Geodesic
Towards a classification of connected components of the strata of $k$-differentials
Documenta mathematica, Tome 27 (2022), pp. 1031-1100.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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 $k$.
Classification : 14H10, 32G15, 14H15
Keywords: \(k\)-differentials, quadratic differentials, hyperelliptic structure, spin parity 
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     title = {Towards a classification of connected components of the strata of \(k\)-differentials},
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Chen, Dawei; Gendron, Quentin. Towards a classification of connected components of the strata of \(k\)-differentials. Documenta mathematica, Tome 27 (2022), pp. 1031-1100. http://geodesic.mathdoc.fr/item/DOCMA_2022__27__a39/