Moduli of flat connections on smooth varieties

This is a report on a joint work with Bertrand To\""{e}n. We study the derived geometry of the moduli of local systems and flat bundles on a smooth but not necessarily proper complex algebraic variety $X$. In the Betti case we show that these moduli carry natural Poisson structures, generalizing the well known case of curves. We also construct symplectic leaves of this Poisson structure by fixing local monodromies at infinity, and show that a new feature, called strictness, appears as soon as the divisor at infinity has non-trivial crossings. In the de Rham case we introduce the notion of a formal boundary of $X$, and explain how to define a restriction to the boundary map $R$ between derived moduli of flat bundles. I will discuss representability results for the geometric fibers of $R$ and will explain why the morphism $R$ comes equipped with a canonical shifted Lagrangian structure.

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