I am readding one proof of Bogomolov-–Sommese vanishing theorem, and they use that for any $s\in \Omega^p_X(D)$ we have $ds=0.$ Here $D$ is an effective divisor of $Y$:= Complex algebraic manifold.
So, I want to know why the logarithmic p-holomorphic forms are closed?
In the following paper arxiv.org/pdf/math/0110051.pdf theorem 2.24 said : By Hodge theory ($X$ being K¨ahler, or even just in $C\mathbb{C}$), the holomorphic $p$-forms $s_i$ ,$i=0,...,p$ on $X$ are closed. And something similar is used for logarithmic p−holomorphic forms.
Any help, is amazing!