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?
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.
