Nash Algebroid: Resolving Singular Foliations

• Nash algebroid is a Lie-algebroid structure arising from the Nash blowup of singular foliations, resolving singularities into Debord-type objects.
• It reformulates irregular foliation behavior by capturing limiting tangent directions and isotropy data through canonical bundle constructions.
• The construction links analytic properties like symbol calculus and Poisson structures with geometric resolutions in differential geometry.

to=arxiv_search.search 大发快三计划 天天中彩票开奖 重庆时时彩杀 code ={'query': 'Nash algebroid singular foliation Nash blowup Lie algebroid', 'max_results': 10} to=arxiv_search.search เน็ตทรู 天天中彩票中了 code ={'query': '(Louis, 2023) OR (Louis, 2024) OR (Louis, 1 Sep 2025)', 'max_results': 10} A Nash algebroid is a Lie-algebroid-theoretic object attached to the Nash blow-up of a singular foliation or of a Lie algebroid. In the singular-foliation formulation, one starts from a finitely generated foliation $(M,\mathcal F)$, forms its Nash blow-up $p:M_{\mathcal F}\to M$, and obtains a Lie algebroid $\mathfrak D_{\mathcal F}\to M_{\mathcal F}$ whose image is the resolved foliation $p^{-1}(\mathcal F)$ (Louis, 1 Sep 2025). In the Lie algebroid formulation, one starts from a Lie algebroid $A\to M$ with basic singular foliation $\mathcal F=\rho(\Gamma(A))$, forms the base blow-up determined by the singular foliation, and constructs $\mathrm{Nash}(A)=p^!A$ together with a canonical quotient Debord algebroid $\mathcal D_{\mathcal F}$ (Louis, 2024). These constructions are rooted in a broader program of Nash resolutions of singular foliations via universal Lie $\infty$-algebroids, where a sequence of blowups $(\widetilde M_i,\pi_i)_{i\in\mathbb N_0}$ is canonically associated to the foliation and, after one blowup, the lifted foliation becomes Debord (Louis, 2023).

1. Formation from Nash resolutions of singular foliations

The immediate precursor of the Nash algebroid is the series of Nash resolutions constructed for a singular foliation $p:M_{\mathcal F}\to M$0 admitting a geometric resolution. In that setting one considers a sequence of vector bundles and bundle maps

$p:M_{\mathcal F}\to M$1

such that the induced sequence on sections is exact and $p:M_{\mathcal F}\to M$2. Given such a geometric resolution, there exists a canonical, up to homotopy, structure of a universal Lie $p:M_{\mathcal F}\to M$3-algebroid with higher brackets $p:M_{\mathcal F}\to M$4, where $p:M_{\mathcal F}\to M$5 is the differential, $p:M_{\mathcal F}\to M$6 is an almost-Lie algebroid bracket, and the higher $p:M_{\mathcal F}\to M$7 encode higher homotopy relations among sections (Louis, 2023).

For each $p:M_{\mathcal F}\to M$8, if $p:M_{\mathcal F}\to M$9 denotes the relevant differential, the $\mathfrak D_{\mathcal F}\to M_{\mathcal F}$0-th Nash blowup $\mathfrak D_{\mathcal F}\to M_{\mathcal F}$1 is defined as the Nash blowup of $\mathfrak D_{\mathcal F}\to M_{\mathcal F}$2. The construction is made by taking the locus where $\mathfrak D_{\mathcal F}\to M_{\mathcal F}$3 has constant rank, mapping a point to the image subspace $\mathfrak D_{\mathcal F}\to M_{\mathcal F}$4 in the appropriate Grassmannian, and then taking the closure. Each $\mathfrak D_{\mathcal F}\to M_{\mathcal F}$5 comes with a projection $\mathfrak D_{\mathcal F}\to M_{\mathcal F}$6, and each singular foliation $\mathfrak D_{\mathcal F}\to M_{\mathcal F}$7 lifts uniquely to a singular foliation $\mathfrak D_{\mathcal F}\to M_{\mathcal F}$8 on $\mathfrak D_{\mathcal F}\to M_{\mathcal F}$9.

This series recovers earlier constructions at low stages. For $p^{-1}(\mathcal F)$0, it recovers a blowup introduced by Sinan Sertöz. For $p^{-1}(\mathcal F)$1, it recovers a notion due to Omar Mohsen. For $p^{-1}(\mathcal F)$2, the blowups are presented as new invariants associated to the singular foliation. A central result is that, after the first blowup $p^{-1}(\mathcal F)$3, the lifted foliation $p^{-1}(\mathcal F)$4 is always a Debord foliation, i.e. projective as a module and the image of the anchor of a Lie algebroid. This establishes the Nash procedure as a canonical pathway from an arbitrary singular foliation with geometric resolution to a Debord-type object.

2. Nash algebroid of a singular foliation

For a singular foliation $p^{-1}(\mathcal F)$5 with $p^{-1}(\mathcal F)$6 locally finitely generated and closed under the Lie bracket, one assumes an anchored bundle $p^{-1}(\mathcal F)$7 satisfying $p^{-1}(\mathcal F)$8. The Nash blow-up is a space $p^{-1}(\mathcal F)$9, described as a blow-up along the foliation, and the resolved foliation $A\to M$0 is the image of a Lie algebroid over $A\to M$1, called the Nash algebroid and denoted $A\to M$2 (Louis, 1 Sep 2025).

