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 覆盖构造为代数栈，其作用类似于复杂几何中厄米线邦的曲率或正特征域上簇范畴中的 Frobenius 映射。应用它，我们可以翻译 Esnault-Viehweg 消失定理。我们得到了具有有理系数的向量束的消失定理。将消失定理扩大到对数极点的方法我们为其准备了更多的代数分析理论，证明了藤田猜想的自交数是否大于。一，除数加上一个大于除数的倍数的标准除数，如果倍数等于维数，则它变得自由。藤田研究了自同构群的作用和归纳论证，当一个充足的除数具有 1 的自交时，使得倍数为维数加 1 的除数不会变得自由。 Esnault-Vehweg 型消失定理发现，如果小数部分的支撑是法向交叉，则数值等价于零的约数而不是线性等价于零的约数因此我们定义数值litaka维数是数值等价类中最大的Iitaka维数。我们提出了这个新维度的 Iitaka-Viehweg 猜想的模拟，假设对数 Iitaka-Viehweg 猜想，我们得到了数值锥。定理类似于 Kawamata-Shokulov 的圆锥定理，然后我们有 Zariski 分解定理，如果存在通过满射对数平滑态射支配对数堆栈的对数堆栈，我们将对数堆栈定义为对数代数。这个概念在最小模型的研究中很有用，我们无需解决奇点就可以得到 Esnault-Viehweg 型消失定理。我们还得到了一个适用于非 Fujiki Less 范畴的消失型定理。