4.4: Solvability of ∂-Problem in the Polydisc

Last updated
Save as PDF

Page ID: 75534

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Let us tackle solvability of the \(\bar{\partial}\)-problem for differential forms. In general, the problem is equivalent to holomorphic convexity, although it is rather involved, and thus we content ourselves with polydiscs. To work with differential forms, we, as before, split the derivatives into the holomorphic and antiholomorphic parts. For higher order forms we work with multi-indices.

Definition: \(\partial\) and \(\overline{\partial}\) Form

Let \(\alpha\) and \(\beta\) range over a finite set of multi-indices. We write a differential form as \[\eta = \sum_{\alpha,\beta} \eta_{\alpha \beta} \, dz^\alpha \wedge d\bar{z}^\beta .\] If the \(\alpha\) and \(\beta\) only range over those multi-indices such that \(|\alpha|=p\) and \(|\beta|=q\), then we call \(\eta\) a \((p,q)\)-form or a differential form of bidegree \((p,q)\). Define \[\partial \eta \overset{\text{def}}{=} \sum_{\alpha,\beta} \, \sum_{j=1}^n \frac{\partial \eta_{\alpha \beta}}{\partial z_j} dz_j \wedge dz^\alpha \wedge d\bar{z}^\beta , \qquad \text{and} \qquad \bar{\partial} \eta \overset{\text{def}}{=} \sum_{\alpha,\beta} \, \sum_{j=1}^n \frac{\partial \eta_{\alpha \beta}}{\partial \bar{z}_j} d\bar{z}_j \wedge dz^\alpha \wedge d\bar{z}^\beta .\]

If \(\eta\) is of bidegree \((p,q)\), then \(\partial \eta\) if of bidegree \((p+1,q)\) and \(\bar{\partial} \eta\) is of bidegree \((p,q+1)\). The total exterior derivative \(d \eta = \partial \eta + \bar{\partial} \eta\) as before.

Exercise \(\PageIndex{1}\)

Prove that \(d \eta = \partial \eta + \bar{\partial} \eta\).

Solvability of the equation \(\bar{\partial} \omega = \eta\) for every \((0,q)\)-form \(\eta\) such that \(\bar{\partial} \eta = 0\) whenever \(1 \leq q \leq n-1\) is equivalent to holomorphic convexity, although the proof of this is beyond the scope of this book, and we prove this result only in the case of a polydisc.

Theorem \(\PageIndex{1}\)

Let \(\Delta \subset \mathbb{C}^n\) be a polydisc, let \(q \geq 1\) be an integer, and let \(\eta\) be a smooth \((0,q)\)-form on \(\Delta\) such that \(\bar{\partial} \eta = 0\), then there exists a \((0,q-1)\)-form \(\omega\) such that \(\bar{\partial} \omega = \eta\).

The theorem follows from solving the problem on a compact subpolydisc, which is usually called the Dolbeault Lemma or Dolbeault-Grothendieck Lemma (Lemma \(\PageIndex{1}\)).

Lemma \(\PageIndex{1}\)

Dolbeault-Grothendieck Lemma

Let \(\Delta_s(w) \subset \Delta_r(w) \subset \mathbb{C}^n\) be polydiscs where \(0 < s < r < \infty\). Let \(q \geq 1\) be an integer, and let \(\eta\) be a smooth \((0,q)\)-form on \(\Delta_r(w)\) such that \(\bar{\partial} \eta = 0\), then there exists a smooth \((0,q-1)\)-form \(\omega\) on \(\Delta_s(w)\) such that \(\bar{\partial} \omega = \eta\).