### Abstract

The apparatus of non-abelian Hodge theory can be made very exlicit in the abelian case (i.e. for the group $\mathrm{GL}_1$). This was carefully made in [1]. It is instructive to put the stress on the non complex nature of the map from the Dolbeault model to the Betti one, and the highly transcendental nature of the holomorphic map form de Rham to Betti, given by integrating differential equations. One can emphasize differences between the three models, e.g. the de Rham model is Stein but has no regular functions, whereas the Betti model is affine. If time allows some discussion of the results in [2] would be interesting.

Date

2018-07-23 11:15 — 12:15

###### Phd student

My research domain is about the arithmetic of differential equations, periods, and exponential sums.