Analytic Subset Research Articles

Sequences with increasing subsequence

We study analytic and Borel subsets defined similarly to the old example of analytic complete set given by Luzin. Luzin's example, which is essentially a subset of the Baire space, is based on the natural partial order on naturals, i.e. division. It consists of sequences which contain increasing subsequence in given order.We consider a variety of sets defined in a similar way. Some of them occurs to be Borel subsets of the Baire space, while others are analytic complete, hence not Borel.In particular, we show that an analogon of Luzin example based on the natural linear order on rationals is analytic complete. We also characterize all countable linear orders having such property.

Analyticity theorems for parameter-dependent plurisubharmonic functions

In this paper, we first show that a union of upper-level sets associated to fibrewise Lelong numbers of plurisubharmonic functions is in general a pluripolar subset. Then we obtain analyticity theorems for a union of sub-level sets associated to fibrewise complex singularity exponents of some special (quasi-)plurisubharmonic functions. As a corollary, we confirm that, under certain conditions, the logarithmic poles of relative Bergman kernels form an analytic subset when the (quasi-)plurisubharmonic weight function has analytic singularities. In the end, we give counterexamples to show that the aforementioned sets are in general non-analytic even if the plurisubharmonic function is supposed to be continuous.

A non-archimedean definable Chow theorem

Peterzil and Starchenko have proved the following surprising generalization of Chow’s theorem: A closed analytic subset of a complex algebraic variety that is definable in an o-minimal structure is in fact an algebraic subset. In this paper, we prove a non-archimedean analogue of this result.

Exceptional values of entire functions of finite order in one of the variables

Let F(z,w) be a holomorphic function in Cn×C of finite order in w with n≥2. Let Ω be the set of points z∈Cn where F(z,w) is a non-constant function omitting a value π(z). Near a finite accumulation point z0 of Ω, we prove in the main result (Theorem 1) that Ω is a local analytic set and π(z) is holomorphic, and show the existence of a proper globally analytic set Δ of Cn such that either Ω⊂Δ or Ω=Cn∖Δ, being possible in the last case to also determine F(z,w) in terms of π(z). We apply this result to several problems. First, we extend a Theorem due to Nishino about exceptional values when near z0 dimension of Ω is n and assure the existence of a meromorphic function α(z) in Cn such that π(z)=α(z) except at points where α(z) has poles or F(z,w) is constant (also being F(z,w) a polynomial in w if α(z) is ∞). After, we prove that Ω is a local analytic set in Cn and the existence of a proper analytic subset E of Cn such that Ω⊂E or Ω=Cn∖E. Finally, we generalize a Lelong-Gruman Theorem about the set of points z where π(z)=0.

Breaking the 80:20 rule in health research using large administrative data sets.

Objective: To explore the application of online analytic processing (OLAP) to improve the efficiency of analytics using large administrative health data sets. Methods: 18 years of administrative health data (1994/95 to 2012/13) were obtained from the Alberta Ministry of Health in Canada. The data sets included hospitalization, ambulatory care and practitioner claims data. Reference files were obtained that provided information including patient demographics, resident postal code, facility, and provider details. Population counts and projections for each year, sex, age were included for rate calculations. These sources were used to develop a data cube using OLAP tools. Results: Time required for analyses was reduced to 5% of that required when comparing run-time for simple queries that did not require linkage of data sets. The data cube negated the need for many intermediary steps for data extraction and analyses for research activities. Conventional methods required over 250GB of server space for multiple analytic subsets, compared to only 10.3GB for the data cube. Conclusions: Cross-training in information technology and health analytics is recommended to provide capacity to better leverage OLAP tools which are available with many common applications.

HOLOMORPHIC EXTENSION

In this paper, we prove that if there exists a holomorphic, proper, surjective map defined on a complex manifold $ X $ into a smooth algebraic curve with parallelizable fibers, then any holomorphic mappings defined on the Hartogs domain $ T $ of $\mathbb{C}^n$ can be extended holomorphically (resp. meromorphically) from $ \Delta ^n \setminus Z $ into $ X $, where $ Z $ is an analytic subset of $\Delta^n $ such that codimension of $ Z $ at least 2.

ON THE JORDAN STRUCTURE OF HOLOMORPHIC MATRICES

Let X ⊂ CN be open, and let A be an n × n matrix of holomorphic functions on X. We call a point ξ ∈ X Jordan stable for A if ξ is not a splitting point of the eigenvalues of A and, moreover, there is a neighborhood U of ξ such that, for each 1 ≤ k ≤ n, the number of Jordan blocks of size k in the Jordan normal forms of A(ζ) is the same for all ζ ∈ U. H. Baumg¨artel [4, S 3.4] proved that there is a nowhere dense closed analytic subset of X, which contains the set of all non-Jordan stable points. We give a new proof of this result. This proof shows that the set of non-Jordan stable points ist not only contained in a nowhere dense closed analytic subset, but it is itself such a set, and can be defined by holomorphic functions, the growth of which is bounded by some power (depending only on n) of the growth of A. Also, this proof applies to arbitrary (possibly non-smooth) reduced complex spaces X.

