# On the moduli space of holomorphic G-connections on a compact Riemann surface

## TL;DR

This paper investigates the moduli space of holomorphic G-connections on a compact Riemann surface, showing it is non-affine and constructing an algebraic symplectic form that aligns with the Goldman form via the Riemann--Hilbert correspondence.

## Contribution

It demonstrates that the moduli space of holomorphic G-connections is non-affine and constructs an algebraic symplectic form compatible with the Goldman form through the Riemann--Hilbert correspondence.

## Key findings

- The moduli space ${
m f M}_X(G)$ is not affine.
- An algebraic symplectic form on ${
m f M}_X(G)$ is constructed.
- The pullback of the Goldman symplectic form matches the constructed algebraic form.

## Abstract

Let $X$ be a compact connected Riemann surface of genus at least two and $G$ a connected reductive complex affine algebraic group. The Riemann--Hilbert correspondence produces a biholomorphism between the moduli space ${\mathcal M}_X(G)$ parametrizing holomorphic $G$--connections on $X$ and the $G$--character variety $${\mathcal R}(G):= \text{Hom}(\pi_1(X, x_0), G)/\!\!/G\, .$$ While ${\mathcal R}(G)$ is known to be affine, we show that ${\mathcal M}_X(G)$ is not affine. The scheme ${\mathcal R}(G)$ has an algebraic symplectic form constructed by Goldman. We construct an algebraic symplectic form on ${\mathcal M}_X(G)$ with the property that the Riemann--Hilbert correspondence pulls back to the Goldman symplectic form to it. Therefore, despite the Riemann--Hilbert correspondence being non-algebraic, the pullback of the Goldman symplectic form by the Riemann--Hilbert correspondence nevertheless continues to be algebraic.