Failure of the invariant cycle theorem over $\mathbb Z$

Abstract

We initiate a study of the local invariant cycle theorem with integral coefficients for 1-parameter semistable families of varieties. We show that it always holds for $H^1$, and it holds for $H^2$ if the general fiber has trivial Albanese variety. The latter generalizes results of Friedman, Griffiths, and Scattone on K3 surfaces and I-surfaces. We construct the first example of a semistable family which fails the local (and global) invariant cycle theorems with integral coefficients. The family has constant period map associated to $H^2$, and its smooth fibers are algebraic surfaces with $p_g=q=1$; in particular, they have non-trivial Albanese varieties. The surfaces in the family have maximal Picard rank and minimal discriminant, and they are closely related to Vinberg's most algebraic K3 surface. Our construction also generalizes the Shioda--Inose construction for rational double covers of K3 surfaces.