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 Fatou-Bieberbach (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 his 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.
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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 Fatou-Bieberbach (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 his 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 his her paper, it shows that the complement of his 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. |
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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 Fatou-Bieberbach (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 his paper ("Fatou-Bieberbach domains with smooth boundary", Annals of Math, vol. 145, 1997, 365-377) FB domains in ${\mathbb C}^2$ whose boundary is boundaries are smooth. If you look at Proposition 3.1 (part v) of his paper, it shows that the complement of his 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. |
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