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

Dawei Chen   ; Quentin Gendron

Documenta mathematica, Tome 27 (2022), pp. 1031-1100

Résumé

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.

Détail

DOI : 10.4171/dm/892

Classification : 14H10, 14H15, 32G15
Mots-clés : k-differentials, quadratic differentials, hyperelliptic structure, spin parity