In this talk I will present a formalism of refined direct images of characteristic classes for flat vector bundles on families of compact Riemann surfaces. This is inspired by Deligne’s functorial approach to Arakelov geometry, where one deals with hermitian vector bundles instead. A key ingredient in his construction is the theory of Bott—Chern secondary classes, which in our framework is replaced by a variant of Chern—Simons theory. The natural setting of application of our formalisme is that of the moduli space of vector bundles with flat relative connections on a family of curves. In this case, our main construction is a relative version of the so-called complex Chern—Simons line bundle. I will present the main properties of the relative Complex—Simons line bundle, and discuss some applications to families of projective structures on Riemann surfaces. This is joint work with D. Eriksson and R. Wentworth.

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