Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Noncommutative geometry of algebraic curves

Author: Igor V. Nikolaev
Journal: Proc. Amer. Math. Soc. 137 (2009), 3283-3290
MSC (2000): Primary 14H10, 46L40, 58F10
Published electronically: May 7, 2009
MathSciNet review: 2515397
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A covariant functor from the category of generic complex algebraic curves to a category of the $ AF$-algebras is constructed. The construction is based on a representation of the Teichmüller space of a curve by the measured foliations due to Douady, Hubbard, Masur and Thurston. The functor maps isomorphic algebraic curves to the stably isomorphic $ AF$-algebras.

References [Enhancements On Off] (What's this?)

  • 1. M. Bauer, A characterization of uniquely ergodic interval exchange maps in terms of the Jacobi-Perron algorithm, Bol. Soc. Bras. Mat. 27 (1996), 109-128. MR 1418928 (98a:58100)
  • 2. L. Bernstein, The Jacobi-Perron Algorithm, Its Theory and Applications, Lect. Notes in Math., 207, Springer, 1971. MR 0285478 (44:2696)
  • 3. A. Douady and J. Hubbard, On the density of Strebel differentials, Inventiones Math. 30 (1975), 175-179. MR 0396936 (53:796)
  • 4. E. G. Effros, Dimensions and $ C^*$-Algebras, Conf. Board of the Math. Sciences Regional Conference Series in Math., No. 46, Amer. Math. Soc., 1981. MR 623762 (84k:46042)
  • 5. E. G. Effros and C. L. Shen, Approximately finite $ C^*$-algebras and continued fractions, Indiana Univ. Math. J. 29 (1980), 191-204. MR 563206 (81g:46076)
  • 6. G. A. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra 38 (1976), 29-44. MR 0397420 (53:1279)
  • 7. J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), 221-274. MR 523212 (80h:30047)
  • 8. M. Kontsevich, XI Solomon Lefschetz Memorial Lecture Series: Hodge structures in non-commutative geometry (Notes by Ernesto Lupercio), in Contemp. Mathematics, 462, Amer. Math. Soc., 2008. MR 2444365
  • 9. Yu. I. Manin, Real multiplication and noncommutative geometry, in ``Legacy of Niels Hendrik Abel'', 685-727, Springer, 2004. MR 2077591 (2006e:11077)
  • 10. H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2) 115 (1982), no. 1, 169-200. MR 644018 (83e:28012)
  • 11. A. Polishchuk and A. Schwarz, Categories of holomorphic vector bundles on noncommutative two-tori, Commun. Math. Phys. 236 (2003), 135-159. MR 1977884 (2004k:58011)
  • 12. Y. Soibelman and V.  Vologodsky, Noncommutative compactifications and elliptic curves, Int. Math. Res. Not. (2003), 1549-1569. MR 1976601 (2004i:14017)
  • 13. W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988), 417-431. MR 956596 (89k:57023)
  • 14. W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2) 115 (1982), no. 1, 201-242. MR 644019 (83g:28036b)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 14H10, 46L40, 58F10

Retrieve articles in all journals with MSC (2000): 14H10, 46L40, 58F10

Additional Information

Igor V. Nikolaev
Affiliation: The Fields Institute for Mathematical Sciences, Toronto, Ontario, M5T 3J1, Canada
Address at time of publication: 101–315 Holmwood Avenue, Ottawa, Ontario, K1S 2R2, Canada

Keywords: Complex algebraic curves, $C^*$-algebras
Received by editor(s): September 5, 2008
Received by editor(s) in revised form: February 13, 2009
Published electronically: May 7, 2009
Additional Notes: The author was partially supported by NSERC
Communicated by: Varghese Mathai
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society