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.
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.