Vector Bundles on Manifolds

流形上的向量束

• 批准号: 13640026
• 负责人: SUMIHIRO Hideyasu
• 金额: $ 1.86万
• 依托单位: Hiroshima University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640026/
• 关键词: 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

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.

本文研究了射影空间P^n（n ≥ 4）上的二秩向量丛的分裂问题，得到了如下结果：1）上同调准则：定理设E是P^n（n ≥ 4）上的二秩向量丛，P是P^n的4维或5维线性子空间，设^^-E = E| P是F对P的限制。则E是线丛的直和当且仅当H^1（P，End（^^-E））= 0。因此，这意味着我们可以将零特征的分裂问题归结为正特征的分裂问题。2）科代拉消失定理和正特征的行列式子簇的几何结构：定理设X是定义在正特征的代数闭域上的非奇异射影簇，L是X上的正样本线丛。则存在以下不等式：dimH^1（X，L^<-1>）[小于或等于] dimH^1（X，O_X）。定理设X是P ^n上与E相关联的具有正特征的行列式子簇. 3）拓扑平凡向量丛（n = 4）的一个分裂定理：一个秩为2的向量丛E称为拓扑平凡的，如果c_1 =α+β，c_2=α·β（α，β∈Z）。定理若α[大于或等于]（（-1+&lt;$&lt;4β-3&gt;）/2）β，则E是线丛的直和.特别是;若1[小于或等于]β[小于或等于]3，则E分裂成线阵。4）Frobenius态射（n = 4）的分裂定理：设F是指数为q = p^n的Frobenius态射。定理设c^2_1-4c_2 &gt; 0.然后我们有以下，a）E| X是博戈莫洛夫的不稳定状态B）E是线丛的直和当且仅当dim H^1（X，F*（End（E））[小于或等于] O（q^1）对大q。