Let $S$ be a reducible compact complex analytic space, thus we have the decomposition $S=\bigcup\_{i=1}^n {V\_i}$ where $V\_i$ is the irreducible component of $S$. Let $L$ be a line bundle on $S$, I wonder if we can give some descriptions about $H^0(S,L)$ and $H^0(V\_i,L)$.


I have seen this question [Globally generated line bundle on reducible curve](https://math.stackexchange.com/questions/3375341/globally-generated-line-bundle-on-reducible-curve "Globally generated line bundle on reducible curve"). For the special case that a reducible variety with two components $Y$ and $Z$ intersecting at a simple node $p$, then one can say the global sections of $L$ is the codimension one subspace of $H^0(L|\_Y) \oplus H^0(L|\_Z)$ consisting of sections with the same value at $p$. But this case is too peculiar.


So, in general case, can we say something about $H^0(S,L)$ and $H^0(V\_i,L)$ or the relationships of their dimensions? One motivation lies in the following theorem by Grauert.



> 
> Let $f: X \rightarrow Y$ be a proper morphism of complex spaces and $\mathcal{S}$ a coherent analytic sheaf on $X$ which is flat with respect to $Y$ (or $f$), which means that the $\mathcal{O}\_{f(x)}$-modules $\mathcal{S}\_{x}$ are flat for all $x \in X .$ Set $\mathcal{S}(y)$ as the analytic inverse image with respect to the embedding $X\_{y}$ of in $X$. Then for any integers $i, d \geq 0$, the set
> $$
> \left\{y \in Y \,\, |\,\, \dim\_{\mathbb{C}} H^{i}\left(X\_{y}, \mathcal{S}(y)\right) \geq d\right\}
> $$
> is an analytic subset of $Y$.
> 
> 
> 


The fibre $X\_y$ surely can be reducible, we write the decomposition $X\_y=\bigcup\_{i=1}^n {V\_i}$. So, now if we know the dimension of $H^{i}\left(X\_{y}, \mathcal{S}(y)\right) $, can we infer whether there exists some $i$, such that we can say something about $H^{i}\left(V\_i, \mathcal{S}|{V\_i}\right)? $



Global sections of a line bundle on a reducible complex space
In any case $H^0(S,L)$ is a subspace in $\oplus H^0(V\_i,L)$, but the way it sits there depends on the way the components $V\_i$ are glued together. In the simplest case where they for a simple normal crossing configuration, there is an exact sequence
$$
0 \to H^0(S,L) \to 
\bigoplus H^0(V\_i,L) \to 
\bigoplus\_{i < j} H^0(V\_i \cap V\_j,L),
$$
where the last map is given by the differences of the restriction maps.


