Number of Jordan blocks of the maximal size for local monodromies

TL;DR

This paper derives formulas for the number of maximal size Jordan blocks in local monodromies of degenerating complex varieties, revealing dependencies on singularity positions and limitations of combinatorial resolution methods.

Contribution

It provides new formulas for maximal Jordan block counts in local monodromies, especially in singular cases, and demonstrates their dependence on singularity configurations.

Findings

Formulas for Jordan block counts in smooth cases.

Extension of formulas to singular cases for cohomology with compact supports.

Dependence of Jordan block counts on singularity positions, limiting combinatorial approaches.

Abstract

We prove formulas for the number of Jordan blocks of the maximal size for local monodromies of one-parameter degenerations of complex algebraic varieties where the bound of the size comes from the monodromy theorem. In case the general fibers are smooth and compact, the proof calculates some part of the weight spectral sequence of the limit mixed Hodge structure of Steenbrink. In the singular case, we can prove a similar formula for the monodromy on the cohomology with compact supports, but not on the usual cohomology. We also show that the number can really depend on the position of singular points in the embedded resolution even in the isolated singularity case, and hence there are no simple combinatorial formulas using the embedded resolution in general.