Parallel transport for vector bundles on $p$-adic varieties

We develop a theory of étale parallel transport for vector bundles with numerically flat reduction on a $p$-adic variety. This construction is compatible with natural operations on vector bundles, Galois equivariant and functorial with respect to morphisms of varieties. In particular, it provides a continuous $p$-adic representation of the étale fundamental group for every vector bundle with numerically flat reduction. The results in the present paper generalize previous work by the authors on curves. They can be seen as a $p$-adic analog of higher-dimensional generalizations of the classical Narasimhan-Seshadri correspondence on complex varieties. Moreover, they provide new insights into Faltings’ $p$-adic Simpson correspondence between small Higgs bundles and small generalized representations by establishing a class of vector bundles with vanishing Higgs field giving rise to actual (not only generalized) representations.