Local monodromy in non-archimedean analytic geometry -- fifth release

On a torsion analogue of the weight-monodromy conjecture

We formulate and study a torsion analogue of the weight-monodromy conjecture for a proper smooth scheme over a non-archimedean local field. We prove it for proper smooth schemes over equal characteristic non-archimedean local fields, abelian varieties, surfaces, varieties uniformized by Drinfeld upper half spaces, and set-theoretic complete intersections in toric varieties. In the equal characteristic case, our methods rely on an ultraproduct variant of Weil II established by Cadoret.

A proof of Néron--Ogg--Shafarevich criterion via its archimedean analogue

In this short note, we deduce the classical Néron--Ogg--Shafarevich criterion on good reduction of abelian varieties from its archimedean analogue: a holomorphic family of abelian varieties over a punctured disc extends to the whole unit disc if and only if the topological monodromy representation is trivial.

Archimedean cohomology revisited

Archimedean cohomology provides a cohomological interpretation for the calculation of the local L-factors at archimedean places as zeta regularized determinant of a log of Frobenius. In this paper we investigate further the properties of the Lefschetz and log of monodromy operators on this cohomology. We use the Connes-Kreimer formalism of renormalization to obtain a fuchsian connection whose residue is the log of the monodromy. We also present a dictionary of analogies between the geometry of a tubular neighborhood of the ``fiber at arithmetic infinity'' of an arithmetic variety and the compl…

Motivic integral of K3 surfaces over a non-archimedean field

We prove a formula expressing the motivic integral (\cite{ls}) of a K3 surface over $\bC((t))$ with semi-stable reduction in terms of the associated limit Hodge structure. Secondly, for every smooth variety over a non-archimedean field we define an analogue of the monodromy pairing, constructed by Grothendieck in the case of Abelian varieties, and prove that our monodromy pairing is a birational invariant of the variety. Finally, we propose a conjectural formula for the motivic integral of maximally degenerated K3 surfaces over an arbitrary non-archimedean field and prove this conjecture for K…

多元阿基米德zeta函数的极轨迹

Polar loci of multivariable archimedean zeta functions

本文确定了多元阿基米德zeta函数的极轨迹为单项支持轨迹，建立了重数与极阶的关系，推广了Barlet的结果，并应用于Bernstein-Sato型理想。

Hasse-Arf filtrations in $p$-adic analytic geometry -- Fourth Release

We the study of the monodromy of local systems with bounded ramification on a punctured disc defined over a non-archimedean valued field of characteristic zero. First, we construct the local Fourier transforms and we establish their main properties. Also, we show that the Fourier transform of a "perverse sheaf" with bounded ramification on the affine line, is again a "perverse sheaf" with bounded ramification (we put that in quotes, because actually the language of perversity is not used). These foundations are then used to exhibit a natural break decomposition for local systems with bounded r…