Vector Bundles on Manifolds

我们已经研究了P^n（n [大于或等于] 4）的投影空间中两级矢量捆绑的分裂问题，并发现了以下两级矢量捆绑：理论让E在P^n（n [大于或等于] 4）， P a 4或5-P^n和let ^^-E = E| P是F到P的限制，然后E是一个直接的和，如果H^1 （P， End（^^-E）） = 0。Hence这一点暗示了我们可以将零性格中的碎片问题减少到积极性格中的一个。（2）Kodaira消失理论和几何结构的确定子变体在积极性格中的几何结构：理论让X是一个非单一的预测变量，定义了一个在积极性格中封闭的领域的积极例子线捆绑。然后存在以下的不平等： dim H^1 （X， L^<-1>）[小于或等于] dim H ^1（X，O_X）。理论让X在P ^n中成为一个确定子变量相关的E。然后我们有一个H^1（X，O_X， ） = H^1（X，Ω^1_X ） = 0.3）拓扑中的临界矢量束（n = 4）的分割理论：两个矢量束E被称为拓扑中的临界矢量束，如果c_1 =α+β， c_2=α·β （α，β μ Z）。定理如果α大于或等于（（-1+√<4β-3>）/2）β，那么E是一个直线捆绑的直接总和。在特别范围内;如果1 [less than or equal] β [less than or equal] 3，然后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 - 4 c_2> 0。然后我们就有了后续行动， a）E| X是Bogomolov的不稳定。b）E是直线捆绑的直接总和，如果只有dim H^1 （X， F* （结束（E））[小于或等于] O（q^1）为大q。

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.