The purpose of the present manuscript is to survey some of the main ideas that appear in recent research of the author on the topic of applying anabelian geometry to construct a “global multiplicative subspace”— i.e., an analogue of the well-known (local) multiplicative subspace of the Tate module of a degenerating elliptic curve. Such a global multiplicative subspace is necessary to apply the ...

We show that an irreducible component of the Hodge locus of a polarizable variation of Hodge structure of weight 0 on a smooth complex variety X is defined over an algebraically closed subfield k of finite transcendence degree if X is defined over k and the component contains a k-rational point. We also prove a similar assertion for the Hodge locus inside the Hodge bundle if the Hodge bundle to...

We introduce an “extended locus of Hodge classes” that also takes into account integral classes that become Hodge classes “in the limit”. More precisely, given a polarized variation of integral Hodge structure of weight zero on a Zariski-open subset of a complex manifold, we construct a canonical analytic space that parametrizes limits of integral classes; the extended locus of Hodge classes is...