Complex structures on product of circle bundles over complex manifolds

Sankaran, Parameswaran; Thakur, Ajay Singh

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

Soient ${\overline{L}}_i\to X_i$ 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 ${\overline{L}}_i$. 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 ${\overline{L}}_i$ sont des ${\left({ℂ}^{*}\right)}^{n_i}$ 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 $c_1\left({\overline{L}}_1\right)$ 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 ${\overline{L}}_i^{\vee }$ son très amples et le cône sur $X_i$ par rapport au plongement projectif défini par ${\overline{L}}_i^{\vee }$ 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 ${\overline{L}}_i\to X_i$ 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 ${\overline{L}}_i$. 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 ${\overline{L}}_i$ are equivariant ${\left({ℂ}^{*}\right)}^{n_i}$-bundles satisfying some additional conditions. The linear type complex structures are constructed assuming $X_i$ are (generalized) flag varieties and ${\overline{L}}_i$ negative ample line bundles over $X_i$. When $H^1(X_1;ℝ)=0$ and $c_1\left({\overline{L}}_1\right)\in H^2(X_1;ℂ)$ 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, ${\overline{L}}_i^{\vee }$ are very ample and the cone over $X_i$ with respect to the projective imbedding defined by ${\overline{L}}_i^{\vee }$ 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 $ℂ$.

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