Consider the blow up of $\mathbb C^2/\mathbb Z_2$ at its singularity $0$.
Since $dz_1\wedge dz_2$ is invariant under $z\mapsto -z$, it passes to a well defined holomorphic form on $(\mathbb C^2/\mathbb Z_2)\setminus 0$. I want to extend it to $Bl_0\mathbb C^2/\mathbb Z_2$.
I think that this can be done due to Hartog's extension theorem. But how can I check, if this extension has zeros on the exceptional divisor?
(This is a follow up question to my previous question)
Try computing in coordinates. You can cover the blowup by two affine patches with $z_1=z_1, z_2=uz_1$ and $z_2=z_2,z_1=vz_2$. On the first patch, the form can be rewritten as $dz_1\wedge d(uz_1) = dz_1\wedge udz_1 + dz_1\wedge z_1du$, which vanishes on the portion of the exceptional divisor in that patch, and similar computations on the second patch show the same thing on that patch.
What's going on here is that to have anything interesting happen when extending a differential form to a blowup, the differential form can't be holomorphic/regular. Think about what holomorphic differential forms are: they're dual to tangent vectors. But the blow-down map sends all tangent vectors on the exceptional divisor to 0, so you can't get anything interesting to happen with a pullback of a holomorphic differential form to the exceptional divisor.