Two finiteness theorem for (a,b)(a,b)-module

We prove the following two results 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

Strongly quasi-proper maps and the f-flattening theorem

We complete and precise the results of [B.13] and we prove a strong version of the semi-proper direct image theorem with values in the space C f n (M) of finite type closed n--cycles in a complex space M. We describe the strongly quasi-proper maps as the class of holomorphic surjective maps which admit a meromorphic family of fibers and we prove stability properties of this class. In the Appendix we give a direct and short proof of D. Mathieu's flattening theorem (see [M.00]) for a strongly quasi-proper map which is easier and more accessible

Meromorphic quotients for some holomorphic G-actions

Using mainly tools from [B.13] and [B.15] we give a necessary and sufficient condition in order that a holomorphic action of a connected complex Lie group $G$ on a reduced complex space $X$ admits a strongly quasi-proper meromorphic quotient. We apply this characterization to obtain a result which assert that, when $G = K.B$ \ with $B$ a closed complex subgroup of $G$ and $K$ a real compact subgroup of $G$, the existence of a strongly quasi-proper meromorphic quotient for the $B-$action implies, assuming moreover that there exists a $G-$invariant Zariski open dense subset in $X$ which is good for the $B-$action, the existence of a strongly quasi-proper meromorphic quotient for the $G-$action on $X$