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Added information about a holomorphic vector field on ℂ^2.
Daniel Asimov
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Which holomorphic curves can be leaves of a non-singular holomorphic foliation of ℂ^2?

It is easy to see that for any entire function f : ℂ → ℂ, its graph G(f) = {(z,f(z)) ∈ ℂ2 | z ∈ ℂ} can be translated by (0,c) for any c ∈ ℂ, so that all the translated graphs {G(f) + (0,c) | c ∈ ℂ} form the leaves of a holomorphic foliation of ℂ2.

If V : ℂ2 → ℂ2 defines a holomorphic and nowhere zero vector field on ℂ2, then there is a unique non-singular holomorphic foliation of ℂ2 by holomorphic curves that are everywhere tangent to V. These are the orbits of the ℂ-action that V induces on ℂ2. It is known that the leaves of such a foliation must be topologically either planes or cylinders.

Is it known exactly which holomorphic curves X ⊂ ℂ2 can be a leaf of a non-singular holomorphic foliation of ℂ2?

Daniel Asimov
  • 2.9k
  • 24
  • 26