Positive-degree Dolbeault cohomology vanishes on polydiscs

If $\eta =0$, take $\omega =0$. Otherwise, write each finite-radius factor as $D(a_j,R_j)$ and set $r_{j,k}=(1-2^{-k})R_j$; for each factor equal to $\mathbb{C}$, use center $a_j=0$ and radius $r_{j,k}=k$. Let $P_k={\prod }_{j=1}^{n}D(a_j,r_{j,k})$. The sequence is increasing, ${\bigcup }_k{P}_k=P$, and $\overline{P_k}\subset P_{k+1}$ by [F15]. Each $\overline{P_k}$ is closed and bounded in ${\mathbb{C}}^n$, hence compact by [F14]. Thus every successive pair satisfies the compact-containment hypothesis of [F2].

Suppose first that $q>1$. By [F2], choose a primitive ${\omega }_1$ on $P_3$ using $\eta$ on $P_4$. Inductively suppose ${\omega }_k$ is a smooth $(p,q-1)$ form on $P_{k+2}$ with $\bar{\mathrm{\partial }}{\omega }_k=\eta$. By [F2], choose another primitive $\sigma$ on $P_{k+3}$ using $\eta$ on $P_{k+4}$. The difference $d={\omega }_k-\sigma$ on $P_{k+2}$ is a closed $(p,q-1)$ form; because $q-1\ge 1$, [F2] gives a $(p,q-2)$ form $v$ on $P_{k+1}$ with $\bar{\mathrm{\partial }}v=d$ there.

Since $\overline{P_k}$ is compact and contained in $P_{k+1}$, apply [F6] to obtain a smooth $\chi$ equal to $1$ near $\overline{P_k}$ with $\mathrm{supp}\chi \subset P_{k+1}$. The support is closed by [F13]. Extend $\chi v$ by zero outside $P_{k+1}$; near each boundary point of $P_{k+1}$ the closed support is absent, so this extension is smooth. Define ${\omega }_{k+1}=\sigma +\bar{\mathrm{\partial }}(\chi v)$ on $P_{k+3}$. By [F3], ${\bar{\mathrm{\partial }}}^2=0$, so $\bar{\mathrm{\partial }}{\omega }_{k+1}=\bar{\mathrm{\partial }}\sigma =\eta$. On $P_k$, $\chi =1$ on a neighborhood, so the product rule [F16] gives $\bar{\mathrm{\partial }}(\chi v)=\bar{\mathrm{\partial }}v=d$ and ${\omega }_{k+1}={\omega }_k$. Full AC [F5] supplies choices for the successive nonempty sets of local primitives and corrections at every finite stage.

Now suppose $q=1$. Use [F2] to choose ${\omega }_1$ on $P_3$ with $\bar{\mathrm{\partial }}{\omega }_1=\eta$ on $P_3$. Given ${\omega }_k$ on $P_{k+2}$, choose $\sigma$ on $P_{k+3}$ with $\bar{\mathrm{\partial }}\sigma =\eta$. On $P_{k+2}$, $\delta =\sigma -{\omega }_k$ is a closed $(p,0)$ form. In its unique expansion $\delta ={\sum }_{\mathrm{\mid }I\mathrm{\mid }=p}{\delta }_{I}d{z}^I$, [F1] and $\bar{\mathrm{\partial }}\delta =0$ imply ${\mathrm{\partial }}_{{\bar{z}}_j}{\delta }_I=0$ for every $I,j$. Each coefficient is smooth, so [F7] and [F8] make it holomorphic; [F9] then makes it continuous and separately holomorphic. Apply [F10] to each of the finitely many coefficients on $P_{k+2}$. Since $\overline{P_k}\subset P_{k+2}$, their Taylor polynomials can be chosen with maximum coefficient error less than $2^{-k}$ on $\overline{P_k}$. Let $Q_k$ be the resulting holomorphic polynomial $(p,0)$ form and set ${\omega }_{k+1}=\sigma -Q_k$ on $P_{k+3}$. Then $\bar{\mathrm{\partial }}{\omega }_{k+1}=\eta$ and every coefficient of ${\omega }_{k+1}-{\omega }_k$ has absolute value below $2^{-k}$ on $\overline{P_k}$. Full AC [F5] supplies choices throughout this countable recursion.

The exact agreement in step 2.1 defines a smooth form $\omega$ on $P$ by $\omega {\mathrm{\mid }}_{P_k}={\omega }_k{\mathrm{\mid }}_{P_k}$. The sets $P_k$ cover $P$ and the definitions agree on each nested overlap. Locally $\omega$ equals a local primitive ${\omega }_k$, so $\bar{\mathrm{\partial }}\omega =\eta$ on all of $P$.

For any compact $K\subset P$, the increasing open cover $\{P_k\}$ has a finite subcover, so $K\subset P_{\mathrm{\ell }}$ for some $\mathrm{\ell }$. If $m>k\ge \mathrm{\ell }$, the coefficient error estimate in step 2.2 gives $\mathrm{\parallel }{\omega }_m-{\omega }_k{\mathrm{\parallel }}_{K,\mathrm{\infty }}\le {\sum }_{j=k}^{m-1}{2}^{-j}<2^{1-k}$, where the norm is the maximum over the finitely many $(p,0)$ coefficients and $K$. Thus the coefficients converge locally uniformly on $P$ to those of a form $\omega$. For each fixed $\mathrm{\ell }$, every coefficient of ${\omega }_m-{\omega }_{\mathrm{\ell }}$ is holomorphic on $P_{\mathrm{\ell }}$ for $m\ge \mathrm{\ell }$, by the closed $(p,0)$ argument in step 2.2. Their locally uniform limit is holomorphic on $P_{\mathrm{\ell }}$ by [F11], and is smooth by [F12]. By [F7], this holomorphic difference has zero $\bar{\mathrm{\partial }}$. Since $\bar{\mathrm{\partial }}{\omega }_{\mathrm{\ell }}=\eta$, it follows that $\omega$ is smooth and $\bar{\mathrm{\partial }}\omega =\eta$ on every $P_{\mathrm{\ell }}$, hence on $P$.

Both cases produce a smooth $\bar{\mathrm{\partial }}$-primitive for every closed $(p,q)$ form on $P$. Therefore every element of the numerator in [F4] belongs to its exact-form denominator, and the quotient is the zero vector space.