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.

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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.
Mot clés : structures complexes et structures complexes generalisées, fibrés holomorphes, integrabilité, groups de Lie, géométrie spéciale complexe
David, Liana 1

1 “Simion Stoilow” Institute of Mathematics of the Romanian Academy Research Unit no. 4 Calea Grivitei 21 Sector 1, Bucharest (Romania) 
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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/

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