We show that torsion in the first homology of the manifold underlying an algebraic variety is ubiquitous and an important part of its complex geometry.

More precisely, let X be a complex algebraic variety of dimension greater than or equal to two. We show that there exists, for any finite abelian group G, a Zariski open subvariety U (for geometers, a Stein manifold dense in X) so that G is a subgroup of the torsion of the first homology of U with integer coefficients. We show how a fortuitous marriage of Kümmer theory and Hodge theory conspire to produce this torsion, and that this torsion is stubborn: it persists in all open subvarieties.

This talk will be accessible to those with a basic understanding of complex manifolds and their homology.