I have questions regarded to the resolution of Kummer surface. You can see the other 2 ones here
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 question regarding the object;
I want to compute $\mathbb{H}^{2}(X,\mathbb{R})$ and $\mathbb{H}^{(1,1)}(X,\mathbb{C})$. it seems to be an application of Mayer-victories but I have some problems in understanding how to formulate te connected sum between $\mathbb{T}^{4}$ and $\mathcal{O}_{\mathrm{p}^{1}}(-2)$. Someone may know that $X$ is a $K3$ surface and he/she can think about the topological invariants of $X$ from the properties of $K3$ surfaces but I prefer direct computations.
Any help would be appreciated.