Non-commutative Hodge structures

Sabbah, Claude

Annales de l'Institut Fourier, Tome 61 (2011) no. 7, pp. 2681-2717.

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*.

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.

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