Analytic normal forms and inverse problems for unfoldings of 2-dimensional saddle-nodes with analytic center manifold

TL;DR

This paper develops normal forms and moduli spaces for generic families of holomorphic vector fields unfolding saddle-node singularities in two complex dimensions, with applications to differential Galois theory.

Contribution

It introduces new normal forms for parametric families of vector fields with saddle-node singularities and solves the realization problem for their moduli spaces.

Findings

Normal forms for generic unfolding families are established.

The realization problem for the moduli space is solved.

Characterization of families with analytically parameter-dependent moduli.

Abstract

We give normal forms for generic k-dimensional parametric families $(Z_\varepsilon)_\varepsilon$ of germs of holomorphic vector fields near $0\in\mathbb{C}^2$ unfolding a saddle-node singularity $Z_0$, under the condition that there exists a family of invariant analytic curves unfolding the weak separatrix of $Z_0$. These normal forms provide a moduli space for these parametric families. In our former 2008 paper, a modulus of a family was given as the unfolding of the Martinet-Ramis modulus, but the realization part was missing. We solve the realization problem in that partial case and show the equivalence between the two presentations of the moduli space. Finally, we completely characterize the families which have a modulus depending analytically on the parameter. We provide an application of the result in the field of non-linear, parameterized differential Galois theory.