Talk:Complex differential form

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Great job, SR!. Charles Matthews 12:44, 18 November 2005 (UTC)Reply

A few things:

• Sorry about the awkward sectioning. I (or someone) will fix it later. Furthermore, I know the intro to the Complex differential forms section is a bit redundant with the article intro. I imported this from differential form.
• Thanks for pointing my in the right direction, Charles. I did a search for complex differential form, but got nothing. Speaking of which, complex differential form is probably a more appropriate title for the article.
• Also, this article, including my own contributions, is prejudiced towards complex forms on complex manifolds. There are cases in which one may want to consider complexified forms where the underlying manifold does not carry a complex structure (e.g., CR manifolds.) Anyway, I'll see if I can find a way to remove this bias unobtrusively.

Best Regards, Silly rabbit 12:58, 18 November 2005 (UTC)Reply

The article states:

By the Cauchy-Riemann equations, one can show that if we used a different holomorphic coordinate system wj, the spaces Ω1,0 and Ω0,1 are stable.

What does the word "stable" mean here? I can guess that maybe its a synonym for "invariant" or "complete". However "stable" also connotes the idea that its immune to small perturbations, and I don't think that that is what is meant. linas 23:13, 18 November 2005 (UTC)Reply

Ok, after nearly eight months... My inclination is to leave the word stable. To be completely precise, one would have to say "stabilized under the biholomorphism group" which is sort of a mouthful. Invariant is not correct, since the spaces are not invariant. They are equivariant, but that opens a whole other can of worms. I've changed the sentence structure and added a bit of clarification. Let me know if it's okay. Thanks for pointing it out. Silly rabbit 21:26, 9 June 2006 (UTC)Reply

What does it mean to "transform tensorially"? I've never seen this terminology used before. 2620:101:F000:700:223:6CFF:FE98:4D1 (talk) 14:26, 8 September 2013 (UTC)Reply

It means "transforms like a tensor", as opposed to something more complicated. Tensors always transform in a characteristic way: you relabel the base-point, and then contract the indexes with a linear transform. 67.198.37.16 (talk) 17:59, 7 May 2019 (UTC)Reply

The redirect D-bar operator has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2024 April 26 § D-bar operator until a consensus is reached. 1234qwer1234qwer4 21:36, 26 April 2024 (UTC)Reply

The last paragraph in the section One-forms begins as follows:

"Let Ω^1,0 be the space of complex differential forms containing only ${\displaystyle dz}$'s and Ω^0,1 be the space of forms containing only ${\displaystyle d{\bar {z}}}$'s. One can show, by the Cauchy–Riemann equations, that the spaces Ω^1,0 and Ω^0,1 are stable under holomorphic coordinate changes. In other words, if one makes a different choice w_i of holomorphic coordinate system, then elements of Ω^1,0 transform tensorially, as do elements of Ω^0,1."

But it is not correct to say that elements of Ω^1,0 "transform" at all. They are independent of any choice of coordinates.

It is the coordinates of the one-forms that transform, not the differential forms themselves.

This is a fundamental conceptual error.

I hope someome familiar with this subject can fix this.

The section The Dolbeault operators begins as follows:

"The usual exterior derivative defines a mapping of sections ${\displaystyle d:\Omega ^{r}\to \Omega ^{r+1}}$ via

${\displaystyle d(\Omega ^{p,q})\subseteq \bigoplus _{r+s=p+q+1}\Omega ^{r,s}}$

The exterior derivative does not in itself reflect the more rigid complex structure of the manifold.'"

But the symbol ${\displaystyle \Omega ^{r}}$ has not yet been defined anywhere in the article.

Because the article has before this point stated:

"E^k is the space of all complex differential forms of total degree k,"

it appears that this symbol ${\displaystyle \Omega ^{r}}$ may mean the same thing as E^k when k = r. But I am not at all sure of this.

I hope someone familiar with this subject can fix this infelicity.