Mixed Hodge modules without slope

TL;DR

This paper investigates morphisms without slope in mixed Hodge modules, establishing key properties like commutativity of certain cycles and compatibility of filtrations, with implications for understanding singularities.

Contribution

It introduces the notion of 'strictly without slope' for mixed Hodge modules and proves its stability under proper direct images, advancing the theory of Hodge modules.

Findings

Proved commutativity of iterated nearby and vanishing cycles for morphisms without slope.

Defined 'strictly without slope' and showed its preservation under proper direct images.

Established compatibility of Hodge and Kashiwara-Malgrange filtrations for specific pure Hodge modules.

Abstract

In this article we are interested in morphisms without slope for mixed Hodge modules. We first show the commutativity of iterated nearby cycles and vanishing cycles applied to a mixed Hodge module in the case of a morphism without slope. Then we define the notion "strictly without slope" for a mixed Hodge module and we show the preservation of this condition under the direct image by a proper morphism. As an application we prove the compatibility of the Hodge filtration and Kashiwara-Malgrange filtrations for some pure Hodge modules with support an hypersurface with quasi-ordinary singularities.