Fibrations sur le cercle et surfaces complexes

Pichon, Anne

Annales de l'Institut Fourier, Tome 51 (2001) no. 2, pp. 337-374.

Nous donnons des conditions nécessaires et suffisantes pour qu’une variété de dimension 3 se réalise comme bord d’une famille dégénérée de courbes complexes, et pour qu’un entrelacs dans une 3-variété se réalise comme bord d’un germe de fonction analytique en un point d’une surface complexe normale. Ces résultats s’appuient sur une étude des objets topologiques fournis par de telles fonctions holomorphes : soit $M$ une variété de Waldhausen et soit $L$ une union finie, éventuellement vide, de fibres de Seifert d’une décomposition de Waldhausen de $M$. Nous classifions topologiquement les fibrations en livre ouvert de reliure $L$ transverses à la décomposition de Waldhausen de $M$. Nous donnons une condition nécessaire et suffisante d’existence d’une telle fibration en fonction d’un système linéaire à coefficients rationnels, et nous décrivons explicitement toutes ces fibrations à partir de la topologie de $(M,L)$. Lorsque $L=\varnothing$, nous montrons qu’il en existe un nombre fini. Lorsque $L\ne \varnothing$, nous caractérisons les cas où il en existe une infinité.

We give necessary and sufficient conditions for the realization of a given 3-manifold as the boundary of a degenerating family of complex curves, and for the realization of a given link in a 3-manifolds as the boundary of a germ of analytic function at a point of a normal complex surface. These results are based on a study of the topological objects given by these holomorphic maps: let $M$ be a Waldhausen manifold and let $L$ be a union of Seifert fibres, possibly empty, in a Waldhausen decomposition of $M$. We topologically classify the open-book fibrations $\Phi :M/L\to {𝕊}^1$ with binding $L$ which are transverse to the Waldhausen decomposition of $M$. We give a necessary and sufficient condition for the existence of such a fibration in terms of a linear system with rational coefficients and we obtain an explicit description of all these fibrations from the topology of $(M,L)$. If $L\ne \varnothing$, we show that there is only a finite number of them. If $L\ne \varnothing$, we show that there is only a finite number of them. If $L=\varnothing$, we characterize the cases for which there exists an infinite number of such fibrations.

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