This article gives a survey of recent results on a generalization of the notion of a Hodge structure. The main example is related to the Fourier-Laplace transform of a variation of polarizable Hodge structure on the punctured affine line, like the Gauss-Manin systems of a proper or tame algebraic function on a smooth quasi-projective variety. Variations of non-commutative Hodge structures often occur on the tangent bundle of Frobenius manifolds, giving rise to a tt* geometry.
Nous donnons un panorama des résultats récents concernant une généralisation de la notion de structure de Hodge. L’exemple principal est celui produit par la transformation de Fourier-Laplace d’une variation de structure de Hodge polarisable sur la droite affine épointée, comme les systèmes de Gauss-Manin de fonctions algébriques propres ou modérées sur une variété quasi-projective lisse complexe. Le fibré tangent d’une variété de Frobenius peut souvent être muni d’une variation de structures de Hodge non-commutatives polarisables, d’où l’on déduit une géométrie spéciale du type tt*.
Keywords: Non-commutative Hodge structure, Fourier-Laplace transformation, Brieskorn lattice
Mot clés : Structure de Hodge non commutative, transformation de Fourier-Laplace, réseau de Brieskorn
@article{AIF_2011__61_7_2681_0, author = {Sabbah, Claude}, title = {Non-commutative {Hodge} structures}, journal = {Annales de l'Institut Fourier}, pages = {2681--2717}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {61}, number = {7}, year = {2011}, doi = {10.5802/aif.2790}, mrnumber = {3112504}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2790/} }
TY - JOUR AU - Sabbah, Claude TI - Non-commutative Hodge structures JO - Annales de l'Institut Fourier PY - 2011 SP - 2681 EP - 2717 VL - 61 IS - 7 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2790/ DO - 10.5802/aif.2790 LA - en ID - AIF_2011__61_7_2681_0 ER -
Sabbah, Claude. Non-commutative Hodge structures. Annales de l'Institut Fourier, Volume 61 (2011) no. 7, pp. 2681-2717. doi : 10.5802/aif.2790. http://www.numdam.org/articles/10.5802/aif.2790/
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