Relative Algebraic Reduction and Relative Albanese Map for a Fiber Space in \mathcal{C}
Relative Algebraic Reduction and Relative Albanese Map for a Fiber Space in \mathcal{C}
复制标题
mathcal{C} 中纤维空间的相对代数约简和相对阿尔巴尼映射
DOI：
10.2977/prims/1195182985
复制
发表时间：
1983
影响因子：
1.2
通讯作者：
A. Fujiki
中科院分区：
文献类型：
--
作者：
A. Fujiki
设/：X-+Y是紧致复流形的纤维空间，即：有连通纤维的满射。设U^Y是Zariski开子集，f在其上是光滑的。然后对于每个y(=U，我们有U上的阿尔巴尼亚式映射Xu-+AlbXu/U，其中Xu=f~\U)。那么本文要处理的主要问题是：什么时候可以将Alb Xu/U紧化为Y上的紧致复流形Alb*X/Y，使得(PU扩张到亚纯映射())‘。X-*Alb*X/Y而不是Y？(在这里，我们不需要A\b*X/Y的任何良好属性；任何压缩都足以满足我们的目的。)在本文中，我们将证明，如果i)全空间X在C中，以及ii)任何光滑纤维xy都是Moishezon，(在可能的限制U之后)。此外，在这种情况下，Alb*X/Y也在C中，并且对(ft，A\b*X/Y)是唯一的，直到双纯等价。我们简称cp为/的相对阿尔巴尼亚语映射。)的一个重要性质是它是Moishezon的，即它是投射态射的双纯。因此，在某种意义上，相对阿尔巴尼亚品种Alb*X/Y可以被认为是普通纤维Moishezon为Moishezon态射的纤维空间的障碍。我们遵循Grothendieck[13]在代数几何中的方法，将Alb Xu/U构造为相对Picard簇Pic((P\c，TXU/U)/U)的某个分支PicrXu/U的某个分支PicrXu/U的相对Picard簇Pic Xu/U。这里，Pic Xu/U(或者至少它的很大一部分)依次被构造为Xn上相对因子的空间Div Xu/U的平坦商。=C是保证Div*X/Y的每个不可约分支是紧的(第2节)所必需的。然后，第二步是将Pic Xu/U完成为复数变种Pic*J^/F除以Y，使得
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