Invariant divisors and equivariant line bundles
Boris Kruglikov, Eivind Schneider
TL;DR
This work develops a global theory of relative invariants for holomorphic Lie algebra actions by introducing the $\mathfrak{g}$-equivariant Picard group $\mathrm{Pic}_{\mathfrak{g}}(M)$ and the invariant-divisor group $\mathrm{Div}_{\mathfrak{g}}(M)$. It constructs a double complex that couples Čech cohomology with Chevalley–Eilenberg cohomology, yielding a hypercohomology description $\mathrm{Pic}_{\mathfrak{g}}(M) \cong \mathbb{H}^1(M,\mathfrak{C}_{\mathfrak{g}})$ and clarifies how invariant divisors give rise to $\mathfrak{g}$-equivariant line bundles. The paper develops criteria for transversal lifts, establishes relations between equivariant and ordinary Picard groups, and applies the theory to affine and jet bundles, deriving polynomial divisors and reducing invariant differential invariants to finite-dimensional data. Through explicit examples on $\mathbb{CP}^1$, $\mathbb{CP}^2$, elliptic curves, and ODE jet spaces, it connects classical differential invariants with a cohesive cohomological framework, enabling systematic computation of global relative invariants. The results provide a principled pathway to classify invariant differential equations and to understand multipliers and weights in a global, geometric setting.
Abstract
Scalar relative invariants play an important role in the theory of group actions on a manifold as their zero sets are invariant hypersurfaces. Relative invariants are central in many applications, where they often are treated locally since an invariant hypersurface may not be a locus of a single function. Our aim is to establish a global theory of relative invariants. For a Lie algebra $\mathfrak{g}$ of holomorphic vector fields on a complex manifold $M$, any holomorphic $\mathfrak{g}$-invariant hypersurface is given in terms of a $\mathfrak{g}$-invariant divisor. This generalizes the classical notion of scalar relative $\mathfrak{g}$-invariant. Any $\mathfrak{g}$-invariant divisor gives rise to a $\mathfrak{g}$-equivariant line bundle, and a large part of this paper is therefore devoted to the investigation of the group $\mathrm{Pic}_{\mathfrak{g}}(M)$ of $\mathfrak{g}$-equivariant line bundles. We give a cohomological description of $\mathrm{Pic}_{\mathfrak{g}}(M)$ in terms of a double complex interpolating the Chevalley-Eilenberg complex for $\mathfrak{g}$ with the Čech complex of the sheaf of holomorphic functions on $M$. We also obtain results about polynomial divisors on affine bundles and jet bundles. This has applications to the theory of differential invariants. Those were actively studied in relation to invariant differential equations, but the description of multipliers (or weights) of relative differential invariants was an open problem. We derive a characterization of them with our general theory. Examples, including projective geometry of curves and second-order ODEs, not only illustrate the developed machinery, but also give another approach and rigorously justify some classical computations. At the end, we briefly discuss generalizations of this theory.