Sub-holonomicity: D-modules & Horizontal Holonomy

• Sub-holonomicity is a term describing weakened or restricted holonomic conditions in both D-module theory and contact/sub-Riemannian geometry.
• In D-module settings, it involves controlling characteristic dimensions and minimal homological bounds that often lead to full holonomicity.
• In geometric contexts, horizontal holonomy exemplifies sub-holonomicity by restricting parallel transport to nonintegrable distributions.

Searching arXiv for recent and relevant papers on sub-holonomicity and adjacent notions. arXiv search query: "sub-holonomicity holonomicity weak holonomicity horizontal holonomy" Sub-holonomicity is not a standardized term across the cited literatures. The surveyed papers repeatedly state that they do not define such a notion explicitly; instead, they develop adjacent concepts that fall into two main patterns. In $D$-module theory, the nearest notions are weakened or intermediate dimension conditions on singular support, characteristic varieties, or homological dimension, with some settings exhibiting genuine weak analogues of holonomicity and others collapsing any such intermediate behavior to full holonomicity. In sub-Riemannian and contact geometry, the closest analogue is horizontal holonomy, namely holonomy generated by horizontal paths for a partial connection on a nonintegrable distribution, together with comparison theorems showing that this “sub-” holonomy is either equal to, or differs by codimension one from, the holonomy of a canonically associated full connection (Ardakov et al., 2019, Galaev, 11 Feb 2025).

1. Terminological status and principal meanings

Across the relevant arXiv literature, “sub-holonomicity” functions more as an interpretive label than as a settled definition. In rigid-analytic $\widehat{\mathcal D}$-module theory, the closest formal notion is weak holonomicity, defined by minimal homological dimension. In several algebraic $D$-module settings—$A$-hypergeometric systems, binomial $D$-modules, and relative characters—the papers do not stop at weaker estimates, but derive full holonomicity from dimension bounds. In contact and sub-Riemannian geometry, the closest formal notion is horizontal holonomy, i.e. holonomy generated by horizontal loops and partial connections on the horizontal bundle (Ardakov et al., 2019, Berkesch et al., 2013, Zamaere et al., 2013, Aizenbud et al., 2015, Grong et al., 31 Jul 2025).

Context	Nearest formal notion	Characteristic feature
Rigid-analytic $\widehat{\mathcal D}$-modules	Weak holonomicity	Minimal dimension $d(M)=\dim X$
Algebraic $D$-modules	Dimension bounds on $SS(M)$ or $\operatorname{Char}(M)$	Often upgraded to full holonomicity
Contact/sub-Riemannian geometry	Horizontal holonomy	Generated by horizontal loops
Arithmetic $\widehat{\mathcal D}$0-adic $\widehat{\mathcal D}$1-modules on curves	Stability of holonomicity without Frobenius	Non-Liouville hypotheses force holonomicity

This suggests that the term is best understood as denoting a family of weakened, restricted, or precursor holonomic conditions rather than a single invariant. In the $\widehat{\mathcal D}$2-module direction, the basic template is to control the size of characteristic data; in the geometric-holonomy direction, the template is to restrict transport to a nonintegrable distribution and compare the resulting subgroup with ordinary holonomy.

2. Dimension-theoretic weakenings in $\widehat{\mathcal D}$3-module settings

The only surveyed paper that introduces a direct weakening of holonomicity is Ardakov–Wadsley’s rigid-analytic theory of coadmissible $\widehat{\mathcal D}$4-modules. For a smooth equidimensional rigid analytic $\widehat{\mathcal D}$5-space $\widehat{\mathcal D}$6, the dimension of a nonzero coadmissible module on an affinoid is defined by

$\widehat{\mathcal D}$7

and the rigid-analytic Bernstein inequality gives

$\widehat{\mathcal D}$8

A coadmissible module is then called weakly holonomic if

$\widehat{\mathcal D}$9

For nonzero modules this is equivalent to the minimal-dimension condition $D$0 (Ardakov et al., 2019).

