Log algebraic stacks and Diophantine Problems

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The summary of research results is as follows.For Diophantine problems of higher dimensional varieties over function fields of characteristic O, we proved a prototype of higher dimensional Shafarevich conjecture. We found view point of birational geometry non effective but we used Kodaira-Nakano vanishing and Kodaira-Spencer deformation theory. We had two proofs for higher dimensional Mordell conjectureover function fields. One method is to use infinitesimal extension of degree one and the canonical model. Another one is obtained by proving that rational points are dense in the fiber of a projective bundle of multiple differential sheaf over a rational point of a given variety. In this case we can estimate the intersection number between a canonical divisor and a curve which is a section of a given fiber space. One expects to apply this to arithmetic fiber spase with a fiber of general type. We construct another logarithmic deformation theory for relatively log smooth morphism. This th … More eory is weaker in rigidity than Kodaira-Spencer deformation theory which is weaker than Kawamata log deformation theoryThis theory controls fibres outside relatively normal crossing divisor defined by log smooth structure. We take the usual dual of the Verdier dual of logarithmic differential sheaf instead of the tangential sheaf. We apply it to the stable fixed components of the canonical divisor in the proof of the Iitaka-Viehweg conjecture. We call it deformation theory of function fields of Kodaira dimension non negative. The weak positivity of direct image of multiple power of relative dualizing sheaf is a great result of Fujita-Kawamata-Viehweg. This role is in part replaced by Mochizuki's pro-p result for Grothendieck conjecture. We can construct birational deformation theory coarseer than log deformation theory. If the open continuous representation of the absolute Galois group of the function field of the base variety into outer automorphism groupof the absolute Galois group of the total variety is trivial then the semi-direct product of the absolute Galois groups of the base variety and geometric generic fiber variety turns to be a direct product. There are many applications to Diophantine problems for higher dimensional varieties.We see that algebraic cycles be found inductively by using the structure of log open subvarieties which is log etale over quasi-projective toric varieties. A minimal model is studied from view point of Kato's log smooth schemes since toroidal embediing is locally etale over toric varieties. A key point is the condition of possibility of blow-down. Without it we proposed a log algebraic stack dominated by log smooth morphism to be taken as a minimal model. Even for a strong minimal model problem the structure of log smooth scheme is available. We apply Fourier Deligne-Sato transformation to reconstruct Hedge theory for complex varieties. We think however it is natural to apply the transformation to p-adic etale cohomologies. The analogous problem of Iitaka-Viehweg conjecture for fiber space of log open varieties can be treated by semi-local ring of height one thanks to Mochizuki's theory. These resuls are published in academic reports T.I.P.and oral communication in Berlin ICM in part. Less