Complete Nonsingular Holomorphic Foliations on Stein Manifolds

Abstract

AbstractLet X be a Stein manifold of complex dimension $$n>1$$ n > 1 endowed with a Riemannian metric $${\mathfrak {g}}$$ g . We show that for every integer k with $$\left[ \frac{n}{2}\right] \le k \le n-1$$ n 2 ≤ k ≤ n - 1 there is a nonsingular holomorphic foliation of dimension k on X all of whose leaves are closed and $${\mathfrak {g}}$$ g -complete. The same is true if $$1\le k<\left[ \frac{n}{2}\right] $$ 1 ≤ k < n 2 provided that there is a complex vector bundle epimorphism $$TX\rightarrow X\times {\mathbb {C}}^{n-k}$$ T X → X × C n - k . We also show that if $${\mathcal {F}}$$ F is a proper holomorphic foliation on $${\mathbb {C}}^n$$ C n $$(n>1)$$ ( n > 1 ) then for any Riemannian metric $${\mathfrak {g}}$$ g on $${\mathbb {C}}^n$$ C n there is a holomorphic automorphism $$\Phi $$ Φ of $${\mathbb {C}}^n$$ C n such that the image foliation $$\Phi _*{\mathcal {F}}$$ Φ ∗ F is $${\mathfrak {g}}$$ g -complete. The analogous result is obtained on every Stein manifold with Varolin’s density property.