Towards a classification of connected components of the strata of $k$-differentials

Abstract

A k-differential on a Riemann surface is a section of the k-th power of the canonical bundle. Loci of k-differentials with prescribed number and multiplicities of zeros and poles form a natural stratification for the moduli space of k-differentials. The classification of connected components of the strata of k-differentials was known for holomorphic differentials, meromorphic differentials and quadratic differentials with at worst simple poles by Kontsevich–Zorich, Boissy and Lanneau, respectively. Built on their work we develop new techniques to study connected components of the strata of k-differentials for general k. As an application, we give a complete classification of connected components of the strata of quadratic differentials with arbitrary poles. Moreover, we distinguish certain components of the strata of kdifferentials by generalizing the hyperelliptic structure and spin parity for higher k. We also describe an approach to determine explicitly parities of k-differentials in genus zero and one, which inspires an amusing conjecture in number theory. A key viewpoint we use is the notion of multi-scale k-differentials introduced by Bainbridge– Chen–Gendron–Grushevsky–Möller for k = 1 and extended by Costantini–Möller– Zachhuber for all k.