Conditions such that taking global sections of line bundles commutes with tensor product?

Question

Let us work with projective algebraic varieties over $k= \mathbb{C}$. If necessary we can also assume smoothness of the varieties.

Of course it is not in general true that given two line bundles $L, M$ on a variety $X$, we have $$ \Gamma(X,L \otimes M) = \Gamma(X,L)\otimes \Gamma(X,M), $$ an easy example is $\mathcal{O}(1)$ and $\mathcal{O}(-1)$ on projective space.

I was wondering if there are conditions that one can place on the variety or the bundles, such that the above equality does hold. Maybe if the bundles admit global sections, or are even generated by global sections?

As a second question, when we know the dimension of $\Gamma(X,L \otimes M)$, can we translate this back into information on $\Gamma(X,L)$ or $\Gamma(X, M)$? (assuming for the moment anything that you wish to assume.)

I know this last question is vague, so as an answer basically any general observation, or anything in a direction of a technique for calculation would be great!

Answer 1

The paper Global sections and tensor products of line bundles over a curve by David C. Butler quotes and proves some interesting results in that direction. For example:

On a smooth projective curve of genus $g$ let $L_1$ be a line bundle of degree $\geq 2g$ and $L_2$ a line bundle of degree $>2g$. Then $\tau : \Gamma(L_1) \otimes \Gamma(L_2) \to \Gamma(L_1 \otimes L_2)$ is surjective (this is due to Mumford). This also holds when $L_1$ and $L_2$ are globally generated and $\mathrm{deg}(L_1) + \mathrm{deg}(L_2) \geq 4g+1$. The paper also contains refinements about the image of $\tau$ (Theorems 1 and Theorem 2 in loc. cit). The proofs use Riemann-Roch.

If you are also interested in vector bundles instead of just line bundles, see the paper On the tensor product of sections of vector bundles on an algebraic curve by M. Baiesi and E. Ballico, and the references there.

I don't know if anything is known beyond curves.

Answer 2

If $X=Spec(A)$ is any affine scheme and if $L,M$ are arbitrary line bundles, then the canonical map $$ \Gamma(X,L)\otimes_{\mathcal O_X(X)} \Gamma(X,M) \to \Gamma(X,L \otimes_{\mathcal O_X} M) \quad (\bigstar)$$ is always an isomorphism.

Indeed $L=\tilde P$ and $M=\tilde{Q}$ are associated to the projective $A$-modules of rank one $P=\Gamma(X,L)$ and $Q=\Gamma(X,M)$.
The morphism $(\bigstar)$ then becomes $$P\otimes_A Q\to \Gamma(X,\tilde P \otimes_{\mathcal O_X} \tilde X) \quad (\bigstar \bigstar)$$
To conclude, it suffices to show that $$\tilde P \otimes_{\mathcal O_X} \tilde X=\widetilde {P\otimes _A Q}\quad (KEY)$$ because then in $(\bigstar \bigstar)$ $$\Gamma(X,\tilde P \otimes_{\mathcal O_X} \tilde M)=\Gamma(X,\widetilde {P\otimes _A Q})=P\otimes_AQ$$ too.
And here is the good news: Dieudonné and Grothendieck proved $(KEY)$ for you half a century ago!
Just check EGA I, Corollaire (1.3.12 (i)), page 88.

A Guess
I am pretty confident that the corresponding result holds for two arbitrary holomorphic line bundles on a Stein manifold.
Edit: June 18th, 2016
The guess above is indeed true.
In fact we have the following incredibly more general result, proved by Otto Forster in Zur Theorie der Steinschen Algebren und Moduln, line -11 on page 403 :

If $\mathcal F,\mathcal G$ are coherent sheaves on the Stein space $X$, then the canonical morphism $$\Gamma(X,\mathcal F)\otimes _{\Gamma(X,\mathcal O_X)} \Gamma(X,\mathcal G) \stackrel {\cong }{\to} \Gamma(X,\mathcal F\otimes_{\mathcal O_X} \mathcal G)$$ is an isomorphism.