| Title:
|
Invariants of complex structures on nilmanifolds (English) |
| Author:
|
Rodríguez Valencia, Edwin Alejandro |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
51 |
| Issue:
|
1 |
| Year:
|
2015 |
| Pages:
|
27-50 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $(N, J)$ be a simply connected $2n$-dimensional nilpotent Lie group endowed with an invariant complex structure. We define a left invariant Riemannian metric on $N$ compatible with $J$ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. In [7], J. Lauret proved that minimal metrics (if any) are unique up to isometry and scaling. This uniqueness allows us to distinguish two complex structures with Riemannian data, giving rise to a great deal of invariants. We show how to use a Riemannian invariant: the eigenvalues of the Ricci operator, polynomial invariants and discrete invariants to give an alternative proof of the pairwise non-isomorphism between the structures which have appeared in the classification of abelian complex structures on 6-dimensional nilpotent Lie algebras given in [1]. We also present some continuous families in dimension 8. (English) |
| Keyword:
|
complex |
| Keyword:
|
nilmanifolds |
| Keyword:
|
nilpotent Lie groups |
| Keyword:
|
minimal metrics |
| Keyword:
|
Pfaffian forms |
| MSC:
|
22E25 |
| MSC:
|
32Q60 |
| MSC:
|
37J15 |
| MSC:
|
53C15 |
| MSC:
|
53C30 |
| idZBL:
|
Zbl 06487019 |
| idMR:
|
MR3338764 |
| DOI:
|
10.5817/AM2015-1-27 |
| . |
| Date available:
|
2015-04-01T12:50:59Z |
| Last updated:
|
2016-04-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144232 |
| . |
| Reference:
|
[1] Andrada, A., Barberis, M.L., Dotti, I.G.: Classification of abelian complex structures on 6-dimensional Lie algebras.J. London Math. Soc. 83 (1) (2011), 232–255. Zbl 1218.17006, MR 2763953, 10.1112/jlms/jdq071 |
| Reference:
|
[2] Ceballos, M., Otal, A., Ugarte, L., Villacampa, R.: Classification of complex structures on 6-dimensional nilpotent Lie algebras.arXiv:math.DG/1111.5873. |
| Reference:
|
[3] Cordero, L.A., Fernández, M., Ugarte, L.: Abelian complex structures on 6-dimensional compact nilmanifolds.Comment. Math. Univ. Carolin. 43 (2) (2002), 215–229. Zbl 1078.53020, MR 1922123 |
| Reference:
|
[4] Dolgachev, I.: Lectures on invariant theory.London Math. Soc. Lecture Note Ser. 296 (2003), 1–220. Zbl 1023.13006, MR 2004511 |
| Reference:
|
[5] Galitski, L.Yu., Timashev, D.A.: On classification of metabelian Lie algebras.J. Lie Theory 9 (1999), 125–156. Zbl 0923.17015, MR 1680007 |
| Reference:
|
[6] Lauret, J.: Minimal metrics on nilmanifolds.Differential geometry and its applications, Matfyzpress, Prague, 2005, pp. 79–97. Zbl 1113.53033, MR 2268923 |
| Reference:
|
[7] Lauret, J.: A canonical compatible metric for geometric structures on nilmanifolds.Ann. Global Anal. Geom. 30 (2006), 107–138. Zbl 1102.53021, MR 2234091, 10.1007/s10455-006-9015-y |
| Reference:
|
[8] Lauret, J.: Rational forms of nilpotent Lie algebras and Anosov diffeomorphisms.Monatsh. Math. 155 (2008), 15–30. Zbl 1153.22008, MR 2434923, 10.1007/s00605-008-0562-0 |
| Reference:
|
[9] Lauret, J.: Einstein solvmanifolds and nilsolitons.Contemp. Math. 491 (2009), 1–35. Zbl 1186.53058, MR 2537049, 10.1090/conm/491/09607 |
| Reference:
|
[10] Lauret, J.: Einstein solvmanifolds are standard.Ann. of Math. (2) 172 (2010), 1859–1877. Zbl 1220.53061, MR 2726101 |
| Reference:
|
[11] Salamon, S.M.: Complex structures on nilpotent Lie algebras.J. Pure Appl. Algebra 157 (2001), 311–333. Zbl 1020.17006, MR 1812058, 10.1016/S0022-4049(00)00033-5 |
| . |
