2011/01/20 by Daniel Barlet, Barlet, Daniel
Mathematics · #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory #Homotopy and Cohomology in Algebraic Topology #math.AG #math.CV
paper · pdf · doi:10.48550/arxiv.1101.3962
54 pages
arxiv created 2011/01/20 · openalex publication_date 2011/01/20 · arxiv updated 2011/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
When we consider a proper holomorphic map f: X → C of a complex manifold X on a smooth complex curve C with a critical value at a point 0 in C, the choice of a local coordinate near this point allows to dispose of an holomorphic function f. Then we may construct, using this function, an (a,b)-modules structure on the cohomology sheaves of the formal completion (in f) of the complex of sheaves (Ker df\bullet,d\bullet). These (a,b)-modules represent a filtered version of the Gauss-Manin connection of f. The most simple example of this construction is the Brieskorn module (see [Br.70]) of a function with an isolated singular point. See [B.08] for the case of a 1-dimensional critical locus. But it is clear that this construction depends seriously on the choice of the function f that is to say on the choice of the local coordinate near the critical point 0 in the complex curve C. The aim of the present paper is to study the behaviour of such constructions when we make a change of local coordinate near the origin. We consider the case of [λ]-primitive frescos, which are monogenic geometric (a,b)-modules corresponding to a minimal filtered differential equation associated to a relative de Rham cohomology class on X (see [B.09-a] and [B.09-b]). An holomorphic parameter is a function on the set of isomorphism classes of frescos which behave holomorphically in an holomorphic family of frescos. In general, an holomorphic parameter is not invariant by a change of variable, but we prove a theorem of stability of holomorphic families of frescos by a change of variable and it implies that an holomorphic parameter gives again an holomorphic parameter by a change of variable. We construct here two different kinds of holomorphic parameters which are (quasi-)invariant by change of variable. The first kind is associated to Jordan blocks of the monodromy with size at least two. The second kind is associated to the semi-simple part of the monodromy and look like some "cross ratio" of eigenvectors. They allow, in the situation describe above, to associate to a given (vanishing) relative de Rham cohomology class some numbers, which will depend holomorphically of our data, and are independant of the choice of the local coordinate near 0 to study the Gauss-Manin connection of this degeneration of compact complex manifolds.