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5

$\def\ZZ{\mathbb{Z}}\def\CC{\mathbb{C}}$ Let $E$ be an elliptic curve with affine chart $\{ (x,y) : y^2 = x+ax^2+bx^3 \}$ . We will write $p$ for the point $(0,0)$ and $\infty$ for the puncture. We want a holomorphic function on $E$ with a simple zero at $p$. We'll write $\omega$ for the nonvanishing holomorphic form $$\omega = \frac{dx}{2y} = ... -1 Let D \subset X be an irreducible divisor. Then there is a standard exact sequence (Hartshorne II.6.5)$$ \mathbb{Z}\cdot D \to Cl(X) \to Cl(X - D) \to 0.  In your case $D$ is the inflection point at infinity. This sequence shows that the Picarg group of the affine elliptic curve is not trivial. In particular, the ideal sheaf of a point is not trivial, ...

2

Here's an easier strategy: Consider the real-analytic vector bundle $\operatorname{Hom}(\mathbb R^n,E)$ where $\mathbb R^n$ denotes the trivial bundle of rank $n=\operatorname{rk}E$. By assumption, it has a smooth section which is a fiberwise isomorphism. Approximate this smooth section by a real-analytic section (this is where we use a hard theorem to ...

2

Here's just a sketch, maybe someone else can fill in the details. Most of the steps will require the use of the nontrivial fact that $C^\omega(M)$ is dense in $C^\infty(M)$. Let's assume $M$ is compact. Step 1: Construct a real-analytic embedding of vector bundles $E\to M\times\mathbb R^N$ for some finite $N$. This gives a real-analytic classifying map ...

4

Considering your second display, you probably mean $\mathrm{hol}^1\colon\pi_2(A)\to\pi_1(S^1)$. In any case, for all closed surfaces $A$ except $S^2$ and $\mathbb{R}\mathrm{p}^2$, you have $\pi_k(A)=0$ for all $k\ge2$, so there's no hope. For proof, just consider the uniformization.

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