Complex structures on product of circle bundles over complex manifolds

Sankaran, Parameswaran; Thakur, Ajay Singh

doi:10.5802/aif.2805

Complex structures on product of circle bundles over complex manifolds
[Structures complexes sur les produit de fibrés en cercles au-dessus des variétés complexes]

Sankaran, Parameswaran ¹ ; Thakur, Ajay Singh ²

¹ The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India
² Indian Statistical Institute, 8th Mile, Mysore Road, RVCE Post Bangalore 560059, India

Annales de l'Institut Fourier, Tome 63 (2013) no. 4, pp. 1331-1366.

Résumé
Abstract

Soient ${\bar{L}}_{i} \to X_{i}$ 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 ${\bar{L}}_{i}$ . On construit des structures complexes sur $S = S_{1} \times S_{2}$ 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 ${\bar{L}}_{i}$ sont des ${(ℂ^{*})}^{n_{i}}$ 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 $H^{1} (X_{1}; ℝ) = 0$ et $c_{1} ({\bar{L}}_{1})$ 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 ${\bar{L}}_{i}^{\lor}$ son très amples et le cône sur $X_{i}$ par rapport au plongement projectif défini par ${\bar{L}}_{i}^{\lor}$ sont Cohen-Macaulay. On applique ces résultats au groupe de Picard de $S$ . Quand $X_{i} = G_{i} / P_{i}$ où $P_{i}$ 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 $ℂ$ .

Let ${\bar{L}}_{i} \to X_{i}$ 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 ${\bar{L}}_{i}$ . We construct complex structures on $S = S_{1} \times S_{2}$ 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 ${\bar{L}}_{i}$ are equivariant ${(ℂ^{*})}^{n_{i}}$ -bundles satisfying some additional conditions. The linear type complex structures are constructed assuming $X_{i}$ are (generalized) flag varieties and ${\bar{L}}_{i}$ negative ample line bundles over $X_{i}$ . When $H^{1} (X_{1}; ℝ) = 0$ and $c_{1} ({\bar{L}}_{1}) \in H^{2} (X_{1}; ℂ)$ 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, ${\bar{L}}_{i}^{\lor}$ are very ample and the cone over $X_{i}$ with respect to the projective imbedding defined by ${\bar{L}}_{i}^{\lor}$ 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 $ℂ$ .

MR Zbl

DOI : 10.5802/aif.2805

Classification : 32L05, 32J18, 32Q55
Keywords: circle bundles, complex manifolds, homogeneous spaces, Picard groups, meromorphic function fields
Mot clés : fibré en cercles, variétés complexes, espaces homogénes, groupes de Picard, corpses des fonctions meromorphes

Affiliations des auteurs :

Sankaran, Parameswaran ¹ ; Thakur, Ajay Singh ²

¹ The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India
² Indian Statistical Institute, 8th Mile, Mysore Road, RVCE Post Bangalore 560059, India

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     title = {Complex structures on product of circle bundles over complex manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {1331--1366},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {63},
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Sankaran, Parameswaran; Thakur, Ajay Singh. Complex structures on product of circle bundles over complex manifolds. Annales de l'Institut Fourier, Tome 63 (2013) no. 4, pp. 1331-1366. doi : 10.5802/aif.2805. https://aif.centre-mersenne.org/articles/10.5802/aif.2805/

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