Monodromy eigenvalues and zeta functions with differential forms

Details Monodromy eigenvalues and zeta functions with differential forms Monodromy eigenvalues and zeta functions with differential forms

2007-01-18

arxiv.org/abs/math/0701502v1

Mathematics

Algebraic Geometry

To appear in Advances in Mathematics. 17 pages

Scientific paper

For a complex polynomial or analytic function f, one has been studying intensively its so-called local zeta functions or complex powers; these are integrals of |f|^{2s}w considered as functions in s, where the w are differential forms with compact support. There is a strong correspondence between their poles and the eigenvalues of the local monodromy of f. In particular Barlet showed that each monodromy eigenvalue of f is of the form exp(a2i\pi), where a is such a pole. We prove an analogous result for similar p-adic complex powers, called Igusa (local) zeta functions, but mainly for the related algebro-geometric topological and motivic zeta functions.

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