|
Twisted homogeneous coordinate rings of abelian surfaces via mirror symmetry
Author(s):
Marco
Aldi
Journal:
Proc. Amer. Math. Soc.
137
(2009),
2741-2747.
MSC (2000):
Primary 53D12;
Secondary 14A22
Posted:
February 11, 2009
Retrieve article in:
PDF
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  . Moreover, we describe a family of embeddings of a given abelian surface into noncommutative projective spaces.
References:
-
- 1.
- M. Abouzaid, Homogeneous coordinate rings and mirror symmetry for toric varieties, Geom. Topol. 10 (2006), 1097-1157 (electronic). MR 2240909 (2007h:14052)
- 2.
- M. Aldi, E. Zaslow, Seidel's mirror map for abelian varieties, Adv. Theor. Math. Phys. 10 (2006), no. 4, 591-602. MR 2259690 (2007d:14074)
- 3.
- M. Artin, J. Tate and M. Van den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift, vol. 1, Birkhäuser, Boston, 1990, 33-85. MR 1086882 (92e:14002)
- 4.
- Ch. Birkenhake and H. Lange, Complex Tori, Progress in Mathematics 177, Birkhäuser, Boston, 1999. MR 1713785 (2001k:14085)
- 5.
- I.V. Cherednik, Some
 -matrices connected with abelian varieties, Soviet Phys. Dokl. 24 (1979), no. 12, 974-976. MR 565088 (81d:14013) - 6.
- M.R. Gonzalez-Dorrego,
 Configurations and geometry of Kummer surfaces in  , Memoirs of the AMS, vol. 107, no. 512 (1994). MR 1182682 (94f:14034) - 7.
- R. Hudson, Kummer's Quartic Surface, Cambridge Univ. Press, 1905.
- 8.
- A. Kapustin, D. Orlov, Lectures on mirror symmetry, derived categories, and D-branes. Russian Math. Surveys 59 (2004), no. 5, 907-940. MR 2125928 (2006a:14068)
- 9.
- M. Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Zurich, 1994). MR 1403918 (97f:32040)
- 10.
- E.K. Sklyanin, Some algebraic structures connected with the Yang-Baxter equation, Functional Anal. Appl. 16 (1982), no. 4, 263-270. MR 684124 (84c:82004)
- 11.
- S.P. Smith, The
 -dimensional Sklyanin algebras, Proceedings of the Conference on Algebraic Geometry and Ring Theory in Honor of Mike Artin, Part I (Antwerp, 1992),  -theory 8 (1994), no. 1, 65-80. MR 1273836 (95i:16016) - 12.
- J. T. Stafford, M. Van den Bergh, Noncommutative curves and noncommutative surfaces, Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 2, 171-216. MR 1816070 (2002d:16036)
- 13.
- E. Zaslow, Seidel's mirror map for the torus, Adv. Theor. Math. Phys. 9 (2005), no. 6, 120-139. MR 2207572 (2006j:14055)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
53D12,
14A22
Retrieve articles in all Journals with MSC
(2000):
53D12,
14A22
Additional Information:
Marco
Aldi
Affiliation:
Department of Mathematics, University of California, Berkeley, 970 Evans Hall \#3840, Berkeley, California 94720-3840
DOI:
10.1090/S0002-9939-09-09817-7
PII:
S 0002-9939(09)09817-7
Received by editor(s):
October 19, 2006,
Received by editor(s) in revised form:
October 27, 2008
Posted:
February 11, 2009
Additional Notes:
This work was partially supported by NSF grant DMS-0072508
Communicated by:
Ted Chinburg
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org