Constructible Differential Forms

• Constructible differential forms are a collection of natural, functorial operations on differential forms in smooth, singular, and discrete settings characterized by locality and compatibility with the exterior derivative.
• They underpin versatile frameworks such as finite-order jet formulations, stratified extensions on singular varieties, and discrete de Rham complexes in numerical analysis.
• Key examples include the wedge product, exterior differentiation, and projection operators in finite element exterior calculus, illustrating both theoretical and computational applications.

Constructible differential forms are not a single universally standardized object. In the literature surveyed here, the label is used for several closely related ideas: natural operations on smooth differential forms characterized purely by functoriality and finite-order locality; systems of honest forms obtained on the strata of singular varieties from reflexive differentials; differential-form dg models for constructible derived categories on stratified spaces; and locally computable discrete de Rham complexes on simplicial, cubical, and polytopal meshes. Across these settings, the common structural themes are naturality, compatibility with exterior differentiation, and locality in either the geometric or computational sense (Navarro et al., 2014, Kebekus, 2012, Balthasar, 2008, Bonaldi et al., 2023).

1. Naturality and the functorial formulation

In the smooth category, the relevant framework begins with the category $\mathrm{Man}$ of smooth manifolds and smooth maps, together with the subcategory $\mathrm{Mann}$ whose morphisms are local diffeomorphisms. For each $p\ge 0$, differential $p$-forms define a contravariant functor

$$\Omega^p:\mathrm{Man}^{op}\to \mathrm{Set},\qquad X\mapsto \Omega^p(X),\qquad f:X\to Y\mapsto f^*:\Omega^p(Y)\to\Omega^p(X).$$

A $k$-linear natural operation of arity $(p_1,\dots,p_k)$ and target degree $q$ is a natural transformation

$$P:\Omega^{p_1}\times\cdots\times \Omega^{p_k}\Rightarrow \Omega^q$$

such that for every smooth map $f:X\to Y$,

$\mathrm{Mann}$0

The smooth theory imposes the Peetre-Slovák regularity condition. Equivalently, when restricted to $\mathrm{Mann}$1, $\mathrm{Mann}$2 must come from a differential operator of finite order on jets. Peetre-Slovák’s theorem then ensures that each $\mathrm{Mann}$3 is locally given by a finite-order differential operator on the forms $\mathrm{Mann}$4. This formulation isolates the precise sense in which an operation is “constructible” from the forms themselves: no auxiliary metric, vector field, connection, or coordinate choice is allowed (Navarro et al., 2014).

2. Classification of natural operations on forms

Navarro–Sancho’s classification theorem states that every regular natural operation between differential forms is obtained from finitely many linear combinations, wedge products, and exterior differentials. More precisely, for $\mathrm{Mann}$5 and $\mathrm{Mann}$6, one introduces formal anti-commuting variables

$\mathrm{Mann}$7

and the free graded-commutative algebra

$\mathrm{Mann}$8

A homogeneous polynomial of total degree $\mathrm{Mann}$9 is a finite sum of monomials whose weighted degree is $p\ge 0$0. The theorem asserts that for any regular natural operation

$p\ge 0$1

there exists a unique homogeneous polynomial

$p\ge 0$2

such that for every manifold $p\ge 0$3 and forms $p\ge 0$4,

$p\ge 0$5

where monomials are evaluated by replacing multiplication with wedge product (Navarro et al., 2014).

The proof proceeds through three reductions. First, regularity and naturality identify $p\ge 0$6 with a finite-order differential operator on jets, and hence with a $p\ge 0$7-equivariant map on jet fibers over $p\ge 0$8. Second, invariance under homotheties forces a precise homogeneity relation, and a homogeneous function argument shows that only finitely many monomials can occur. Third, invariant theory for $p\ge 0$9 implies that the only surviving equivariant linear maps are skew-symmetrizations, producing wedges of either $p$0 or $p$1. Uniqueness is obtained by constructing explicit forms on $p$2 realizing each monomial (Navarro et al., 2014).

