When is the space of holomorphic sections of the tensor product of two line bundles given by the span of the tensor product of the basis?

Question

Let $S$ be a compact complex manifold and $L_1, L_2 \longrightarrow S$ be two holomorphic line bundles. Under what conditions (hopefully something that is easy to check) on $L_1$ and $L_2$ is the following fact true:

Let $f_1, f_2 \ldots, f_n$ be a basis for $H^0(L_1)$ and $g_1, g_2, \ldots g_m$ be a basis for $H^0(L_2)$. Then every element $s \in H^0(L_1 \otimes L_2)$ can be expressed as a $\mathbb{C}$-linear combination of $f_{i} \otimes g_j$, i.e. any $h \in H^0(L_1 \otimes L_2)$ is of the form $$ h = \Sigma c_{ij} f_{i} \otimes g_j $$ for some complex numbers $c_{ij}$.

Note that I am not asking for this expression to be unique (i.e. I am not asking for the collection $\{f_{i} \otimes g_j\} $ to be linearly independent in $H^0(L_1 \otimes L_2)$).

$\textbf{Remark:}$ 1) In general this is not true. Take $S:= \mathbb{CP}^N$, $L_1:= \mathcal{O}(2)$ and $L_2:= \mathcal{O}(-1)$.

I was wondering if it is it true if for instance when $L_1$ and $L_2$ are very ample?

2) If this question has been asked in too much generality, is anything known if $L_1= L_2 = L$? The statement does happen to be true if $L:= \mathcal{O}(1) \longrightarrow \mathbb{CP}^N$.

$\textbf{Remark:}$ I had meant to ask is the statement true when $L_1$ and $L_2$ are very ample (as opposed to ample). By this I mean the Kodiara map from $S$ to the projectivization of the dual of $H^0(L)$ is a well defined map and an embedding.

Answer 1

This issue arises often and in general it is false. For a simple example, take $S$ to be an elliptic curve, $L$ a line bundle of degree two. Then, you can easily check that the map $H^0(L)\otimes H^0(L)\to H^0(L^2)$ is not surjective.

Answer 2

There are ample line bundles without any non-zero global sections. E.g., on any smooth projective non-rational non-hyperelliptic curve $X$ and pairwise different points $P,Q,R\in X$, the line bundle $\mathscr O_X(P+Q-R)$ is ample but has no global sections. Then let $\mathscr A$ be such a line bundle and $n\in \mathbb N$ the largest integer such that $H^0(X,\mathscr A^{2^n})=0$. Finally, let $\mathscr L=\mathscr A^{2^n}$. Then $H^0(X,\mathscr L)\otimes H^0(X,\mathscr L)=0$, while $H^0(X,\mathscr L^{2})\neq 0$.

In your example of $X=\mathbb P^n$, it works, because all line bundles of positive degree on $\mathbb P^n$ are very ample and give a projectively normal embedding.