Parallel transport for vector bundles on 𝑝-adic varieties

Abstract

We develop a theory ofétale 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 theétale 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. MSC: 14J20, 11G252 Numerically flat bundles Throughout this paper, a variety over a field k is a geometrically integral separated scheme of finite type over k. Recall that a scheme is called integral if it is irreducible and reduced.We want to study the following class of vector bundles which was introduced in [DMOS82] (2.34).Definition 2.1 Let X be a connected and complete scheme over a perfect field k. A vector bundle E over X is called Nori-semistable if for all k-morphisms f : C → X from smooth connected projective curves C over k into X the pullback f * E is semistable of degree zero.A related but different notion is due to Nori [Nor82] who considers only pullbacks to embedded smooth projective curves. In [DMOS82] the bundles in Definition 1 are simply called semistable. Since this can lead to misunderstandings, the name Nori-semistable has been adopted by several authors.

The p-adic Corlette–Simpson correspondence for abeloids

“…Indeed, given any two choices of an exponential, the resulting equivalences on indecomposable objects only differ by twists with analytic torsion line bundles. Related work Towards establishing a p-adic Corlette-Simpson correspondence for representations of the étale fundamental group, Deninger and the third author have investigated the case of Higgs bundles with vanishing Higgs field, in which case they show that one can attach representations to vector bundles with numerically flat reduction [12,13]. Würthen [41] has extended this functor to the rigid analytic case and has shown that for analytic vector bundles, the notion of numerically flat reduction is closely related to being pro-finite-étale, i.e.…”

“…Lemma 4. 13 Let K be any non-archimedean field over Q p and T a finite free Z pmodule. Consider V := T ⊗ Z p G a as a vector group.…”

Diamantine Picard functors of rigid spaces

“…In the case of vanishing Higgs field, this question has been studied by Deninger-Werner [DW20] and Würthen [Wü20,§3], whose results suggest that in the special case of vanishing Higgs field, vector bundles that are trivialised by pro-finite-étale covers give the correct subcategory of Higgs bundles. We argued in [Heu20a,§4] that it is also the right condition in the case of rank one, and used it to construct the Simpson correspondence for line bundles.…”

The $p$-adic Corlette-Simpson correspondence for abeloids

“…A major open question in the field is which Higgs bundles correspond to representations: Deninger and the third author have investigated the case of numerically flat vector bundles with trival Higgs field in [DW05b] and [DW20]. Würthen [Wü20] has shown that for analytic vector bundles, the notion of numerical flatness is closely related to being pro-finite-étale.…”