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Although I tend to shy away from list questions, this is more fun to contemplate than the pile of exams on my desk right now. So take a compact Riemann surface $X$ with genus $g\ge 2$. A nonconstant holomorphic self map $f:X\to X$ is necessarily an isomorphism. Proof: Surjectivity is automatic by, for example, the open mapping theorem. If $f$ were not injective, then it would have a degree $d>1$. But the Riemann-Hurwitz formula would give $g-1\ge d(g-1)$ which is impossible. |
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Although I tend to shy away from list questions, this is more fun to contemplate than the pile of exams on my desk right now. So take a compact Riemann surface $X$ with genus $g\ge 2$. A nonconstant holomorphic map $f:X\to X$ is necessarily an isomorphism. Proof: Surjectivity is automatic by, for example, the open mapping theorem. If $f$ were not injective, then it would have a degree $d>1$. But the Riemann-Hurwitz formula would give $g-1\ge d(g-1)$ which is impossible. |
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