The dlt motivic zeta function is not well defined
type: Item
instance of: scholarly article
title: The dlt motivic zeta function is not well defined
zbMATH Open document ID: 1547.14033
author: Johannes Nicaise
author: Naud Potemans
author: Willem Veys
published in: Michigan Mathematical Journal
publication date: 2024-10-07T00:00:00Z
review text: As its title explicitly suggests, the object of the article under review is to show that the dlt motivic zeta function introduced by \textit{C. Xu} [Mich. Math. J. 65, No. 1, 89--103 (2016; Zbl 1346.14062)] is \textit{not} well defined, hence contradicting the main claim of the aforementioned paper.\N\NMore precisely, the definition in [\textit{C. Xu}, Mich. Math. J. 65, No. 1, 89--103 (2016; Zbl 1346.14062)] \textit{does} depend on the choice of a so-called \textit{dlt modification}.\N\NThis dlt motivic zeta function, associated with a regular function \(f\) on a smooth variety X over a field of characteristic zero, is a variant of the zeta function introduced in [\textit{J. Denef} and \textit{F. Loeser}, J. Algebr. Geom. 7, No. 3, 505--537 (1998; Zbl 0943.14010)].\N\NAfter a set of preliminaries (dlt pairs, differents, stringy motives...) extracted for example from [\textit{W. Veys}, Adv. Stud. Pure Math. 43, 529--572 (2006; Zbl 1127.14004)] and [\textit{V. V. Batyrev}, J. Eur. Math. Soc. (JEMS) 1, No. 1, 5--33 (1999; Zbl 0943.14004)], the authors present a number of counterexamples to Hu's claim, starting from an explicit Newton nondegenerate polynomial\N\[\Nf = x^4 +x^2y^2 +y^6 +z^3\N\]\Nand constructing two very explicit blow-ups (with pictures of the corresponding fans for the reader's convenience) which are two different dlt modifications for the pair \(( \mathbb{A}^3 , Z ( f ) ) \) giving two different values for the motivic zeta function. Plus, another dlt modification is obtained via a flop, providing a second counter-example.\N\NThe end of the paper is devoted to an analysis of three incorrect proofs (the ones of Proposition 3.1, Proposition 3.2 and Theorem 1.2) appearing in \textit{C. Xu}'s paper [Mich. Math. J. 65, No. 1, 89--103 (2016; Zbl 1346.14062)], and to comments about how one could modify or exploit Hu's definition of the dlt motivic zeta function. Very roughly, the conclusion of the paper seems to be that a correct and exploitable definition (i…
reviewed by: Loïs Faisant
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14E18
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 11S40
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 52B20
zbMATH DE Number: 7922236
zbMATH Keywords: motivic dlt zeta function
zbMATH Keywords: dlt modifications
zbMATH Keywords: counter-examples
zbMATH Keywords: divisorial log terminal
MaRDI profile type: Publication
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DOI: 10.1307/MMJ/20216148
description: scientific article; zbMATH DE number 7922236