Research of algebro-analytic varieties of higher dimension

众所周知，Kummer-Kawamata覆盖是证明Kawamata消失定理的关键。我们推广了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而使谱序列消失。循环覆盖(分别为Kawamata Covering)在微分几何中扮演着线丛的曲率的角色。我们将Kummer覆盖实现为无奇点的1-代数Champ。因此，我们可以将它应用于正特征的完全非奇异型，它可以提升到特征零，其中Hodge谱序列的退化是由Deligne-Illusie得到的。作为代数积的Kummer覆盖使我们能够得到一个自同态，它满足Serre的论文《Riemann猜想的Kahler模拟》的假设。此外，傅里叶-德莱恩变换适用于这种自同态，这将导致复代数几何中Hodge谱序列的退化。这是一个纯粹的代数证明。选择一定的Grothendieck拓扑，代数Champs的上同调理论表明Hodge-Kodaira分解定理和消失定理是等价的。环形拓朴上的Fiberd方案范畴的Gerbe通过将其提升到Gerbe的分类Topos来形成一个相对方案。因此，与代数空间相关的环拓扑图上的Fiberd格式范畴的G是代数的。我们期望将其推广到无限代数冠军。这与分析冠军的情况是一样的。Zariski拓扑中具有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GERBE。在Gerbe的分类拓扑点上取最大根扩张，得到了其Hodge分解。我们准备了二次几何基本猜想的证明和函数域上高维Shafarevitch猜想的一个类似的证明。即将公布的是一种模拟

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