[Paper Review] Two dimensional neighborhoods of elliptic curves: formal classification and foliations

Frank Loray, Olivier Thom+1 more|arXiv (Cornell University)|Apr 18, 2017

Algebraic Geometry and Number Theory9 references3 citations

TL;DR

This paper provides a formal classification of two-dimensional neighborhoods of an elliptic curve $C$ with torsion normal bundle, using a pencil of formal foliations tangent to $C$. The key result is that such neighborhoods are completely determined by the holonomy of a pair of formal foliations, and the moduli space of analytic equivalence classes is infinite-dimensional for each formal model.

ABSTRACT

We classify two dimensional neighborhoods of an elliptic curve C with torsion normal bundle, up to formal equivalence. The proof makes use of the existence of a pair (indeed a pencil) of formal foliations having C as a common leaf, and the fact that neighborhoods are completely determined by the holonomy of such a pair. We also discussanalytic equivalence and show, for each formal model, that the corresponding moduli space is infinite dimensional.

Motivation & Objective

• To classify two-dimensional neighborhoods of an elliptic curve $C$ with torsion normal bundle up to formal equivalence.
• To understand the role of foliations in determining the structure of such neighborhoods.
• To analyze the analytic classification and show that the corresponding moduli space is infinite-dimensional.
• To extend known results on linearizability in the torsion case, particularly in contrast to the non-torsion case where diophantine conditions govern analytic linearization.
• To investigate the interplay between holonomy representations, diffeomorphism germs, and the geometry of the neighborhood via bifoliated structures.

Proposed method

• Use of a pencil of formal foliations $\hat{\mathcal{F}}_t$ tangent to $C$, with $C$ as a common leaf.
• Leverage the fact that the neighborhood is completely determined by the holonomy of such a pair of foliations.
• Apply results from one-dimensional dynamics (Yoccoz, Pérez-Marco) to analyze holonomy representations $\pi_1(C) \to \mathrm{Diff}(\mathbb{C},0)$.
• Construct formal models $U_{\lambda,\Lambda}$ parametrized by invariants $\lambda$ and $\Lambda$, with $k = \mathrm{utype}(U,C)$ controlling the order of tangency.
• Use conjugacy via polynomial diffeomorphisms $\phi \in \mathrm{Diff}(\mathbb{C},0)$ of degree $k+1$ to match holonomy up to order $k$.
• Analyze the action of $\mathrm{Aut}(C)$ on the normal forms, particularly for $q = 2,3,4,6$, and describe symmetry-induced equivalences.

Experimental results

Research questions

• RQ1 How can two-dimensional neighborhoods of an elliptic curve with torsion normal bundle be formally classified?
• RQ2 What is the role of a pencil of formal foliations in determining the structure of such neighborhoods?
• RQ3 Under what conditions is a formal neighborhood analytically equivalent to its linear model?
• RQ4 What is the structure of the moduli space of analytic equivalence classes for such neighborhoods?
• RQ5 How do holonomy representations of the fundamental group of $C$ relate to the classification and deformation of neighborhoods?

Key findings

• The formal classification of neighborhoods with torsion normal bundle is completely determined by the holonomy of a pair of formal foliations having $C$ as a common leaf.
• For each formal model, the moduli space of analytic equivalence classes is infinite-dimensional, as shown by realizing arbitrary holonomy representations via Yoccoz’s methods.
• When the normal bundle is torsion and both foliations $\mathcal{F}_{t_1}, \mathcal{F}_{t_2}$ are convergent, the moduli space can be explicitly described via independent deformations of holonomy representations.
• The action of $\mathrm{Aut}(C)$ on the formal models induces diffeomorphisms between normal forms, with explicit conjugacy maps given by $ (x,y) \mapsto (-x, \xi y) $, $ \xi^k = -1 $, preserving the Ueda type.
• In exceptional cases with $q = 3,4,6$, the conjugacy maps are more complex, but the bifoliated method still allows description of equivalent normal forms.
• The paper establishes a complete classification up to formal equivalence, with explicit normal forms $U(a_1,a_\tau,\lambda,\Lambda)$ parametrized by monodromy data and formal invariants $\lambda, \Lambda$.