Complex structures on product of circle bundles over complex manifolds

TL;DR

This paper constructs various complex structures on products of circle bundles over complex manifolds, demonstrating non-Kähler examples and analyzing their cohomological and function-theoretic properties.

Contribution

It introduces scalar, diagonal, and linear type complex structures on circle bundle products, expanding the understanding of non-Kähler manifolds and their algebraic properties.

Findings

Scalar type structures always exist.

Certain diagonal structures require equivariant bundle conditions.

Linear structures over flag varieties lead to purely transcendental meromorphic functions.

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-K\"ahler 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 $\bc$.