Gendron, Quentin; Tahar, Guillaume Isoresidual fibration and resonance arrangements. (English) Zbl 1492.14049 Lett. Math. Phys. 112, No. 2, Paper No. 33, 36 p. (2022). Consider a meromorphic differential \(\omega\) on \(\mathbb C\mathbb P^1\) that has a unique zero of order \(a\) and \(n\) distinct poles of order \(b_1, \ldots, b_n\) where \(a - \sum_{i=1}^n b_i + 2 = 0\). Given a tuple \((\lambda_1, \ldots, \lambda_n)\) such that \(\sum_{i=1}^n \lambda_i = 0\), what is the number of \(\omega\) with the prescribed type of zeros and poles such that the residues of \(\omega\) at the poles are given by the \(\lambda_i\)? In this paper the authors study this natural and interesting question. It turns out the answer depends on whether the \(\lambda_i\) are contained in certain resonance hyperplanes \(A_I\colon \sum_{i\in I} \lambda_i = 0\) where \(I\) is a proper subset of \(\{1,\ldots, n\}\). The authors solve the problem for generic residues (i.e., not contained in any resonance hyperplane) as well as when \(I^{c}\) is a singleton. Moreover, the authors compute the monodromy of the residue map for the case of at most three poles. In order to obtain these results, the authors use the correspondence between (meromorphic) differentials and translation surfaces (of infinite area), degenerate the residues to be of real value, and enumerate certain combinatorial structures called decorated trees that arise in this process. Reviewer: Dawei Chen (Chestnut Hill) Cited in 4 Documents MSC: 14H15 Families, moduli of curves (analytic) 14N20 Configurations and arrangements of linear subspaces 30F30 Differentials on Riemann surfaces Keywords:isoresidual fibration; translation surfaces; meromorphic \(1\)-forms; resonance arrangements × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Arnold, V., Real Algebraic Geometry (2013), Berlin: Springer, Berlin · Zbl 1268.14001 · doi:10.1007/978-3-642-36243-9 [2] Aguiar, M.; Mahajan, S., Topics in Hyperplane Arrangements (2017), Providence: American Mathematical Society, Providence · Zbl 1388.14146 · doi:10.1090/surv/226 [3] Bainbridge, M.; Chen, D.; Gendron, Q.; Grushevsky, S.; Möller, M., Compactification of strata of abelian differentials, Duke Mathematical Journal, 167, 12, 2347-2416 (2018) · Zbl 1403.14058 · doi:10.1215/00127094-2018-0012 [4] Bainbridge, M.; Chen, D.; Gendron, Q.; Grushevsky, S.; Möller, M., Strata of \(k\)-differentials, Algebr. 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