Algebraic Intersection Theory on Singular Varieties

奇异簇的代数交集理论

• 批准号: 09640041
• 负责人: SUMIHIRO Hideyasu
• 金额: $ 2.11万
• 依托单位: HIROSHIMA UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640041/
• 关键词: Chow groups; Bivariant sheaves; Alexander schemes; Vector Bundles; Determinantal varieties; Rigid cohomologies; 特異多様体; 代数的サイクル

In this project, we studied algebraic intersection theory on singular varieties by the following two methods and obtained the following results :1) Bivariant sheaf theory. If the Chow group of a singular algebraic variety X has a ring structure, then X is called an Alexander scheme. We constructed the topos C of algebraic varieties with the Grothendieck topology which is obtained by proper morphisms between algebraic varieties. Using this topos C, we introduced the Bivariant sheaves for algebraic varieties. It is showned that an algebraic variety X is an Alexander scheme if and only if H'(X, A) =0, where A is the Bivariant sheaf on X.In addition, we have started to study the higher cohomologies of Bivarinat sheaves in order to generalize the above result which might concern the problem on finite dimensionality of Motives that is the most important problem in the field of algebraic cycles and introduced the theory of Hyper-Covering to compute the higher cohomologies of Bivariant sheaves concretely.2) Splitting of Vector Bundles. As for the splitting problem for rank two vector bundles on projective spaces which is one of the most important problem in the field of algebraic vector bundles, we obtained the following two results. (1) Let E be a rank two very ample vector bundle on P^n (n*4) and X an determinantal variety defined by global sections of E.Analyzing the structure of the Hilbert scheme of those determinantal varieties, it is shown that E splits into line bundles if and only if H^1 (P, End(E))=0, where P is a 4- or 5- dimensional linear subspace of P^n. (2) E is a direct sum of line bundles if and only if dimH^1(X, O_x(r-Z)) *O(r^1)(r*0) and diinH ^k(X, O_x(-Rz-_sH)) * P_k (s) (r, s>O) (l*k*dimX-l), where Z and H are specific effective divisors on the determinantal variety X and P_k (s) is a polynomial on s which is independent of r.

在本项目中，我们用以下两种方法研究了奇异簇上的代数交理论，得到了以下结果：1）双变层理论。如果奇异代数簇X的Chow群具有环结构，则称X为亚历山大概型.利用代数簇之间的真态射得到的Grothendieck拓扑构造了代数簇的拓扑C。利用这个拓扑C，我们引入了代数簇的双变层。证明了代数簇X是亚历山大概型当且仅当H '（X，A）=0，其中A是X上的双变层。我们开始研究Bivarinat层的高阶上同调，以推广上述结果，从而可能涉及到代数圈领域中最重要的Motives的有限维问题，并引入了Hyper-复盖法具体计算了双变层的高阶上同调。2）向量丛的分裂。对于代数向量丛领域中最重要的问题之一--射影空间上秩为2的向量丛的分裂问题，我们得到了以下两个结果。(1)设E是P^n（n*4）上的秩为2的非常充足向量丛，X是由E的整体截面定义的行列式簇.通过分析这些行列式簇的Hilbert格式的结构，证明了E分裂成线丛的充要条件是H^1（P，End（E））=0，其中P是P^n的4维或5维线性子空间. (2)E是线丛直和的充要条件是dimH^1（X，O_x（r-Z））*O（r^1）（r*0）和diinH ^k（X，O_x（-Rz-_sH））* P_k（s）（r，s>O）（l*k* dimX-1），其中Z和H是行列式簇X上的特定有效因子，P_k（s）是s上的多项式，且与r无关.

项目成果

Hideyasu Sumihiro: "Determinantel varieties associated to rank two vector bundles on projective Spaces and splitting theorems" Hiroshima Math.J.(1999)

Hideyasu Sumihiro：“与射影空间上的两个向量丛的排序相关的行列式簇和分裂定理”Hiroshima Math.J.（1999）

N.Tsuzuki: "Slope filtration of quasi-unipotent overconvergent F-isocrystals" Ann.Inst.Fourier, Greunoble.48. 379-412 (1998)

N.Tsuzuki：“准单能过收敛 F 等晶体的斜率过滤”Ann.Inst.Fourier，Greunoble.48。

TSUZUKI Nobuo: "Slope filtration of quasi-unipotant averconvergent Fisocystals" Ann,Institut Fourier., Greuobal. 48. (1998)

TSUZUKI Nobuo：“准单能均聚 Fisocystals 的斜率过滤”Ann，Institut Fourier，Greuobal。

Nobuo Tsuzuki: "Finite local monoclromg of overconvergent unit-root F-isocrystals on a curve" Amer.J.Math.120. 1165-1190 (1998)

Nobuo Tsuzuki：“曲线上过收敛单位根 F 等晶体的有限局部单色”Amer.J.Math.120。

H.Sumihiro: "Determinantal varieties associated to rank two vector bundles on projective spaces and splitting theorems" Hiroshima Jour.of Math. (to appear). (1999)

H.Sumihiro：“与射影空间和分裂定理上的两个向量束排序相关的行列式簇”《广岛数学杂志》。