Given a smooth geometrically connected variety X defined over a number field K and an & eacute;tale torsor V-* U over a Zariski-open U of X, we investigate the problem of which adelic points of X can be approximated by adelic points that lift to a (twist of a) V. The question has long been investigated in the literature when U = X, but less so in the general case. We introduce a Brauer-Manin obstruction to the problem, and provide an example where this obstruction is nontrivial and purely transcendental. This answers in the negative a question posed by Harari at a 2019 workshop. Our example is also an explicit example of a nontrivial transcendental Brauer-Manin obstruction on a smooth compactification of a quotient SLn/G, with G constant metabelian.
In local relative p-adic Hodge theory, we show that the Galois cohomology of a finite-height crystalline representation (up to a twist) is essentially computed via the (Fontaine-Messing) syntomic complex with coefficients in the associated F-isocrystal. In global applications, for smooth (p-adic formal) schemes, we establish a comparison between the syntomic complex with coefficients in a locally free Fontaine-Laffaille module and the p-adic nearby cycles of the associated & eacute;tale local system on the (rigid) generic fibre.
We prove an injectivity theorem for the cohomology of the Du Bois complexes of varieties with isolated singularities. We use this to deduce vanishing statements for the cohomologies of higher Du Bois complexes of such varieties. Besides some extensions and conjectures in the non-isolated case, we also provide analogues for intersection complexes.
A real biextension is a real mixed Hodge structure that is an extension of R(0) by a mixed Hodge structure with weights -1 and -2. A unipotent real biextension over an algebraic manifold is a variation of mixed Hodge structure over it, each of whose fibers is a real biextension and whose weight graded quotients are do not vary. We show that if a unipotent real biextension has non abelian monodromy, then its “general fiber” does not split. This result is a tool for investigating the boundary behaviour of normal functions and is applied in arXiv:2408.07809 to study the boundary behaviour of the normal function of the Ceresa cycle.
