Research of higher dinendional algebraic varaieties

本研究的目的是调查更高维度的结构，我们考虑了从数字几何的视点中得出的后续问题，以及通过对交集数字的计算对变量的区分研究。（I）什么样的几何结合：大自然中的非线性元素可以从自然中获得哪些是线性元素？（II）在给定变量中，存在着一些重要的因素，在现实向量空间中，分配的数字类。关于这些硬币的属性，我们能说些什么呢？对于（I），我们考虑到下面的富士塔投影仪。让X成为一个以X.富士高田东京技术研究所投影的线性系统为N和H的平滑投影变体|MH + Kx|如果m >= n + 1，如果m >= n + 2很好，如果m >= n + 2。这个概念从下面的更强的概念中遵循： （1）如果（H^n） > n^n和（H^d·W） >= n^d保留任何尺寸d的子变量，然后|H + Kx|是fr。 ... More ee ; （2） （H^n） > （n + 1）^n和（H^d·W） >= （n + 1）^d保留任何尺寸d的子变量W，然后|H + Kx|这是个很好的例子。例如，一个人可以很容易地检查X = P^n的上面的概念。如果Riemann-Roch定理中的n = 1，如果Reider的结果中的n = 2，我认为自由的概念（概念的第一部分）是正确的，因为自由基点定理的论点在我之前可以使用到这个问题，并且得到了以下的结果：如果更强的概念是真实的，如果n = 3，如果weaker一个则n = 4。n = 3的结果是基于Ein-Lazarsfeld和Fujita的先前结果，以及Ein-Lazarsfeld和Helmke的结果。在上面结果的证明课程中，对日志canonical Singularities的最小单一中心的分析是重要的。我证实了这个问题与规范分流器的转换过程有关，并且证明，如果它的编码在大多数2上使用Knudsen的模块空间的点稳定曲线的确定性。对于（II），我为Calabi-Yau纤维空间的分流器确定性。连接到最小模型几何，其中已在尺寸3中验证。任何代数变体X 0都有一个带有代数光纤空间的生物模型X * S这样的X * S，其中规范分支Kx被表示为Kx = *^<*> H用于S上的一个Q-divisor H示例。该光纤空间 * ： X * S是一个Calabi-Yau光纤空间示例。这是一种通用的习惯，它在理论物理学中也是重要的。我认为在任何dim X = 3的情况下，在X相对于S的情况下，在X = 3的情况下，分配员的数值相等分类。来自有限维度真实向量空间N^1 （X/S）的此类类别。样本的数字类别（答复。可移动，大） divisors from a cone A （X/S） （resp. M （X/S）、B （X/S）） inside N^1 （X/S）。A （X/S）与X的生物契约或光纤空间结构的对应面，以及M （X/S）与A （X'/S）的分离与不同的纳米模型的X'是与X相似的，受到Calabi-Yau Manifolds的镜像对称性几何的启发，D. Morrison提出了下面的物体：A （X/S）的表面上存在着足够多的相等类别，其中A（X/S）的作用下，and there exist only finitely many equivalence classes of subcones A （X/S） of M （X/S） under the action of the birational automorphism group Bir （X/S）。我以前研究了这些建议，在一张纸上用3维尺寸规范性Singularity及其应用到表面的程度，Math，127 （1988）， 93- 163.在论文[10]中，我介绍了一个有标记的最小模型的概念，并为圆锥B （X/S）中的部分提供了一个完美的理论，该理论由在上面的纸张和用于代数3-文件的较后纸张终止的日志翻转， Intl。J. Math 3 （1992年）， 653-659 Moreover，我在主要理论中给出了一个积极的答案，对上面的概念在什么dim S > 0的情况下。作为一个合唱团，我只提出了足够多的最小模型，使之成为一个固定的生物均衡类的代数3-文件夹与积极的Kodaira尺寸。关于上述结果的报告在联合王国沃里克大学的国际会议上得到了很大的赞赏，在美国约翰霍普金斯大学的国际会议上得到了很大的赞赏。Less（低）

The purpose of this research was to investigate the structure of higher dimensio We considered the following questions from the view point of numerical geometry, which studies divisors on varieties by the calculation of intersection numbers.(I) What kind of geometric conclusions which are non-linear in nature can be obtained from the numerical conditions which are linear in nature?(II)There are several important cones in the real vector space of numerical classes of divisors on a given variety. What can we say about the properties of these cones?For (I), we considered the following Fujita conjectures. Let X be a smooth projective variety of dimension n and H an ample divisor on X.Takao Fujita of Tokyo Institute of Technology conjectured that the linear system |mH + Kx|is free if m >= n + 1, and very ample if m >= n + 2. This conjecture follows from the following stronger conjecture : (1) If (H^n) > n^n and (H^d・W) >= n^d holds for any subvariety W of any dimension d, then |H + Kx|is fr … More ee ; (2) (H^n) > (n + 1)^n and (H^d・W) >= (n + 1)^d holds for any subvariety W of any dimension d, then |H + Kx|is very ample. For example, one can easily check the above conjectures for X = P^n. They are true if n = 1 by the Riemann-Roch theorem, and if n = 2 by Reider's results.I considered