2022/08/24 by Quentin Gendron, Gendron, Quentin, Guillaume Tahar +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric Topology (math.GT) #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2208.11654
openalex publication_date 2022/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the local invariants that a meromorphic k-differential on a Riemann surface of genus g ≥ 0 can have for k ≥ 3. These local invariants include the orders of zeros and poles, as well as the k-residues at the poles. We show that for a given pattern of orders of zeros, there exists, with a few exceptions, a primitive holomorphic k-differential having zeros of these orders. In the meromorphic case, for genus g ≥ 1, every expected tuple appears as a configuration of k-residues. On the other hand, for certain strata in genus zero, finitely many tuples (up to simultaneous scaling) do not occur as configurations of k-residues for a k-differential.