2020/02/24 by Shapiro, Boris, Tahar, Guillaume
#Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT) #Primary 30F30 #Secondary 31A05
paper · doi:10.48550/arxiv.2002.10280
In this paper, motivated by the classical notion of a Strebel quadratic differential on a compact Riemann surfaces without boundary we introduce the notion of a quasi-Strebel structure for a meromorphic differential of an arbitrary order. It turns out that every differential of even order k exceeding 2 satisfying certain natural conditions at its singular points admits such a structure. The case of differentials of odd order is quite different and our existence result involves some arithmetic conditions. We discuss the set of quasi-Stebel structures associated to a given differential and introduce the subclass of positive k-differentials. Finally, we provide a family of examples of positive rational differentials and explain their connection with the classical Heine-Stieltjes theory of linear differential equations with polynomial coefficients.