Log algebraic stacks and Diophantine Problems

主要研究成果如下：对于特征为O的函数域上的高维簇丢番图问题，证明了高维Shafarevich猜想的一个原型.我们发现双有理几何的观点是无效的，但我们使用Kodaira-Nakano消失和Kodaira-Spencer变形理论。对于高维Mordell连续函数域，我们有两个证明。一种方法是使用一次无穷小扩张和典范模型。另一个是通过证明有理点在给定簇的有理点上的重微分层的投射丛的纤维中是稠密的而得到的。在这种情况下，我们可以估计一个典型因子和一条曲线之间的交集，这条曲线是给定纤维空间的一部分。人们期望将这一点应用于具有一般类型的纤维的算术纤维空间。我们构造了相对对数光滑态射的另一个对数变形理论。这个 ...更多信息 该理论的刚性弱于Kodaira-Spencer变形理论，Kodaira-Spencer变形理论的刚性弱于Kawamata原木变形理论，该理论将纤维控制在原木光滑结构所定义的相对正常交叉因子之外。我们取对数微分层的Verdier对偶的通常对偶而不是切向层。在证明Iitaka-Viehweg猜想的过程中，我们将其应用于标准因子的稳定固定分量。我们称之为非负科代拉维函数场的形变理论。相对对偶层的重幂直象的弱正性是Fujita-Kawamata-Viehweg的一个重要结果。这个角色部分被望月的pro-p结果所取代。我们可以构造比对数变形理论更粗的双有理变形理论。如果基簇的函数域的绝对伽罗瓦群到全簇的绝对伽罗瓦群的外自同构群的开连续表示是平凡的，则基簇的绝对伽罗瓦群与几何一般纤维簇的绝对伽罗瓦群的半直积是直积.高维簇的丢番图问题有许多应用，我们看到，利用拟投射复曲面簇上的对数开子簇的结构，可以归纳地找到代数圈。本文从Kato的对数光滑格式的观点研究了一个极小模型，因为环面嵌入在环面簇上是局部稳定的。关键点是放空可能性的条件。没有它，我们提出了一个对数代数堆栈占主导地位的对数光滑态射作为一个最小的模型。即使对于强极小模型问题，对数光滑格式的结构也是可用的。我们应用Fourier Deligne-Sato变换来重建复品种的Hedge理论。然而，我们认为将变换应用于p-adic etale上同调是很自然的。利用Mochizuki的理论，对数开簇的纤维空间的Iitaka-Viehweg猜想的类似问题可以用高度为1的半局部环来处理。这些结果部分发表在柏林ICM的T. I. P.和口语交流学术报告中。少

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