Vector Bundles on Manifolds

2003 Fiscal Year Final Research Report Summary

• Project/Area Number: 13640026
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year 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
• Keywords: Vector bundles / Hartshorne conjecture / Hilbert schemes / Determinantal varieties / Frobenius morphism / Splitting theorem for vector bundles

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 5-dimensional linear subspace of P^n and let ^^-E = E|P 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 non-singular 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+√<4β-3>)/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_1-4c_2 > 0. Then we have the following, a)E|X 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.