Proper holomorphic embeddings with small limit sets

Abstract

Let X X be a Stein manifold of dimension n ≥ 1 n\ge 1 . Given a continuous positive increasing function h h on R + = [ 0 , ∞ ) \mathbb {R}_+=[0,\infty ) with lim t → ∞ h ( t ) = ∞ \lim _{t\to \infty } h(t)=\infty , we construct a proper holomorphic embedding f = ( z , w ) : X ↪ C n + 1 × C n f=(z,w):X\hookrightarrow \mathbb {C}^{n+1}\times \mathbb {C}^n satisfying | w ( x ) | > h ( | z ( x ) | ) |w(x)|>h(|z(x)|) for all x ∈ X x\in X . In particular, f f may be chosen such that its limit set at infinity is a linearly embedded copy of C P n \mathbb {CP}^n in C P 2 n \mathbb {CP}^{2n} .