The dlt motivic zeta function is not well defined. (English) Zbl 07922236
As its title explicitly suggests, the object of the article under review is to show that the dlt motivic zeta function introduced by C. Xu [Mich. Math. J. 65, No. 1, 89–103 (2016; Zbl 1346.14062)] is not well defined, hence contradicting the main claim of the aforementioned paper.
More precisely, the definition in [C. Xu, Mich. Math. J. 65, No. 1, 89–103 (2016; Zbl 1346.14062)] does depend on the choice of a so-called dlt modification.
This 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 [J. Denef and F. Loeser, J. Algebr. Geom. 7, No. 3, 505–537 (1998; Zbl 0943.14010)].
After a set of preliminaries (dlt pairs, differents, stringy motives...) extracted for example from [W. Veys, Adv. Stud. Pure Math. 43, 529–572 (2006; Zbl 1127.14004)] and [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 \[ f = x^4 +x^2y^2 +y^6 +z^3 \] and 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 \(( \mathbf A^3 , Z ( f ) ) \) giving two different values for the motivic zeta function. Plus, another dlt modification is obtained via a flop, providing a second counter-example.
The 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 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 (in particular, regarding the motivic analogue of the Monodromy Conjecture) remains to be found in general.
More precisely, the definition in [C. Xu, Mich. Math. J. 65, No. 1, 89–103 (2016; Zbl 1346.14062)] does depend on the choice of a so-called dlt modification.
This 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 [J. Denef and F. Loeser, J. Algebr. Geom. 7, No. 3, 505–537 (1998; Zbl 0943.14010)].
After a set of preliminaries (dlt pairs, differents, stringy motives...) extracted for example from [W. Veys, Adv. Stud. Pure Math. 43, 529–572 (2006; Zbl 1127.14004)] and [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 \[ f = x^4 +x^2y^2 +y^6 +z^3 \] and 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 \(( \mathbf A^3 , Z ( f ) ) \) giving two different values for the motivic zeta function. Plus, another dlt modification is obtained via a flop, providing a second counter-example.
The 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 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 (in particular, regarding the motivic analogue of the Monodromy Conjecture) remains to be found in general.
Reviewer: Loïs Faisant (Klosterneuburg)
MSC:
| 14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |
| 14E18 | Arcs and motivic integration |
| 14E30 | Minimal model program (Mori theory, extremal rays) |
| 11S40 | Zeta functions and \(L\)-functions |
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
| 52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |
References:
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| [2] | D. A. Cox, J. B. Little, and H. K. Schenck, Toric varieties. Graduate Studies in Mathematics, 124, American Mathematical Society, Providence, RI, 2011. · Zbl 1223.14001 |
| [3] | J. Denef and F. Loeser, Motivic Igusa zeta functions. J. Algebraic Geom. 7 (1998), no. 3, 505-537. · Zbl 0943.14010 |
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| [9] | N. Potemans, Dlt zeta functions. PhD thesis, KU Leuven, 2022. |
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| [11] | W. Veys, Arc spaces, motivic integration and stringy invariants. In Singularity theory and its applications. Adv. Stud. Pure Math., 43, Mathematical Society of Japan, Tokyo, pp. 529-572, 2006. · Zbl 1127.14004 |
| [12] | C. Xu, Motivic zeta function via dlt modification. Michigan Math. J. 65 (2016), no. 1, 89-103. · Zbl 1346.14062 |
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