The monodromy conjecture for zeta functions associated to ideals in dimension two

Van Proeyen, Lise; Veys, Willem

Annales de l'Institut Fourier, Tome 60 (2010) no. 4, pp. 1347-1362.

La conjecture de la monodromie prédit que chaque pôle de la fonction zêta topologique (ou analogue) induit une valeur propre de la monodromie. Cette conjecture a déjà beaucoup été étudiée ; toutefois elle est prouvée en général seulement pour des fonctions zêta associées à un polynôme en deux variables. Dans cet article nous traitons des fonctions zêta associées à un idéal. En dimension quelconque nous obtenons une formule (semblable à celle d’A’Campo) qui calcule les valeurs propres de la “monodromie de Verdier”. Pour des idéaux en deux variables, nous prouvons ensuite une conjecture généralisée de la monodromie.

The monodromy conjecture states that every pole of the topological (or related) zeta function induces an eigenvalue of monodromy. This conjecture has already been studied a lot. However in full generality it is proven only for zeta functions associated to polynomials in two variables.

In this article we work with zeta functions associated to an ideal. First we work in arbitrary dimension and obtain a formula (like the one of A’Campo) to compute the “Verdier monodromy” eigenvalues associated to an ideal. Afterwards we prove a generalized monodromy conjecture for arbitrary ideals in two variables.

Classification : 14E15, 32S40, 14H20
Keywords: Zeta functions for ideals, Verdier monodromy, monodromy conjecture
Mot clés : fonctions zêta pour idéaux, monodromie de Verdier, conjecture de la monodromie

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