Hodge modules and singular hermitian metrics

The purpose of this paper is to study certain notions of metric positivity called “minimal extension property” for the lowest nonzero piece in the Hodge filtration of a Hodge module. Let X be a complex manifold and let \(\mathcal {M}\) be a polarized pure Hodge module on X with strict support X. Let \(F_p\mathcal {M}\) be the smallest nonzero piece in the Hodge filtration. Assume that \(\mathcal {M}\) is smooth outside a closed analytic subset Z and let \(j:X\setminus Z \hookrightarrow X\) be the open embedding. Let h be the smooth hermitian metric on \(F_p\mathcal {M}|_{X\setminus Z}\) induced by the polarization. We show that the canonical morphism of \(\mathcal {O}_X\)-modules $$\begin{aligned} F_p\mathcal {M}\rightarrow j_{*}(F_p\mathcal {M}|_{X\setminus Z}) \end{aligned}$$induces an isomorphism between \(F_p\mathcal {M}\) and the subsheaf of \(j_{*}(F_p\mathcal {M}|_{X\setminus Z})\) consisting of sections which are locally \(L^2\) near Z with respect to h and the standard Lebesgue measure on X. In particular, h extends to a singular hermitian metric on \(F_p\mathcal {M}\) with minimal extension property.

Concavity Property of Minimal $$L^2$$ Integrals with Lebesgue Measurable Gain IV: Product of Open Riemann Surfaces

In this article, we present characterizations of the concavity property of minimal $$L^2$$ integrals degenerating to linearity in the case of products of analytic subsets on products of open Riemann surfaces. As applications, we obtain characterizations of the holding of equality in optimal jets $$L^2$$ extension problem from products of analytic subsets to products of open Riemann surfaces, which implies characterizations of the product versions of the equality parts of Suita conjecture and extended Suita conjecture, and the equality holding of a conjecture of Ohsawa for products of open Riemann surfaces.

On subvarieties of singular quotients of bounded domains

Let X $X$ be a quotient of a bounded domain in C n $\mathbb {C}^n$ . Under suitable assumptions, we prove that every subvariety of X $X$ not included in the branch locus of the quotient map is of log-general type in some orbifold sense. This generalizes a recent result by Boucksom and Diverio, which treated the case of compact, étale quotients. Finally, in the case where X $X$ is compact, we give a sufficient condition under which there exists a proper analytic subset of X $X$ containing all entire curves and all subvarieties not of general type (meant this time in in the usual sense as opposed to the orbifold sense).

On the p-essential normality of principal submodules of the Bergman module on strongly pseudoconvex domains

In this paper, we show that under a mild condition, a principal submodule of the Bergman module on a bounded strongly pseudoconvex domain with smooth boundary in Cn is p-essentially normal for all p>n. This improves a previous result by the first author and K. Wang, in which it was shown that any polynomial-generated principal submodule of the Bergman module on the unit ball Bn is p-essentially normal for all p>n. As a consequence, we show that the submodule of La2(Bn) consisting of functions vanishing on an analytic subset of pure codimension 1 is p-essentially normal for all p>n.

Point counting and Wilkie’s conjecture for non-Archimedean Pfaffian and Noetherian functions

We consider the problem of counting polynomial curves on analytic or definable subsets over the field C((t)) as a function of the degree r. A result of this type could be expected by analogy with the classical Pila–Wilkie counting theorem in the Archimedean situation. Some non-Archimedean analogues of this type have been developed in the work of Cluckers, Comte, and Loeser for the field Qp, but the situation in C((t)) appears to be significantly different. We prove that the set of polynomial curves of a fixed degree r on the transcendental part of a subanalytic set over C((t)) is automatically finite, but we give examples that show their number may grow arbitrarily quickly even for analytic sets. Thus no analogue of the Pila–Wilkie theorem can be expected to hold for general analytic sets. On the other hand, we show that if one restricts to varieties defined by Pfaffian or Noetherian functions, then the number grows at most polynomially in r, thus showing that the analogue of the Wilkie conjecture does hold in this context.

Capacitability for Co-Analytic Sets

It follows from a theorem of Davies (1952) that if A is an analytic subset of the Cantor middle third set, λ is positive and the Hausdorff s-measure of A is greater than λ, then there is a compact subset C of A such that the Hausdorff s-measure of C is greater than λ. We exhibit a counterpoint to Davies’s theorem: In Gödel’s universe of sets, there is a co-analytic subset B of the Cantor set which has full Hausdorff dimension such that if C is a closed subset of B then C is countable.

