[Des fonctions de Morse aux fibrations de Lefschetz sur les fibrés cotangents]
We prove that, for any Morse function on a compact manifold and any adapted gradient satisfying the Morse–Smale condition, there is a homotopically unique complex-valued symplectic Lefschetz fibration on the cotangent bundle whose restriction to the zero-section is the given function, whose imaginary part is the evaluation of covectors on the gradient, and which is equivariant under the actions of the fiberwise antipodal involution and the complex conjugation. Then we study the topology and symplectic geometry of the regular fibers of this fibration, which are well-defined Weinstein manifolds.
On démontre que, pour toute fonction de Morse sur une variété compacte et tout gradient adapté, vérifiant la condition de Morse-Smale, il existe sur le fibré cotangent une fibration de Lefschetz symplectique à valeurs complexes, unique à homotopie près, dont la restriction à la section nulle est la fonction donnée, dont la partie imaginaire est l’évaluation des covecteurs sur le gradient et qui est équivariante sous les actions de l’involution antipodale dans les fibres et de la conjugaison complexe. On étudie ensuite la topologie et la géométrie symplectique des fibres régulières de cette fibration, qui sont des variétés de Weinstein bien définies.
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Keywords: Symplectic Lefschetz fibrations, cotangent bundles, Morse functions, Morse–Smale gradients, Weinstein manifolds, vanishing cycles, exact Lagrangian submanifolds
Mots-clés : Fibrations de Lefschetz symplectiques, fibrés cotangents, fonctions de Morse, gradients de Morse-Smale, variétés de Weinstein, cycles évanescents, sous-variétés lagrangiennes exactes
Emmanuel Giroux 1
CC-BY 4.0
Emmanuel Giroux. From Morse functions to Lefschetz fibrations on cotangent bundles. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 1547-1573. doi: 10.5802/jep.354
@article{JEP_2026__13__1547_0,
author = {Emmanuel Giroux},
title = {From {Morse} functions to {Lefschetz} fibrations on~cotangent~bundles},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1547--1573},
year = {2026},
publisher = {\'Ecole polytechnique},
volume = {13},
doi = {10.5802/jep.354},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.354/}
}
TY - JOUR AU - Emmanuel Giroux TI - From Morse functions to Lefschetz fibrations on cotangent bundles JO - Journal de l’École polytechnique — Mathématiques PY - 2026 SP - 1547 EP - 1573 VL - 13 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.354/ DO - 10.5802/jep.354 LA - en ID - JEP_2026__13__1547_0 ER -
%0 Journal Article %A Emmanuel Giroux %T From Morse functions to Lefschetz fibrations on cotangent bundles %J Journal de l’École polytechnique — Mathématiques %D 2026 %P 1547-1573 %V 13 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.354/ %R 10.5802/jep.354 %G en %F JEP_2026__13__1547_0
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