Moscow Mathematical Journal

Volume 19, Issue 2, April–June 2019  pp. 357–392.

Two-Dimensional Neighborhoods of Elliptic Curves: Formal Classification and Foliations

Authors:  Frank Loray (1), Olivier Thom (1), and Frédéric Touzet (1)
Author institution:(1) Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

Summary:

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 discuss analytic equivalence and for each formal model, we show that the corresponding moduli space is infinite dimensional.

2010 Math. Subj. Class. 32G13, 37F75, 34M40.

Keywords: Elliptic curves, formal neighborhoods, foliations.