Parallel transport for vector bundles on p-adic varieties

TL;DR

This paper develops a theory of étale parallel transport for vector bundles on p-adic varieties, generalizing previous work and connecting to p-adic Simpson correspondence, providing a new framework for understanding p-adic representations.

Contribution

It introduces a new construction of étale parallel transport for vector bundles with flat reduction on p-adic varieties, extending prior results from curves to higher dimensions.

Findings

Provides a continuous p-adic representation of the étale fundamental group.

Generalizes the Narasimhan-Seshadri correspondence to p-adic higher-dimensional varieties.

Establishes a link between flat vector bundles and p-adic Simpson correspondence.

Abstract

We develop a theory of \'etale 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 \'etale 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.