Project/Area Number  09640041 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
Algebra

Research Institution  HIROSHIMA UNIVERSITY 
Principal Investigator 
SUMIHIRO Hideyasu Hiroshima Univ., Math.Depart., Professor, 理学部, 教授 (60068129)

CoInvestigator(Kenkyūbuntansha) 
KIMURA Shunichi Hiroshima Univ., Math.Depart., Assist.Professor, 理学部, 講師 (10284150)
TSUZUKI Nobuo Hiroshima Univ., Math.Depart., Assist, 理学部, 助手 (10253048)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥3,300,000 (Direct Cost : ¥3,300,000)
Fiscal Year 1998 : ¥1,100,000 (Direct Cost : ¥1,100,000)
Fiscal Year 1997 : ¥2,200,000 (Direct Cost : ¥2,200,000)

Keywords  Chow groups / Bivariant sheaves / Alexander schemes / Vector Bundles / Determinantal varieties / Rigid cohomologies / チャウ群 / 双変層 / アレキサンダー・スキーム / ベクトル束 / 行列式多様体 / リジッド・コホモロジー / 特異多様体 / 代数的サイクル 
Research Abstract 
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 HyperCovering 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(rZ)) *O(r^1)(r*0) and diinH ^k(X, O_x(Rz_sH)) * P_k (s) (r, s>O) (l*k*dimXl), 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.
