RESEARCH OF ARITHMETIC VARIETIES AND ALGEBRAIC/ANALYTIC STACKS

算术簇和代数/解析栈的研究

基本信息

批准号：
06640088
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：
1994
资助国家：
日本
起止时间：
1994 至 1996
项目状态：
已结题

项目摘要

We construct Kummer-Kawamata coverings as algebraic stacks, which play roles such as the curvatures of Hermitian line bundies in complex geometries or the Frobenius maps in the category of varieties over fields of positive characteristics. Applying it, we can translate Esnault-Viehweg vanishing theorems into those in the form of Kawamata vanishings. We obtain vanishing theorems for vector bundles with rational coefficients. We find a general method for enlarging vanishing theorems into those with logarithmic poles. We find divisors on Kummer coverings as algebraic stacks which are assumed to exist in the paper with title Kaehler analogue of Weil conjecture by Serre. However we are not satisfied with it since the theory is not purely algebraic. We prepare more algebro-analytic theory for it. We prove one of Fujita conjecture ; if an ample divisor has the self-intersection number greater than one, the divisor adding a canonical divisor with multiple of the ample divisor more than the dim … More ension of a variety becomes very ample. If the multiple is equal to the dimension, it becomes free. We make use of Kummer coverings and the part of the actions of the automorphism group and induction argument. Fujita found the counter example such that when an ample divisor has the self-intersection of one, the divisor such that the multiple is the dimension plus one does not become free. Studing Esnault-Vehweg type vanishing theorem we find divisors of numerically equivalent to zero are essential instead of those of linearly equivalent to zero if the supports of fractional parts are normal crossing. Hence we define numerical litaka dimension is the maximal Iitaka dimension among the numerical equivalence class. We propose an analogue of Iitaka-Viehweg conjecture for this new dimension. Assuming logarithmic Iitaka-Viehweg conjecture, we obtain the numerical cone theorem as an analogue of Kawamata-Shokulov's Cone theorem and then we have a Zariski decomposition theorem. We define log-stacks to be log-algebraic if there exists a log-scheme dominating the log-stack by surjective log-smooth morphism. We hope this concept is useful in the research of minimal models. We obtain Esnault-Viehweg type vanishing theorems without the resolution of singularities conjecture in the category of varieties over fields of positive characteristics. Also we obtain a vanishing type theorem which works in the category of non Fujiki manifolds. Less

我们将kummer-kawamata构建为代数堆栈，它们在复杂的几何形状中扮演着诸如赫尔米尼线弯曲的曲线或在积极特征领域的变化类别中的frobenius地图。应用它，我们可以将Esnault-Viewweg的定理变成川田消失的形式。我们获得具有合理兼容性的向量束的消失定理。我们发现一种将消失的定理提高到具有对数极点的通用方法。我们发现，Kummer覆盖物上的除数为代数堆栈，这些堆栈假定在论文中以Serre的标题为Weil的标题Kaehler类似物。但是，我们对此不满意，因为该理论不是纯粹的代数。我们为此准备了更多的代数分析理论。我们证明了藤田的一种猜想。如果一个足够的除数的自我交流数大于一个，则除数添加了一个规范的分隔线，而多个分隔线却比昏暗的多个……多样性的兴趣变得非常丰富。如果倍数等于尺寸，则将变得自由。我们利用Kummer覆盖物以及自动形态群体和归纳论点的一部分。藤田找到了反示例，使得当一个充分的鸿沟具有一个人的自我干预时，除数是多个尺寸，即尺寸加一个尺寸，一个人不会变得自由。研究Esnault-Vehweg类型消失的定理，我们发现数值等同于零的划分是必不可少的，而不是线性等同于零的划分，如果分数部分的支撑是正常的交叉。因此，我们定义数值litaka维度是数值等效类别之间的最大ITAKA维度。我们为这个新维度提出了一个itaka-viewweg的类似物。假设对数itaka-viewweg的猜想，我们将数值锥定理作为川田 - s-Shokulov的锥体定理的类似物，然后我们具有zariski分解定理。如果存在通过周围的日志平滑形态，我们将日志堆栈定义为对数堆的日志式堆栈。我们希望这个概念在最小模型的研究中很有用。我们获得了esnault-viewweg类型消失的定理，而没有分辨出奇异性的猜想，而在阳性特征领域的品种类别中。同样，我们获得了一个消失的类型定理，该定理在非藤基歧管类别中起作用。较少的