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I am wondering if there exists a complex analytic map $\pi:X\rightarrow Y$ with $g(Y)=g(X)-1.$ By Riemann-Hurwitz formula, I think the simplest of such maps is a complex analytic map between compact Riemann surfaces of degree 1 with one branch point of index $3,$ or with two branch points of indices $2, 2.$ So I am wondering

Does there exist a complex analytic map of compact Riemann surfaces of degree $1$ with one branch point of index $3,$ or with two branch points of indices $2, 2?$

The closest example I found is the map $z\rightarrow z^2$ on the Riemann sphere, which has two branch points, both of index $2,$ but it is of degree $2,$ not $1.$

More generally, out of curiosity, does there exist a complex analytic map of compact Riemann surfaces of given degree with a given number of branch points, of given indices?
Thanks for any help in advance.

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1 Answer 1

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Too long for a comment.

Let me assume that your question is about non-constant holomorphic (aka complex anlytic) mappings between compact Riemann surfaces.

There certainly exist compact Riemann surfaces $X$ and $Y$ with $g(Y)=g(X)-1$ for which there exists a non-constant holomorphic mapping $X\rightarrow Y$. Here, $g(-)$ denotes the genus of a surface. As an example, consider any lattice $\Gamma\subsetneq \mathbb{C}$. Then let $X\colon=\mathbb{C}/\Gamma$ be the corresponding complex torus and let $Y\colon=\mathbb{P}^1$ be the Riemann sphere. Then, the Weierstrass $\wp$-function $$\wp(z,\Gamma)=\frac{1}{z^2}+\sum_{\omega\in\Gamma\setminus\{0\}}\Big(\frac{1}{(z-\omega)^2}-\frac{1}{\omega^2}\Big)$$ defines a meromorphic doubly-periodic (with respect to the lattice $\Gamma$) function on $\mathbb{C}$. Thus, it extends to a non-constant holomorphic mapping $X\rightarrow Y$.


Regarding your other questions:

  1. Any non-constant holomorphic mapping of degree $1$ (between compact Riemann surfaces) is a biholomorphism, thus in particular a homeomorphism. Homeomorphisms preserve the genus. Hence, there cannot exist a holomorphic mapping $X\rightarrow Y$ of degree $1$ if the genera of $X$ and $Y$ differ.
  2. Additionally, there does not exist a holomorphic mapping $F\colon X\rightarrow Y$ of degree $1$ (between compact Riemann surfaces $X$ and $Y$) which has a ramification point (equivalently a branch point), say $x\in X$, since in this case we would have $1=\operatorname{deg}(F)=\sum_{p\in F^{-1}(\{F(x)\})}\operatorname{mult}_p(F)\geq 2$.
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