[Paper Review] The Galoisian envelope of a germ of foliation: the quasi-homogeneous case

Emmanuel Paul|ArXiv.org|Dec 11, 2006

Algebraic Geometry and Number Theory3 references3 citations

TL;DR

This paper establishes geometric and algorithmic criteria for the existence of a proper Galois closure (Galois envelope) of a germ of codimension one quasi-homogeneous foliation in $\mathbb{C}^2$. By analyzing the relative holonomy group and formal normal forms, it proves that Galois reducibility is equivalent to the existence of a finite-length Godbillon-Vey sequence (at most three forms) and links this to the monodromy of the Milnor fiber, with a key result being a realizability theorem for abelian holonomy groups via vector field normalization.

ABSTRACT

We give geometric and algorithmic criterions in order to have of a proper Galois closure for a codimension one germ of quasi-homogeneous foliation. We recall this notion recently introduced by B. Malgrange, and describe the Galois envelope of a group of germs of analytic diffeomorphisms. The geometric criterions are obtained from transverse analytic invariants, whereas the algorithmic ones make use of formal normal forms.

Motivation & Objective

• To determine necessary and sufficient conditions for the existence of a proper Galois closure (Galois envelope) of a germ of codimension one quasi-homogeneous foliation.
• To relate the Galois reducibility of such foliations to the existence of a finite-length Godbillon-Vey sequence of meromorphic one-forms.
• To connect the geometric invariant (relative holonomy group) with the algorithmic invariant (formal normal form) in the context of quasi-homogeneous singularities.
• To establish a realizability result: any non-exceptional abelian subgroup of $\mathrm{Diff}_1(\mathbb{C},0)$ arises as the relative holonomy group of some quasi-homogeneous foliation.
• To lay foundations for extending the Galois-theoretic approach to integrability beyond the codimension one case, particularly in the context of formal normal forms and summability.

Proposed method

• The paper uses the desingularization of the quasi-homogeneous vector field $X_h$ to analyze the structure of the exceptional divisor and the behavior of the foliation on the Milnor fiber.
• It defines the relative holonomy group of the foliation as the monodromy action on the Milnor fiber, which is shown to be abelian and determined by the periods of a closed one-form $\eta_c$.
• The formal normal form of the foliation is constructed using a vector field of the form $X = \alpha X_h + \left(\sum c_i f_i\right) \delta(h) R$, where $f_i$ are holomorphic functions and $R$ is the quasi-homogeneous Euler vector field.
• The algorithmic invariant $\mathcal{L}(\mathcal{F})$ is derived from the coefficients $c_i$ in the normal form, and its dimension corresponds to the transverse rank of the Galois envelope.
• The relationship between the periods $T_i$ of the holonomy and the coefficients $c_i$ is encoded in a matrix equation $T = M \cdot C$, where $M$ is the period matrix of the cohomology basis.
• The paper proves that the Galois envelope is proper (i.e. the foliation is Galois reducible) if and only if the relative holonomy group is abelian and the periods $T_i$ are rationally related, which corresponds to the existence of a finite Godbillon-Vey sequence of length at most three.

Experimental results

Research questions

• RQ1 Under what geometric and algebraic conditions is the Galois envelope of a quasi-homogeneous codimension one foliation proper, i.e. strictly smaller than the full automorphism groupoid?
• RQ2 How is the existence of a finite-length Godbillon-Vey sequence (of length at most three) related to the Galois reducibility of the foliation?
• RQ3 What is the precise relationship between the relative holonomy group of the foliation and the formal normal form of its vector field?
• RQ4 Can any non-exceptional abelian subgroup of $\mathrm{Diff}_1(\mathbb{C},0)$ be realized as the relative holonomy group of a quasi-homogeneous foliation?
• RQ5 What is the geometric meaning of the $k$-summability of the final normal forms, and how does it relate to the Galois closure?

Key findings

• The Galois envelope of a codimension one quasi-homogeneous foliation is proper if and only if the relative holonomy group is abelian and the periods of the associated one-form $\eta_c$ are rationally related.
• The existence of a finite Godbillon-Vey sequence of length at most three is equivalent to Galois reducibility, with the transverse rank of the Galois envelope equal to the minimal length of such a sequence.
• A realization theorem is proven: for any non-exceptional abelian subgroup $H \subset \mathrm{Diff}_1(\mathbb{C},0)$, there exists a quasi-homogeneous foliation whose relative holonomy group is $H$, via a suitable choice of coefficients $c_i$ in the normal form.
• The matrix $M = (m_{i,j})$ with entries $m_{i,j} = \int_{\Gamma_i} \eta_j$ is invertible, ensuring that the coefficients $c_i$ can be uniquely solved from the desired periods $T_i$, establishing a one-to-one correspondence between holonomy data and normal form parameters.
• The Galois envelope is trivial (i.e. the full automorphism groupoid) if and only if the relative holonomy group is non-abelian or the periods are incommensurable, indicating non-reducibility.
• The paper establishes that the algorithmic invariant $\mathcal{L}(\mathcal{F})$ is one-dimensional if and only if the foliation is Liouvillian, linking the Galois-theoretic notion of integrability to the existence of first integrals in a specific class of transcendental extensions.