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Monodromy and the Lefschetz fixed point formula

2015; Société Mathématique de France; Volume: 48; Issue: 2 Linguagem: Inglês

10.24033/asens.2246

1873-2151

Ehud Hrushovski, François Loeser,

Homotopy and Cohomology in Algebraic Topology

We give a new proof - not using resolution of singularities - of a formula of Denef and the second author expressing the Lefschetz number of iterates of the monodromy of a function on a smooth complex algebraic variety in terms of the Euler characteristic of a space of truncated arcs. Our proof uses l-adic cohomology of non-archimedean spaces, motivic integration and the Lefschetz fixed point formula for finite order automorphisms. We also consider a generalization due to Nicaise and Sebag and at the end of the paper we discuss connections with the motivic Serre invariant and the motivic Milnor fiber.