Invariant divisors and equivariant line bundles

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 is not necessarily the locus of a single function. Our aim is to outline a global theory of relative invariants in the complex analytic setting. 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. Since any $\mathfrak{g}$-invariant divisor gives rise to a $\mathfrak{g}$-equivariant line bundle, we investigate the group $\mathrm{Pic}_{\mathfrak{g}}(M)$ of $\mathfrak{g}$-equivariant line bundles. A cohomological description of $\mathrm{Pic}_{\mathfrak{g}}(M)$ is given 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 $M$. In the end we will discuss applications of the theory to jet spaces and differential invariants.
The talk is based on joint work with Boris Kruglikov (arXiv:2404.19439).