Can the holomorphic image of $(\mathbb{C}^*)^n$ be open but not dense - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T01:18:13Z http://mathoverflow.net/feeds/question/96378 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96378/can-the-holomorphic-image-of-mathbbcn-be-open-but-not-dense Can the holomorphic image of $(\mathbb{C}^*)^n$ be open but not dense Charles Staats 2012-05-08T21:36:54Z 2012-07-15T17:55:24Z <blockquote> <p>Let $M$ be a compact complex connected [but <em>not necessarily kähler</em>] $n$-manifold, and suppose we have a holomorphic map $$(\mathbb{C}^*)^n \to M$$ such that the image is open. Is the image necessarily dense in $M$?</p> </blockquote> <p>Motivation: My intuition (which comes from the algebraic world) says that the answer ought to be "yes." On the other hand, I know that many properties of smooth algebraic varieties do not hold for complex manifolds in general. Knowing whether this statement has a counterexample would improve my intuition about the complex world.</p> http://mathoverflow.net/questions/96378/can-the-holomorphic-image-of-mathbbcn-be-open-but-not-dense/96395#96395 Answer by Misha for Can the holomorphic image of $(\mathbb{C}^*)^n$ be open but not dense Misha 2012-05-09T00:52:20Z 2012-05-09T12:31:25Z <p>An example exists already for $M={\mathbb C}P^2$, furthermore, there exists an injective holomorphic map $f: {\mathbb C}^2\to {\mathbb C}^2\subset {\mathbb C}P^2$ whose image is open but not dense. Recall that a domain $\Omega$ in ${\mathbb C}^2$ is called a <em>Fatou-Bieberbach</em> (FB) domain if $\Omega\ne {\mathbb C}^2$ and there exists a biholomorphic map $f: {\mathbb C}^2\to \Omega$. First examples of FB domains were constructed by Fatou and Bieberbach and it is a bit of an industry to construct FB domains with interesting properties. For instance, B. Stensönes constructed in her paper ("Fatou-Bieberbach domains with smooth boundary", Annals of Math, vol. 145, 1997, 365-377) FB domains in ${\mathbb C}^2$ whose boundaries are smooth. If you look at Proposition 3.1 (part v) of her paper, it shows that the complement of her FB domain has nonempty interior. I am pretty sure that one can find earlier examples as well. Now, if you want the domain of $f$ to be $({\mathbb C}^\times)^2$, just restrict the above holomorphic map. </p> http://mathoverflow.net/questions/96378/can-the-holomorphic-image-of-mathbbcn-be-open-but-not-dense/102303#102303 Answer by John Hubbard for Can the holomorphic image of $(\mathbb{C}^*)^n$ be open but not dense John Hubbard 2012-07-15T17:55:24Z 2012-07-15T17:55:24Z <p>Fatou-Bieberbach domains were first constructed by Fatou in 1927; the construction was improved by Bieberbach in 1932. The idea is linearizing coordinates at attracting cycles.</p> <p>Let $f:\bf C^2 \to C^2$ be an analytic automorphism, such that $(0,0)$ is an attracting fixed point, for instance the Hénon mapping $$f: (x,y) \mapsto (x^2-y,x/2).$$ Denote by $L$ the derivative of $f$ at the origin.</p> <p>There is then (usually, in particular in the case above) a unique analytic mapping $\phi:\bf C^2\to C^2$ with $\phi(0,0)=(0,0)$, $D\phi(0,0)=id$ and $L\circ \phi= \phi\circ f$.</p> <p>In this case (and in many others), $\phi^{-1}$ is defined on the basin of attraction of the origin by the formula $$\phi(x,y)= \lim_{m \to \infty} L^{-\circ m} \circ f^{\circ m} (x,y).$$</p> <p>The map $\phi$ is injective, but its image is certainly not dense. The image is the basin of attraction of the origin. There are lots of points not attracted to the origin, for instance the set $$U={(x,y) |\ |x|\ge 2, |y|\le x}.$$ it is easy to see that $f(U)\subset U$, and that if you iterate $f$ in $U$ all orbits tend to infinity.</p> <p>Showing that the limit above exists and defines an analytic map is not hard; I will write a proof if you are interested.</p> <p>There is an enormous literature about linearization at periodic points.</p>