The standard examples are immediate. The wedge product arises from $p$3, the identity from $p$4, and the exterior differential from $p$5. Iterated expressions such as $p$6 correspond to the polynomial $p$7. The same theorem excludes additional multilinear diffeomorphism-invariant operations: contraction with a fixed vector and the Lie derivative are not natural in this sense because they require extra data. The result generalises work by Palais and Freed-Hopkins and gives a definitive “no-other-operations” statement for regular natural operations on forms (Navarro et al., 2014).

3. Natural forms associated with connections

The same classification machinery yields a corollary for principal connections. Let $p$8 be a Lie group with Lie algebra $p$9, and let

$$\Omega^p:\mathrm{Man}^{op}\to \mathrm{Set},\qquad X\mapsto \Omega^p(X),\qquad f:X\to Y\mapsto f^*:\Omega^p(Y)\to\Omega^p(X).$$0

where $$\Omega^p:\mathrm{Man}^{op}\to \mathrm{Set},\qquad X\mapsto \Omega^p(X),\qquad f:X\to Y\mapsto f^*:\Omega^p(Y)\to\Omega^p(X).$$1 is a principal $$\Omega^p:\mathrm{Man}^{op}\to \mathrm{Set},\qquad X\mapsto \Omega^p(X),\qquad f:X\to Y\mapsto f^*:\Omega^p(Y)\to\Omega^p(X).$$2-connection on $$\Omega^p:\mathrm{Man}^{op}\to \mathrm{Set},\qquad X\mapsto \Omega^p(X),\qquad f:X\to Y\mapsto f^*:\Omega^p(Y)\to\Omega^p(X).$$3. A $$\Omega^p:\mathrm{Man}^{op}\to \mathrm{Set},\qquad X\mapsto \Omega^p(X),\qquad f:X\to Y\mapsto f^*:\Omega^p(Y)\to\Omega^p(X).$$4-form naturally associated to the connection is a rule

$$\Omega^p:\mathrm{Man}^{op}\to \mathrm{Set},\qquad X\mapsto \Omega^p(X),\qquad f:X\to Y\mapsto f^*:\Omega^p(Y)\to\Omega^p(X).$$5

commuting with base-change and bundle isomorphisms, with $$\Omega^p:\mathrm{Man}^{op}\to \mathrm{Set},\qquad X\mapsto \Omega^p(X),\qquad f:X\to Y\mapsto f^*:\Omega^p(Y)\to\Omega^p(X).$$6 horizontal and $$\Omega^p:\mathrm{Man}^{op}\to \mathrm{Set},\qquad X\mapsto \Omega^p(X),\qquad f:X\to Y\mapsto f^*:\Omega^p(Y)\to\Omega^p(X).$$7-equivariant on $$\Omega^p:\mathrm{Man}^{op}\to \mathrm{Set},\qquad X\mapsto \Omega^p(X),\qquad f:X\to Y\mapsto f^*:\Omega^p(Y)\to\Omega^p(X).$$8.

The corollary identifies all such forms. For each $$\Omega^p:\mathrm{Man}^{op}\to \mathrm{Set},\qquad X\mapsto \Omega^p(X),\qquad f:X\to Y\mapsto f^*:\Omega^p(Y)\to\Omega^p(X).$$9, every naturally associated $k$0-form is of Chern–Weil type: there exists a unique $k$1-invariant linear map

$k$2

such that, if $k$3 is the curvature $k$4-form,

$k$5

Moreover, all odd-degree natural forms vanish identically. The same statement holds for horizontal, $k$6-equivariant $k$7-valued $k$8-forms, with $k$9 required to be $(p_1,\dots,p_k)$0-equivariant (Navarro et al., 2014).

This recovers a theorem originally due to Kolář. Conceptually, the result says that once one insists on functoriality under base change, bundle isomorphism invariance, horizontality, and $(p_1,\dots,p_k)$1-equivariance, the curvature and invariant polynomials exhaust the possibilities. No additional odd-degree constructions survive these constraints (Navarro et al., 2014).

