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

  • Indranil BiswasEmail author
Research Article


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.


Holomorphic connection Character variety Riemann–Hilbert correspondence Holomorphic symplectic form Affine variety 

Mathematics Subject Classification

14H60 14D20 53B15 



The author is very grateful to the referee for helpful comments. He heartily thanks Gautam Bharali and Subhojoy Gupta for very helpful comments. Thanks are also due to the Indian Institute of Science for hospitality while the work was carried out.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of MathematicsTata Institute of Fundamental ResearchMumbaiIndia

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