2024/01/01 by Pavel Etingof, Etingof, Pavel, Alexander Varchenko +1 · 1 citation
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2401.00636
openalex publication_date 2024/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a new notion of a periodic pencil of flat connections on a smooth algebraic variety X. This is a family ∇(s1,...,sn) of flat connections on a trivial vector bundle on X depending linearly on parameters s1,...,sn and generically invariant, up to isomorphism, under the shifts si↦ si+1 for all i. If in addition ∇ has regular singularities, we call it a quasi-motivic pencil. We use tools from complex analysis to establish various remarkable properties of such pencils over \mathbb C. For example, we show that the monodromy of a quasi-motivic pencil is defined over the field of algebraic functions in e2πisj, and that its singularities are constrained to an arrangement of hyperplanes with integer normal vectors. Then we show that many important examples of families of flat connections, such as Knizhnik-Zamolodchikov, Dunkl, and Casimir connections, are quasi-motivic and thus periodic pencils. Besides being interesting in its own right, the periodic property of a pencil of flat connections turns out to be very useful in computing the eigenvalues of the p-curvature of its reduction to positive characteristic. This will be done in our forthcoming paper.