Global sections of a line bundle on a reducible complex space

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 $$H^{i}\left(V_i, \mathcal{S}|{V_i}\right)?$$

1 Answer

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.