Author:

Keywords:

K3 surface, monodromy property, Calabi-Yau variety, triple-point-free

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 induce monodromy eigenvalues on the cohomology of X. Let k be an algebraically closed field of characteristic 0, and set K = k((t)). In this thesis, we focus on K3 surfaces over K 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 the motivic zeta function for a K3 surface X allowing a triple-point-free model and a volume form 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. We prove that there are 63 possibilities for the special fiber of the Crauder-Morrison model of a K3 surface X allowing a triple-point-free model that does not satisfy the monodromy property.