Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink

Abstract

We prove a motivic analogue of Steenbrink's conjecture [25, Conjecture 2.2] on the Hodge spectrum (proved by M. Saito in [21]). To achieve this, we construct and compute motivic iterated vanishing cycles associated with two functions. We are also led to introduce a more general version of the convolution operator appearing in the motivic Thom-Sebastiani formula. Throughout the article we use the framework of relative equivariant Grothendieck rings of varieties endowed with an algebraic torus action