Vector Bundles on Manifolds
Project/Area Number 
13640026

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Hiroshima University 
Principal Investigator 
SUMIHIRO Hideyasu Hiroshima University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (60068129)

Project Period (FY) 
2001 – 2003

Project Status 
Completed (Fiscal Year 2003)

Budget Amount *help 
¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2002: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2001: ¥1,400,000 (Direct Cost: ¥1,400,000)

Keywords  Vector bundles / Hartshorne conjecture / Hilbert schemes / Determinantal varieties / Frobenius morphism / Splitting theorem for vector bundles / 射影空聞上のベクトル束 / 小平の消滅定理 / Frobenius写像 / Cartierの同型定理 / 射影空間 / ベクトル束の分解問題 / 消滅定理 
Research Abstract 
We have studied the splitting problem of rank two vector bundles on projective space P^n(n 【greater than or equal】4) and obtained the following. 1)Cohomological criterion : Theorem Let E be a rank two vector bundle on P^n(n【greater than or equal】4), P a 4 or 5dimensional linear subspace of P^n and let ^^E = EP be the restriction of F to P. Then E is a direct sum of line bundles if and only if H^1 (P, End(^^E)) = 0. Hence it implies that we can reduce the splitting problem in zero characteristic to the one in positive characteristic. 2)Kodaira vanishing theorem and geometric structures of determinantal subvarieties in positive characteristic : Theorem Let X be a nonsingular projective variety defined over an algebraically closed field of positive characterristic and L a positive ample line bundle on X. Then there exists the following inequality : dim H^1 (X, L^<1>)【less than or equal】 dimH^1(X, Ο_X). Theorem Let X be a determinantal subvariety associated to E on P ^n in positive characteristic. Then we have H^1(X, Ο_X, ) = H^1(X,Ω^1_X ) = 0. 3)A splitting theorem for topologically trivial vector bundles (n = 4) : A rank two vector bundle E is called topologically trivial if c_1 =α+β, c_2=α・β (∃α,β∈Z). Theorem If α【greater than or equal】((1+√<4β3>)/2)β, then E is a direct sum of line bundles. In particular ; if 1【less than or equal】β【less than or equal】3, then E splits into line bundles. 4)A splitting theorem via Frobenius morphism (n = 4) : Let F be the Frobenius morphism with the exponent q = p^n. Theorem Assume that c^2_14c_2 > 0. Then we have the following, a)EX is Bogomolov's unstable. b)E is a direct sum of line bundles if and only if dim H^1 (X, F* (End(E))【less than or equal】Ο (q^1) for large q.

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