The ddbar lemma implies Frölicher E1 degeneration

Let $\alpha \in A^{p,q}(X)$ satisfy $\bar{\mathrm{\partial }}\alpha =0$. Then $u:=\mathrm{\partial }\alpha$ has pure type $(p+1,q)$, is $d$-exact since $u=d\alpha$, and is $d$-closed since $du=d\mathrm{\partial }\alpha =-\mathrm{\partial }\bar{\mathrm{\partial }}\alpha =0$. By [F2], $u=\mathrm{\partial }\bar{\mathrm{\partial }}\beta$ for $\beta \in A^{p,q-1}(X)$; when $q=0$ or $p=n$, the required bidegree is absent and [F2] forces $u=0$. Therefore ${\alpha }^{\mathrm{\prime }}:=\alpha -\bar{\mathrm{\partial }}\beta$ (or ${\alpha }^{\mathrm{\prime }}=\alpha$ at those boundary bidegrees) satisfies $\bar{\mathrm{\partial }}{\alpha }^{\mathrm{\prime }}=0$ and $\mathrm{\partial }{\alpha }^{\mathrm{\prime }}=u-\mathrm{\partial }\bar{\mathrm{\partial }}\beta =0$. It differs from $\alpha$ by a $\bar{\mathrm{\partial }}$-exact form, so it represents the same Dolbeault class.

We prove vanishing of the differentials inductively. At page $r=1$, every $E_1^{p,q}$ class has, by step 1.1, a representative ${\alpha }^{\mathrm{\prime }}$ of type $(p,q)$ with $d{\alpha }^{\mathrm{\prime }}=0$. Suppose $d_1,\dots ,d_{r-1}$ vanish. The canonical identifications then give $E_r=E_1$; in particular every class in every possible source term of $d_r$ is represented by such a $d$-closed ${\alpha }^{\mathrm{\prime }}$. The spectral-sequence differential is induced by the first component of $d{\alpha }^{\mathrm{\prime }}$ that could survive to page $r$, and that component is zero because $d{\alpha }^{\mathrm{\prime }}=0$. Hence $d_r=0$. This proves the induction for every $r$, including all possible incoming differentials to each target class. Since $0\le p\le n$, the filtration has finitely many steps in each total degree, so it converges and $E_1=E_{\mathrm{\infty }}$.