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 ZX,ω(T) 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 ZX,ω(T) for a
K3 surface X allowing a triple-point-free model and a volume form ! on X.
We show that ZX,ω(T) 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.
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 ZX,ω(T) 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 ZX,ω(T) for a
K3 surface X allowing a triple-point-free model and a volume form ! on X.
We show that ZX,ω(T) 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.
| Originalsprog | Engelsk |
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| Status | Udgivet - 2017 |
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