Dimension formulas for period spaces via motives and species
Annette Huber, Martin Kalck
TL;DR
The paper develops a unified, representation-theoretic framework for bounding and computing the dimensions of period spaces of motives. By translating motives into finite dimensional algebras via Morita theory and species, it expresses dim_F P⟨M⟩ in terms of Ext data and the underlying quiver/species, yielding explicit upper bounds that are sharp in hereditary cases. It shows equality with the Period Conjecture in settings where the motive category is hereditary (notably 1-motives and Mixed Tate Motives); it also provides a hereditary saturation procedure to reduce general cases to the hereditary situation. These results encompass classical dimension estimates for multiple zeta values, Deligne–Goncharov’s Mixed Tate framework, and extend to explicit non-hereditary examples, clarifying the structural role of 2-extensions in the period relations. Collectively, the work connects motive theory with detailed path-algebra computations, offering concrete tools for understanding the algebra-geometry of periods and their fundamental constraints.
Abstract
We apply the structure theory of finite dimensional algebras in order to deduce dimension formulas for spaces of period numbers, i.e., complex numbers defined by integrals of algebraic nature. We get a complete and conceptually clear answer in the case of $1$-periods, generalising classical results like Baker's theorem on the logarithms of algebraic numbers and partial results in Huber--W{ü}stholz \cite{huber-wuestholz}. The application to the case of Mixed Tate Motives (i.e., Multiple Zeta Values) recovers the dimension estimates of Deligne--Goncharov \cite{deligne-goncharov}.