Reparamétrisation universelle de familles f-analytiques de cycles et théorème de f-aplatissement géométrique

Daniel Barlet (1, 2)

1 IECN - Institut Élie Cartan de Nancy
2 IUF - Institut universitaire de France

This article is a new presentation of the main results of D. Mathieu [M.00] on meromorphic equivalence relations. We introduce the space of finite type cycles (closed analytic cycles with finitely many irreducible components) of a given finite dimensional complex space and a natural topology on this space, in order to avoid the "regularity" condition for analytic families of cycles introduced in loc. cit. and also the two notions of "escape to infinity" which are here incoded in a natural way in our framework. Then the results are slightly stronger and much simple to state and to use. This contains, in an other language, a clean and more general version of the works of H. Grauert [G.83] and [G.86] and of B. Siebert [S.93] and [S.94] on meromorphic equivalence relations. G.83 Grauert, H. Set theoretic equivalence relations, Math. Ann. 265 (1983), p.137-148. G.86 Grauert, H. On meromorphic equivalence relations, Aspect Math. E9 (1986), p.115-147. M.00 Mathieu, D.Universal reparametrization..., Ann. Inst. Fourier (Grenoble) t.50 fasc.4 (2000), p.1155-1189 S.93 Siebert, B. Fiber cycles of holomorphic maps I. Local flattening, Math. Ann. 296 (1993), p.328-370. S.94 Siebert, B. Fiber cycle space and canonical flattening II, Math. Ann. 300 (1994), p.243-271.

Mots clés: Analytic and meromorphic equivalence relations; cycles; geometric flattening; universal reparametrization; meromorphic quotients

Domaines: Variables complexes [math.CV]

Daniel Barlet. Reparamétrisation universelle de familles f-analytiques de cycles et théorème de f-aplatissement géométrique. Commentarii Mathematici Helvetici, 2008, 83, p. 869-888. ⟨hal-00019111v2⟩