Vector Bundles on Manifolds

点击翻译按钮获取中文摘要

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.