Supplement to 'On the Levi-flats in Complex Tori of Dimension Two'

The purpose of this note is to describe certain Levi-flats in [O] more in detail, by showing that they are natural generalization of Nemirovski’s example in [N]. Let C be a compact Riemann surface, let L → C be a holomorphic line bundle, and let s be a meromorphic section of L whose zeros and poles are all simple. Let C ′ ⊂ C be the complement of the set of zeros and poles of s, let L′ ⊂ L be the complement of the zero section, and let R′ = R\{0}. Let a > 1, and let Z act on L′ by fiberwise multiplication by a for m ∈ Z. Then it is easy to see that the closure of R′s(C ′)/Z in L′/Z is a Levi-flat whose complement is Stein if C ′ = C (cf. [N]). This construction is immediately generalized to produce Levi-flats with Stein complements in higher dimension, by considering smooth ample divisors and the associated line bundles with canonical sections. On the other hand, since not all elliptic principal bundles arise as quotients of C∗-bundles, it may not be so immediate to see how one can generalize Nemirovski’s construction to the elliptic principal bundles. However, it is actually immediate if one regards d log s as a meromorphic connection of the bundle L′/Z as we shall see below. Let C be as above, let E0 be an elliptic curve, i.e. a compact 1-dimensional complex Lie group, and let E π → C be a principal E0-bundle. Let g be the Lie algebra of E0. We shall regard E0 as a quotient space of g by the exponential map. The kernel of the exponential map will be denoted by gZ and E0 will be identified with g/gZ in what follows. Given a closed subgroup Γ ⊂ E0 and a finite subset Σ ⊂ C, let Ω(Γ,Σ) be the set of meromorphic connections ω of E with at most simple poles in Σ