4. Reflexive and stratified forms on singular varieties

On singular complex varieties, constructibility appears in a different sense. If $(p_1,\dots,p_k)$2 is a normal complex variety with smooth locus $(p_1,\dots,p_k)$3, the sheaf of reflexive $(p_1,\dots,p_k)$4-forms is

$(p_1,\dots,p_k)$5

A global section of $(p_1,\dots,p_k)$6 is therefore an ordinary holomorphic $(p_1,\dots,p_k)$7-form on $(p_1,\dots,p_k)$8 extending across codimension-$(p_1,\dots,p_k)$9 sets in the double-dual sense.

When $q$0 is Kawamata log terminal, Kebekus constructs, for every morphism $q$1 of normal varieties, a uniquely determined $q$2-linear sheaf map

$q$3

that is functorial in $q$4 and compatible with ordinary pull-back on the smooth loci. The construction exists even when $q$5. Its universal characterization is that it agrees with the usual pull-back wherever the latter is defined on a dense smooth open set, and it satisfies

$q$6

for composable morphisms (Kebekus, 2012).

This has an immediate stratified consequence. If

$q$7

is the singularity filtration and $q$8, then each $q$9 is smooth and locally closed. Any reflexive form

$$P:\Omega^{p_1}\times\cdots\times \Omega^{p_k}\Rightarrow \Omega^q$$0

induces a genuine holomorphic form

$$P:\Omega^{p_1}\times\cdots\times \Omega^{p_k}\Rightarrow \Omega^q$$1

on every stratum. The same argument applies to any Whitney stratification $$P:\Omega^{p_1}\times\cdots\times \Omega^{p_k}\Rightarrow \Omega^q$$2, so that a single reflexive form on $$P:\Omega^{p_1}\times\cdots\times \Omega^{p_k}\Rightarrow \Omega^q$$3 yields a system of honest forms on all strata (Kebekus, 2012).

A concrete example is the cone over a smooth plane conic in $$P:\Omega^{p_1}\times\cdots\times \Omega^{p_k}\Rightarrow \Omega^q$$4, whose singular locus is the vertex $$P:\Omega^{p_1}\times\cdots\times \Omega^{p_k}\Rightarrow \Omega^q$$5. A nowhere-zero reflexive $$P:\Omega^{p_1}\times\cdots\times \Omega^{p_k}\Rightarrow \Omega^q$$6-form on $$P:\Omega^{p_1}\times\cdots\times \Omega^{p_k}\Rightarrow \Omega^q$$7 pulls back under the inclusion $$P:\Omega^{p_1}\times\cdots\times \Omega^{p_k}\Rightarrow \Omega^q$$8 to

$$P:\Omega^{p_1}\times\cdots\times \Omega^{p_k}\Rightarrow \Omega^q$$9

hence $f:X\to Y$0. This illustrates how constructible stratified forms behave at a zero-dimensional singular stratum (Kebekus, 2012).

5. Dg-algebras of constructible forms on stratified spheres

A third usage arises in the constructible derived category of a stratified space. For the sphere $f:X\to Y$1 stratified by a point and its complement,

$f:X\to Y$2

the bounded constructible derived category is generated by the constant sheaf $f:X\to Y$3 and the skyscraper sheaf $f:X\to Y$4. Instead of working abstractly with injective resolutions, one can model the resulting endomorphism dg-algebra by differential forms.

Choose a small closed geodesic ball $f:X\to Y$5 containing $f:X\to Y$6. With $f:X\to Y$7 and $f:X\to Y$8, let $f:X\to Y$9 be the sheaf of smooth $\mathrm{Mann}$00-valued de Rham forms. The geometric dg-algebra is the $\mathrm{Mann}$01 matrix algebra

$\mathrm{Mann}$02

whose product is induced by matrix multiplication together with wedge product, and whose differential is the de Rham differential applied componentwise. Here the upper-right corner consists of forms on $\mathrm{Mann}$03 with compact support in $\mathrm{Mann}$04, and the lower entries involve forms on $\mathrm{Mann}$05 (Balthasar, 2008).

