Complex structures on product of circle bundles over complex manifolds
Annales de l'Institut Fourier, Volume 63 (2013) no. 4, p. 1331-1366

Let L ¯ i X i be a holomorphic line bundle over a compact complex manifold for i=1,2. Let S i denote the associated principal circle-bundle with respect to some hermitian inner product on L ¯ i . We construct complex structures on S=S 1 ×S 2 which we refer to as scalar, diagonal, and linear types. While scalar type structures always exist, the more general diagonal but non-scalar type structures are constructed assuming that L ¯ i are equivariant ( * ) n i -bundles satisfying some additional conditions. The linear type complex structures are constructed assuming X i are (generalized) flag varieties and L ¯ i negative ample line bundles over X i . When H 1 (X 1 ;)=0 and c 1 (L ¯ 1 )H 2 (X 1 ;) is non-zero, the compact manifold S does not admit any symplectic structure and hence it is non-Kähler with respect to any complex structure.

We obtain a vanishing theorem for H q (S;𝒪 S ) when X i are projective manifolds, L ¯ i are very ample and the cone over X i with respect to the projective imbedding defined by L ¯ i are Cohen-Macaulay. We obtain applications to the Picard group of S. When X i =G i /P i where P i are maximal parabolic subgroups and S is endowed with linear type complex structure with “vanishing unipotent part” we show that the field of meromorphic functions on S is purely transcendental over .

Soient L ¯ i X i des fibrés en droites holomorphes sur des variétés complexes compactes, pour i=1,2. Soit S i le fibré en cercles associé par rapport à un produit scalaire hermitienne sur L ¯ i . On construit des structures complexes sur S=S 1 ×S 2 dites de type scalaire, diagonal, ou linéaire. Bien que des structures de type scalaire existent toujours, on construit des structures plus générales de type diagonal mais non-scalaire dans le cas où les L ¯ i sont des ( * ) n i fibrés équivariants qui vérifient certaines hypothèses supplémentaires. Les structures complexes de type linéaire sont des variétés des drapeaux (généralisées) et les L i sont des fibrés en droites amples négatifs. Lorsque H 1 (X 1 ;)=0 et c 1 (L ¯ 1 ) est non-nulle la variété compacte S n’admet pas de structure symplectique et donc elle est non-Kählerienne par rapport à toute structure complexe.

On montre que H q (S;𝒪 S ) s’annule quand les X i sont des variétés projectives, les L ¯ i son très amples et le cône sur X i par rapport au plongement projectif défini par L ¯ i sont Cohen-Macaulay. On applique ces résultats au groupe de Picard de S. Quand X i =G i /P i P i sont les sousgroupes paraboliques maximaux et la variété S est munie d’une structure complexe du type linéaire avec «  la partie unipotente nulle  » on montre que le corps des fonctions méromorphes sur S est purement transcendental sur .

DOI : https://doi.org/10.5802/aif.2805
Classification:  32L05,  32J18,  32Q55
Keywords: circle bundles, complex manifolds, homogeneous spaces, Picard groups, meromorphic function fields
@article{AIF_2013__63_4_1331_0,
     author = {Sankaran, Parameswaran and Thakur, Ajay Singh},
     title = {Complex structures on product of circle bundles over complex manifolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {63},
     number = {4},
     year = {2013},
     pages = {1331-1366},
     doi = {10.5802/aif.2805},
     mrnumber = {3137357},
     zbl = {06359591},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2013__63_4_1331_0}
}
Sankaran, Parameswaran; Thakur, Ajay Singh. Complex structures on product of circle bundles over complex manifolds. Annales de l'Institut Fourier, Volume 63 (2013) no. 4, pp. 1331-1366. doi : 10.5802/aif.2805. http://www.numdam.org/item/AIF_2013__63_4_1331_0/

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