arXiv subjects

Tobias Kreutz

Publications and source records attributed to Tobias Kreutz.

5 recordsLinked to original sources

arXiv2026

De Rham-Betti classes with coefficients

Let $K$ and $L$ be algebraic extensions of the rational numbers inside the field of complex numbers. An $L$-de Rham-Betti class on a smooth projective variety $X$ over $K$ is a class in the Betti cohomology with $L$-coefficients of the analytification of $X$ that descends to a class in the algebraic de Rham cohomology of $X$ via the period comparison isomorphism. The period conjecture of Grothendieck implies that $L$-de Rham-Betti classes should be $L$-linear combinations of algebraic cycle classes. We prove that $L$-de Rham-Betti classes on products of elliptic curves are $L$-linear combinations of algebraic classes, provided $L$ contains at most one of the CM fields associated with the CM elliptic curves involved in the product. A key step consists in establishing a version of the analytic subgroup theorem with $L$-coefficients. Moreover, building on results of Deligne and André regarding the Kuga-Satake correspondence, we show that codimension-2 $L$-de Rham-Betti classes on hyper-Kähler varieties of known deformation type are $L$-linear combinations of motivated cycles, and we obtain a global de Rham-Betti Torelli theorem for K3 surfaces defined over the algebraic numbers.

math.AG↗

arXiv2023

Hodge structures not coming from geometry

Hodge theory associates to a smooth projective variety over $\mathbb{C}$ a piece of linear algebra information, called a $\mathbb{Q}$-Hodge structure. Conversely, it is a natural question which abstract $\mathbb{Q}$-Hodge structures arise from the cohomology of a smooth projective complex variety, or more generally, from a pure motive over $\mathbb{C}$. By a classical argument involving Griffiths transversality and a Baire category argument, it is well known that there are many Hodge structures which do not come from geometry in this sense. However, the argument is not constructive, and does not seem to give a criterion to decide whether a given Hodge structure comes from geometry. We formulate an intrinsic condition on a $\mathbb{Q}$-Hodge structure that we expect to be satisfied for all Hodge structures coming from geometry. We prove that this expectation follows from the conjunction of two fundamental conjectures in Hodge theory and transcendence theory: the conjecture that Hodge cycles are motivated and André's generalized Grothendieck period conjecture. By doing so, we exhibit explicit examples of $\mathbb{Q}$-Hodge structures which should not come from geometry.

math.AG↗

arXiv2022

Absolutely special subvarieties and absolute Hodge cycles

We introduce the notion of dR-absolutely special subvarieties in motivic variations of Hodge structure as special subvarieties cut out by (de Rham-)absolute Hodge cycles and conjecture that all special subvarieties are dR-absolutely special. This is implied by Deligne's conjecture that all Hodge cycles are absolute Hodge cycles, but is a much weaker conjecture. We prove our conjecture for subvarieties satisfying a simple monodromy condition introduced by Klingler-Otwinowska-Urbanik. We study applications to typical respectively atypical intersections and $\bar{\mathbb{Q}}$-bialgebraic subvarieties. Finally, we show that Deligne's conjecture as well as ours can be reduced to the case of special points in motivic variations.

math.AG↗

arXiv2022

$\ell$-Galois special subvarieties and the Mumford-Tate conjecture

We introduce $\ell$-Galois special subvarieties as an $\ell$-adic analog of the Hodge-theoretic notion of a special subvariety. The Mumford-Tate conjecture predicts that both notions are equivalent. We study some properties of these subvarieties and prove this equivalence for subvarieties satisfying a simple monodromy condition. As applications, we show that the $\ell$-Galois exceptional locus is a countable union of algebraic subvarieties and, if the derived group of the generic Mumford-Tate group of a family is simple, its part of positive period dimension coincides with the Hodge locus of positive period dimension. We use this to prove that for $n$ and $d$ sufficiently large, the absolute Mumford-Tate conjecture in degree $n$ holds on a dense open subset of the moduli space of smooth projective hypersurfaces of degree $d$ in $\mathbb{P}^{n+1}$, with the exception of hypersurfaces defined over number fields. Finally, we show that the Mumford-Tate conjecture for abelian varieties is equivalent to a conjecture about the local structure of $\ell$-Galois special subvarieties in $\mathcal{A}_g$.

math.NT↗

arXiv2022

Rapoport-Zink uniformization for the moduli space of polarized K3 surfaces

We compute the image of the $p$-adic period map for polarized K3 surfaces with supersingular reduction. This gives rise to a Rapoport-Zink type uniformization of their moduli space by an explicit open rigid analytic subvariety of a local Shimura variety of orthogonal type. In contrast to the case of Rapoport-Zink uniformization of Shimura varieties and in analogy to the complex case, the uniformizing domain does not carry an action of a $p$-adic Lie group, but only of a discrete subgroup. We briefly sketch how the same arguments can be applied to obtain a uniformization for the moduli space of smooth cubic fourfolds with supersingular reduction.

math.AG↗