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edited Sep 21, 2023 at 1:59

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Uniformization of $\mathbb{CP}^2-\cup\bigcup C_i$, where $C_i$ are Riemann surfaces intersecting generically

Consider $X=\mathbb{CP}^2-\cup C_i$$X=\mathbb{CP}^2-\bigcup C_i$ where $C_i$ are Riemann surfaces intersecting generically.

How to compute the fundamental group of this space and what is the universal cover?

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edited Sep 19, 2023 at 18:02

YCor

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Uniformization of $\mathbb{CP}^2-\cup C_i$,$C_i$ - where $C_i$ are Riemann surfaces intersecting generically

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edited Sep 18, 2023 at 21:48

0x11111

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Uniformization of $X=\mathbb$\mathbb{CP}^2\text{^2-Riemann surface}$\cup C_i$,$C_i$ - Riemann surfaces intersecting generically

Is there a geometric model for the universal coverConsider $\tilde{X}$ of the space$X=\mathbb{CP}^2-\cup C_i$ where $X=\mathbb{CP}^2\text{-Riemann surface}$?

The$C_i$ are Riemann surface is given by $\{(x,y)\mid P(x,y) = 0\}$ for generic polynomial $P$, in local coordinatessurfaces intersecting generically.

I am looking for 2d analogyHow to compute the fundamental group of 1d situation. In 1d $\mathbb{CP}^1-\{x_i\}$ is uniformized by upper half plane without points $a_i=0,...,n$.

I think this is some well studied space, probably well covered in textbooks, I just do not know and what is the name.universal cover?

fixing the conspicuous typographical error

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edited Sep 18, 2023 at 19:24

Michael Hardy

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asked Sep 17, 2023 at 16:09

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Uniformization of $\mathbb{CP}^2-\cup\bigcup C_i$, where $C_i$ are Riemann surfaces intersecting generically

Uniformization of $\mathbb{CP}^2-\cup C_i$, where $C_i$ are Riemann surfaces intersecting generically

Uniformization of $\mathbb{CP}^2-\bigcup C_i$, where $C_i$ are Riemann surfaces intersecting generically

Uniformization of $\mathbb{CP}^2-\cup C_i$,$C_i$ - where $C_i$ are Riemann surfaces intersecting generically

Uniformization of $X=\mathbb$\mathbb{CP}^2\text{^2-Riemann surface}$\cup C_i$,$C_i$ - Riemann surfaces intersecting generically

Uniformization of $X=\mathbb{CP}^2\text{-Riemann surface}$

Uniformization of $\mathbb{CP}^2-\cup C_i$,$C_i$ - Riemann surfaces intersecting generically