Logarithmic good reduction, monodromy and the rational volume

TL;DR

This paper investigates Nicaise's conjecture relating the rational volume of a smooth, proper variety over a local field to the trace of tame monodromy, proving it for varieties with logarithmic good reduction using logarithmic geometry.

Contribution

The paper proves Nicaise's conjecture for varieties with logarithmic good reduction in any dimension, extending previous results beyond curves.

Findings

Proved the conjecture for varieties with logarithmic good reduction.

Established a link between rational volume and monodromy trace in this class.

Applied logarithmic geometry techniques to study monodromy and reduction properties.

Abstract

Let $R$ be a strictly local ring complete for a discrete valuation, with fraction field $K$ and residue field of characteristic $p > 0$. Let $X$ be a smooth, proper variety over $K$. Nicaise conjectured that the rational volume of $X$ is equal to the trace of the tame monodromy operator on $\ell$-adic cohomology if $X$ is cohomologically tame. He proved this equality if $X$ is a curve. We study his conjecture from the point of view of logarithmic geometry, and prove it for a class of varieties in any dimension: those having logarithmic good reduction.