Monodromy and the Lefschetz fixed point formula

Abstract

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.

38 pages; correction of misprints and addition of details; in section 8 statements have been slightly modified

References in corpus (4)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• A trace formula for rigid varieties, and motivic Weil generating series for formal schemes
• Proofs of the integral identity conjecture over algebraically closed fields
• Cohomology of locally-closed semi-algebraic subsets

Cited by in corpus (8)

• Proofs of the integral identity conjecture over algebraically closed fields
• Cohomology of locally-closed semi-algebraic subsets
• Additive invariants in o-minimal valued fields
• The motivic Thom-Sebastiani theorem for regular and formal functions
• Integration in algebraically closed valued fields with sections
• Topology of nonarchimedean analytic spaces and relations to complex algebraic geometry
• A proof of the $\ell$-adic version of the integral identity conjecture for polynomials
• Motivic Serre invariants modulo the square of $\mathbb{L}-1$