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A nonclosed dbar form cannot have a potential

Statement refuted

Let $g = {\overset{ˉ}{z}}_{1} d {\overset{ˉ}{z}}_{2}$ on $C^{2}$ . For every nonempty open $U \subseteq C^{2}$ , the restricted form $g ∣_{U}$ is not of the form $\overset{ˉ}{\partial} u$ for any smooth function $u : U \to C$ .

Facts & Assumptions

Given: The nonempty open set $U \subseteq C^{2}$ and the smooth $(0, 1)$ -form $g ∣_{U} = {\overset{ˉ}{z}}_{1} d {\overset{ˉ}{z}}_{2}$ .

[F1]

The coordinate formula for $\overset{ˉ}{\partial}$ differentiates form coefficients in each ${\overset{ˉ}{z}}_{j}$ direction and wedges the result with $d {\overset{ˉ}{z}}_{j}$ (Bigraded complex forms and the Dolbeault operators).

[F2]

For every smooth complex-valued form, ${\overset{ˉ}{\partial}}^{2} = 0$ (The d, partial and dbar identities).

Counterexample

technique · counterexample

1.1

The proposed witness has a nonzero $\overset{ˉ}{\partial}$ derivative at every point. [F1, given, algebra] Applying the coefficient formula [F1] to $g = {\overset{ˉ}{z}}_{1} d {\overset{ˉ}{z}}_{2}$ differentiates its coefficient once in each barred coordinate. Only the $j = 1$ derivative is nonzero, and it equals $1$ , so $\overset{ˉ}{\partial} g = d {\overset{ˉ}{z}}_{1} \land d {\overset{ˉ}{z}}_{2} .$ Because the two coordinate covectors are distinct members of the local wedge basis, this $(0, 2)$ -form is nonzero at every point of $C^{2}$ ; hence $g ∣_{U}$ is not $\overset{ˉ}{\partial}$ -closed on any nonempty open $U$ .

2.1

Nonclosedness contradicts the necessary condition for having a potential. [F2, step 1.1, given, algebra] If a smooth $u : U \to C$ satisfied $\overset{ˉ}{\partial} u = g ∣_{U}$ , applying $\overset{ˉ}{\partial}$ and using [F2] would give $0 = {\overset{ˉ}{\partial}}^{2} u = \overset{ˉ}{\partial} g ∣_{U} = d {\overset{ˉ}{z}}_{1} \land d {\overset{ˉ}{z}}_{2}$ . The last form is nonzero at every point of the nonempty set $U$ by step 1.1, a contradiction. Thus no such $u$ exists. ∎

Depends on

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Sources

Lebl, Tasty Bits of Several Complex Variables, v4.4, Chapter 4 §4.2 (standard reference, not scraped)