164 research outputs found
Jumps and monodromy of abelian varieties
We prove a strong form of the motivic monodromy conjecture for abelian
varieties, by showing that the order of the unique pole of the motivic zeta
function is equal to the size of the maximal Jordan block of the corresponding
monodromy eigenvalue. Moreover, we give a Hodge-theoretic interpretation of the
fundamental invariants appearing in the proof.Comment: Section 5 rewritten, Section 6 expande
Motivic zeta functions of degenerating Calabi-Yau varieties
We study motivic zeta functions of degenerating families of Calabi-Yau
varieties. Our main result says that they satisfy an analog of Igusa's
monodromy conjecture if the family has a so-called Galois-equivariant Kulikov
model; we provide several classes of examples where this condition is verified.
We also establish a close relation between the zeta function and the skeleton
that appeared in Kontsevich and Soibelman's non-archimedean interpretation of
the SYZ conjecture in mirror symmetry.Comment: New result on existence of Kulikov models for abelian varieties added
in section 5.
A relative Hilbert-Mumford criterion
We generalize the classical Hilbert-Mumford criteria for GIT (semi-)stability
in terms of one parameter subgroups of a linearly reductive group G over a
field k, to the relative situation of an equivariant, projective morphism X ->
Spec A to a noetherian k-algebra A. We also extend the classical projectivity
result for GIT quotients: the induced morphism X^ss/G -> Spec A^G is
projective. As an example of applications to moduli problems, we consider
degenerations of Hilbert schemes of points.Comment: v4: minor correction
Stable reduction of curves and tame ramification
We study stable reduction of curves in the case where a tamely ramified base
extension is sufficient. If X is a smooth curve defined over the fraction field
of a strictly henselian discrete valuation ring, there is a criterion, due to
T. Saito, that describes precisely, in terms of the geometry of the minimal
model with strict normal crossings of X, when a tamely ramified extension
suffices in order for X to obtain stable reduction. For such curves we
construct an explicit extension that realizes the stable reduction, and we
furthermore show that this extension is minimal. We also obtain purely
geometric proof of Saito's criterion, avoiding the use of vanishing cycles.Comment: 19 page
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