Its basic structural relation is the exact sequence

$A\to M$3

where $A\to M$4 is the tautological bundle over the relevant Grassmannian and $A\to M$5 is a quotient of the pulled-back anchored bundle. The fiber $A\to M$6 at a point $A\to M$7 consists of the possible limits of the regular tangent spaces to the leaves passing through regular points approaching $A\to M$8. In this way, the Nash blow-up replaces singular behavior by a bundle-theoretic record of limiting tangent data.

This formulation emphasizes that the Nash algebroid is not merely an auxiliary vector bundle. It is the object that realizes the resolved foliation as the image of a Lie algebroid after passage to the Nash blow-up. The construction therefore globalizes limit tangent information into a geometric object that supports further differential-geometric and analytic structures.

3. Nash-type blow-up of a Lie algebroid

If $A\to M$9 is a Lie algebroid with anchor $\mathcal F=\rho(\Gamma(A))$0, its image $\mathcal F=\rho(\Gamma(A))$1 is a singular foliation. Let $\mathcal F=\rho(\Gamma(A))$2 be the constant dimension of $\mathcal F=\rho(\Gamma(A))$3 on the open dense regular subset $\mathcal F=\rho(\Gamma(A))$4. On $\mathcal F=\rho(\Gamma(A))$5, the kernels $\mathcal F=\rho(\Gamma(A))$6 define a section of the Grassmann bundle $\mathcal F=\rho(\Gamma(A))$7, and the Nash blow-up of the base is

$\mathcal F=\rho(\Gamma(A))$8

The projection $\mathcal F=\rho(\Gamma(A))$9 is the restriction of the Grassmannian projection (Louis, 2024).

The corresponding Nash-algebroid construction is

$\mathrm{Nash}(A)=p^!A$0

Here $\mathrm{Nash}(A)=p^!A$1 is a Lie algebra bundle, given by the restriction of the tautological subbundle; its fiber at $\mathrm{Nash}(A)=p^!A$2 is $\mathrm{Nash}(A)=p^!A$3. The quotient $\mathrm{Nash}(A)=p^!A$4 is a Debord Lie algebroid whose anchor is injective on the dense open subset $\mathrm{Nash}(A)=p^!A$5. The construction is described as functorial and as depending only on the basic singular foliation.

In this language, the term “Nash algebroid” may refer either to $\mathrm{Nash}(A)=p^!A$6 or, in the singular-foliation formulation, to the quotient Lie algebroid $\mathrm{Nash}(A)=p^!A$7. The two usages are closely related: both are built over the same blow-up geometry, both encode limiting kernel or tangent data, and both separate a Lie algebra bundle part from a Debord-type part. The Lie algebroid version also extends, following Mohsen’s approach, to singular subalgebroids in the sense of Androulidakis–Zambon.

4. Debordization, isotropy, and geometric meaning

A defining feature of the Nash procedure is the passage from arbitrary singular foliations to Debord objects. In the foliation-resolution framework, after the first blowup the lifted foliation $\mathrm{Nash}(A)=p^!A$8 is Debord, equivalently projective as a module and equal to the image of the anchor of a Lie algebroid whose anchor is injective on a dense open subset (Louis, 2023). In the Lie algebroid framework, the quotient $\mathrm{Nash}(A)=p^!A$9 in the short exact sequence above is explicitly a Debord Lie algebroid with injective anchor on a dense open subset (Louis, 2024).

This does not mean that the blow-up space itself is necessarily smooth. The singular-foliation formulation allows $\mathcal D_{\mathcal F}$0 to be “possibly singular, sometimes a manifold,” while the Lie algebroid formulation states that the leaves of the new foliation are smooth manifolds even if the blow-up space itself is singular. The gain is therefore not a blanket smoothing of the base but a controlled reorganization of the foliation and isotropy into Lie-algebroid data.

The isotropy content is also explicit. For the first-stage blowup in the foliation-resolution setting, one has

$\mathcal D_{\mathcal F}$1

where $\mathcal D_{\mathcal F}$2 is the rank of the regular leaf, $\mathcal D_{\mathcal F}$3 is the leaf through $\mathcal D_{\mathcal F}$4, and $\mathcal D_{\mathcal F}$5 is the isotropy Lie algebra. For each $\mathcal D_{\mathcal F}$6, its image in $\mathcal D_{\mathcal F}$7 is a Lie subalgebra. In the Lie algebroid formulation, the blow-up replaces a singular point by the limiting positions of $\mathcal D_{\mathcal F}$8 as $\mathcal D_{\mathcal F}$9 through regular points. These descriptions show that the Nash algebroid records not only tangent directions to leaves but also limiting isotropy configurations.

A special case clarifies the role of Debord objects: if the original singular foliation is already projective, i.e. already a Debord foliation, then all blowups are trivial, $\infty$0 for all $\infty$1. The Nash construction is therefore sensitive to precisely the non-projective part of the singularity.

