Hodge structure on the fundamental group and its application to p-adic integration

We study the unipotent completion $\Pi^{DR}_{un}(x_0, x_1, X_K)$ of the de Rham fundamental groupoid [De] of a smooth algebraic variety over a local non-archimedean field K of characteristic 0. We show that the vector space $\Pi^{DR}_{un}(x_0, x_1, X_K)$ possesses a distinguished element. In the other words, given a vector bundle E on $X_K$ together with a unipotent integrable connection, we have {\sf a canonical} isomorphism $E_{x_0}\simeq E_{x_1}$ between the fibers. The latter construction is a generalization of Colmez's p-adic integration (rk E=2) and Coleman's p-adic iterated integrals ($X_K$ is a curve with good reduction). In the second part we prove that, if $X_{K_0}$ is a smooth variety over an unramified extension of $\mathbb{Q}_p$ with good reduction and $r \leq \frac{p-1}{2}$ then there is a canonical isomorphism $\Pi^{DR}_{r}(x_0, x_1, X_{K_0})\otimes B_{DR} \simeq \Pi^{et}_{r}(x_0, x_1, X_{\overline K_0}) \otimes B_{DR}$ compatible with the action of Galois group (Here $\Pi^{DR}_{r}(x_0, x_1, X_{K_0})$ is the level r quotient of $\Pi^{DR}_{un}(x_0, x_1, X_K)$). In particularly, it implies the Crystalline Conjecture for the fundamental group [Shiho] (for $r \leq \frac{p-1}{2}$) .