Let $X$ be a smooth projective algebraic surface over $\mathbb C$, and let $L\_1, L\_2$ be ample line bundles satisfying $H^i(X,L\_j) = 0$ for $i > 0$.


Do we necessarily have $(h^0(L\_1) + h^0(L\_2))^2 \geq h^0(L\_1 \otimes L\_2)$, or are there examples where this inequality is violated?



Inequalities of dimensions of global sections for line bundles on surfaces
Here is a counterexample. Let $X = C \times \mathbb{P}^1$ where $C$ is a curve of genus $g \geq 2$ and let $L\_1 = L\_2 = p\_1^\*L \otimes p\_2^\*\mathcal{O}(1)$ where $L$ s a generic line bundle of degree $g-1$. Then $H^0(L) = 0$ since the map $\mathrm{Sym}^{g-1}(C) \to \mathrm{Pic}^{g-1}(C)$ is not surjective for dimension reasons, and $\chi(L) = 0$ by Riemann-Roch so $L$ is an ample line bundle on $C$ with no sections and no cohomology.


In particular, $L\_1 = L\_2$ is ample as its the sum of pullbacks of amples, and by Künneth formula, we deduce that $h^0 = h^1 = h^2 = 0$. Thus $L\_1$ and $L\_2$ satisfy the conditions and the left hand side of the inequality is $0$. On the other hand,
$$
L\_1 \otimes L\_2 = \pi\_1^\*L^{\otimes 2} \otimes \pi\_2^\*\mathcal{O}(2)
$$
which has sections by Riemann-Roch + Künneth again.