the freeness conjectures (the first part of the conjectures) because the argument of the base point free theorem which I proved before can be used to this problem, and obtained the following results : the stronger conjecture is true if n = 3, and the weaker one if n = 4. The result for n = 3 is based on the previous results by Ein-Lazarsfeld and Fujita, and uses results of Ein-Lazarsfeld and Helmke.In the course of the proof of the above results, the analysis of the singularities of the minimal center of log canonical singularities is important. I realized that this problem is related to the adjunction process of the canonical divisors, and proved that the minimal center has only log terminal singularties if its codimension is at most 2 by using the moduli space of pointed stable curves by Knudsen.For (II), I investigated certain cones of divisors for Calabi-Yau fiber spaces.Accrding to the Minimal Model Conjecture, which is already verified in dimension 3, any algebraic variety X0 has a birational model X with an algebraic fiber space * : X * S such that the canonical divisor Kx is expressed as Kx = *^<**> H for an ample Q-divisor H on S.This fiber space * : X * S is an example of a Calabi-Yau fiber space. It is a generalized notion of a Calabi-Yau manifold which is also important in theoretical physics.I considered numerical equivalence classed of divisors on X relatively over S in the case in which dim X = 3. Such classes from a finite dimensional real vector space N^1 (X/S). The numerical classes of ample (resp. movable, big) divisors from a cone A (X/S) (resp. M (X/S), B (X/S)) inside N^1 (X/S). The faces of A (X/S) correspond to birational contractions or fiber space structures of X,and the decomposition of M (X/S) into subcones A (X'/S) corresponds to different ninimal models X'which are birationally equivalent to X.Inspired by the Mirror Symmetry Conjecture for Calabi-Yau manifolds, D.Morrison raised the following conjecture : there exist only finitely many equivalence classes of faces of A (X/S) under the action of the biregular automorphism group Aut (X/S), and there exist only finitely many equivalence classes of subcones A (X/S) of M (X/S) under the action of the birational automorphism group Bir (X/S). I studied these cones previously in a paper Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. 127 (1988), 93-163.In the paper [10], I introduced a concept of marked minimal models, and proved a finiteness theorem for the part inside the cone B (X/S) by using some results in the above paper and a later paper Termination of log-flips for algebraic 3-folds, Intl. J.Math. 3 (1992), 653-659. Moreover, I gave an affirmative answer to the above conjecture in the case in which dim S > 0 as the main theorem. As a corollary, I proved that there exist only finitely many minimal models up to isomorphisms in a fixed birational equivalence class of algebraic 3-folds with positive Kodaira dimension.The reports of the above results were much appreciated at the international conferences held at the University of Warwick in the United Kingdom, and at the Johns Hopkins University in the United States. Less