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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: enter image description here


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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    $\begingroup$ I believe that for a closed complex curve (which $T^2$ with a choice of complex structure is) the second Chern class of the holomorphic tangent bundle vanishes. There is an expository paper by Iskander Taimanov which may be relevant to the first question arxiv.org/pdf/1708.05967.pdf $\endgroup$ – Aknazar Kazhymurat Mar 12 at 16:05
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    $\begingroup$ also the formula in the second point is somewhat hard to understand. $c_2$ is presumably a cohomology class, and to get a well-defined integral, you need to integrate against a homology class (say $[D_i]$). I am not sure what is meant by volume of a divisor but it probably is not an invariant of the homology class. $\endgroup$ – Aknazar Kazhymurat Mar 12 at 16:09
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    $\begingroup$ What are these $z_i$ and $D_i$?? $\endgroup$ – abx Mar 12 at 22:33
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    $\begingroup$ The notation $c_{2i}$ is confusing, maybe $c_2^i$ is better or $c_{2,i}$. In any case, your equation involving the volume of $D_i$ is false : $c_2$ is a cohomology class, not a number ! Saying, you have formulas for $c_2(X\times Y)$ involving $c_1(X), c_2(X), c_1(Y),c_2(Y)$, see on wikipedia Chern classes $\endgroup$ – Bleuderk Mar 13 at 11:53
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    $\begingroup$ Sorry, but "setting $z_i=0$" does not make any sense. $\endgroup$ – abx Mar 13 at 11:56

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