Lefschetz numbers of Verdier monodromy and the motivic monodromy conjecture for toric varieties

Abstract

We give an expression of the Lefschetz number of iterates of the Verdier monodormy associated to an ideal on a complex algebraic variety in terms of the Euler characteristic of a space of truncated arcs. This extends the result of Denef and Loeser in the case of principal ideal on a smooth variety, and motivates a definition of motivic Milnor fiber for ideals. A discussion of the monodromy conjecture for motivic zeta functions of torus-invariant ideals on toric varieties is also included.