This dg-algebra is formal. The proof constructs a finite-dimensional dg-subalgebra

$\mathrm{Mann}$06

spanned by $\mathrm{Mann}$07, where $\mathrm{Mann}$08 is a volume-form generator of $\mathrm{Mann}$09 and $\mathrm{Mann}$10 satisfies $\mathrm{Mann}$11. Inside $\mathrm{Mann}$12,

$\mathrm{Mann}$13

and the only nontrivial products are those allowed by degree. The resulting cohomology is

$\mathrm{Mann}$14

with $\mathrm{Mann}$15 and $\mathrm{Mann}$16. Since the inclusion $\mathrm{Mann}$17 and the projection $\mathrm{Mann}$18 are quasi-isomorphisms, $\mathrm{Mann}$19 is quasi-isomorphic to its cohomology algebra, and the constructible derived category is correspondingly formal (Balthasar, 2008).

In this setting, constructible differential forms are not merely local forms on strata; they provide a dg model for a constructible sheaf category. The stratification enters through the matrix decomposition, while the de Rham algebra controls the homotopy-theoretic structure (Balthasar, 2008).

6. Discrete and finite element exterior-calculus constructions

In numerical exterior calculus, “constructible” refers to discrete form complexes built from local polynomial data, local traces, and commuting interpolation or projection operators. On simplicial meshes, one standard framework introduces polynomial differential-form spaces $\mathrm{Mann}$20 and the “minus” family

$\mathrm{Mann}$21

where $\mathrm{Mann}$22 is the Koszul contraction with the radial vector field. The homotopy identity

$\mathrm{Mann}$23

on homogeneous polynomial $\mathrm{Mann}$24-forms yields exactness of the local polynomial de Rham complex and of the Koszul complex. Global spaces are assembled by face-wise moment degrees of freedom, and the resulting projections commute with $\mathrm{Mann}$25, producing subcomplexes of the de Rham complex suited to finite element exterior calculus (Arnold, 2012).

On cubical meshes, Arnold and Awanou define serendipity spaces

$\mathrm{Mann}$26

prove that this is a direct sum, and show the subcomplex property

$\mathrm{Mann}$27

The associated face-moment functionals are unisolvent, the assembled sequence is exact,

$\mathrm{Mann}$28

and the local interpolators satisfy

$\mathrm{Mann}$29

In two dimensions these spaces include the serendipity finite elements and rectangular BDM elements, while in three dimensions they include 3-D serendipity, new $\mathrm{Mann}$30 and $\mathrm{Mann}$31 spaces on cubes, and discontinuous $\mathrm{Mann}$32 in top degree (Arnold et al., 2012).

For general polytopal meshes, the DDR framework constructs discrete $\mathrm{Mann}$33-form spaces

$\mathrm{Mann}$34

with all degrees of freedom expressed as moments of the Hodge-star of traces. The discrete exterior derivative $\mathrm{Mann}$35 is defined cellwise by a Stokes-inspired integration-by-parts formula, and one proves

$\mathrm{Mann}$36

The construction also has full polynomial consistency, exactness on a zero-average subcomplex, and cohomology isomorphic to that of the continuous de Rham complex (Bonaldi et al., 2023).

A further generalization treats $\mathrm{Mann}$37-form-valued $\mathrm{Mann}$38-forms on simplices. Hu and Lin define $\mathrm{Mann}$39, the “minus” spaces $\mathrm{Mann}$40, symmetry-reduced kernel spaces associated with the BGG operators $\mathrm{Mann}$41 and $\mathrm{Mann}$42, and several conforming trace regimes based on standard pull-back $\mathrm{Mann}$43 and generalized trace $\mathrm{Mann}$44. Their interpolation operators satisfy

$\mathrm{Mann}$45

and the resulting discrete BGG diagrams recover, in a unified framework, Whitney forms, Regge finite elements, TDNNS and HHJ-type spaces, MCS elements, and higher-order analogues. The stated applications include strain and stress tensors in continuum mechanics and metric and curvature tensors in differential geometry in any dimension (Hu et al., 5 Mar 2025).

These numerical constructions are discrete rather than smooth or sheaf-theoretic. Even so, they retain the characteristic features that motivate the broader terminology: local constructibility, exterior-derivative compatibility, exactness or cohomological control, and a systematic replacement of ad hoc formulas by functorial or diagrammatic structure (Arnold, 2012, Bonaldi et al., 2023, Hu et al., 5 Mar 2025).