2003/12/21 by Shi-shyr Roan, Roan, Shi-shyr
Mathematics · #11G #14E #14J #14K #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT) #math.AG #math.NT #msc:11G #msc:14E #msc:14J #msc:14K
paper · pdf · doi:10.48550/arxiv.math/0312399
LaTeX 24 pages
arxiv created 2003/12/21 · openalex publication_date 2003/12/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We determine all the Kummer-surface-type Calabi-Yau (CY) 3-folds, i.e., those 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= \ZZ3 and T being the triple-product of 1-torus carrying an order 3 automorphism, the other with G= \ZZ7 and T being the Jacobian of Klein quartic curve. These CY 3-folds T/G are all rigid, hence no complex structure deformation for each of these two varieties. We further investigate problems of \PZ1-curves C in 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 T/G is obtained for both cases. In the case G=\ZZ3, we have derived an effective method of constructing C in T/G, and obtained the explicit forms of rational curves for some other d by this procedure.