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

TL;DR

This paper explores the local invariant cycle theorem with integral coefficients in algebraic geometry, proving it in some cases and providing the first example of its failure in others, especially for complex surfaces.

Contribution

It demonstrates the theorem's validity for certain cohomology groups and constructs a novel counterexample involving algebraic surfaces with specific properties.

Findings

Invariant cycle theorem holds for H^1 with integral coefficients.

It holds for H^2 if the fiber's Albanese variety is trivial.

Constructs the first example of failure with integral coefficients in a semistable family.

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.