The area is a good enough metric
Annales de l'Institut Fourier, Volume 74 (2024) no. 3, pp. 1017-1059.

In the first part we extend the construction of the smooth normal-crossing divisors compactification of projectivized strata of abelian differentials given by Bainbridge, Chen, Gendron, Grushevsky and Möller to the case of k-differentials. Since the generalized construction is closely related to the original one, we mainly survey their results and justify the details that need to be adapted in the more general context.

In the second part we show that the flat area provides a canonical hermitian metric on the tautological bundle over the projectivized strata of finite area k-differentials whose curvature form represents the first Chern class. This result is useful in order to apply Chern–Weil theory tools. It has already been used as an assumption in the work of Sauvaget for abelian differentials and is also used in a paper of Chen, Möller and Sauvaget for quadratic differentials.

Dans la première partie de cet article nous étendons aux k-différentielles la construction d’une compactification lisse avec un bord a croisements normaux des strates projectivisées de différentielles abéliennes introduite par Bainbridge, Chen, Gendron, Grushevsky et Moeller. Comme cette construction est très liée à la construction originale, nous ne présentons qu’un survol de celle-ci en soulignant les points qui nécessitent des modifications dans ce contexte plus général.

Dans la deuxième partie nous démontrons que l’aire fournit une métrique canonique hermitienne sur le fibré tautologique au-dessus des strates projectivisées dont la courbure représente la première classe de Chern. Ce résultat est utile pour pouvoir appliquer les outils de la théorie de Chern–Weyl. Ce résultat a déjà été utilisé comme une hypothèse dans les travaux de Sauvaget sur les différentielles abéliennes et dans les travaux de Chen, Moeller et Sauvaget sur les différentielles quadratiques.

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Accepted:
Online First:
Published online:
DOI: 10.5802/aif.3592
Classification: 14H99, 53C07
Keywords: Abelian differentials, multi-scale differentials, flat surfaces, Chern–Weil theory, hermitian metric.
Mot clés : Différentielles abéliennes, différentielles multi-échelles, surfaces planes, théorie de Chern–Weil, métrique hermitienne.

Costantini, Matteo 1; Möller, Martin 2; Zachhuber, Jonathan 2

1 Institut für Mathematik Universität Duisburg-Essen 45117 Essen (Germany)
2 Institut für Mathematik Goethe–Universität Frankfurt Robert-Mayer-Str. 6–8 60325 Frankfurt am Main (Germany)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Costantini, Matteo; Möller, Martin; Zachhuber, Jonathan. The area is a good enough metric. Annales de l'Institut Fourier, Volume 74 (2024) no. 3, pp. 1017-1059. doi : 10.5802/aif.3592. https://aif.centre-mersenne.org/articles/10.5802/aif.3592/

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