no. 1
On cotangent manifolds, complex structures and generalized geometry
David, Liana1
1 “Simion Stoilow” Institute of Mathematics of the Romanian Academy Research Unit no. 4 Calea Grivitei 21 Sector 1, Bucharest (Romania)
Annales de l'Institut Fourier, Volume 66 (2016) no. 1, pp. 1-28.

We develop various properties of symmetric generalized complex structures (in connection with their holomorphic space and B-field transformations), which are analogous to the well-known results of Gualtieri on skew-symmetric generalized complex structures. Given an adapted (symmetric or skew-symmetric) generalized complex structure 𝒥 and a linear connection D on a manifold M, we construct an almost complex structure J 𝒥,D on the cotangent manifold T * M and we study its integrability. For 𝒥 skew-symmetric, we relate the Courant integrability of 𝒥 with the integrability of J 𝒥,D . We consider in detail the case when M=G is a Lie group and 𝒥, D are left-invariant. We also show that our approach unifies and generalizes various known results from special complex geometry.

Nous développons plusieurs propriétés des structures complexes généralisées symétriques (en relation avec leurs espaces holomorphes et les “B-field” transformations), qui sont similaires aux résultats bien connus de Gualtieri sur les structures complexes généralisées anti-symétriques. Étant données une structure complexe généralisée adaptée (symétrique ou anti-symétrique) 𝒥 et une connexion D sur une variété M, nous construisons une structure presque-complexe J 𝒥,D sur la variété cotangente T * M et nous étudions son intégrabilité. Pour 𝒥 anti-symétrique, nous relions l’intégrabilité de 𝒥 au sens de Courant avec l’intégrabilité de J 𝒥,D . Nous considérons en détail le cas où M=G est un groupe de Lie et 𝒥,D sont invariants à gauche. Nous montrons aussi que notre approche unifie et généralise plusieurs résultats connus en géométrie spéciale complexe.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3003
Classification: 53C15, 53D18, 53C55
Keywords: complex and generalized complex structures, holomorphic bundles, integrability, Lie groups, special complex geometry.
David, Liana 1

1 “Simion Stoilow” Institute of Mathematics of the Romanian Academy Research Unit no. 4 Calea Grivitei 21 Sector 1, Bucharest (Romania) 
@article{AIF_2016__66_1_1_0,
     author = {David, Liana},
     title = {On cotangent manifolds, complex structures and generalized geometry},
     journal = {Annales de l'Institut Fourier},
     pages = {1--28},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {66},
     number = {1},
     year = {2016},
     doi = {10.5802/aif.3003},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3003/}
}
TY  - JOUR
AU  - David, Liana
TI  - On cotangent manifolds, complex structures and generalized geometry
JO  - Annales de l'Institut Fourier
PY  - 2016
SP  - 1
EP  - 28
VL  - 66
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3003/
UR  - https://doi.org/10.5802/aif.3003
DO  - 10.5802/aif.3003
LA  - en
ID  - AIF_2016__66_1_1_0
ER  - 
%0 Journal Article
%A David, Liana
%T On cotangent manifolds, complex structures and generalized geometry
%J Annales de l'Institut Fourier
%D 2016
%P 1-28
%V 66
%N 1
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.3003
%R 10.5802/aif.3003
%G en
%F AIF_2016__66_1_1_0
David, Liana. On cotangent manifolds, complex structures and generalized geometry. Annales de l'Institut Fourier, Volume 66 (2016) no. 1, pp. 1-28. doi : 10.5802/aif.3003. https://aif.centre-mersenne.org/articles/10.5802/aif.3003/

[1] Alekseevsky, D. V.; Cortés, V.; Devchand, C. Special complex manifolds, J. Geom. Phys., Volume 42 (2002) no. 1-2, pp. 85-105 | DOI

[2] Alekseevsky, Dmitri V.; David, Liana Invariant generalized complex structures on Lie groups, Proc. Lond. Math. Soc. (3), Volume 105 (2012) no. 4, pp. 703-729 | DOI

[3] Bourbaki, Nicolas Lie Groups and Lie Algebras. Chapters 4–6, Elements of Math. (Berlin), Springer-Verlag, New York, 2002, xii+300 pages

[4] Gauduchon, Paul Hermitian connections and Dirac operators, Boll. Un. Mat. Ital. B (7), Volume 11 (1997) no. 2, suppl., pp. 257-288

[5] Gray, Alfred; Hervella, Luis M. The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. (4), Volume 123 (1980), pp. 35-58 | DOI

[6] Gualtieri, Marco Generalized Complex Geometry, University of Oxford (2003) (Ph. D. Thesis)

[7] Helgason, Sigurdur Differential Geometry and Symmetric Spaces, Graduate Studies in Mathematics, 34, AMS Chelsea Publishing, American Mathematical Society, Providence, Rhode Island, 2001

[8] Hitchin, Nigel Generalized Calabi-Yau manifolds, Q. J. Math., Volume 54 (2003) no. 3, pp. 281-308 | DOI

[9] Knapp, Anthony W. Lie Groups Beyond an Introduction, Progress in Mathematics, 140, Birkhäuser Boston, Inc., Boston, MA, 1996, xvi+604 pages | DOI

[10] Nannicini, Antonella Almost complex structures on cotangent bundles and generalized geometry, J. Geom. Phys., Volume 60 (2010) no. 11, pp. 1781-1791 | DOI

[11] Onishchik, Arkadij L.; Vinberg, Ernest B Lie Groups and Lie Algebras III, Encyclopaedia of Mathematical Sciences, 41, Springer-Verlag, Berlin, 1994, iv+248 pages

Cited by Sources: