# Gerard Freixas i Montplet: Functorial intersection theory and moduli spaces of flat vector bundles on curves

**Time: **
Tue 2021-11-30 14.30 - 15.30

**Location: **
Institut Mittag-Leffler, Seminar Hall Kuskvillan (alt. Zoom, meeting ID: 921 756 1880)

**Participating: **
Gerard Freixas i Montplet (Institut de Mathématiques de Jussieu - Paris Rive Gauche)

**Abstract:** 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 formalism 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 Chern–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.