We know that a Hodge structure that appears in the cohomology of an algebraic variety must be polarizable. Instead we do not know a single example of a polarizable Hodge structure which does not appear in the cohomology of an algebraic variety nor an extra condition that such a Hodge structure should satisfy. A recent paper of Tobias Kreutz investigates this question and gives explicit conditions and examples based on the following geometric inputs: field of definition, periods and Griffiths transversality.

The moduli space of genus g curves occupies a central position in the study of special loci associated with variations of Hodge structures. In this talk, I will explore various interpretations of what it means for a curve to be "special," drawing inspiration from algebraic geometry, number theory, and Teichmüller theory. These perspectives are unified by the sentiment that special curves are rare. I will discuss classical and recent results, as well as conjectures, concerning the distribution of special loci in Mg​ and the Hodge bundle over it.

Given a l-adic local system on a variety over a number field, one can define its toric locus, which is the analogue of the CM locus of a variation of Hodge structure. I will discuss a work in progress on the sparcity of points of bounded degree in the toric locus of those local systems arising from geometry.

I will describe several appearances of analogues of Hilbert polynomials in Arakelov geometry and geometry of numbers, with applications to fundamental groups and to diophantine geometry.

In this short talk I want to discuss the problem in the title.

Let X be a smooth algebraic variety over the field of algebraic numbers. Simpson conjectured that a vector bundle with connection on X whose associated local system is also defined over the algebraic numbers has motivic origin. I will present a beautiful unreleased work by Tobias dealing with this question for unipotent connections on a punctured projective line and general connections on an elliptic curve. Among other things, the proofs rely on the construction of the motivic unipotent fundamental group by Deligne-Goncharov, as well as irrationality and transcendence theorems by Apéry, Baker, and Wüstholz.

Given a family of smooth projective varieties defined over a finitely generated field of characteristic zero, 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. 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 these objects and the results obtained by Tobias in his thesis: the l-Galois exceptional locus is indeed a countable union of algebraic subvarieties (analog to the celebrated Cattani-Deligne-Kaplan result for the Hodge locus); and, under a mild condition on the generic Mumford-Tate group, the Hodge locus and the l-Galois exceptional locus coincide if one restricts to their positive dimensional parts.

In 1852, in his last published paper, Eisenstein proved that if a function f(z) is algebraic, then its Taylor expansion at a point has coefficients lying in a finitely-generated Z-algebra. I will discuss joint work with Daniel Litt which studies the converse of this theorem, which is moreover closely related to the Grothendieck-Katz p-curvature conjecture. More precisely, we conjecture that if f(z) satisfies a (possibly non-linear!) algebraic ODE, non-singular at 0, and its Taylor expansion has coefficients lying in a finitely-generated Z-algebra, then f(z) is algebraic. For linear ODEs, we prove this conjecture when (A) f(z) satisfies a Picard-Fuchs equation, with initial condition the class of an algebraic cycle. For non-linear ODEs, we prove it when f(z) satisfies an “isomonodromy” ODE with Picard-Fuchs initial condition.

When one wants to study the variation of periods in families one is usually constrained by the phenomenon of Griffiths transversality: This forbids, for example, nontrivial families of extensions of Z by Z(n) when n>1. Still, Zagier's conjecture predicts that the periods of such extensions can be expressed in terms of special values of interesting special functions; in this case, the polylogarithm. I will explain a notion of "relative motivic cohomology" that comes with "relative regulator maps" and so that one can have nontrivial families of extensions of Z by Z(n) for n>1 in relative motivic cohomology. In fact there is a canonical example whose relative regulator gives the polylogarithm. I will discuss in particular the example of the dilogarithm and give a new perspective on its domain and codomain. I will also explain the five-term relation as an identity of relative motivic cohomology classes.

For a curve of genus at least four which is either very general or very general hyperelliptic, we classify all ways in which a power of its Jacobian can be isogenous to a product of Jacobians of curves. We use this to show that for a very general principally polarized abelian variety of dimension at least four, or the intermediate Jacobian of a very general cubic threefold, no power is isogenous to a Jacobian of a curve. This confirms various cases of the Coleman-Oort conjecture and has some relation to the question whether cubic threefolds are stably irrational. Joint work with Olivier de Gaay Fortman.

We define a natural bi-$\overline{\mathbb{Q}}$-structure on the tangent space at a CM point on a Hermitian locally symmetric domain. We prove that this bi-$\overline{\mathbb{Q}}$-structure decomposes into the direct sum of 1-dimensional bi-$\overline{\mathbb{Q}}$-subspaces, and make this decomposition explicit for the moduli space of abelian varieties $A_g$ . We propose an "Hyperbolic Analytic Subspace" Conjecture, which is the analogue of Wüstholz’s Analytic Subgroup Theorem in this context. We show that this conjecture, applied to $A_g$ , implies that all quadratic $\overline{\mathbb{Q}}$-relations among the holomorphic periods of CM abelian varieties arise from elementary ones. We then show that the elementary quadratic relations between CM periods are at the heart of the theory: For any CM abelian variety A, there exists an abelian variety B such that all the algebraic relations among CM periods on $A\times B$, induced by Hodge cycles, are generated by these elementary quadratic relations.

We explain how to prove asymptotic upper and lower bounds on the degree of the Noether-Lefschetz locus (or Hodge locus) in terms of the Hodge norm of the Hodge vectors which define the associated components. Our methods reduce the problem to counting rational points in Siegel set orbits and an explicit analysis of such orbits.

For smooth projective varieties defined over the rational numbers, the de Rham-Betti conjecture is the analogue of the Hodge conjecture, except that one replaces the Hodge filtration on the algebraic de Rham cohomology with its $\mathbb{Q}$-vector space structure. It is a special case of the Grothendieck Period Conjecture. I will present joint work with Tobias Kreutz and Mingmin Shen, where we explore the validity of (stronger forms of) the de Rham-Betti conjecture for abelian varieties and hyper-Kähler varieties.