I need some help in order to clarify some topological data of a $K3\times T^{2}$ Calabi Yau manifold in which $K3$ part has been obtained as a resolution of a $T^{4}/ \mathbb{Z_{2}}$ orbifold .

EDIT: A bit of notation: I am following mostly this formalism. The divisors $D_{i}$ are the $3$ hypersurfaces of complex dimension $2$ coming from setting each of the complex (non-projective) coordinates $z_{1},z_{2}$ of the K3 surface and the $z_{3}$ coordinate of the $T^{2}$to $0$ inside the manifold: i.e for example let´s define $f$ to be the 2-plane $z_{1}=0$ in $\mathbb{C}^{3}$ . Then the divisor would be $D_{1}=f\cap ( K3 \times T^{2})$.

Also, the divisors and their dual curves $C_{i}$ form an integral basis. We call $\hat{D_{i}}$ to the $(1,1)$-forms associated to $D_{i}$ and $\hat{C_{i}}$ to the $(2,2)$-forms associated to $C_{i}$, such that:

**1) Number of divisors $=19$?** I understand that we will have $16$ exceptional divisors (one coming from resolving each singularity) and $3$ ordinary ones. Is this true?

**2) Second Chern Classes** I am interested in $$c_{2i}=\int_{D_{i}} \hat{c_{2}}\left ( K3 \times T^{2} \right ).$$

I think I must expand $c_{2}(K3 \times T^{2})\sim \sum _{i,j}\hat{D_{i}} \hat{D_{j}}$ which result in a (2,2)-form which can be integrated over the 2-cycle $D_{i}$, but I do not know the form of that expansion.

**3) Intersection numbers** $\kappa _{ijk}=\int _{K3 \times T^{2}}D_{i}D_{j}D_{k}$. I just know that $\kappa _{ij}=\int _{K3}D_{i}D_{j}=2\delta _{i}^{j}$ for the exceptional divisors, but I do not know how to proceed to calculate the triple intersection numbers.

Any help or reference is welcome

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