I am trying to understand what exactly is $\Omega_X (log D)$ in a particularly case. More precisely, I am looking for conditions on a smooth surface and an effective divisor on it to obtain $\Omega_X (log D)\cong\Omega(D)$ and I am wondering if it's enough to ask that our divisor is a bunch of rational curves ( -2-curves).

Here's my attempt :

I tried to write both of these sheaves in an open set around a point where two curves meet : ( D is snc) We have : $$\Omega_X(logD)_{\vert U_P}=<\frac{dz_1}{z_1}, \frac{dz_2} {z_2}>\mathcal{O}_{X_{\vert U_P}}$$ & $$\Omega_X(D)_{\vert U_P}=<\frac{dz_1}{z_1z_2}, \frac{dz_2}{z_1z_2}>\mathcal{O}_{X_{\vert U_P}}$$ Taking quotient we obtain : $$<\frac{1}{z_2} ,\frac{1}{z_1} >\mathcal{O}_{X_{\vert_{U_P}}}$$ And now i am quite of stuck here, I also wrote the local coordinates around a point on a the curve of my divisor "far" from an intersection point but I still dont see why these sheaves would be equal even if the curves are rational. Some help or any hint would be grateful. PS : If anyone know some literature about this long complex sheaf I would be interested ( other than Griffiths and Harris)


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