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
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by Marco Aldi

Proc. Amer. Math. Soc. 137 (2009), 2741-2747

DOI: https://doi.org/10.1090/S0002-9939-09-09817-7

Published electronically: February 11, 2009

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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.

References

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, DOI 10.2140/gt.2006.10.1097
Marco Aldi and Eric Zaslow, Seidel’s mirror map for abelian varieties, Adv. Theor. Math. Phys. 10 (2006), no. 4, 591–602. MR 2259690
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
Christina Birkenhake and Herbert Lange, Complex tori, Progress in Mathematics, vol. 177, Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1713785, DOI 10.1007/978-1-4612-1566-0
I. V. Čerednik, Some $S$-matrices connected with abelian varieties, Dokl. Akad. Nauk SSSR 249 (1979), no. 5, 1095–1098 (Russian). MR 565088
Maria R. Gonzalez-Dorrego, $(16,6)$ configurations and geometry of Kummer surfaces in $\textbf {P}^3$, Mem. Amer. Math. Soc. 107 (1994), no. 512, vi+101. MR 1182682, DOI 10.1090/memo/0512
R. Hudson, Kummer’s Quartic Surface, Cambridge Univ. Press, 1905.
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, DOI 10.1070/RM2004v059n05ABEH000772
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
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
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, DOI 10.1007/BF00962090
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, DOI 10.1090/S0273-0979-01-00894-1
Eric Zaslow, Seidel’s mirror map for the torus, Adv. Theor. Math. Phys. 9 (2005), no. 6, 999–1006. MR 2207572