I am trying to understand this paper. Let $M$ be a compact Kaehler manifold of dimension $n$, $X$ is a holomorphic vector field, $i_X$ the contraction operator, i.e. for $\alpha$ a $p$-form, then $i_X(\alpha) = \alpha(X, \ldots)$ a $(p-1)$-forms. Let $\mathcal{O}_M$ be the sheaf of holomorphic functions and $\Omega^p$ be the sheaf of holomorphic $p$-form. I have a complex

$$ 0 \rightarrow \Omega^n \xrightarrow{i_X} \Omega^{n-1} \xrightarrow{i_X} \ldots \xrightarrow{i_X} \Omega \xrightarrow{i_X} \mathcal{O}_M \rightarrow 0 $$

I then construct the Cartan-Eilenberg resolution $\mathcal{I}^{\bullet,\bullet}$, then apply the global section functor to get $H^0(M, \mathcal{I}^{\bullet,\bullet})$ and get two spectral sequences from the filtered total complex. Consider the vertical one. It has the first page $E^{p,q}_1 = H^q(M, \Omega^p)$, since each column $\mathcal{I}^{p, \bullet}$ is an injective resolution of $\Omega^p$. The author then claimed that the wedge product $\Omega^a \times \Omega^b \rightarrow \Omega^{a+b}$ induces natural product $E^{a,b}_r \times E^{c,d}_r \rightarrow E^{a+c, b+d}_r$ and I don't know why.

We do have the wedge product on the first page $$H^{q_1}(M, \Omega^{p_1}) \times H^{q_2}(M, \Omega^{p_2}) \rightarrow H^{q_1+q_2}(M, \Omega^{p_1+p_2}) $$ thanks to the isomorphism: $$H^{q}(M, \Omega^p) = H^{p,q}_{\bar{\partial}}(M)$$ but normally to define the multiplication we need the multiplication on the total complex $T^{\bullet}$ which respects the filtration i.e. $$F_aT^{b} \times F_cT^{d} \rightarrow F_{a+c}T^{b+d}$$

but the complex consists of $H^0(M, \mathcal{I}^{\bullet, \bullet})$ and I don't know how to extend the multiplication on either ${I}^{\bullet, \bullet}$ or $H^0(M, \mathcal{I}^{\bullet, \bullet})$. One attempt is to consider the resolution $0 \rightarrow \Omega^p \rightarrow \mathcal{I}^{p, \bullet}$ then we have a complex $\Omega^p \times \Omega^q \rightarrow \mathcal{J}^{\bullet}$, where $\mathcal{J}^n = \bigoplus_{i+j=n} \mathcal{I}^{p,i} \times \mathcal{I}^{q,j}$. I would like to induce a morphism $\mathcal{J}^{\bullet} \rightarrow \mathcal{I}^{p+q, \bullet}$ from $\Omega^p \times \Omega^q \rightarrow \Omega^{p+q}$, but there is no guarantee that the complex $\mathcal{J}^{\bullet}$ is exact to do that.