Graded supermanifolds and homogeneity
Katarzyna Grabowska, Janusz Grabowski
TL;DR
The paper develops a differential-geometric framework of homogeneity supermanifolds by equipping a supermanifold with a global weight vector field $\nabla$, allowing real-valued degrees and avoiding the traditional ringed-space approach to graded geometry.It systematically builds tangent/cotangent lifts, homogeneous distributions, and homogeneity Lie supergroups, proving foundational results such as the homogeneous Poincaré Lemma, the homogeneous Frobenius theorem, and a homogeneous Darboux theorem.The approach provides a unifying, coordinate-based treatment of graded structures, including vector superbundles and double/bi-homogeneous objects, with broad applicability to symplectic and Poisson-type geometries within a purely differential-geometric setting.By leveraging weight-preserving transitions and canonical lifts, the work offers a robust, flexible foundation for further development of homogeneous principal bundles, connections, and higher-graded geometric structures.
Abstract
We introduce the concept of a homogeneity supermanifold, which is, roughly speaking, a supermanifold equipped with a privileged atlas whose coordinates carry prescribed (real) homogeneity degrees. This structure defines a sheaf of graded algebras on the supermanifold, regarded as an additional geometric structure. The guiding principle of this approach is that grading is ultimately related to homogeneity. Assigning homogeneity degrees to coordinates in a consistent way is equivalent to fixing a global vector field, the weight vector field. This approach is simple and substantially more general than most existing approaches to graded manifolds. In particular, the homogeneity degrees may be arbitrary real numbers, and the resulting category includes compact supermanifolds. We systematically study homogeneity submanifolds, homogeneity Lie supergroups, tangent and cotangent lifts of homogeneity structures, homogeneous distributions and codistributions, as well as related notions such as double homogeneity. The main achievements of this framework include proofs of the homogeneous Poincaré Lemma, the homogeneous Frobenius Theorem, and the homogeneous symplectic Darboux Theorem, results that are of independent interest even in the purely even case.