A proof of the stratified Morse inequalities for singular complex algebraic curves using the Witten deformation

Ludwig, Ursula

Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 1749-1777.

Soit $M$ une variété Riemannienne compacte et soit $f:M\to \mathbf{R}$ une fonction de Morse sur $M$. La méthode de Witten utilise une déformation du complexe de de Rham pour démontrer les inegalités de Morse. Le but de cette note est d’étendre cette méthode au cas des courbes algébriques singulières et aux fonctions de Morse stratifiées au sens de la théorie de Goresky/MacPherson.

Dans une note précédente, l’auteur a donné une généralisation de la méthode de Witten pour le cas modèle d’une courbe à singularités coniques et des fonctions de Morse admissibles. Ici on présente les méthodes et arguments nécessaires pour étendre la théorie au courbes équipées de la métrique induite par la métrique de Fubini-Study de l’espace ambiant et à toutes les fonctions de Morse stratifiées.

The Witten deformation is an analytic method proposed by Witten which, given a Morse function $f:M\to \mathbf{R}$ on a smooth compact manifold $M$, allows to prove the Morse inequalities. The aim of this article is to generalise the Witten deformation to stratified Morse functions (in the sense of stratified Morse theory as developed by Goresky and MacPherson) on a singular complex algebraic curve. In a previous article the author developed the Witten deformation for the model of an algebraic curve with cone-like singularities and a certain class of functions called admissible Morse functions. The perturbation arguments needed to understand the Witten deformation on the curve with its metric induced from the Fubini-Study metric of the ambient projective space and for any stratified Morse function are presented here.

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