Motivic and arithmetic aspects of exceptional loci

Contact: Daniel Huybrechts

Given a family of smooth projective varieties defined over a number field, one often studies the Hodge locus (resp. the l-Galois exceptional locus), defined as the set of points in the base where exceptional Hodge (resp. l-adic Tate) classes occur in the cohomology of the fiber. The Hodge and Tate conjectures state that Hodge and Tate classes should be classes of algebraic cycles, and therefore make the following predictions about these loci: Both loci should in fact be equal, and moreover form a countable union of algebraic subvarieties of the base. In this talk, I want to discuss some recent progress on these questions. We prove unconditionally that the l-Galois exceptional locus is indeed a countable union of algebraic subvarieties. The corresponding result for the Hodge locus, obtained by Cattani-Deligne-Kaplan in 1995, is often viewed as important evidence in favor of the Hodge conjecture. In addition, under a mild condition on the generic Mumford-Tate group, we prove that the expected equality between the Hodge locus and the l-Galois exceptional locus holds true if one restricts to their positive dimensional parts.