On the monodromies and the limit mixed Hodge structures of families of algebraic varieties

TL;DR

This paper investigates the monodromies and limit mixed Hodge structures of algebraic variety families, expressing motivic nearby fibers via Newton polyhedra, leading to explicit descriptions of Hodge numbers and monodromy Jordan forms.

Contribution

It provides explicit formulas for limit mixed Hodge numbers and monodromy Jordan forms using Newton polyhedra, advancing understanding of degenerations in algebraic geometry.

Findings

Explicit formulas for limit mixed Hodge numbers.

Descriptions of Jordan normal forms of monodromies.

Connection between motivic nearby fibers and Newton polyhedra.

Abstract

We study the monodromies and the limit mixed Hodge structures of families of complete intersection varieties over a punctured disk in the complex plane. For this purpose, we express their motivic nearby fibers in terms of the geometric data of some Newton polyhedra. In particular, the limit mixed Hodge numbers and some part of the Jordan normal forms of the monodromies of such a family will be described very explicitly.