Research of higher dinendional algebraic varaieties

更高维代数簇的研究

• 批准号: 07454004
• 负责人: KAWAMATA Yujiro
• 金额: $ 3.46万
• 依托单位: the University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1995
• 资助国家: 日本
• 起止时间: 1995 至 1996
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07454004/
• 关键词: Calabi-Yau; higher dimensional algebraic varieties; numerical geometry; divisors; intersection numbers; cones; Fujita conjectures; Mirror synmetriy; 豊富因子の錘体; 可動因子; 代数多様体; 射影多様体; 因子; 固定点; 標準特異点; 有理特異点; 安定曲線

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

本文从数值几何的角度研究了高维数的结构，即通过计算交数来研究簇上的因子。(I)从线性的数值条件可以得到什么样的非线性几何结论？（II）在给定簇上因子的数值类的真实的向量空间中存在几个重要的锥。关于这些锥体的性质，我们能说些什么呢？对于（I），我们考虑了以下藤田模型。设X是n维光滑射影簇，H是X上的充分因子，东京工业大学的Takao Fujita证明了线性系统|mH + Kx|是自由的，如果m>= n +1，非常充足，如果m>= n +2。（1）若（H ^n）> n ^n且（H ^d·W）>= n ^d对任意维数d的任意子簇W成立，则|H + Kx|为fr ...更多信息 ee;（2）（H ^n）>（n +1）^n和（H ^d·W）>=（n +1）^d对任意维数d的任意子簇W成立，则|H + Kx|非常充足。例如，我们可以很容易地检验上面关于X = P ^n的公式。根据Riemann-Roch定理，当n = 1时成立，根据Reider的结果，当n = 2时成立.考虑自由度定理（第一部分），因为我以前证明的基点自由定理的论证可用于此问题，并得到如下结果：强猜想在n = 3时成立，弱猜想在n = 4时成立. n = 3时的结果是基于Ein-Lazarsfeld和Fujita的结果，并利用了Ein-Lazarsfeld和黑尔姆克的结果.在证明上述结果的过程中，对对数正则奇点的极小中心的奇点性的分析是很重要的.利用Knudsen的稳定点曲线的模空间，证明了当余维数不超过2时，极小中心只有对数终端奇异性.对于（II），我们研究了Calabi-Yau纤维空间的某些因子锥，根据已在3维中得到验证的极小模型猜想，任何代数簇X0都有一个双有理模型X，它有一个代数纤维空间 *：X * S，使得对于S上的一个充分Q-因子H，标准因子Kx被表示为Kx =*^<**> H。这个纤维空间 *：X * S是卡-丘纤维空间的一个例子。它是Calabi-Yau流形的一个推广概念，在理论物理中也很重要。本文考虑了在dim X = 3的情形下，相对于S上X上的因子的数值等价类。这样的类来自有限维真实的向量空间N^1（X/S）。大量的数值类（分别）。可移动的，大的）因子从锥A（X/S）（分别为M（X/S），B（X/S））在N^1（X/S）内部。A（X/S）的面对应于X的双有理压缩或纤维空间结构，而M（X/S）分解成子锥A（X '/S）对应于不同的与X双有理等价的九次模型X '。受Calabi-Yau流形镜像对称猜想的启发，D.莫里森提出了如下猜想：在双正则自同构群Aut（X/S）的作用下，A（X/S）的面只存在2个等价类，在双有理自同构群Bir（X/S）作用下，M（X/S）的子锥A（X/S）只存在2个等价类.我以前在一篇论文Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces（Ann.of Math.127（1988），93 - 163）中研究了这些锥。在论文[10]中，我引入了标记极小模型的概念，并利用上述文献及后文《代数三重对数翻转的终止》中的结果证明了锥B（X/S）内部分的有限性定理。J. Math.3（1992），653 - 659.并以dim S> 0为主要定理，对上述猜想给出了肯定的回答。作为推论，我证明了，在一个固定的双有理等价类的代数3倍与积极的科代拉dimension.The报告上述结果非常赞赏在英国的沃里克大学举行的国际会议上，并在美国的约翰霍普金斯大学的同构只存在100多个最小模型。少

项目成果

T.KATSURA: "Multicamonical systems of illipti surfaces in small charactorstics" Conpositio Math.97. 119-134 (1995)

T.KATSURA：“小特征中的 illipti 表面的多重系统”Conpositio Math.97。

KAWAMATA Yujiro: "Subadjunction of log canonical divisors for a subvarety of codimension2" "Contemporary Mathematics". (to appear). (1997)

KAWAMATA Yujiro：“余维数 2 的子变体的对数正则除数的子结合”“当代数学”。

中村 博昭: "副有限基本群のガロア剛性" 数学. 47. 1-17 (1995)

Hiroaki Nakamura：“亚有限基本群的伽罗瓦刚度” 数学 47. 1-17 (1995)

KAWAMATA Yujiro: "Divisorial Contractions to 3-dimensional terminal quotient singularities" "Higher Dimensional Complex Varieties". 241-246 (1996)

KAWAMATA Yujiro：“除数收缩到 3 维终端商奇点”“高维复杂品种”。

川又雄二郎: "Divisorial Contractions to 3-dimensional terminal guotient singularities" Higher Dimensional Complex Variefies. 241-246 (1996)

Yujiro Kawamata：“除数收缩到 3 维终端 guotient 奇点”，Higher Dimensional Complex Variefies 241-246 (1996)。