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Grothendieck proved that there is an analytification functor $X \mapsto X^{an}$ from schemes locally of finite type over $\mathbb C$ to the category of (non-reduced!) analytic spaces, which is fully …

No. On $\mathbb P^1=\mathbb P^1(\mathbb C)$ we have $\Gamma(\mathbb P^1,\mathcal O_{\mathbb P^1}(-1))=\Gamma(\mathbb P^1,\mathcal O_{\mathbb P^1}(-2)=0$, but $O_{\mathbb P^1}(-1)$ and $O_{\mathbb P^1 …

Let $X$ be a complex manifold of dimension $n$ and $Y\subset X$ a closed submanifold of codimension $k$. a) Say that $Y$ is a complete intersection if the ideal $I(Y)\subset \mathcal O(X)$ of global h …

The answer is Yes for complex manifolds of dimension one. Indeed for any open subset $U\subset X$ the long exact cohomology sequence associated to the exponential sequence you mention yields the fr …

I interpret your line bundle $L$ to be differentiable rather than holomorphic, else the statement that line bundles are parametrized by their Chern class would be completely false. There always e …