Mirror quintics, discrete symmetries and Shioda maps
Authors:
Gilberto Bini, Bert van Geemen and Tyler L. Kelly
Journal:
J. Algebraic Geom. 21 (2012), 401-412
DOI:
https://doi.org/10.1090/S1056-3911-2011-00544-4
Published electronically:
May 11, 2011
MathSciNet review:
2914798
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Abstract |
References |
Additional Information
Abstract: In a recent paper, Doran, Greene and Judes considered one parameter families of quintic threefolds with finite symmetry groups. A surprising result was that each of these six families has the same Picard–Fuchs equation associated to the holomorphic $3$-form. In this paper we give an easy argument, involving the family of Mirror Quintics, which implies this result. Using a construction due to Shioda, we also relate certain quotients of these one-parameter families to the family of Mirror Quintics. Our constructions generalize to degree $n$ Calabi–Yau varieties in $(n-1)$-dimensional projective space.
References
- G. Bini, Quotients of Hypersurfaces in Weighted Projective Space, eprint arXiv:0905.2099.
- Charles Doran, Brian Greene, and Simon Judes, Families of quintic Calabi-Yau 3-folds with discrete symmetries, Comm. Math. Phys. 280 (2008), no. 3, 675–725. MR 2399610, DOI https://doi.org/10.1007/s00220-008-0473-x
- B. R. Greene and M. R. Plesser, Duality in Calabi-Yau moduli space, Nuclear Phys. B 338 (1990), no. 1, 15–37. MR 1059831, DOI https://doi.org/10.1016/0550-3213%2890%2990622-K
- B. R. Greene, M. R. Plesser, and S.-S. Roan, New constructions of mirror manifolds: probing moduli space far from Fermat points, Essays on mirror manifolds, Int. Press, Hong Kong, 1992, pp. 408–448. MR 1191435
- M. Harris, N. Shepherd-Barron, R. Taylor, A family of Calabi–Yau varieties and potential automorphy, available on: http://www.math.harvard.edu/$\sim$rtaylor/ .
- Tetsuji Shioda, An explicit algorithm for computing the Picard number of certain algebraic surfaces, Amer. J. Math. 108 (1986), no. 2, 415–432. MR 833362, DOI https://doi.org/10.2307/2374678
References
- G. Bini, Quotients of Hypersurfaces in Weighted Projective Space, eprint arXiv:0905.2099.
- C. Doran, B. Greene, S. Judes, Families of quintic Calabi–Yau 3-folds with discrete symmetries, Comm. Math. Phys. 280 (2008) 675–725. MR 2399610 (2009g:14044)
- B.R. Greene, M.R. Plesser, Duality in Calabi–Yau moduli space, Nucl. Phys. B 338 (1990) 15–37. MR 1059831 (91h:32018)
- B.R. Greene, M.R. Plesser, S.S. Roan, New constructions of mirror manifolds: Probing moduli space far from Fermat points, in: Essays on Mirror Manifolds, editor S-T Yau, International Press 1992, 408–450. MR 1191435 (94c:32018)
- M. Harris, N. Shepherd-Barron, R. Taylor, A family of Calabi–Yau varieties and potential automorphy, available on: http://www.math.harvard.edu/$\sim$rtaylor/ .
- T. Shioda, An explicit algorithm for computing the Picard number of certain algebraic surfaces, Amer. J. Math. 108 (1986) 415–432. MR 833362 (87g:14033)
Additional Information
Gilberto Bini
Affiliation:
Dipartimento di Matematica, Università di Milano, Via Saldini 50, I-20133 Milano, Italia
Email:
gilberto.bini@unimi.it
Bert van Geemen
Affiliation:
Dipartimento di Matematica, Università di Milano, Via Saldini 50, I-20133 Milano, Italia
MR Author ID:
214021
Email:
lambertus.vangeemen@unimi.it
Tyler L. Kelly
Affiliation:
Department of Mathematics, University of Pennsylvania, David Rittenhouse Lab., 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395
MR Author ID:
874289
Email:
tykelly@math.upenn.edu
Received by editor(s):
May 19, 2009
Received by editor(s) in revised form:
July 17, 2009
Published electronically:
May 11, 2011