## Algebra & Number Theory

### Logarithmic good reduction, monodromy and the rational volume

Arne Smeets

#### 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 $ℓ$-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.

#### Article information

Source
Algebra Number Theory, Volume 11, Number 1 (2017), 213-233.

Dates
Revised: 6 July 2016
Accepted: 10 August 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842718

Digital Object Identifier
doi:10.2140/ant.2017.11.213

Mathematical Reviews number (MathSciNet)
MR3602769

Zentralblatt MATH identifier
1361.14017

#### Citation

Smeets, Arne. Logarithmic good reduction, monodromy and the rational volume. Algebra Number Theory 11 (2017), no. 1, 213--233. doi:10.2140/ant.2017.11.213. https://projecteuclid.org/euclid.ant/1510842718

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