Why $H^0(X,L\otimes \mathcal{I}_{\{x\}})$ are global sections vanish at $x$ Let $X$ be a complex manifold and $L$ be the holomorphic line bundle, we have the short exact sequence $$0\to \mathcal{I}_{\{x\}}\otimes L \to L \to L(x)\to 0 $$ where $L$ is the line bundle and $L(x)$ is the fiber , $\mathcal{I}_{\{x\}}$ be the ideal sheaf.
I am not sure if I interpret it correct, I treat $L$ be sheaf of section ,and $L(x)$ be the skyscraper sheaf with value $L(x)$.
The question is why $H^0(X,L\otimes \mathcal{I}_{\{x\}})$ are sections vanish at $x$, how can I see this?
 A: Start off by the exact sequence of sheaves on $X$ $$
 0 \to \mathcal{I} \to \mathcal{O}_X \to i_{x,\ast}\mathbb{C} \to 0
$$
where $\mathcal{I}$ is the ideal sheaf defining the point $\{x\}$ and $i_{x,\ast}\mathcal{\mathbb{C}}$ is the skyscraper sheaf concentrated at $x$ - explicitly, if you view $X$ locally around the point $x$, this exact sequence is just $$
0 \to \{\text{holomorphic functions vanishing at }x\} \to \{\text{holomorphic functions defined near }x\} \to \mathbb{C} \to 0
$$ where the last map simply sends $f : U \to \mathbb{C}$ to the complex number $f(x)$.
Since the sheaf of sections of $L$ is invertible, tensoring with $L$ preserves exactness of this sequence: $$
 0 \to \mathcal{I} \otimes_{\mathcal{O}_X}L \to L \to L\mid_{\{x\}} = i_{x,\ast}\mathbb{C} \to 0
$$ where the right-most term is again the skyscraper sheaf concentrated at $x$ by the projection formula, if you like - explicitly, the restriction of the sheaf of holomorphic sections of $L$ to $x$ is again just a copy of the complex numbers, by definition of $L$ being a line-bundle.
The global sections functor $H^0(X,-) : \{\text{sheaves of }\mathcal{O}_X - \text{modules}\} \to \{\Gamma(X,\mathcal{O}_X)-\text{modules}\}$
(= complex vector spaces if $X$ is compact) is left exact (simply because injectivity for sheaves can be checked either at stalks or at the level of sections) and we thus get an exact sequence of $\mathbb{C}$-vector spaces $$
   0 \to H^0(X,\mathcal{I} \otimes_{\mathcal{O}_X}L) \to H^0(X, L) \to \mathbb{C}
$$ which identifies $H^0(X,\mathcal{I} \otimes_{\mathcal{O}_X}L)$ with a sub-vector space of the family of global sections of $L$. Since the map $L \to \mathbb{C} = L\mid_{\{x\}}$ is simply given by evaluating local sections of $L$ at the point $x$, such is true at the level of global sections as well (in this last exact sequence) and thus the subspace $H^0(X,\mathcal{I}\otimes_{\mathcal{O}_X}L) \subseteq H^0(X,L)$ is precisely given by the kernel of this evaluation map, by using exactness in the middle term :)
Sorry if my description was a little too explicit; I hope this helps!