5. Helffer–Nourrigat cone, symbols, and Poisson geometry

The analytic significance of the Nash algebroid appears in the description of longitudinal differential operators on a singular foliation. For a singular foliation $\infty$2, the Helffer–Nourrigat cone $\infty$3 is defined as the closure of the union of regular conormal spaces inside $\infty$4. There is a canonical identification

$\infty$5

so the dual of the Nash algebroid serves as a desingularization of the Helffer–Nourrigat cone (Louis, 1 Sep 2025).

For each leaf $\infty$6 of $\infty$7, there exists a transitive holonomy Lie algebroid $\infty$8. The paper states that the Helffer–Nourrigat cone is a union of symplectic leaves of the canonical Poisson structures on the duals of these holonomy Lie algebroids, and more generally a union of symplectic leaves in the dual spaces carrying the canonical linear Poisson structures. This links the Nash algebroid directly to Poisson geometry and to the organization of singular symbol spaces by symplectic leaves.

The symbol calculus is formulated in two compatible ways. First, for $\infty$9, where $(\widetilde M_i,\pi_i)_{i\in\mathbb N_0}$0 is the universal enveloping algebra in the sense of Lie–Rinehart, the symbol is a family of functions on $(\widetilde M_i,\pi_i)_{i\in\mathbb N_0}$1 for all leaves $(\widetilde M_i,\pi_i)_{i\in\mathbb N_0}$2. Second, for a longitudinal operator $(\widetilde M_i,\pi_i)_{i\in\mathbb N_0}$3, one pulls back to the Nash blow-up to obtain $(\widetilde M_i,\pi_i)_{i\in\mathbb N_0}$4 on $(\widetilde M_i,\pi_i)_{i\in\mathbb N_0}$5, whose symbol is a polynomial function on $(\widetilde M_i,\pi_i)_{i\in\mathbb N_0}$6. The realization map

$(\widetilde M_i,\pi_i)_{i\in\mathbb N_0}$7

is in general not injective. The compatibility statement is

$(\widetilde M_i,\pi_i)_{i\in\mathbb N_0}$8

for any $(\widetilde M_i,\pi_i)_{i\in\mathbb N_0}$9 realizing $p:M_{\mathcal F}\to M$00. A longitudinal differential operator $p:M_{\mathcal F}\to M$01 is longitudinally elliptic if and only if its symbol on the Nash algebroid,

$p:M_{\mathcal F}\to M$02

is strictly positive outside the zero section. The paper presents this as the precise extension of ellipticity for the Nash blow-up Lie algebroid and as a generalization of previous treatments in the literature.

6. Examples and special cases

Several examples delimit the range of the construction and illustrate how closely it parallels classical blow-up theory.

Setting	Nash construction	Outcome
Debord foliation	Any $p:M_{\mathcal F}\to M$03	$p:M_{\mathcal F}\to M$04
Vector fields on $p:M_{\mathcal F}\to M$05 vanishing at $p:M_{\mathcal F}\to M$06	$p:M_{\mathcal F}\to M$07	Classical blowup of $p:M_{\mathcal F}\to M$08 at the origin
$p:M_{\mathcal F}\to M$09	$p:M_{\mathcal F}\to M$10	Blowup of the singular locus of $p:M_{\mathcal F}\to M$11
Euler vector field on $p:M_{\mathcal F}\to M$12	$p:M_{\mathcal F}\to M$13	Blowup at $p:M_{\mathcal F}\to M$14, then subsequent blowups trivial
Adjoint foliation on $p:M_{\mathcal F}\to M$15	$p:M_{\mathcal F}\to M$16	Blowup along the singular locus characterized by the drop of rank in the centralizer

In the Lie algebroid setting, a particularly explicit example is the action Lie algebroid

$p:M_{\mathcal F}\to M$17

over $p:M_{\mathcal F}\to M$18, with foliation generated by the vector fields $p:M_{\mathcal F}\to M$19, i.e. the foliation by all vector fields vanishing at the origin (Louis, 2024). Here $p:M_{\mathcal F}\to M$20, the Nash blow-up $p:M_{\mathcal F}\to M$21 is the classical projective blow-up at the origin, and the fiber over $p:M_{\mathcal F}\to M$22 is $p:M_{\mathcal F}\to M$23. At each point $p:M_{\mathcal F}\to M$24 above the origin, the fiber $p:M_{\mathcal F}\to M$25 consists of pairs $p:M_{\mathcal F}\to M$26 such that $p:M_{\mathcal F}\to M$27 is an eigenspace of $p:M_{\mathcal F}\to M$28. On each chart of the blowup, the pullback foliation is generated by vector fields tangent to the exceptional divisor.

Taken together, these examples show that the Nash algebroid interpolates between classical blow-ups of algebraic or analytic geometry and Lie-algebroid resolutions of singular foliations. The construction recovers ordinary blow-ups in standard model cases, is trivial on already projective foliations, and otherwise isolates limiting tangent and isotropy data in a canonical algebroid over the blow-up space.