Algebra & Number Theory

Logarithmic good reduction, monodromy and the rational volume

Arne Smeets

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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 -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

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

Received: 17 March 2016
Revised: 6 July 2016
Accepted: 10 August 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14F20: Étale and other Grothendieck topologies and (co)homologies
Secondary: 11G25: Varieties over finite and local fields [See also 14G15, 14G20] 11S15: Ramification and extension theory

étale cohomology logarithmic geometry monodromy nearby cycles rational points


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.

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