Cohomology of the differential fundamental group of algebraic curves

TL;DR

This work proves that for a smooth projective curve $X$ of genus $g\ge 1$ over a field of characteristic zero, the group cohomology $\mathrm{H}^{i}(\pi(X),V)$ of the differential fundamental group $\pi(X)$ is canonically isomorphic to the de Rham cohomology $\mathrm{H}^{i}_{\mathrm{dR}}(X,(\mathcal{V},\nabla))$ for all $i\ge 0$ and all $\mathrm{MIC}^{\mathrm{ind}}(X)$-objects $(\mathcal{V},\nabla)$ with fiber $V$ at a base point. This yields $\mathrm{H}^{i}(\pi(X),V)=0$ for $i\ge 3$ and, in genus 1, a decomposition $\pi(X)\cong\pi^{\mathrm{uni}}(X)\times\pi^{\mathrm{diag}}(X)$ with $\pi^{\mathrm{uni}}(X)$ described as a one-relator pro-unipotent group of rank $2g$. The paper further computes the cohomology of the unipotent quotient $\pi^{\mathrm{uni}}(X)$, applying Lubotsky–Magid theory, and gives an explicit description of $\pi(X)$ for elliptic curves, including the structure of the diagonal part in terms of line bundles with connections. These results place smooth projective curves in characteristic $0$ into the framework of de Rham $k(\pi,1)$-spaces, connecting Tannakian duality, $\mathrm{MIC}$-inductive limits, and topological intuition from Riemann surfaces.

Abstract

Let X be a smooth projective curve over a field k of characteristic zero. The differential fundamental group of X is defined as the Tannakian dual to the category of vector bundles with (integrable) connections on X. This work investigates the relationship between the de Rham cohomology of a vector bundle with connection and the group cohomology of the corresponding representation of the differential fundamental group of X . Consequently, we obtain some vanishing and non-vanishing results for the group cohomology.

Theorems & Definitions (16)

• Theorem 1.1: Theorem \ref{['theorem_delta_i']}
• Remark 3.1
• Remark 3.2
• Theorem 4.1
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Lemma 4.4
• proof