Research of algebro-analytic varieties of higher dimension

Kummer-Kawamata覆盖是证明Kawamata消失定理的关键。本文推广了Esnault-Viehweg的结果，即Hodge谱序列的退化通过循环覆盖和去奇异化蕴涵了零谱。循环覆盖（cyclic covering）Kawamata覆盖）在微分几何中扮演了线丛曲率的角色。我们把一个库默覆盖实现为一个没有奇点的1-代数冠军。因此，我们可以把它应用到提升到特征零的正特征的完全非奇异簇，其中Hodge谱序列的退化是由Deligne-Illusie得到的。库默覆盖作为一个代数冠军，使我们能够取一个满足Serre的论文“Riemann猜想的Kahler模拟”中假设的自同态。此外，Fourier-Deligne变换适用于这种自同态，这将导致Hodge谱序列在复代数几何中的退化。这是一个纯代数证明。在一定的Grothendieck拓扑下，代数champs的上同调理论证明了Hodge-Kodaira分解定理与消失定理是等价的.环拓扑上的纤维范畴的格贝通过提升到格贝的分类拓扑上形成一个相对格贝。因此，在与代数空间相关联的环形拓扑上的fiberd范畴的概型的gerbe是代数的。我们期望将其推广到无限代数冠军。这与分析冠军相同。Zebraki拓扑中正特征的完全非奇异簇到长度为2的Witt环的局部提升成为一个特例。在Gerbe的分类拓扑中取极大根扩张，得到Hodge分解。我们准备证明的基本猜想的双有理几何和一个类似的高维Shafarevitch猜想的功能域。即将发表的是，

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