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

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 $${{\mathscr {M}}}_X(G)$$ parametrizing holomorphic G-connections on X and the G-character variety While $${{\mathscr {R}}}(G)$$ is known to be affine, we show that $${{\mathscr {M}}}_X(G)$$ is not affine. The scheme $${{\mathscr {R}}}(G)$$ has an algebraic symplectic form constructed by Goldman. We construct an algebraic symplectic form on $${{\mathscr {M}}}_X(G)$$ with the property that the Riemann–Hilbert correspondence pulls back 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.