Relative Algebraic Reduction and Relative Albanese Map for a Fiber Space in $\mathcal C$

Abstract

Let /: X-+Y be a fiber space of compact complex manifolds, i. e., / is surjective with connected fibers. Let U^Y be a Zariski open subset over which / is smooth. Then for each y(=U we have the Albanese map Xu-+AlbXu/U over U where Xu=f~\U). Then the main problem to be treated in this paper is the following: When can we compactify Alb Xu/U to a compact complex manifold Alb*X/Y over Y so that (pu extends to a meromorphic map (])'. X-*Alb*X/Y over Y? (Here we do not require any good property for A\b*X/Y; any compactification is enough for our purpose.) We shall show in this paper that this is the case if i) the total space X is in C, and ii) any smooth fiber Xy is Moishezon, (after a possible restriction of U}. Moreover it turns out that in this case Alb*X/Y is again in C and the pair (ft, A\b*X/Y) is unique up to bimeromorphic equivalences. We call cp briefly the relative Albanese map for /. One notable property of <]) we prove is that it is Moishezon in the sense that it is bimeromorphic to a projective morphism. Thus, in a sense, the relative Albanese variety Alb*X/Y may be considered as the obstruction for a fiber space with general fiber Moishezon to be a Moishezon morphism. We follow the method of Grothendieck [13] in algebraic geometry, constructing Alb Xu/U as a component of the relative Picard variety Pic ((P\c,TXu/U)/U) of some component PicrXu/U of the relative Picard variety Pic Xu/U of Xu over U. Here Pic Xu/U (or at least a good part of it) in turn is constructed as a flat quotient of the space Div Xu/U of relative divisors on Xn over U. Our first step is thus to construct a natural completion Div*X/Y of Div Xu/U over Y, where the assumption that X<=C is essential to guarantee that each irreducible component of Div*X/Y is compact (Section 2). The second step is then to complete Pic Xu/U to a complex variety Pic*J^/F over Y such that the