CR-manifolds of infinite type in the sense of Bloom and Graham

An analogue of the Bloom–Graham theorem for germs of real analytic CR-manifolds of infinite type is devised, and a certain standard form to which they can be transformed (a reduced form) is described. The concept of Bloom–Graham type is refined (as a stratified type). The refined type is also holomorphically invariant. The concept of a quasimodel surface is introduced and it is shown that for biholomorphically equivalent manifolds such surfaces are quasilinearly equivalent. A criterion for the Lie algebra of infinitesimal holomorphic automorphisms to be finite-dimensional is obtained in the case when the type is uniformly infinite (that is, infinite at all points). In combination with the criterion of a finite-dimensional automorphism algebra for manifolds of finite type almost everywhere, this yields a complete criterion for this algebra to be finite-dimensional. The sets of fixed Blooom–Graham type are shown to be semi-analytic and the type of a generic point (lying outside a proper analytic subset) is minimal in a certain sense.

Borel complexity of the family of attractors for weak IFSs

This paper is an attempt to measure the difference between the family of iterated function systems attractors and a broader family, the set of attractors for weak iterated function systems. We discuss Borel complexity of the set wIFS\(^d\) of attractors for weak iterated function systems acting on \([0,1]^d\) (as a subset of the hyperspace \(K([0,1]^d)\) of all compact subsets of \([0,1]^d\) equipped with the Hausdorff metric). We prove that wIFS\(^d\) is \(G_{\delta\sigma}\)-hard in \(K([0,1]^d)\), for all \({d\in\mathbb{N}}\). In particular,wIFS\(^d\) is not \(F_{\sigma\delta}\) (in contrast to the family IFS\(^d\) of attractors for classical iterated function systems acting on \([0,1]^d\), which is \(F_{\sigma}\)). Moreover, we show that in the one-dimensional case, wIFS\(^1\) is an analytic subset of \(K([0,1])\).

On a Spectral Version of Cartan’s Theorem

For a domain \(\Omega \) in the complex plane, we consider the domain \(S_n(\Omega )\) consisting of those \(n\times n\) complex matrices whose spectrum is contained in \(\Omega \). Given a holomorphic self-map \(\Psi \) of \(S_n(\Omega )\) such that \(\Psi (A)=A\) and the derivative of \(\Psi \) at A is identity for some \(A\in S_n(\Omega )\), we investigate when the map \(\Psi \) would be spectrum-preserving. We prove that if the matrix A is either diagonalizable or non-derogatory then for most domains \(\Omega \), \(\Psi \) is spectrum-preserving on \(S_n(\Omega )\). Further, when A is arbitrary, we prove that \(\Psi \) is spectrum-preserving on a certain analytic subset of \(S_n(\Omega )\).

Lelong Number and the Log Canonical Thresholds of Plurisubharmonic Functions on Analytic Subsets

Lelong Number and the Log Canonical Thresholds of Plurisubharmonic Functions on Analytic Subsets

-modules on rigid analytic spaces III: weak holonomicity and operations

AbstractWe develop a dimension theory for coadmissible$\widehat {\mathcal {D}}$-modules on rigid analytic spaces and study those which are of minimal dimension, in analogy to the theory of holonomic$\mathcal {D}$-modules in the algebraic setting. We discuss a number of pathologies contained in this subcategory (modules of infinite length, infinite-dimensional fibres). We prove stability results for closed immersions and the duality functor, and show that all higher direct images of integrable connections restricted to a Zariski open subspace are coadmissible of minimal dimension. It follows that the local cohomology sheaves$\underline {H}^{i}_Z(\mathcal {M})$with support in a closed analytic subset$Z$of$X$are also coadmissible of minimal dimension for any integrable connection$\mathcal {M}$on$X$.

Deformations of Dolbeault cohomology classes for Lie algebra with complex structures

In this paper, we study deformations of complex structures on Lie algebras and its associated deformations of Dolbeault cohomology classes. A complete deformation of complex structures is constructed in a way similar to the Kuranishi family. The extension isomorphism is shown to be valid in this case. As an application, we prove that given a family of left-invariant deformations $$\{M_t\}_{t\in B}$$ of a compact complex manifold $$M=(\Gamma \setminus G, J)$$ where G is a Lie group, $$\Gamma$$ a sublattice and J a left-invariant complex structure, the set of all $$t\in B$$ such that the Dolbeault cohomology on $$M_t$$ may be computed by left-invariant tensor fields is an analytic open subset of B.

Geometric Arveson-Douglas conjecture for the Hardy space and a related compactness criterion

We consider a class of analytic subsets M˜ of an open neighborhood of the closed unit ball in Cn. Such an M˜ gives rise to a submodule R and a quotient module Q of the Hardy module H2(S) on the unit sphere S⊂Cn. We show that, as predicted by the geometric Arveson-Douglas conjecture, the quotient module Q is p-essentially normal for p>d=dimCM˜. We further show that, more interestingly, the quotient module Q exhibits a behavior that is only found on the Bergman space and the Fock space: an operator A in the Toeplitz algebra on Q is compact if and only if its Berezin transform vanishes near M˜∩S.