Complex structures on product of circle bundles over complex manifolds

Abstract

Let $\bar{L}_i\lr X_i$ 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 {\em 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 $(\bc^*)^{n_i}$-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 $H^1(X_1;\br)=0$ and $c_1(\bar{L}_1)\in H^2(X_1;\br)$ is non-zero, the compact manifold $S$ does not admit any symplectic structure and hence it is non-Kähler with respect to {\em any} complex structure. We obtain a vanishing theorem for $H^q(S;{\mathcal{O}}_S)$ when $X_i$ are projective manifolds, ${\bar{L}}_i^{\vee }$ are very ample and the cone over $X_i$ with respect to the projective imbedding defined by ${\bar{L}}_i^{\vee }$ 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 $\bc$.

29 pages, To appear in Annales de L'Institut Fourier

Cited by in corpus (2)

• Group actions, non-Kähler complex manifolds and SKT structures
• Hermitian non-Kähler structures on products of principal $S^1$-bundles over complex flag manifolds and applications in Hermitian geometry with torsion