Analogues of Hyperlogarithm Functions on Affine Complex Curves

Abstract

For $C$ a smooth affine complex curve, there is a unique minimal unital subalgebra $A_C$ of the algebra ${\mathcal{O}}_{\mathrm{hol}}(\tilde{C})$ of holomorphic functions on its universal cover $\tilde{C}$, which is stable under all the operations $f\mapsto \int f\omega$, for $\omega$ in the space $\Omega (C)$ of regular differentials on $C$. We identify $A_C$ with the image of the iterated integration map $I_{x_0}:\mathrm{Sh}(\Omega (C))\to {\mathcal{O}}_{\mathrm{hol}}(\tilde{C})$ 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 ${\mathcal{O}}_{\mathrm{hol}}(\tilde{C})$ 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 ${\mathrm{H}}_{\mathrm{dR}}^1(C{)}^{\ast }$ gives rise to an isomorphism of $A_C$ with $\mathcal{O}(C)\otimes \mathrm{Sh}({\mathrm{H}}_{\mathrm{dR}}^1(C))$, 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 $\mathrm{Sh}({\mathrm{H}}_{\mathrm{dR}}^1(C))$) to any MC element. We also associate an MC element $J_{\sigma }$ to each section $\sigma$ of the projection $\Omega (C)\to {\mathrm{H}}_{\mathrm{dR}}^1(C)$; when $C$ has genus zero, we exhibit a particular section ${\sigma }_0$ for which ${\mathcal{H}}_C(J_{{\sigma }_0})$ is the algebra of hyperlogarithm functions (Poincaré, Lappo-Danilevsky).