Log algebraic stacks and Diophantine Problems
Project/Area Number  09640076 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  TOKYO INSTITUTE OF POLYTECHNICS 
Principal Investigator 
MAEHARA Kazuhisa TOKYO INSTITUTE OF POLYTECHNICS, 工学部, 助教授 (10103160)

Project Period (FY) 
1997 – 2000

Project Status 
Completed(Fiscal Year 2000)

Budget Amount *help 
¥2,100,000 (Direct Cost : ¥2,100,000)
Fiscal Year 2000 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1999 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1998 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1997 : ¥600,000 (Direct Cost : ¥600,000)

Keywords  logarithmic smooth scheme / birational deformation theory / higher dimensional Mordell conjecture / IitakaViehweg Conjecture / 代数多様体 / 双有理同型群 / 飯高・フィーヴェックの予想 / ディオファントス / 対数代数堆積 / ティト予想 / ホッジ理論 / ディオファントス問題 / トロイダル埋入 / 双有理幾何学 / 高次元多様体の分類 / 飯高予想 / フィーベック予想 
Research Abstract 
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 KodairaNakano vanishing and KodairaSpencer 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 KodairaSpencer deformation theory which is weaker than Kawamata log deformation theory This 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 IitakaViehweg 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 FujitaKawamataViehweg. This role is in part replaced by Mochizuki's prop 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 semidirect 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 quasiprojective 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 blowdown. 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 DeligneSato transformation to reconstruct Hedge theory for complex varieties. We think however it is natural to apply the transformation to padic etale cohomologies. The analogous problem of IitakaViehweg conjecture for fiber space of log open varieties can be treated by semilocal 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

Report
(5results)
Research Products
(15results)