That theory is explicitly presented as a weakening of classical holonomicity rather than a full replacement. The paper proves that weakly holonomic modules form an abelian subcategory, are stable under subobjects, quotients, extensions, kernels, cokernels, duality, and closed immersions, and that higher direct images of integrable connections under Zariski open embeddings remain coadmissible and weakly holonomic. At the same time, it exhibits decisive pathologies: weakly holonomic modules may have infinite length, may have infinite-dimensional fibres, are not stable under arbitrary pushforward, and are generally not stable under pullback. A central example on the unit disc constructs a finitely presented weakly holonomic module $D$1 with arbitrarily large finite-length quotients, hence not of finite length, and with infinite-dimensional fibre $D$2 (Ardakov et al., 2019).

The arithmetic $D$3-adic curve case supplies a different kind of near-holonomic phenomenon. Caro does not define a weaker category, but proves that coherent $D$4-complexes on smooth curves satisfying the Christol–Mebkhout non-Liouville condition $D$5 are in fact holonomic even without Frobenius structure. In module form, a coherent $D$6-module that is also $D$7-coherent and satisfies $D$8 is holonomic (Caro, 2010). This does not define sub-holonomicity, but it identifies a weaker-looking class—overconvergent isocrystal-type objects with non-Liouville control—that collapses to full holonomicity in dimension one.

3. Microlocal bounds that become full holonomicity

Several algebraic papers are relevant precisely because they show that putative “sub-holonomic” estimates are already strong enough to force holonomicity. For $D$9-hypergeometric systems, Berkescht–Griffeth–Miller prove that the comparison ring

$A$0

has dimension $A$1, via the commutative-algebra statement

$A$2

Since $A$3 surjects onto $A$4, this yields

$A$5

and hence holonomicity for every parameter $A$6. The same method proves that the family

$A$7

forms a holonomic family over parameter space (Berkesch et al., 2013). The paper explicitly notes that it does not define sub-holonomicity; its relevance lies in the fact that the proof proceeds through dimension control on characteristic data and then reaches the optimal bound.

Binomial $A$8-modules make the absence of an intermediate regime especially sharp. For

$A$9

Dickenstein–Matusevich–Miller prove three equivalent characterizations: holonomicity, properness of the singular locus, and $D$0-holonomicity for some—or equivalently every—projective weight vector $D$1. Moreover, if the module is not holonomic, then $D$2, and for every projective $D$3, $D$4 has a component in $D$5 of dimension $D$6 (Zamaere et al., 2013). This leaves very little room for a meaningful intermediate notion based on proper singular locus, finite rank, or favorable $D$7-characteristic behavior.

A related pattern appears in the theory of relative characters. For a real reductive group $D$8, spherical subgroups $D$9, and relative character $\widehat{\mathcal D}$0, Aizenbud–Gourevitch–Sahi show

$\widehat{\mathcal D}$1

where

$\widehat{\mathcal D}$2

and prove the geometric theorem

$\widehat{\mathcal D}$3

The resulting upper bound on singular-support dimension then implies that every relative character is a holonomic distribution, with analytic restriction to a Zariski open dense subset (Aizenbud et al., 2015). Here again, a “sub-holonomic” estimate appears only as the penultimate step in a proof of full holonomicity.

4. Closure properties, support conditions, and minimal extension

In several categories, the most natural content associated with sub-holonomicity is not a weaker dimension bound but stability under passage to subobjects, quotients, supports, and geometric functors. For holonomic modules over internal skein algebras $\widehat{\mathcal D}$4, the holonomic subcategory is explicitly stated to be thick, meaning closed under subobjects, quotients, and extensions. The same paper proves that transfer functors arising from skein transfer bimodules preserve holonomicity, and that the support-defined submodule

$\widehat{\mathcal D}$5

of a holonomic module is holonomic over the smaller algebra $\widehat{\mathcal D}$6 (Jordan et al., 26 Sep 2025). In that precise sense, holonomicity admits a robust “submodule-stable” behavior.

For sheaves of Cherednik algebras $\widehat{\mathcal D}$7, Bellamy–Etingof–Thompson define holonomicity by isotropic singular support in $\widehat{\mathcal D}$8 and develop a support-theoretic formalism strongly reminiscent of holonomic $\widehat{\mathcal D}$9-modules. Open pushforward $d(M)=\dim X$0, closed pullback $d(M)=\dim X$1, and both $d(M)=\dim X$2 and $d(M)=\dim X$3 for arbitrary $d(M)=\dim X$4-melys morphisms preserve holonomicity. Duality preserves holonomicity, extraordinary functors exist on the holonomic derived categories, and irreducible holonomic modules are classified as minimal extensions of integrable connections on locally closed strata, with

$d(M)=\dim X$5

as the minimal-extension functor (Bellamy et al., 2024). This framework does not define sub-holonomicity, but it gives a precise theory of how holonomic objects behave under restriction to subvarieties and re-extension from locally closed supports.

