Research of algebro-analytic varieties of higher dimension

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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