula: see text] defines the subring [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] has no isolated points, then there is a discrete ring [Formula: see text] contained in [Formula: see text] and naturally isomorphic to [Formula: see text], such that the ring of functions [Formula: see text] which take values in [Formula: see text] is definable in [Formula: see text].

Extension of holomorphic functions defined on singular complex hypersurfaces with growth estimates in strictly pseudoconvex domains of $\mathbb{C}^n$

Let D be a strictly pseudoconvex domain and X be a singular analytic set of pure dimension n − 1 in C n such that X ∩ D = ∅ and X ∩ bD is transverse. We give sufficient conditions for a function holomorphic on D ∩ X to admit a holomorphic extension which belongs to L q (D), q ∈ [1, +∞[, or to BM O(D). The extension is given by mean of integral representation formulas and residue currents.

Anti-commuting varieties

We study the anti-commuting variety which consists of pairs of anti-commuting $n\times n$ matrices. We provide an explicit description of its irreducible components and their dimensions. The GIT quotient of the anti-commuting variety with respect to the conjugation action of $GL_n$ is shown to be of pure dimension $n$. We also show the semi-nilpotent anti-commuting variety (in which one matrix is required to be nilpotent) is of pure dimension $n^2$ and describe its irreducible components.

Double lines on quadric hypersurfaces

We study double line structures in projective spaces and quadric hypersurfaces, and investigate the geometry of irreducible components of Hilbert scheme of curves and moduli of stable sheaves of pure dimension 1 on a smooth quadric threefold.

Global Koppelman Formulas on (Singular) Projective Varieties

Let i:Xrightarrow {mathbb {P}}^N be a projective manifold of dimension n embedded in projective space {mathbb {P}}^N, and let L be the pullback to X of the line bundle {mathcal {O}}_{{mathbb {P}}^N}(1). We construct global explicit Koppelman formulas on X for smooth (0,*)-forms with values in L^s for any s. The same construction works for singular, even non-reduced, X of pure dimension, if the sheaves of smooth forms are replaced by suitable sheaves {mathcal {A}}_X^* of (0,*)-currents with mild singularities at X_{sing}. In particular, if sge mathrm{reg,}X -1, where mathrm{reg,}X is the Castelnuovo–Mumford regularity, we get an explicit representation of the well-known vanishing of H^{0,q}(X, L^{s-q}), qge 1. Also some other applications are indicated.

The bar{partial }-equation on a non-reduced analytic space

Let X be a, possibly non-reduced, analytic space of pure dimension. We introduce a notion of overline{partial }-equation on X and prove a Dolbeault–Grothendieck lemma. We obtain fine sheaves mathcal {A}_X^q of (0, q)-currents, so that the associated Dolbeault complex yields a resolution of the structure sheaf mathscr {O}_X. Our construction is based on intrinsic semi-global Koppelman formulas on X.