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In this paper we consider holomorphic families of frescos (i.e. filtereddifferential equations with a regular singularity) and we construct a locallyversal holomorphic family for every fixed Bernstein polynomial. We constructalso several holomorphic parameters (a holomorphic parameter is a functiondefined on a set of isomorphism classes of frescos) which are quasi-invariantby changes of variable. This is motivated by the fact that a fresco isassociated to a relative de Rham cohomology class on a one parameterdegeneration of compact complex manifolds, up to a change of variable in theparameter. Then the value of a quasi-invariant holomorphic parameter on suchdata produces a holomorphic (quasi-)invariant of such a situation.

Contruction of quasi-invariant holomorphic parameters for the Gauss-Manin connection of a holomorphic map to a curve (second version)