Arithmetic of Degenerating Principal Variations of Hodge Structure: Examples Arising From Mirror Symmetry and Middle Convolution

Jr., Genival da Silva ; Kerr, Matt ; Pearlstein, Gregory

Canadian journal of mathematics, Tome 68 (2016) no. 2, pp. 280-308

Résumé

We collect evidence in support of a conjecture of Griffiths, Green, and Kerr on the arithmetic of extension classes of limiting mixed Hodge structures arising from semistable degenerations over a number field. After briefly summarizing how a result of Iritani implies this conjecture for a collection of hypergeometric Calabi–Yau threefold examples studied by Doran and Morgan, the authors investigate a sequence of (non-hypergeometric) examples in dimensions $1\,\le \,d\,\le \,6$ arising from Katz's theory of the middle convolution. A crucial role is played by the Mumford-Tate group (which is ${{G}_{2}}$ ) of the family of 6-folds, and the theory of boundary components of Mumford–Tate domains.

Détail

DOI : 10.4153/CJM-2015-020-4

Mots-clés : 14D07, 14M17, 17B45, 20G99, 32M10, 32G20, variation of Hodge structure, limiting mixed Hodge structure, Calabi–Yau variety, middle convolution, Mumford–Tate group

Bibliographie

[An] [An] André, Y., Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part. Compositio Math. 82(1992), no. 1,1–24. Google Scholar

[CDLNT] [CDLNT] Clingher, A., Doran, C., Lewis, J., Novoseltsev, A., and Thompson, A., The 14th case VHS via K3fibrations. arxiv:1312.6433 Google Scholar

[CK] [CK] Cox, D. A. and Katz, S., Mirror symmetry and algebraic geometry. Mathematical Surveys and Monographs, 68, American Mathematical Society, Providence, RI, 1999. http://dx.doi.Org/10.1090/surv/068 Google Scholar

[De] [De] Deligne, P., Milne, J. S., Ogus, A., and Shih, K.-Y., Hodge cycles, Motives, and Shimura varieties. Lecture Notes in Math., 900, Springer-Verlag, New York, 1982. Google Scholar

[D] [D] Dettweiler, M., On the middle convolution of local systems. http://arxiv:0810.3334 Google Scholar

[DR1] [DR1] Dettweiler, M. and Reiter, S., Rigid local systems and motives of type G. Compos. Math. 146(2010), no. 4, 929–963. http://dx.doi.Org/10.1112/S0010437X10004641 Google Scholar

[DR2] [DR2] Dettweiler, M. and Reiter, S., On exceptional rigid local systems. arxiv:math/0609142 Google Scholar

[DS] [DS] Dettweiler, M. and Sabbah, C., Hodge theory of the middle convolution. Publ. Res. Inst. Math. Sci. 49(2013), no. 4, 761–800. http://dx.doi.Org/10.4171/PRIMS/119 Google Scholar

[DK] [DK] Doran, C. and Kerr, M., Algebraic cycles and local quantum cohomology. Commun. Number Theory Phys. 8(2014), no. 4, 703–727. Google Scholar | DOI

[DM] [DM] Doran, C. F. and Morgan, J., Mirror symmetry and integral variations of Hodge structure underlying one-parameter families of Calabi-Yau threefolds. In: Mirror Symmetry V: Proceedings of the BIRS Workshop on CY Varieties and Mirror Symmetry (Dec. 2003), AMS/IP Stud. Adv. Math., 38, American Mathematical Society, Providence, RI, 2006, pp. 517–537. Google Scholar

[GGK] [GGK] Green, M., Griffiths, P., and Kerr, M., Neron models and boundary components for degenerations of Hodge structures of mirror quintic type. In: Curves and Abelian varieties, Contemp. Math, 465, American Mathematical Society, Providence, RI, 2008, pp. 71–145. http://dx.doi.Org/10.1090/conm/465/09101 Google Scholar

[GGK2] [GGK2] Green, M., Griffiths, P., and Kerr, M., Mumford-Tate groups and domains: their geometry and arithmetic. Annals of Mathematics Studies, 183, Princeton University Press, Princeton, NJ, 2012. Google Scholar

[GL] [GL] Greene, B. and Lazaroiu, C., Collapsing D-branes in Calabi-Yau moduli space. I. Nuclear Phys. B 604(2001), no.1-2, 181–255. Google Scholar | DOI

[Ir] [Ir] Iritani, H., An integral structure in quantum cohomology and mirror symmetry for toric orbifolds. Adv. Math. 222(2009), no. 3,1016–1079. http://dx.doi.Org/10.1016/j.aim.2009.05.016 Google Scholar

[Ka] [Ka] Katz, N., Rigid local systems. Annals of Mathematics Studies, 139, Princeton Univ. Press, Princeton, 1996. Google Scholar

[KP] [KP] Kerr, M. and Pearlstein, G., Boundary Components of Mumford–Tate domains. http://arxiv:121O.53O1 Google Scholar

[Le] [Le] Levine, M., Motivic tubular neighborhoods. Doc. Math. 12(2007), 71–146. Google Scholar

[MATH] [MATH] Wolfram Research, Inc., Mathematica. Version 6.0, Champaign, IL, 2007. Google Scholar

[Mo] [Mo] Morrison, D. R., Mathematical aspects of mirror symmetry, In: Complex algebraic geometry, IAS/Park City Math. Ser., 3, American Mathematical Society, Providence, RI, 1997, pp. 265–340. Google Scholar

[Ro] [Ro] Robles, C., Principal Hodge representations. In: Hodge theory, complex geometry, and representation theory, Contemp. Math., 608, American Mathematical Society, Providence, RI, 2014. Google Scholar | DOI

[Sc] [Sc] Schmid, W., Variation of Hodge structure: the singularities of the period mapping. Invent. Math. 22(1973), 211–319. Google Scholar | DOI