Two finiteness theorem for $(a,b)$-modules

Daniel Barlet (1, 2)

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

1. For a proper holomorphic function \ $ f : X \to D$ \ of a complex manifold \ $X$ \ on a disc such that \ $\{ df = 0 \} \subset f^{-1}(0)$, we construct, in a functorial way, for each integer \ $p$, a geometric (a,b)-module \ $E^p$ \ associated to the (filtered) Gauss-Manin connexion of \ $f$.\\ This first theorem is an existence/finiteness result which shows that geometric (a,b)-modules may be used in global situations. 2. For any regular (a,b)-module \ $E$ \ we give an integer \ $N(E)$, explicitely given from simple invariants of \ $E$, such that the isomorphism class of \ $E\big/b^{N(E)}.E$ \ determines the isomorphism class of \ $E$.\\ This second result allows to cut asymptotic expansions (in powers of \ $b$) \ of elements of \ $E$ \ without loosing any information.

Mots clés: (a; b)-module or Brieskorn modules; Gauss-Manin connexion; vanishing cycles

Domaines: Variables complexes [math.CV]

Daniel Barlet. Two finiteness theorem for $(a,b)$-modules. 2008. ⟨hal-00207543⟩