Rational Curves in Rigid Calabi-Yau Three-folds

Abstract

We determine all the Kummer-surface-type Calabi-Yau (CY) 3-folds, i.e., those $\hat{T/G}$ which are resolutions of 3-torus-orbifolds $T/G$ with only isolated singularities. There are only two such CY spaces: one with $G= \ZZ_3$ and $T$ being the triple-product of 1-torus carrying an order 3 automorphism, the other with $G= \ZZ_7$ and $T$ being the Jacobian of Klein quartic curve. These CY 3-folds $\hat{T/G}$ are all rigid, hence no complex structure deformation for each of these two varieties. We further investigate problems of $\PZ^1$-curves $C$ in $\hat{T/G}$ not contained in exceptional divisors, by considering the counting number $d$ of elements in $C$ meeting exceptional divisors in a certain manner. We have obtained the constraint of $d$. With the smallest number $d$, the complete solution of $C$ in $\hat{T/G}$ is obtained for both cases. In the case $G=\ZZ_3$, we have derived an effective method of constructing $C$ in $\hat{T/G}$, and obtained the explicit forms of rational curves for some other $d$ by this procedure.