The monodromy property for K3 surfaces allowing a triple-point-free model

TL;DR

This thesis investigates the monodromy property for K3 surfaces with triple-point-free models, showing it holds for certain degenerations and analyzing the poles of their motivic zeta functions.

Contribution

It provides explicit pole computations for K3 surfaces with triple-point-free models and proves the monodromy property for specific degeneration types.

Findings

Motivic zeta functions can have multiple poles for these K3 surfaces.

Monodromy property holds for K3 surfaces with flowerpot degenerations.

Further investigation needed for chain degenerations.

Abstract

The aim of this thesis is to study under which conditions $K3$ surfaces allowing a triple-point-free model satisfy the monodromy property. This property is a quantitative relation between the geometry of the degeneration of a Calabi-Yau variety $X$ and the monodromy action on the cohomology of $X$: a Calabi-Yau variety $X$ satisfies the monodromy property if poles of the motivic zeta function $Z_{X,\omega}(T)$ induce monodromy eigenvalues on the cohomology of $X$. In this thesis, we focus on $K3$ surfaces allowing a triple-point-free model, i.e., $K3$ surfaces allowing a strict normal crossings model such that three irreducible components of the special fiber never meet simultaneously. Crauder and Morrison classified these models into two main classes: so-called flowerpot degenerations and chain degenerations. This classification is very precise, which allows to use a combination of geometrical and combinatorial techniques to check the monodromy property in practice. The first main result is an explicit computation of the poles of $Z_{X,\omega}(T)$ for a $K3$ surface $X$ allowing a triple-point-free model and a volume form $\omega$ on $X$. We show that the motivic zeta function can have more than one pole. This is in contrast with previous results: so far, all Calabi-Yau varieties known to satisfy the monodromy property have a unique pole. We prove that $K3$ surfaces allowing a flowerpot degeneration satisfy the monodromy property. We also show that the monodromy property holds for $K3$ surfaces with a certain chain degeneration. We don't know whether all $K3$ surfaces with a chain degeneration satisfy the monodromy property, and we investigate what characteristics a $K3$ surface $X$ not satisfying the monodromy property should have.