Duality theory of tautological systems
Paul Görlach, Christian Sevenheck
TL;DR
This work develops a unified holonomic duality framework for tautological systems arising from group actions on varieties, using a Chevalley–Eilenberg type complex $\hat{T}^\bullet(\rho,\overline{Y},\beta)$ to compute duals. It shows that the holonomic dual $\mathbb{D}\hat{\tau}(\rho,\overline{Y},\beta)$ is captured by $H^{n-m}\hat{T}(\rho,\overline{Y},\tilde{\beta})$ (with $n=\dim\overline{Y}$, $m=\dim G$) and specializes to an actual tautological system when $\dim G=\dim\overline{Y}$, with $\tilde{\beta}$ determined by the data; in the Gorenstein case the shift is explicit. The theory extends to equivariant and Lie algebroid contexts, providing general duality formulas for $\mathcal{U}(\mathcal{E})$-modules and tying to classical CE homology in the toric/GKZ limit. Applications include linear free divisors and homogeneous spaces, where duality yields Hodge-theoretic structure and often irreducible monodromy for the associated tautological systems. Overall, the paper unifies duality phenomena across GKZ-like systems, equivariant $\mathscr{D}$-modules, and geometric settings, offering practical criteria for when duals remain tautological and how twists transform under duality.
Abstract
We discuss the holonomic dual of tautological systems, with a view towards applications to linear free divisors and to homogeneous spaces. As a technical tool, we consider a Chevalley--Eilenberg type complex, generalizing Euler--Koszul technology from the GKZ theory, and show equivariance and holonomicity of it.