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

TY - BOOK

T1 - The monodromy property for K3 surfaces allowing a triple-point-free model

AU - Jaspers, Annelies Kristien J

PY - 2017

Y1 - 2017

N2 - The aim of this thesis is to study under which conditions K3 surfaces allowinga triple-point-free model satisfy the monodromy property. This property is aquantitative relation between the geometry of the degeneration of a Calabi-Yauvariety X and the monodromy action on the cohomology of X: a Calabi-Yau variety X satisfies the monodromy property if poles of the motivic zetafunction 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-freemodel, i.e., K3 surfaces allowing a strict normal crossings model such that threeirreducible components of the special fiber never meet simultaneously. Crauderand Morrison classified these models into two main classes: so-called flowerpotdegenerations and chain degenerations. This classification is very precise, whichallows to use a combination of geometrical and combinatorial techniques tocheck the monodromy property in practice.The first main result is an explicit computation of the poles of ZX,ω(T) for aK3 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 withprevious results: so far, all Calabi-Yau varieties known to satisfy the monodromyproperty have a unique pole.We prove that K3 surfaces allowing a flowerpot degeneration satisfy themonodromy property. We also show that the monodromy property holdsfor K3 surfaces with a certain chain degeneration. We don’t know whetherall K3 surfaces with a chain degeneration satisfy the monodromy property,and we investigate what characteristics a K3 surface X not satisfying themonodromy property should have. We prove that there are 63 possibilities forthe special fiber of the Crauder-Morrison model of a K3 surface X allowing atriple-point-free model that does not satisfy the monodromy property.

AB - The aim of this thesis is to study under which conditions K3 surfaces allowinga triple-point-free model satisfy the monodromy property. This property is aquantitative relation between the geometry of the degeneration of a Calabi-Yauvariety X and the monodromy action on the cohomology of X: a Calabi-Yau variety X satisfies the monodromy property if poles of the motivic zetafunction 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-freemodel, i.e., K3 surfaces allowing a strict normal crossings model such that threeirreducible components of the special fiber never meet simultaneously. Crauderand Morrison classified these models into two main classes: so-called flowerpotdegenerations and chain degenerations. This classification is very precise, whichallows to use a combination of geometrical and combinatorial techniques tocheck the monodromy property in practice.The first main result is an explicit computation of the poles of ZX,ω(T) for aK3 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 withprevious results: so far, all Calabi-Yau varieties known to satisfy the monodromyproperty have a unique pole.We prove that K3 surfaces allowing a flowerpot degeneration satisfy themonodromy property. We also show that the monodromy property holdsfor K3 surfaces with a certain chain degeneration. We don’t know whetherall K3 surfaces with a chain degeneration satisfy the monodromy property,and we investigate what characteristics a K3 surface X not satisfying themonodromy property should have. We prove that there are 63 possibilities forthe special fiber of the Crauder-Morrison model of a K3 surface X allowing atriple-point-free model that does not satisfy the monodromy property.

M3 - Ph.D. thesis

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

ER -