1COMPLEX STRUCTURES ON PRODUCT OF CIRCLE BUNDLES OVER COMPLEX MANIFOLDS STRUCTURES COMPLEXES SUR LES PRODUIT DE FIBRE ́ EN

Abstract: Let L̄i − → Xi be a holomorphic line bundle over a compact complex manifold for i = 1, 2. Let Si denote the associated principal circle-bundle with respect to some hermitian inner product on L̄i. We construct complex structures on S = S1×S2 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 L̄i are equivariant (C∗)ni-bundles satisfying some additional conditions. The linear type complex structures are constructed assuming Xi are (generalized) flag varieties and L̄i negative ample line bundles over Xi. When H 1(X1;R) = 0 and c1(L̄1) ∈ H2(X1;R) 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 Hq(S;OS) when Xi are projective manifolds, L̄∨i are very ample and the cone over Xi with respect to the projective imbedding defined by L̄ i are Cohen-Macaulay. We obtain applications to the Picard group of S. When Xi = Gi/P