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Esterov, Alexander; Lemahieu, Ann; Takeuchi, Kiyoshi

On the monodromy conjecture for non-degenerate hypersurfaces. (English) Zbl 1506.14106

J. Eur. Math. Soc. (JEMS) 24, No. 11, 3873-3949 (2022).
The authors verify the monodromy conjecture [J. Denef and F. Loeser, J. Am. Math. Soc. 5, No. 4, 705–720 (1992; Zbl 0777.32017)] about topological zeta functions for a large class of singularities that are non-degenerate with respect to their Newton polyhedra, including all such singularities of functions depending on four variables. They explain the novelty of their approach as follows. Thus, it is known that in the case of three variables all singularities close to a non-degenerate one are non-degenerate as well [A. Lemahieu and L. Van Proeyen, Trans. Am. Math. Soc. 363, No. 9, 4801–4829 (2011; Zbl 1248.14012)]. Next, in the setting of E. Artal Bartolo et al. [Quasi-ordinary power series and their zeta functions. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1095.14005)], all singularities close to a quasi-ordinary one are also quasi-ordinary. However, in contrast to these papers, a new phenomenon arises in the four-dimensional case: there are degenerate singularities arbitrarily close to a nonisolated non-degenerate singularity; this is one of the main difficulties of the proof pointed out by the authors.
The paper is divided into several parts. In the introduction, the authors give a detailed review of a number of previously obtained results and the corresponding extensive bibliography, explain the essence of their approach, and formulate the main statements. Then the monodromy conjecture for the topological zeta function and the main properties of Newton polyhedra are discussed, and configurations of faces of the Newton polyhedron that do not ensure the existence of the corresponding pole of the topological zeta function are studied. After that, the authors study face configurations, which, on the contrary, always nontrivially contribute to the multiplicity of the expected monodromy eigenvalue and prove the conjecture for a certain class of non-degenerate singularities in arbitrary dimension. The last section contains a proof of the monodromy conjecture for non-degenerate singularities of functions depending on four variables. In the appendix, it is briefly discussed some basic concepts related to the geometry of the lattice, and the necessary results used in the article.
Reviewer: Aleksandr G. Aleksandrov (Moskva)

Cited in 2 Documents

MSC:

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)

Keywords:

nonisolated singularities; non-degenerate singularities; topological zeta functions; monodromy conjecture; toric varieties; Newton polytopes; non-convenient Newton polyhedra; eigenvalues of monodromy; corners; hypermodular function; lattice geometry

Citations:

Zbl 0777.32017; Zbl 1248.14012; Zbl 1095.14005
PDFBibTeX XMLCite
\textit{A. Esterov} et al., J. Eur. Math. Soc. (JEMS) 24, No. 11, 3873--3949 (2022; Zbl 1506.14106)
Full Text: DOI arXiv

References:

[1] Vol Z ./ D 2 and Vol Z ./ D 1. In the first case the sought statement follows from Lemma 8.28 (2). Suppose now that
[2] A’Campo, N.: La fonction zêta d’une monodromie. Comment. Math. Helv. 50, 233-248 (1975) Zbl 0333.14008 MR 371889 · Zbl 0333.14008
[3] Artal Bartolo, E., Cassou-Noguès, P., Luengo, I., Melle Hernández, A.: Monodromy conjecture for some surface singularities. Ann. Sci. École Norm. Sup. (4) 35, 605-640 (2002) Zbl 1020.32025 MR 1981174 · Zbl 1020.32025
[4] Artal Bartolo, E., Cassou-Noguès, P., Luengo, I., Melle Hernández, A.: Quasi-ordinary power series and their zeta functions. Mem. Amer. Math. Soc. 178, no. 841, vi+85 pp. (2005) Zbl 1095.14005 MR 2172403 · Zbl 1095.14005
[5] Blanco, G., Budur, N., van der Veer, R.: Monodromy conjecture for semi-quasihomogeneous hypersurfaces. Math. Nachr., to appear; arXiv:2106.11015
[6] Bories, B., Veys, W.: Igusa’s p-adic local zeta function and the monodromy conjecture for non-degenerate surface singularities. Mem. Amer. Math. Soc. 242, no. 1145, vii+131 pp. (2016) Zbl 1397.14020 MR 3498149 · Zbl 1397.14020
[7] Budur, N., Mustaţȃ, M., Teitler, Z.: The monodromy conjecture for hyperplane arrangements. Geom. Dedicata 153, 131-137 (2011) Zbl 1227.32035 MR 2819667 · Zbl 1227.32035
[8] Denef, J.: Report on Igusa’s local zeta function. In: Séminaire Bourbaki, Vol. 1990/91, Astérisque 201-203, exp. 741, 359-386 (1991) Zbl 0749.11054 MR 1157848 · Zbl 0749.11054
[9] Denef, J.: Degree of local zeta functions and monodromy. Compos. Math. 89, 207-216 (1993) Zbl 0932.11073 MR 1255694 · Zbl 0932.11073
[10] Denef, J., Loeser, F.: Caractéristiques d’Euler-Poincaré, fonctions zêta locales et modifica-tions analytiques. J. Amer. Math. Soc. 5, 705-720 (1992) Zbl 0777.32017 MR 1151541 · Zbl 0777.32017
[11] Denef, J., Loeser, F.: Motivic Igusa zeta functions. J. Algebraic Geom. 7, 505-537 (1998) Zbl 0943.14010 MR 1618144 · Zbl 0943.14010
[12] Denef, J., Loeser, F.: Geometry on arc spaces of algebraic varieties. In: European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math. 201, Birkhäuser, Basel, 327-348 (2001) Zbl 1079.14003 MR 1905328 · Zbl 1079.14003
[13] Dimca, A.: Sheaves in Topology. Universitext, Springer, Berlin (2004) Zbl 1043.14003 MR 2050072 · Zbl 1043.14003
[14] Esterov, A. Tropical nearby monodromy eigenvalues. Submitted; arXiv:1807.00609
[15] Fulton, W.: Introduction to Toric Varieties. Ann. of Math. Stud. 131, Princeton Univ. Press, Princeton, NJ (1993) Zbl 0813.14039 MR 1234037 · Zbl 0813.14039
[16] González Villa, M., Lemahieu, A.: The monodromy conjecture for plane meromorphic germs. Bull. London Math. Soc. 46, 441-453 (2014) Zbl 1348.14008 MR 3210699 · Zbl 1348.14008
[17] Hotta, R., Takeuchi, K., Tanisaki, T.: D-modules, Perverse Sheaves, and Representation Theory. Progr. Math. 236, Birkhäuser Boston, Boston, MA (2008) Zbl 1136.14009 MR 2357361 · Zbl 1136.14009
[18] Kouchnirenko, A. G.: Polyèdres de Newton et nombres de Milnor. Invent. Math. 32, 1-31 (1976) Zbl 0328.32007 MR 419433 · Zbl 0328.32007
[19] Lemahieu, A., Van Proeyen, L.: Monodromy conjecture for nondegenerate surface singular-ities. Trans. Amer. Math. Soc. 363, 4801-4829 (2011) Zbl 1248.14012 MR 2806692 · Zbl 1248.14012
[20] Lemahieu, A., Veys, W.: Zeta functions and monodromy for surfaces that are general for a toric idealistic cluster. Int. Math. Res. Notices 2009, 11-62 Zbl 1161.14017 MR 2471295 · Zbl 1161.14017
[21] Loeser, F.: Fonctions d’Igusa p-adiques et polynômes de Bernstein. Amer. J. Math. 110, 1-21 (1988) Zbl 0644.12007 MR 926736 · Zbl 0644.12007
[22] Loeser, F.: Fonctions d’Igusa p-adiques, polynômes de Bernstein, et polyèdres de Newton. J. Reine Angew. Math. 412, 75-96 (1990) Zbl 0713.11083 MR 1079002 · Zbl 0713.11083
[23] Mustaţȃ, M.: Bernstein-Sato polynomials for general ideals vs. principal ideals. arXiv:1906.03086 (2019)
[24] Némethi, A., Veys, W.: Generalized monodromy conjecture in dimension two. Geom. Topol. 16, 155-217 (2012) Zbl 1241.14004 MR 2872581 · Zbl 1241.14004
[25] Nicaise, J.: An introduction to p-adic and motivic zeta functions and the monodromy conjecture. In: Algebraic and Analytic Aspects of Zeta Functions and L-functions, MSJ Mem. 21, Math. Soc. Japan, Tokyo, 141-166 (2010) Zbl 1275.11147 MR 2647606 · Zbl 1275.11147
[26] Oda, T.: Convex Bodies and Algebraic Geometry. Ergeb. Math. Grenzgeb. (3) 15, Springer, Berlin (1988) Zbl 0628.52002 MR 922894 · Zbl 0628.52002
[27] Oka, M.: Non-degenerate Complete Intersection Singularity. Actualités Math., Hermann, Paris (1997) Zbl 0930.14034 MR 1483897 · Zbl 0930.14034
[28] Rodrigues, B., Veys, W.: Holomorphy of Igusa’s and topological zeta functions for homogeneous polynomials. Pacific J. Math. 201, 429-440 (2001) Zbl 1054.11061 MR 1875902 · Zbl 1054.11061
[29] Van Proeyen, L., Veys, W.: The monodromy conjecture for zeta functions associated to ideals in dimension two. Ann. Inst. Fourier (Grenoble) 60, 1347-1362 (2010) Zbl 1211.14021 MR 2722244 · Zbl 1211.14021
[30] Varchenko, A. N.: Zeta-function of monodromy and Newton’s diagram. Invent. Math. 37, 253-262 (1976) Zbl 0333.14007 MR 424806 · Zbl 0333.14007
[31] Veys, W.: Poles of Igusa’s local zeta function and monodromy. Bull. Soc. Math. France 121, 545-598 (1993) Zbl 0812.14014 MR 1254752 · Zbl 0812.14014
[32] Veys, W.: Determination of the poles of the topological zeta function for curves. Manuscripta Math. 87, 435-448 (1995) Zbl 0851.14012 MR 1344599 · Zbl 0851.14012
[33] Veys, W.: Vanishing of principal value integrals on surfaces. J. Reine Angew. Math. 598, 139-158 (2006) Zbl 1106.14010 MR 2270570 · Zbl 1106.14010
[34] Yuran, A.: Newton polytopes of nondegenerate quadratic forms. arXiv:1910.06135 (2019)
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