At first, I am describing a resolution of Kummer surface:
Get a lattice of rank 4 ; $\Gamma$ on $ \mathbb{C}^2$. The quotient $\mathbb{T}^4:\mathbb{C}^2 /\Gamma$ would be a complex tori .
Now consider the quotient $\mathbb{T}^4/\mathbb{Z}_2$ where $\mathbb{Z}_2 = \{id , -id \}$ (it's named Kummer surface) the quotient has $2^{4}$ singularities (there are the fixpoint of the $\mathbb{Z}_2$ action). we can describe the resolution of the singularities as fallow: the singularities are as the singularities of $\mathbb{C}^2/\mathbb{Z}^2$. consider the map $(z_{1}, z_{2})\mapsto (z_{1}^{2}, z_{1} z_{2}, z_{2}^{2})\in \mathbb{C}^{3}$ it make a bijection between $\mathbb{C}^2/\mathbb{Z}^2$ and the hypersurface defined by $v^{2}=uw$ in $\mathbb{C}^{3}$. We can blow up the origin of $\mathbb{C}^{3}$ to desingular the hypersurface (what we get is $\mathcal{O}_{\mathrm{p}^{1}}(-2)$ ). Now we can identify a neighborhood of the singular points of $\mathbb{C}^2/\mathbb{Z}^2$ with a neighborhood of the origin of the hypersurface, so, the blow-up will describe how we get rid of the singularities. let's say $ \pi : X \mapsto \mathbb{T}^4/\mathbb{Z}_2$ be the resolution we have got.
This is my questions regarding the object;
$\mathbb{T}^{4}$ has a holomorphic (2,0)-form $dz_1 dz_2$ inherited from $\mathbb{C}^2$. It induce a holomorphic (2,0)-form on $\mathbb{T}^4/\mathbb{Z}_2 - \{singularities\}$ (lets name it $dz_1 dz_2$ again). the question is to extend $dz_1 dz_2$ to a nonvanishing holomorphic (2,0)-form on $X$. someone can use her\his knowledge in complex algebraic geometry to say $X$ is a $K3$ surface and the resolution is called a crepant resolution and my question becomes trivial for him\her. but I don't have a good background on complex algebraic geometry and I prefer a differential\analytical approach to the problem. (the question is equivalent to saying $X$ is a Calabi-Yau manifold)
Any help would be appreciated.