Research of algebro-analytic varieties of higher dimension

Kummer-Kawamata覆盖是证明Kawamata消失定理的关键。我们推广了Esnault-Viehweg的结论，即Hodge谱序列的退化意味着通过循环覆盖和去广域化使索普勒消失。循环覆盖(如：川俣覆盖（Kawamata cover）在微分几何中扮演线束曲率的角色。我们将Kummer覆盖实现为无奇点的1-代数冠军。因此，我们可以将其应用于可提升到特征零点的正特征的完全非奇异变化，并利用delign - illusie方法得到了霍奇谱序列的退化。Kummer盖作为代数champ使我们能够取一个满足Serre论文“Kahler模拟黎曼猜想”假设的自同态。此外，傅里叶-德里涅变换适用于这种自同态，这将导致复杂代数几何中Hodge谱序列的退化。这是一个纯粹的代数证明。选取一定的Grothendieck拓扑，代数阵的上同论意味着Hodge-Kodaira分解和消失定理是等价的。环形拓扑上的纤维类格式的gerbe通过将其提升为gerbe的分类拓扑而形成相对格式。因此，与代数空间相关的环拓扑上的光纤类方案是代数的。我们期望将其推广到无穷代数冠军。这和分析冠军是一样的。在Zariski拓扑中，正特征的完全非奇异变异对长度为2的Witt环的局部提升成为gerbe。在gerbe的分类拓扑上取最大的根扩展，得到Hodge分解。本文给出了二维几何基本猜想的证明，并给出了函数场上高维Shafarevitch猜想的类似证明。这是要发表的一个类比

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