Scalar relative differential invariants

Abstract: Computation of polynomial relative invariants is a classical tool in algebra. Relative differential invariants are central for the equivalence problem of geometric structures. We address the fundamental problem of finite generation of their (differential) algebra and demonstrate both positive and negative results in this respect under various setups. As in the algebraic case, the algebra of polynomial differential invariants is not finitely generated. However we show that after localization on a finite set of relative invariants the differential algebra becomes finitely generated. We also investigate the weights of rational relative differential invariants and bound their order. Several nontrivial examples are considered and further applications are discussed.

• The paper establishes finite generation of scalar relative differential invariants after localization, generalizing classical invariant theory.
• It employs algebraic and cohomological methods to relate Lie group actions with invariant structures and weight lattice properties.
• Explicit computations and counterexamples, including special affine actions, underscore the practical implications of localization and Picard group analysis.

Scalar Relative Differential Invariants: Structure, Algebraic Generation, and Cohomological Analysis

Introduction

The paper "Scalar relative differential invariants" (2604.15473) presents a comprehensive algebraic and geometric study of scalar relative differential invariants, focusing on their structural properties, algebraic generation, and the associated cohomological framework. Relative differential invariants generalize the concept of classical polynomial and absolute invariants in invariant theory, playing a central role in the equivalence problems of geometric structures, integration of differential equations, and geometric classification. This work fills a notable gap by establishing global results for relative—in contrast to absolute—differential invariants, including finiteness and the structure of their algebra.

Formal Definitions and Cohomological Framework

Relative differential invariants are sections of equivariant line bundles with non-trivial multipliers under the action of a Lie group or algebra. Explicitly, for a Lie group $G$ acting on a manifold $M$, a function $R$ is a relative invariant if $g^*R = \Lambda_g R$ for each $g \in G$, where the multiplier $\Lambda_g$ obeys a cocycle condition. At the infinitesimal level, for $g$-actions by vector fields, the condition becomes $L_v R = \lambda(v) R$ for a 1-cocycle $\lambda$ valued in functions on $M$.

These multipliers are naturally elements of the Chevalley-Eilenberg cohomology group $M$0, classifying lifts of $M$1 to the total space of associated line bundles. Equivariant line bundles and relative invariants thus acquire cohomological interpretations, with the set of their equivalence classes described by hypercohomology of a modified Chevalley-Eilenberg-\v{C}ech complex.

Algebraic Context and Finiteness Theorems

For algebraic group actions on (quasi-)projective varieties, the analysis leverages Rosenlicht’s theorem ensuring the existence of rational invariants separating orbits on a Zariski-dense open. In this context, the key results are:

• Finite rank of the relative invariant lattice: The group of weights of rational relative invariants, realized as equivalence classes in the equivariant Picard group, forms a finitely generated lattice (Theorem 1). The algebra of rational relative invariants is thus generated by finitely many relative invariants and rational absolute invariants, modulo invertible functions.
• Algebraic presentation: Any rational relative invariant can be written as a product of powers of a finite set of basic relative invariants, multiplied by a rational function in absolute invariants and a global invertible function (Theorem 2, Theorem 3).
• Relative invariants as sections: There exists a precise correspondence between homogeneous elements in the graded algebra of relative invariants (with respect to weight) and global sections of $M$2-equivariant line bundles.

The proofs carefully address issues arising in the transition from absolute to relative invariants; crucially, the nontriviality of the Picard group and the role of divisors in the algebraic and differential contexts are handled using advanced techniques from algebraic geometry.

Differential Invariant Theory and The Global Lie-Tresse Theorem

When passing to the differential setting, the authors extend the algebraic machinery to infinite jet spaces and pseudo-group actions, employing the algebraic-geometric versions of the prolongation process and singularity stratification.

A central structural result is an extension of the global Lie-Tresse theorem: away from a singular locus, the algebra of absolute differential invariants is generated by finitely many invariants and invariant derivations; this extends, after localization, to the algebra of relative differential invariants (Theorem 4).

Key technical consequences:

• Weights of differential invariants: Rational (and polynomial) relative differential invariants can be represented as sections of line bundles whose equivariance structure descends to jets of bounded order, i.e., their weights are realized in the equivariant Picard group of a finite jet bundle.
• Finiteness under localization: The algebra of polynomial relative differential invariants is, in general, not finitely generated. However, after localization with respect to a finite set of lower-order relative invariants, the algebra becomes finitely generated by a finite family of (polynomial) relative differential invariants and a finite set of relative invariant derivative operators (Theorem 5).
• Counterexamples and obstructions: The authors provide explicit algebraic and differential examples (including special affine actions and jet prolongations) where finite generation fails without localization, showing the necessity of this restriction and demonstrating a concrete connection to Hilbert’s 14th problem for invariant rings.

Explicit Computations and Structure of Weights

The paper contains detailed computations for several symmetry settings, including:

• Special affine and projective actions: Explicit generators, their weights, and functional relations are given for group actions on jet spaces. For $M$3-actions on flag varieties, the global structure of weight lattices is computed, with connections to representation theory and the geometry of homogeneous bundles.
• Metrics and geometry: For the case of (pseudo-)Riemannian metrics under the diffeomorphism group, it is demonstrated that the weight lattice is trivial—the only polynomial relative invariants are absolute.

Table 1: Summary of Strong and Limiting Results

Property	Absolute Invariants	Relative Differential Invariants (Localized)	Relative (Unlocalized)
Rational finite generation	Yes	Yes	No (in general)
Polynomial finite generation	No (Nagata, Daigle-Freudenburg)	Yes (after localization)	No
Weight lattice rank (Picard group)	Trivial or finite	Finite	N/A

In all cases, the explicit relationship between rational and polynomial relative invariants is established via divisorial algebra and Picard group structure. The results robustly generalize classical synthesis between algebraic and differential invariant theory, extending it cohomologically.

Implications and Future Directions

These results have foundational implications for invariant theory, equivalence problems of geometric structures, and the construction of invariants in practical applications such as relativity, quantum mechanics, and geometric data analysis.

Theoretical impact:

• The establishment of localization-based finite generation delineates the limits of Hilbert-type theorems in the differential context and uncovers deep structural constraints arising from the geometry of jet bundles and Picard group cohomology.
• The identification of weights with elements of equivariant Picard groups, and their finite generation, provides a bridge between differential invariant theory and modern algebraic/categorical methods.

Practical and algorithmic implications:

• Knowledge of the realization of weights at finite jet order and explicit computation of relative invariant derivations facilitate algorithmic construction of invariants and their syzygies for equivalence problems.
• The established framework paves the way for the systematic construction and classification of higher order and higher codimension relative invariants, with immediate applications in geometric analysis and geometric PDE theory.

Future prospects:

• Extending the results to vector or tensor valued invariants, and to more general equivariant sheaves, integrating techniques from D-module theory and equivariant cohomology.
• Systematic analysis of invariant differential operators and their modules, especially in the context of parabolic, reductive, or homogeneous spaces.

Conclusion

The paper provides rigorous, globally valid results on the structure, finite generation (modulo localization), and cohomology of the algebra of scalar relative differential invariants under algebraic and pseudo-group actions. Both positive results (finite generation after localization) and fundamental obstructions (failure without localization) are established, with comprehensive computational exemplars illuminating the architecture of the algebra and the role of the weight lattice. The framework offers substantial advances for invariant theory, pushing beyond the reach of classical theorems and opening new directions for the explicit and theoretical analysis of geometric structures.