Complex structures on product of circle bundles over complex manifolds

Details Complex structures on product of circle bundles over complex manifolds Complex structures on product of circle bundles over complex manifolds

2010-12-03

arxiv.org/abs/1012.0668v1

Mathematics

Complex Variables

26 pages

Scientific paper

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, diagonal 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-K\"ahler with respect to {\em any} complex structure. In the case of diagonal type complex structures on $S$, we determine their Picard groups and the field of meromorphic function when $X_i=G_i/P_i$ where $G_i$ are simple and $P_i$ maximal parabolic subgroups.

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