Twisted homogeneous coordinate rings of abelian surfaces via mirror symmetry

Aldi, Marco

doi:10.1090/S0002-9939-09-09817-7

Twisted homogeneous coordinate rings of abelian surfaces via mirror symmetry

Author: Marco Aldi
Journal: Proc. Amer. Math. Soc. 137 (2009), 2741-2747
MSC (2000): Primary 53D12; Secondary 14A22
DOI: https://doi.org/10.1090/S0002-9939-09-09817-7
Published electronically: February 11, 2009
MathSciNet review: 2497487
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study Seidel's mirror map for abelian and Kummer surfaces. We find that mirror symmetry leads in a very natural way to the classical parametrization of Kummer surfaces in $\mathbb{P}^3$ . Moreover, we describe a family of embeddings of a given abelian surface into noncommutative projective spaces.

1. Mohammed Abouzaid, Homogeneous coordinate rings and mirror symmetry for toric varieties, Geom. Topol. 10 (2006), 1097–1156. [Paging previously given as 1097–1157]. MR 2240909, https://doi.org/10.2140/gt.2006.10.1097
2. Marco Aldi and Eric Zaslow, Seidel’s mirror map for abelian varieties, Adv. Theor. Math. Phys. 10 (2006), no. 4, 591–602. MR 2259690
3. M. Artin, J. Tate, and M. Van den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 33–85. MR 1086882
4. Christina Birkenhake and Herbert Lange, Complex tori, Progress in Mathematics, vol. 177, Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1713785
5. I. V. Čerednik, Some 𝑆-matrices connected with abelian varieties, Dokl. Akad. Nauk SSSR 249 (1979), no. 5, 1095–1098 (Russian). MR 565088
6. Maria R. Gonzalez-Dorrego, (16,6) configurations and geometry of Kummer surfaces in 𝑃³, Mem. Amer. Math. Soc. 107 (1994), no. 512, vi+101. MR 1182682, https://doi.org/10.1090/memo/0512
7. R. Hudson, Kummer's Quartic Surface, Cambridge Univ. Press, 1905.
8. A. N. Kapustin and D. O. Orlov, Lectures on mirror symmetry, derived categories, and D-branes, Uspekhi Mat. Nauk 59 (2004), no. 5(359), 101–134 (Russian, with Russian summary); English transl., Russian Math. Surveys 59 (2004), no. 5, 907–940. MR 2125928, https://doi.org/10.1070/RM2004v059n05ABEH000772
9. Maxim Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 120–139. MR 1403918
10. E. K. Sklyanin, Some algebraic structures connected with the Yang-Baxter equation, Funktsional. Anal. i Prilozhen. 16 (1982), no. 4, 27–34, 96 (Russian). MR 684124
11. S. P. Smith, The four-dimensional Sklyanin algebras, Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part I (Antwerp, 1992), 1994, pp. 65–80. MR 1273836, https://doi.org/10.1007/BF00962090
12. J. T. Stafford and M. van den Bergh, Noncommutative curves and noncommutative surfaces, Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 2, 171–216. MR 1816070, https://doi.org/10.1090/S0273-0979-01-00894-1
13. Eric Zaslow, Seidel’s mirror map for the torus, Adv. Theor. Math. Phys. 9 (2005), no. 6, 999–1006. MR 2207572