(I have asked the question Surjectivity of multiplicative map. I ask here the more specific case.)

Let $S$ be a smooth complex algebraic surface, and $D$ be a divisor on $S$ such that $D^2>0$ and $H^i(S, \mathcal{O}(nD)) = 0 $ for all $i > 0, n \in \mathbb{Z}_{\geq 1}$ ($D$ is not necessarily ample , but nef & big).

Write $W = S \times S$ and the coherent sheaf $\mathcal{G} = \mathcal{O}_{S}(nD) \boxtimes \mathcal{O}_{S}(nD)$ on $W$. Suppose that we have an exact sequence

$$ 0 \rightarrow H^0(W, \mathcal{G} \otimes I_{\Delta}^{n}) \rightarrow H^0(W, \mathcal{G}) \xrightarrow{\varphi} H^0(W, \mathcal{G} \otimes \mathcal{O}_W/I_{\Delta}^{n}) \rightarrow H^1(W, \mathcal{G} \otimes I_{\Delta}^{n}) \rightarrow 0.$$ where $\Delta$ is a diagonal embedding $S \hookrightarrow S \times S = W$ and $I_{\Delta}$ is an idela sheaf of $\Delta$.

By the Kunneth formula, we have $H^0(W, \mathcal{G}) = H^0(S, \mathcal{O}_S(nD)) \otimes H^0(S, \mathcal{O}_S(nD)).$ So, $\varphi$ can be regarded as a multiplicative map followed by some inflation into infinitesimal neighborhood.

I want to show the surjectivity of the map $\varphi$, or equivalently, the vanishing of $H^1(W, \mathcal{G} \otimes I_{\Delta}^{n})$ (under the action of $\mathfrak{S}_2$ in practice) for sufficiently large $n$.

The sheaf $\mathcal{O}_W/I_{\Delta}^{n}$ has a filtration by $I^k_{\Delta}/I^{n}_{\Delta}$ with associated graded factors $$ I^k_{\Delta}/I^{k+1}_{\Delta} \simeq Sym^k(\Omega_S), \ 0 \leq k \leq n-1. $$

So, I think we have to check the surjectivity on each $H^0(S, \mathcal{O}(2nD)\otimes Sym^{k}(\Omega_S))$ (is it right?). I have shown $(H^i(S, \mathcal{O}_S(2nD) \otimes Sym^k(\Omega_S))^{\mathfrak{S}_2} = 0$ for all $i > 0$ and $0 \leq k \leq n-1$.

I found the paper of Mark L. Green Koszul cohomology and the geometry of projective varieties(I read part of it) which says that:

Corollary(4.e.4)(The Explicit $H^0$ Lemma). Let $C$ be a smooth curve of genus $g$, and $L \rightarrow C$ and $M \rightarrow C$ analytic line bundles. Assume that $\deg L \leq \deg M$ and that $|L|$ is base-point free. If either $$\deg L + \deg M \geq 4g + 2$$ or $$\deg M = 2g+1, \deg L = 2g$$ then the multiplication map $$H^0(C, L) \otimes H^0(C, M) \rightarrow H^0(C, L \otimes M)$$ is surjectictive.

My question is that can we generalize this statement to the case of surfaces? How can I prove the surjectivity of $\varphi$ ?

I appreciate any advice, answers, or references.