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

Daniel Barlet (1, 2)

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

In this paper we consider holomorphic families of frescos (i.e. filtered differential equations with a regular singularity) and we construct a locally versal holomorphic family for every fixed Bernstein polynomial. We construct also several holomorphic parameters (a holomorphic parameter is a function defined on a set of isomorphism classes of frescos) which are quasi-invariant by changes of variable. This is motivated by the fact that a fresco is associated to a relative de Rham cohomology class on a one parameter degeneration of compact complex manifolds, up to a change of variable in the parameter. Then the value of a quasi-invariant holomorphic parameter on such data produces a holomorphic (quasi-)invariant of such a situation.

Mots clés: vanishing period; Theme; Fresco; filtered Gauss-Manin connection; Relative de Rham cohomology classes; Degenerating family of complex manifolds; geometric (a; asymptotic expansion; b)-module

Domaines: Géométrie algébrique [math.AG]; Variables complexes [math.CV]

Daniel Barlet. Contruction of quasi-invariant holomorphic parameters for the Gauss-Manin connection of a holomorphic map to a curve (second version). 2012. ⟨hal-00683071⟩