On a Holomorphic Fiber Bundle with Meromorphic Structure

Abstract

Let / : X-+Y be a proper surjective morphism of compact complex manifolds. Let U^Y be a Zariski open subset over which / is smooth. Let Xu=f~ (U) and let fn: Xn^U be the induced morphism. Assume that fn is a holomorphic fiber bundle with typical fiber F and the structure group H. Let Gu-*U be the holomorphic fiber bundle associated with fn with typical fiber H with the adjoint action of H on itself so that Gn acts naturally on Xn over U. Let lu-^U be the principal //-bundle associated to fuThen GU acts naturally on In over U also. We say that fn is a holomorphic fiber bundle with meromorphic structure if there exists a compact complex space G* (resp. /*) over Y containing GU (resp. In) as a Zariski open subset such that the action of Gn on Xn (resp. In) extends 'meromorphically' to that of G* on X (resp. /*). Then in this paper we shall prove the following: Suppose that fu is a holomorphic fiber bundle with meromorphic structure for some G* and /* as above. Then 1) there exists a 'generic quotients' X/G* of X by G* over Y, and 2) X/G* is bimeromorphic to the product space (F/H)xY where F/H is a generic quotient of F by H.* Actually in this paper, these results are obtained in a more general setting of comparing two proper morphisms fi: Xi->Y', i=l, 2, over Y having isomorphic general fibers (cf. Theorems 1 and 2); the above special case corresponds to the case where one of the ft is isomorphic to the projection p: FxY-*Y. (This generalization is in a sense parallel with Grothendieck's generalization [7] of the theory of fiber bundles to the theory of general fiber spaces with structure sheaf.) Section 1 is preliminary, and in Section 2 we prove Theorems 1 and 2 mentioned above. Then in Section 3 we shall give some general examples which appear naturally in the study of the structure of compact complex manifolds in C [5]; indeed, the application to these examples is the principal motivation for this paper. Finally in Section 4, as a reference for [5], we gather some results