We study irrational pencils with isolated critical points on compact aspherical complex manifolds. We prove that if the set of critical points is nonempty, the homology of the kernel of the morphism induced by the pencil on fundamental groups is not finitely generated. This generalizes a result by Dimca, Papadima and Suciu. By considering self-products of the Cartwright–Steger surface, this allows us to build new examples of smooth projective varieties whose fundamental group has a non-finitely generated homology.
Nous étudions les pinceaux irrationnels à points critiques isolés sur les variétés complexes compactes et asphériques. Nous prouvons que si un tel pinceau possède au moins un point critique, alors l’homologie du noyau du morphisme induit entre groupes fondamentaux n’est pas de type fini. Ceci généralise un résultat de Dimca, Papadima et Suciu. En considérant le produit de plusieurs copies de la surface de Cartwright–Steger, ceci nous permet de donner de nouveaux exemples de variétés projectives lisses dont le groupe fondamental a un groupe d’homologie qui n’est pas de type fini.
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Nicolás, Francisco 1 ; Py, Pierre 1
CC-BY 4.0
@article{AFST_2023_6_32_1_55_0,
author = {Nicol\'as, Francisco and Py, Pierre},
title = {Irrational pencils and {Betti} numbers},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {55--67},
year = {2023},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 32},
number = {1},
doi = {10.5802/afst.1727},
language = {en},
url = {https://www.numdam.org/articles/10.5802/afst.1727/}
}
TY - JOUR AU - Nicolás, Francisco AU - Py, Pierre TI - Irrational pencils and Betti numbers JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2023 SP - 55 EP - 67 VL - 32 IS - 1 PB - Université Paul Sabatier, Toulouse UR - https://www.numdam.org/articles/10.5802/afst.1727/ DO - 10.5802/afst.1727 LA - en ID - AFST_2023_6_32_1_55_0 ER -
%0 Journal Article %A Nicolás, Francisco %A Py, Pierre %T Irrational pencils and Betti numbers %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2023 %P 55-67 %V 32 %N 1 %I Université Paul Sabatier, Toulouse %U https://www.numdam.org/articles/10.5802/afst.1727/ %R 10.5802/afst.1727 %G en %F AFST_2023_6_32_1_55_0
Nicolás, Francisco; Py, Pierre. Irrational pencils and Betti numbers. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 32 (2023) no. 1, pp. 55-67. doi: 10.5802/afst.1727
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