Absolutely special subvarieties and absolute Hodge cycles

TL;DR

This paper introduces the concept of dR-absolutely special subvarieties in Hodge structures, conjectures their equivalence to all special subvarieties, and proves this under certain monodromy conditions, with implications for Hodge cycles and algebraic intersections.

Contribution

It defines dR-absolutely special subvarieties, formulates a weaker conjecture related to Hodge cycles, and proves it for cases with specific monodromy conditions, advancing understanding of Hodge theory.

Findings

Proposed the notion of dR-absolutely special subvarieties.

Conjectured all special subvarieties are dR-absolutely special.

Proved the conjecture under a monodromy condition.

Abstract

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.