Let $D=D_1 \cup D_2$ be a reduce hypersurface on $X$, where $D_1$ and $D_2$ are reduced hipersurfaces and $C=D_1 \cap D_2$ is a reduced complete intersection. Also, let $h_1$, $h_2$ and $h = h_1 h_2$ be the equations which defines $D_1,D_2$ and $D$, respectively, on an open subset $U \subset X$.
I'm reading a paper with these notations, and I'm noticing that the author uses frequently that, for an exemple, $w \in \Omega_X^1(\log D_1) \Rightarrow dh_2 \wedge w$ is an holomorphic 2-form, and I don't understand why. Can someone explain me? I think that's consequence of $(h_1,h_2)$ being a regular sequence, but I'm not sure.
($\Omega_X^1(\log D_1)$ is the notation for the sheaf of $\cal{O}$$_X$-modules of logarithmic 1-forms along $D$)