Research of algebro-analytic varieties of higher dimension

高维代数解析簇的研究

基本信息

批准号：
03640105
负责人：
MAEHARA Kazuhisa
金额：
$ 1.02万
依托单位：
Tokyo Institute of Polytechnics
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1991
资助国家：
日本
起止时间：
1991 至 1993
项目状态：
已结题

项目摘要

It has been known that Kummer-Kawamata covering is a key to prove Kawamata vanishing theorem. We generalize Esnault-Viehweg's result that the degeneration of Hodge spectral sequence implies vanishing thoprems through cyclic covering and desingularization. Cyclic covering (resp. Kawamata covering) takes a role of the curvature of a line bundle in differential geometry. We realize a Kummer covering as a 1-algebraic champ without singular points. Hence we can apply it to the complete non singular variety of positive characteristic which is liftable to characteristic zero, where the degeneration of Hodge spectral sequence is obtained by Deligne-Illusie. The Kummer cover as an algebraic champ enable us to take an endomorphism which satisfies the assumption of Serre's paper "Kahler analogue of Riemann conjecture". Furthermore Fourier-Deligne transformation is applicable to this endomorphism, which should induce the degeneration of Hodge spectral sequence in complex algebraic geometry. It is a pure algebraic proof. Chosen certain Grothendieck topologies, the cohomology theory of algebraic champs implies that Hodge-Kodaira decomposition and vanishing theorems are equivalent. The gerbe of the fiberd category of schemes over the ringed topos forms a relative scheme by lifting it to the classifying topos of the gerbe. Thus the gerbe of the fiberd category of schemes over the ringed topos associated to an algebraic space is algebraic. We expect to extend it to the infinite algebraic champs. It is the same as for analytic champs. The local liftings of a complete non singular variety of positive characteristic in Zariski topology to the Witt ring of length two becomes a gerbe. Taking the maximal radical extention in the classifying topos of the gerbe we obtain the Hodge decompositon. We prepare the proof of the fundamental conjecture of the birational geometry and an analogue of higher dimensional Shafarevitch conjecture over function fields. It is to be published that an analog

众所周知，Kummer-Kawamata 覆盖是证明 Kawamata 消失定理的关键。我们概括了 Esnault-Viehweg 的结果，即 Hodge 谱序列的退化意味着通过循环覆盖和去奇异化来消失 thoprems。循环覆盖（分别为川俣覆盖）在微分几何中扮演线束曲率的角色。我们实现了库默覆盖作为一个没有奇点的 1 代数冠军。因此，我们可以将其应用于可提升到特征零的完全非奇异正特征变，其中Hodge谱序列的退化由Deligne-Illusie获得。库默作为代数冠军的封面使我们能够采用满足塞尔论文“黎曼猜想的卡勒类似物”假设的自同态。此外，Fourier-Deligne变换适用于这种自同态，这应该会导致复杂代数几何中Hodge谱序列的退化。这是一个纯代数证明。选择某些格洛腾迪克拓扑，代数冠军的上同调理论意味着 Hodge-Kodaira 分解和消失定理是等价的。环状拓扑上的纤维类别方案的非洲菊通过将其提升到非洲菊的分类拓扑而形成相对方案。因此，与代数空间相关的环形拓扑上的纤维范畴的方案的 gerbe 是代数的。我们希望将其扩展到无限代数冠军。这与分析冠军相同。 Zariski拓扑中正特性的完全非奇异簇局部提升到长度为二的维特环成为gerbe。取非洲菊分类拓扑中的最大根扩展，我们得到 Hodge 分解。我们准备了双有理几何基本猜想的证明以及函数域上高维沙法列维奇猜想的类比。即将发布的是一个模拟