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

### Dawei Chen

Department of Mathematics, Boston College, Chestnut Hill, MA 02467, USA### Quentin Gendron

Instituto de Matemáticas de la UNAM, Ciudad Universitaria, CDMX, 04510, México

## 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 $k$-differentials 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 $k$.

## Cite this article

Dawei Chen, Quentin Gendron, Towards a classification of connected components of the strata of $k$-differentials. Doc. Math. 27 (2022), pp. 1031–1100

DOI 10.4171/DM/892