I am studying proper holomorphic maps: one result is that there are no proper holomorphic maps from $B_n$ to $\Delta^n$, i.e. from the open ball in $\mathbb C^n$ in the open polydisc. What if I increase the dimension on the right?

I can't come to an answer: for what I know, embedding theorems could hold, such that every ball can be embedded holomorphically into some polydisc in great dimension.

Thank you in advance for any suggestion.

  • $\begingroup$ Why no proper holomorphic map from $B_n$ to $\Delta^n$?, isn't the inclusion $B_n\to \Delta^n$ such a map? $\endgroup$ – chan kifung Jul 12 '17 at 19:21
  • $\begingroup$ Well, the inclusion is not surjective; so pick a compact in $\Delta^n$ that intersects $B_n$ and its boundary (so it has points in and out of $B_n$): its preimage is not compact in $B_n$, since it approaches the boundary. Am I right? $\endgroup$ – W. Rether Jul 12 '17 at 19:26
  • $\begingroup$ Yes, you are right! $\endgroup$ – Georges Elencwajg Jul 12 '17 at 20:47

Yes, every ball can be holomorphically embedded into some polydisc of sufficiently high dimension.

More generally, Løw has proved that every strictly psudoconvex open subset $\Omega \subset \mathbb C^n$ with $C^2$ boundary can be properly holomorphically embedded into a polydisc $P\subset \mathbb C^m$ of sufficiently high dimension $m\gt n$.
This result is Corollary 1, page 453 of Løw's Theorem 1 in this 1985 Mathematische Zeitschrift article.

A complement
Although it was not asked in the question let me mention that the result does not hold in the other direction:
If $m\gt1$ and $r\gt0$ there is no holomorphic proper map of a polydisc $P \subset \mathbb C^m$ into an open ball $B\subset \mathbb C^r.$
This is part of Theorem 15.2.4, pages 306-307, of Rudins Function Theory in the Unit Ball of $\mathbb C ^n$.


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