Good bad reduction


An ever popular theme in arithmetic algebraic geometry is the study of good reduction properties of algebraic varieties. The aim of this proposal is to push the study of "good'' and "bad'' reduction further in several new directions. The common denominator of these is the philosophy that what one should call good reduction, or maybe "good bad reduction", really depends on the context and on the applications one has in mind.

I propose three subprojects which aim to answer the following questions for some of the simplest classes of algebraic varieties - curves, abelian varieties, K3 surfaces, hyperkähler manifolds, ... What geometric or cohomological properties guarantee "good bad reduction'' of a certain type for a particular class of varieties? What are the potential obstructions? Finally, how "good'' can "good bad'' be, i.e. what are the nicest possible models?

In the first subproject, I study the interaction between different reduction types and the world of tame and wild ramification; for example, what base change does one need to do for a wildly ramified curve to acquire semistable or logarithmic good reduction? The aim of the second subproject is to make significant progress on an old question of Lang on compactifications of Néron models of abelian varieties in the semistable case. Finally, in the third project I draw inspiration from recent work of Liedtke-Matsumoto on good reduction of K3 surfaces to study similar questions for their higher-dimensional siblings, the so-called hyperkähler manifolds.





Dr. A.J.B. Smeets

Verbonden aan

Radboud Universiteit Nijmegen, Faculteit der Natuurwetenschappen, Wiskunde en Informatica, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP)


01/09/2017 tot 31/08/2020