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

We 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 $X$ .

Analytic deformations of pencils and integrable one-forms having a first integral

We consider integrable analytic deformations of codimension one holomorphic foliations near an initially singular point. Such deformations are of two possible types. The first type is given by an analytic family [Formula: see text] of integrable one-forms [Formula: see text] defined in a neighborhood [Formula: see text] of the initial singular point, and parametrized by the disc [Formula: see text]. The initial foliation is defined by [Formula: see text]. The second type, more restrictive, is given by an integrable holomorphic one-form [Formula: see text] defined in the product [Formula: see text]. Then, the initial foliation is defined by the slice restriction [Formula: see text]. In the first part of this work, we study the case where the starting foliation has a holomorphic first integral, i.e. it is given by [Formula: see text] for some germ of holomorphic function [Formula: see text] at the origin [Formula: see text]. We assume that the germ [Formula: see text] is irreducible and that the typical fiber of [Formula: see text] is simply-connected. This is the case if outside of a dimension [Formula: see text] analytic subset [Formula: see text], the analytic hypersurface [Formula: see text] has only normal crossings singularities. We then prove that, if cod sing [Formula: see text] then the (germ of the) developing foliation given by [Formula: see text] also exhibits a holomorphic first integral. For the general case, i.e. cod sing [Formula: see text], we obtain a dimension two normal form for the developing foliation. In the second part of the paper, we consider analytic deformations [Formula: see text], of a local pencil [Formula: see text], for [Formula: see text]. For dimension [Formula: see text] we consider [Formula: see text]. For dimension [Formula: see text] we assume some generic geometric conditions on [Formula: see text] and [Formula: see text]. In both cases, we prove: (i) in the case of an analytic deformation there is a multiform formal first integral of type [Formula: see text] with some properties; (ii) in the case of an integrable deformation there is a meromorphic first integration of the form [Formula: see text] with some additional properties, provided that for [Formula: see text] the axes remain invariant for the foliations [Formula: see text].

Turing degrees in Polish spaces and decomposability of Borel functions

We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (e.g. the Shore–Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative results on the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Additionally we prove results about the transfinite version as well as the computable version of the Decomposability Conjecture.

A dichotomy of sets via typical differentiability

We obtain a criterion for an analytic subset of a Euclidean space to contain points of differentiability of a typical Lipschitz function: namely, that it cannot be covered by countably many sets, each of which is closed and purely unrectifiable (has a zero-length intersection with every $C^1$ curve). Surprisingly, we establish that any set failing this criterion witnesses the opposite extreme of typical behaviour: in any such coverable set, a typical Lipschitz function is everywhere severely non-differentiable.

DISTINGUISHING PERFECT SET PROPERTIES IN SEPARABLE METRIZABLE SPACES

All spaces are assumed to be separable and metrizable. Our main result is that the statement “For every space X, every closed subset of X has the perfect set property if and only if every analytic subset of X has the perfect set property” is equivalent to b > ω1 (hence, in particular, it is independent of ZFC). This, together with a theorem of Solecki and an example of Miller, will allow us to determine the status of the statement “For every space X, if every Γ subset of X has the perfect set property then every Γ′ subset of X has the perfect set property” as Γ, Γ′ range over all pointclasses of complexity at most analytic or coanalytic.Along the way, we define and investigate a property of independent interest. We will say that a subset W of 2ω has the Grinzing property if it is uncountable and for every uncountable Y ⊆ W there exists an uncountable collection consisting of uncountable subsets of Y with pairwise disjoint closures in 2ω. The following theorems hold.(1)There exists a subset of 2ω with the Grinzing property.(2)Assume MA + ¬CH. Then 2ω has the Grinzing property.(3)Assume CH. Then 2ω does not have the Grinzing property.The first result was obtained by Miller using a theorem of Todorčević, and is needed in the proof of our main result.

ONE-DIMENSIONAL LIE FOLIATIONS WITH GENERIC SINGULARITIES IN COMPLEX DIMENSION THREE

We prove that a germ of a one-dimensional holomorphic foliation with a generic singularity in dimension two or three that exhibits a Lie group transverse structure in the complement of some codimension one analytic subset is logarithmic, that is, given by a system of closed meromorphic one-forms with simple poles. In the global context, we prove that a foliation by curves in a three-dimensional complex manifold with generic singularities and a Lie group transverse structure off a codimension one analytic subset is logarithmic; that is, it is given by a system of closed meromorphic one-forms with simple poles.

On relatively analytic and Borel subsets

Define to be the smallest cardinality of a function f: X→Y with I, X, Y, ⊆ 2ω such that there is no Borel function g ⊇ f. In this paper we prove that it is relatively consistent with ZFC to have b < where b is, as usual, smallest cardinality of an unbounded family in Ωω. This answers a question raised by Zapletal.We also show that it is relatively consistent with ZFC that there exists X ⊆ 2ω such that the Borei order of X is bounded but there exists a relatively analytic subset of X which is not relatively coanalytic. This answers a question of Mauldin.

Analytic Subsets of Products of Infinite-Dimensional Projective Spaces

Let 𝑉𝑖, 1 ≤ 𝑖 ≤ 𝑠, be complex topological vector spaces with 𝑉1 infinite-dimensional and 𝑌 a closed analytic subset of finite codimension of 𝐏(𝑉1) × . . . × 𝐏(𝑉𝑠). Here we show that 𝑌 is algebraic (at least if each 𝑉𝑖 is a Banach space) and that any two points of 𝑌 may be connected by a chain of 𝑠 + 3 lines contained in 𝑌.