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

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 MX(G) is known to be affine, we show that MX(G) is not affine. The scheme R(G) has an algebraic symplectic form constructed by Goldman. We construct an algebraic symplectic form on MX(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.