Lars Halvard Halle

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

Some Open Problems

{X \rightarrow \,{\rm Spec}\,\, A}

Component Groups and Non-Archimedean Uniformization

X→SpecA to a noetherian k-algebra A. We also extend the classical projectivity result for GIT quotients: the induced morphism

Cohomological Interpretation of the Motivic Zeta Function

{X^{ss} /G \rightarrow \,{\rm Spec}\,\, A^G}

The Base Change Conductor and the Artin Conductor

Xss/G→SpecAG is projective. As an example of applications to moduli problems, we consider degenerations of Hilbert schemes of points.

Models of Curves and the Néron Component Series of a Jacobian

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.