Analogues of hyperlogarithm functions on affine complex curves
Benjamin Enriquez, Federico Zerbini
TL;DR
The paper constructs a minimal stable subalgebra A_C inside the holomorphic functions on the universal cover of a smooth affine complex curve and proves A_C is the image of the iterated integral map associated with Maurer–Cartan data, identifying A_C with O(C) ⊗ Sh(H^1_{dR}(C)). It develops a MC-based framework that yields isomorphisms between A_C and a shuffle algebra extension, and shows A_C is a union of unipotent modules for the action of π_1(C) within a subalgebra of moderate growth. A C‑type filtration theory is developed, including two differential filtrations F^δ and F^μ, and it is shown that A_C coincides with the stable subalgebras generated by MC data; the work introduces filtered formality for HACAs and proves filtered formality for the associated HACA ((CΓ_C)', F_∞ 𝒪_{mod}( ilde{C})). In genus 0, the theory recovers hyperlogarithm algebras as a special case, connecting to Chen’s π_1–de Rham framework and providing a robust arithmetic-analytic generalization to higher genus and more general affine curves.
Abstract
For $C$ a smooth affine complex curve, there is a unique minimal subalgebra $A_C$ of the algebra $\mathcal O_{hol}(\tilde C)$ of holomorphic functions on its universal cover $\tilde C$, which is stable under all the operations $f\mapsto \int fω$, for $ω$ in the space $Ω(C)$ of regular differentials on $C$. We identify $A_C$ with the image of the iterated integration map $I_{x_0} : \mathrm{Sh}(Ω(C))\to\mathcal O_{hol}(\tilde C)$ based at any point $x_0$ of $\tilde C$ (here $\mathrm{Sh}(-)$ denotes the shuffle algebra of a vector space), as well as with the unipotent part, with respect to the action of $\mathrm{Aut}(\tilde C/C)$, of a subalgebra of $\mathcal O_{hol}(\tilde C)$ of moderate growth functions. We show that any regular Maurer-Cartan (MC) element $J$ on $C$ with values in the topologically free Lie algebra over $\mathrm H^1_{\mathrm{dR}}(C)^*$ gives rise to an isomorphism of $A_C$ with $\mathcal O(C) \otimes\mathrm{Sh}(\mathrm H^1_{\mathrm{dR}}(C))$, where $\mathcal O(C)$ is the algebra of regular functions on $C$, leading to the assignment of a subalgebra $\mathcal H_C(J)$ of $A_C$ (isomorphic to $\mathrm{Sh}(\mathrm H^1_{\mathrm{dR}}(C))$) to any MC element. We also associate a MC element $J_σ$ to each section $σ$ of the projection $Ω(C)\to \mathrm H^1_{\mathrm{dR}}(C)$; when $C$ has genus $0$, we exhibit a particular section $σ_0$ for which $\mathcal H_C(J_{σ_0})$ is the algebra of hyperlogarithm functions (Poincaré, Lappo-Danilevsky).