## Monodromy of Picard-Fuchs differential equations for Calabi-Yau threefolds.(English)Zbl 1153.34055

Summary: We are concerned with the monodromy of Picard-Fuchs differential equations associated with one-parameter families of Calabi-Yau threefolds. Our results show that in the hypergeometric cases the matrix representations of monodromy relative to the Frobenius bases can be expressed in terms of the geometric invariants of the underlying Calabi-Yau threefolds. This phenomenon is also verified numerically for other families of Calabi-Yau threefolds in the paper. Furthermore, we discover that under a suitable change of bases the monodromy groups are contained in certain congruence subgroups of Sp$$(4, \mathbb Z)$$ of finite index and whose levels are related to the geometric invariants of the Calabi-Yau threefolds.

### MSC:

 34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 32Q25 Calabi-Yau theory (complex-analytic aspects)
Full Text:

### References:

 [1] DOI: 10.1016/S0550-3213(98)00020-0 · Zbl 0896.14025 [2] DOI: 10.1007/BF02101841 · Zbl 0843.14016 [3] Beukers F., Astérisque 147 pp 271– (1987) [4] DOI: 10.1007/BF01393900 · Zbl 0663.30044 [5] Beukers F., Math. 351 pp 42– (1984) [6] Borcea Ciprian, AMS/IP Stud. Adv. Math. 1 pp 717– (1997) [7] DOI: 10.1016/0550-3213(91)90292-6 · Zbl 1098.32506 [8] DOI: 10.1016/j.aim.2003.07.012 · Zbl 1122.11087 [9] DOI: 10.1016/S0196-8858(02)00034-9 · Zbl 1013.33010 [10] Guillera Jesús, Experiment. Math. 12 (4) pp 507– (2003) [11] Katz Sheldon, Adv. Th. Math. Phys. 3 (5) pp 1445– (1999) [12] DOI: 10.1016/0550-3213(93)90289-2 [13] DOI: 10.1142/S0217732394001660 · Zbl 1020.32507 [14] DOI: 10.1007/BF02099367 · Zbl 0867.14017 [15] DOI: 10.1155/S1073792893000030 · Zbl 0789.14005 [16] DOI: 10.1023/A:1001847914402 · Zbl 0974.14026 [17] Stiller P., Mem. Amer. Math. Soc. 49 (299) pp 116– (1984) [18] DOI: 10.1007/BF02684341 · Zbl 0374.57002 [19] DOI: 10.1007/s00209-003-0573-4 · Zbl 1108.11040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.