Noncommutative geometry of algebraic curves
Author:
Igor V. Nikolaev
Journal:
Proc. Amer. Math. Soc. 137 (2009), 32833290
MSC (2000):
Primary 14H10, 46L40, 58F10
Published electronically:
May 7, 2009
MathSciNet review:
2515397
Fulltext 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 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 algebras.
 1.
Max
Bauer, A characterization of uniquely ergodic interval exchange
maps in terms of the JacobiPerron algorithm, Bol. Soc. Brasil. Mat.
(N.S.) 27 (1996), no. 2, 109–128. MR 1418928
(98a:58100), http://dx.doi.org/10.1007/BF01259355
 2.
Leon
Bernstein, The JacobiPerron algorithm—Its theory and
application, Lecture Notes in Mathematics, Vol. 207, SpringerVerlag,
BerlinNew York, 1971. MR 0285478
(44 #2696)
 3.
A.
Douady and J.
Hubbard, On the density of Strebel differentials, Invent.
Math. 30 (1975), no. 2, 175–179. MR 0396936
(53 #796)
 4.
Edward
G. Effros, Dimensions and 𝐶*algebras, CBMS Regional
Conference Series in Mathematics, vol. 46, Conference Board of the
Mathematical Sciences, Washington, D.C., 1981. MR 623762
(84k:46042)
 5.
Edward
G. Effros and Chao
Liang Shen, Approximately finite 𝐶*algebras and continued
fractions, Indiana Univ. Math. J. 29 (1980),
no. 2, 191–204. MR 563206
(81g:46076), http://dx.doi.org/10.1512/iumj.1980.29.29013
 6.
George
A. Elliott, On the classification of inductive limits of sequences
of semisimple finitedimensional algebras, J. Algebra
38 (1976), no. 1, 29–44. MR 0397420
(53 #1279)
 7.
John
Hubbard and Howard
Masur, Quadratic differentials and foliations, Acta Math.
142 (1979), no. 34, 221–274. MR 523212
(80h:30047), http://dx.doi.org/10.1007/BF02395062
 8.
Maxim
Kontsevich, XI Solomon Lefschetz Memorial Lecture series: Hodge
structures in noncommutative geometry, Noncommutative geometry in
mathematics and physics, Contemp. Math., vol. 462, Amer. Math. Soc.,
Providence, RI, 2008, pp. 1–21. Notes by Ernesto Lupercio. MR 2444365
(2009m:53236), http://dx.doi.org/10.1090/conm/462/09058
 9.
Yu.
I. Manin, Real multiplication and noncommutative geometry (ein
Alterstraum), The legacy of Niels Henrik Abel, Springer, Berlin,
2004, pp. 685–727. MR 2077591
(2006e:11077)
 10.
Howard
Masur, Interval exchange transformations and measured
foliations, Ann. of Math. (2) 115 (1982), no. 1,
169–200. MR
644018 (83e:28012), http://dx.doi.org/10.2307/1971341
 11.
A.
Polishchuk and A.
Schwarz, Categories of holomorphic vector bundles on noncommutative
twotori, Comm. Math. Phys. 236 (2003), no. 1,
135–159. MR 1977884
(2004k:58011), http://dx.doi.org/10.1007/s0022000308139
 12.
Yan
Soibelman and Vadim
Vologodsky, Noncommutative compactifications and elliptic
curves, Int. Math. Res. Not. 28 (2003),
1549–1569. MR 1976601
(2004i:14017), http://dx.doi.org/10.1155/S1073792803205080
 13.
William
P. Thurston, On the geometry and dynamics of
diffeomorphisms of surfaces, Bull. Amer. Math.
Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596
(89k:57023), http://dx.doi.org/10.1090/S027309791988156856
 14.
William
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), http://dx.doi.org/10.2307/1971391
 1.
 M. Bauer, A characterization of uniquely ergodic interval exchange maps in terms of the JacobiPerron algorithm, Bol. Soc. Bras. Mat. 27 (1996), 109128. MR 1418928 (98a:58100)
 2.
 L. Bernstein, The JacobiPerron 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), 175179. MR 0396936 (53:796)
 4.
 E. G. Effros, Dimensions and 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 algebras and continued fractions, Indiana Univ. Math. J. 29 (1980), 191204. MR 563206 (81g:46076)
 6.
 G. A. Elliott, On the classification of inductive limits of sequences of semisimple finitedimensional algebras, J. Algebra 38 (1976), 2944. MR 0397420 (53:1279)
 7.
 J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), 221274. MR 523212 (80h:30047)
 8.
 M. Kontsevich, XI Solomon Lefschetz Memorial Lecture Series: Hodge structures in noncommutative 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'', 685727, Springer, 2004. MR 2077591 (2006e:11077)
 10.
 H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2) 115 (1982), no. 1, 169200. MR 644018 (83e:28012)
 11.
 A. Polishchuk and A. Schwarz, Categories of holomorphic vector bundles on noncommutative twotori, Commun. Math. Phys. 236 (2003), 135159. MR 1977884 (2004k:58011)
 12.
 Y. Soibelman and V. Vologodsky, Noncommutative compactifications and elliptic curves, Int. Math. Res. Not. (2003), 15491569. MR 1976601 (2004i:14017)
 13.
 W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988), 417431. 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, 201242. 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
Email:
igor.v.nikolaev@gmail.com
DOI:
http://dx.doi.org/10.1090/S0002993909099171
PII:
S 00029939(09)099171
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.
