Notation in Chern's book "Complex manifolds without potential theory"

Question

Page 15 from the book of Chern, one can read : $ " d\theta^k = 0$ mod $\theta^j"$. Here the $\theta^k$ are (1,0) form. I don't understand the meaning of this equation : $\theta^j$ are 1-form, $d\theta^k$ is a 2-form. Chern seems to use this expression as "$d\theta$ does not contains any term in $\overline{\theta}^j \wedge \overline{\theta}^k$. If anyone has a clear explanation about it I would be interested.

This condition seems to be important so I want to be sure to understand what is its rigourous meaning.

score 1 · Accepted Answer · answered Oct 04 '16 at 17:24

1

The $\theta^j$'s generate an ideal in the space of all differential forms (under wedge product), and the statement is that $d\theta^k$ is in that ideal. In other words, every term in the expansion of $d\theta^k$ contains at least one $\theta^j$ factor.

answered Oct 04 '16 at 17:24

Jack Lee

46,803

I see, thanks a lot !! (By the way your books are wonderful :-) ) – guest1380138 Oct 04 '16 at 17:27
@guest1380138: Thank you!! I'm glad they're helpful. – Jack Lee Oct 04 '16 at 18:36

Notation in Chern's book "Complex manifolds without potential theory"

1 Answers1