Towards a classification of connected components of the strata of -differentials
Dawei ChenDepartment of Mathematics, Boston College, Chestnut Hill, MA 02467, USA
Quentin GendronInstituto de Matemáticas de la UNAM, Ciudad Universitaria, CDMX, 04510, México
A -differential on a Riemann surface is a section of the -th power of the canonical bundle. Loci of -differentials with prescribed number and multiplicities of zeros and poles form a natural stratification for the moduli space of -differentials. The classification of connected components of the strata of -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 -differentials for general . 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 -differentials by generalizing the hyperelliptic structure and spin parity for higher . We also describe an approach to determine explicitly parities of -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 -differentials introduced by Bainbridge-Chen-Gendron-Grushevsky-Möller for and extended by Costantini-Möller-Zachhuber for all .
Cite this article
Dawei Chen, Quentin Gendron, Towards a classification of connected components of the strata of -differentials. Doc. Math. 27 (2022), pp. 1031–1100DOI 10.4171/DM/892