The category of pro-étale vector bundles on a proper rigid-analytic variety X over C is equivalent to the category of Higgs bundles on the eh-site of X.
Foundations of Rigid Geometry I
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abstract
In this research oriented manuscript, foundational aspects of rigid geometry are discussed, putting emphasis on birational side of formal schemes and topological feature of rigid spaces. Besides the rigid geometry itself, topics include the general theory of formal schemes and formal algebraic spaces, based on a theory of complete rings which are not necessarily Noetherian (cf. introduction). The manuscript is encyclopedic and almost self-contained, and contains plenty of new results. A discussion on relationship with J. Tate's rigid analytic geometry, V. Berkovich's analytic geometry and R. Huber's adic spaces is also included. As a model example of applications, a proof of Nagata's compactification theorem for schemes is given in the appendix. 5th version (Feb. 28, 2017): minor changes.
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A $p$-adic Simpson correspondence for singular rigid-analytic varieties
The category of pro-étale vector bundles on a proper rigid-analytic variety X over C is equivalent to the category of Higgs bundles on the eh-site of X.