Global sections of a line bundle on a reducible complex space

Question

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. 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)? $

Answer 1

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.