The motivic fundamental groupoid at tangential basepoints

We give a general construction of the motivic fundamental groupoid at tangential basepoints, extending previous works of P. Deligne, A. B. Goncharov, and M. Levine, which were limited to ordinary basepoints or to specific varieties. Given a smooth variety over a field endowed with a simple normal crossings divisor, we encode its tangential basepoints using the language of logarithmic geometry. Building on the recent construction by F. Binda, D. Park, and P. A. {\O}stv{\ae}r of a stable $\infty$-category of $\mathbb{A}^1$-invariant logarithmic motives and its comparison with the usual $\infty$-category of motives, we define in a functorial manner the associated motivic pointed path spaces. In the presence of a motivic $t$-structure, truncating yields the motivic fundamental groupoid. In general, we construct Betti and de Rham realization functors for logarithmic motives (linearizing the construction of F. Binda, D. Park and P. A. {\O}stv{\ae}r for the Betti case) and we show that the periods of the motivic fundamental groupoid are given by regularized iterated integration of logarithmic differential $1$-forms, thus yielding a general version of Chen's theorem with tangential basepoints.