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 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 $L:= \mathcal{O}(1) \longrightarrow \mathbb{CP}^N$.

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.

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 $L_1$ and $L_2$ are 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 $L:= \mathcal{O}(1) \longrightarrow \mathbb{CP}^N$.

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 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 $L:= \mathcal{O}(1) \longrightarrow \mathbb{CP}^N$.