Geometric monodromies, mixed Hodge numbers of motivic Milnor fibers and Newton polyhedra

TL;DR

This paper develops a theory connecting monodromies, motivic Milnor fibers, and Newton polyhedra, revealing new insights into their interplay in algebraic geometry and Hodge theory.

Contribution

It introduces a comprehensive framework linking local and global monodromies with motivic Milnor fibers and Newton polyhedra using modern cohomological tools.

Findings

Equivariant mixed Hodge numbers are expressed via Newton polyhedra.

The theory relates monodromies to toric geometry.

New methods for analyzing motivic Milnor fibers are proposed.

Abstract

We introduce the theory of local and global monodromies of polynomials in cohomology groups in various geometric situations, focusing on its relations with toric geometry and motivic Milnor fibers, and moreover in the modern languages of nearby and vanishing cycle functors. Equivariant mixed Hodge numbers of motivic Milnor fibers will be described in terms of Newton polyhedra of polynomials.