RESEARCH OF ARITHMETIC VARIETIES AND ALGEBRAIC/ANALYTIC STACKS

我们将Kummer-Kawamata覆盖构造为代数堆栈，它起到了诸如Hermite线束在复杂几何中的曲率或Frobenius映射在正特征域上簇范畴中的作用。应用它，我们可以将Esnault-Viehweg消失定理转化为Kawamata消失定理的形式。我们得到了有理系数向量丛的零化定理。我们找到了一种将消失定理推广到对数极点的一般方法。我们发现Kummer覆盖上的因子是代数堆积，这是Serre的一篇题为Kaehler模拟Weil猜想的文章中假定存在的。然而，我们并不满足于它，因为它不是纯粹的代数理论。我们为它准备了更多的代数分析理论。证明了Fujita猜想中的一个猜想：如果一个充分因子的自交数大于1，则该因子加的一个标准因子的充分因子的倍数大于暗…多种多样的维度变得非常丰富。如果该倍数等于该尺寸，则该尺寸变为自由。我们利用Kummer覆盖、自同构群的部分作用和归纳引理。Fujita发现了一个反例，当一个充足的除数有1的自交时，使倍数是维度加1的除数不会变得自由。研究了Esnault-Vehweg型消失定理，发现当分数部分的支撑点为正交点时，数值等价于零的因子是本质的，而不是线性等价于零的因子是本质的。因此，我们定义数值Litaka维是数值等价类中的最大Iitaka维。对于这个新的维度，我们提出了一个与Iitaka-Viehweg猜想类似的猜想。假设对数Iitaka-Viehweg猜想，我们得到了类似于Kawamata-Shokulov锥定理的数值锥定理，然后我们得到了Zariski分解定理。如果存在一个通过满射对数光滑态射控制对数堆栈的对数方案，我们定义对数堆栈是对数代数的。我们希望这一概念对极小模型的研究有所帮助。在正特征域上的变元范畴中，我们得到了不需要解决奇点猜想的Esnault-Viehweg型消失定理。得到了一个适用于非Fujiki流形范畴的消失型定理。较少

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