Liouville's theorem for $\bar\partial$ closed forms

Question

Is there a Liouville type equivalent theorem for $\bar\partial$ closed forms in $\mathbb{C}^n$. So under which conditions of asymptotic behaviour, could we conclude that the form is constant ?

Answer 1

Counterexample: In ${\mathbb C}^2$ take

$\frac{1}{{|z_1|}^2+{|z_2|}^2+1} d\bar{z}_1 \wedge d\bar{z}_2$

That's dbar closed, the coefficient is bounded, but not constant. In fact, there is no condition on that function in front.

Ok, you say that was a (0,2)-form, that was cheating. Let's make a 1-form, say a (0,1)-form for example: Take $\omega = f d\bar{z}_1 + g d \bar{z}_2$. The dbar of that is: $(g_{\bar{z}_1} - f_{\bar{z}_2}) d \bar{z}_1 \wedge d\bar{z}_2$. Make say $g = \frac{1}{{|z_2|}^2+1}$ and $f = \frac{1}{{|z_1|}^2+1}$, and you get $\omega$ is dbar closed, coefficients bounded, but not constant. In a similar way, you can make all sorts of other counterexamples of all sorts of degrees $(k,\ell)$ as long as $\ell$ is not 0.

If you have a $(k,0)$-form, then we're good. Because then if it is dbar closed, then the coefficients must have been holomorphic to begin with.