A plausible implication is that, in these categories, the phrase “sub-holonomic” is often best interpreted operationally: not as a separate intrinsic class, but as the fact that holonomic objects remain well controlled under support restriction, subquotients, localization, and closed-immersion functors.

5. Horizontal holonomy as sub-holonomy in contact and sub-Riemannian geometry

In contact and sub-Riemannian geometry, the nearest formal counterpart of sub-holonomicity is horizontal holonomy. For a contact sub-Riemannian manifold $d(M)=\dim X$6 with horizontal distribution $d(M)=\dim X$7, the Schouten connection is a horizontal Levi-Civita connection

$d(M)=\dim X$8

and its horizontal holonomy group is generated by parallel transport along piecewise smooth horizontal loops. Galaev shows that for a $d(M)=\dim X$9-contact sub-Riemannian manifold the horizontal holonomy is either equal to the holonomy of the adapted connection $D$0, or a codimension-one normal subgroup of it; moreover, $D$1 is the holonomy group of the Levi-Civita connection of some Riemannian manifold (Galaev, 11 Feb 2025).

A complementary normalization theorem is provided by effective normalization of sub-Riemannian connections with constant symbol. Given a partial affine connection on the horizontal bundle $D$2, there is a unique extension to both an $D$3-grading and a strongly compatible full affine connection satisfying the extension condition

$D$4

and for the resulting Cartan connection one has

$D$5

Equivalently, the ordinary holonomy of the full normalized connection coincides with horizontal holonomy (Grong et al., 31 Jul 2025). This realizes the “sub-” holonomy as a full holonomy generated entirely by horizontal data.

The pseudo-Hermitian case refines this picture further. For a pseudo-Hermitian manifold $D$6, the adapted connection is the Tanaka–Webster connection, while the Schouten connection yields intrinsic horizontal holonomy. Under a Codazzi condition on the sub-torsion, Schouten and adapted holonomy are either equal or differ by codimension one. In the torsion-free case the adapted holonomy is Riemannian; in the torsionful pseudo-Hermitian case the only possibilities are

$D$7

with the Schouten holonomy always either equal to the adapted one or obtained by dropping the central $D$8 direction (Galaev et al., 29 Oct 2025). Within this literature, “sub-holonomy” is therefore an actual geometric invariant attached directly to a nonintegrable distribution, not merely an informal weakening of ordinary holonomy.

6. Stability under limits and general significance

A different but related theme is persistence of holonomy constraints under deformation. For metric connections with locally Lipschitz coefficients converging in compact-open $D$9, holonomy inclusion in a fixed closed subgroup $SS(M)$0 is preserved in the limit. The same statement holds for restricted holonomy, and for Riemannian metrics this yields lower semicontinuity of the conjugacy class of the restricted holonomy group with respect to the $SS(M)$1-topology: $SS(M)$2 is lower semicontinuous (Götzfried, 20 Oct 2025). The paper emphasizes that this is strictly one-sided: holonomy may become more special in the limit, but not more generic.

Taken together, the surveyed works support a broad encyclopedic conclusion. In $SS(M)$3-module theory, “sub-holonomicity” usually names, or would naturally name, one of three adjacent phenomena: minimal-dimension weakenings such as weak holonomicity, precursor dimension bounds on singular support, or support-stable holonomic subcategories. In many algebraic settings those precursor estimates already force full holonomicity, so no substantial intermediate class remains. In contact and sub-Riemannian geometry, by contrast, the genuinely intrinsic object is horizontal holonomy, and there the “sub-” prefix refers to restriction to a horizontal distribution and to horizontal loops rather than to a weakened finiteness condition. The common structural theme is restriction: either restriction of characteristic data to minimal size, or restriction of parallel transport to a distinguished nonintegrable subbundle.