Log algebraic stacks and Diophantine Problems

对数代数栈和丢番图问题

基本信息

批准号：
09640076
负责人：
MAEHARA Kazuhisa
金额：
$ 1.34万
依托单位：
TOKYO INSTITUTE OF POLYTECHNICS
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 2000
项目状态：
已结题

项目摘要

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

研究成果总结如下：对于特征O函数域上的高维簇的丢番图问题，我们证明了高维Shafarevich猜想的原型，我们发现双有理几何的观点是无效的，但我们使用了Kodaira-Nakano消失和。 Kodaira-Spencer 变形理论。我们对函数域上的高维 Mordell 猜想有两种证明。一种方法是使用一阶无穷小扩展和规范模型。另一种方法是通过证明有理点在给定品种的有理点上的多重微分束的射影丛中是密集的。在这种情况下，我们可以估计正则除数和曲线之间的数交点。我们期望将其应用于具有一般类型的纤维的算术纤维空间，该理论的刚性比对数光滑态射要弱。弱于 Kawamata 对数变形理论的 Kodaira-Spencer 变形理论该理论控制由对数平滑结构定义的相对正常交叉除数之外的纤维，我们采用对数微分束的 Verdier 对偶来代替切向束。 Iitaka-Viehweg 猜想证明中正则除数的稳定固定分量我们称之为小平函数域变形理论。相对二元束的多重幂的直接图像的弱正性是Fujita-Kawamata-Viehweg 的一个伟大结果，这个作用部分地被望月的Grothendieck 猜想的pro-p 结果所取代。我们可以构造双有理变形理论。如果将基变函数域的绝对伽罗瓦群的开连续表示转化为总的绝对伽罗瓦群的外自同构群。簇是微不足道的，那么基簇和几何通用纤维簇的绝对伽罗瓦群的半直积就变成了直积。对于高维簇的丢番图问题有很多应用。我们看到代数循环可以归纳地找到。通过使用对数开子变体的结构，即准投影环面变体上的对数 etale，从 Kato 的对数平滑方案的角度研究了最小模型，因为环形嵌入是局部 etale。一个关键点是，如果没有它，我们提出了一个以对数平滑态射为主导的对数代数堆栈，即使对于强最小模型问题，对数平滑的结构也是如此。我们应用 Fourier Deligne-Sato 变换来重建复杂簇的 Hedge 理论，但是我们认为将变换应用于 p-adic etale 上同调是很自然的。由于望月的理论，原木开放品种的纤维空间可以用高度为1的半局部环来处理，这些结果部分发表在柏林ICM的学术报告T.I.P和口头交流中。