Study of Vector Bundles on Manifolds

研究了代数闭域k上n维射影空间P^n（n ≠ 4）上秩2向量丛的分裂问题（p = chark > 0），得到了如下结果：1）Bogomolov分解的变形研究：设E是P^4上的秩二向量丛，满足c_1^2-4c_2 <$0（c_1是E的第i个Chern数）.设X是一个与E和Z相关联的行列式曲面，Z^* 是X上与E相关联的因子。此外，让我们用E^<（q）>表示|X是向量丛E的逆像|其中F是X上次数为q = p^n的Frobenius态射。然后我们看到E^<（q）>| X ∈ H^1（X，O（-q（Z + Z^*））和E^<（q）>的任意变形G ∈ H^1（X，O（-q（Z + Z^*））|X在博戈莫洛夫意义上是不稳定的。G有如下的Bogomolov分解：0→O（qC+rZ）→G→I叉积O（（2 q-r）Z））→ 0.定理：E是线丛的直和当且仅当对大q r <$q. 2）用Frobenius态射研究向量丛的直象的稳定性：设X是定义在代数闭域k（p = chark > 0）上的非奇异射影曲面，F是X的Frobenius态射.定理：设X为非奇异射影曲面，H为X上的数值正定线丛，则X上的正向线丛的正像是一个正的正像。假设Ω_x^1关于H是半稳定的，且K_xH > 0。则对X上的任意线丛L，其直像F *（L）关于H是半稳定的。特别地，如果X是一般型的非奇异极小曲面，其Ω x ^1关于K x是半稳定的，则对任意线丛L，直像F *（L）关于K x是半稳定的.进一步设X是一个非奇异射影曲面，使得K_x在数值上是平凡的，Ω_x^1关于X上的一个数值上为正数的线丛H是半稳定的。则对任意线丛L，我们看到F_*（L）关于H是半稳定的。

We have studied splitting problem of rank two vector bundles on n-dimensional projective space P^n ( n ≧ 4 ) defined over an algebraically closed field k ( p = chark > 0 ) and obtained the following.1) A study of deformation of Bogomolov decomposition : Let E be a rank two vector bundle on P^4 satisfying c_1^2-4c_2 ≧ 0 ( c_1 being the i-th Chern number of E ). Let X be a determinantal surface associated to E and Z, Z^* divisors on X associated to E. In addition, let us denote by E^<(q)> | X the inverse image of the vector bundle E | X where F is the Frobenius morphism of degree q = p^n on X. Then we see that E^<(q)> | X ∈ H^1(X, O(-q(Z + Z^*)) and any deformation G ∈ H^1(X, O(-q(Z + Z^*)) of E^<(q)> | X is unstable in the sense of Bogomolov. G has the following Bogomolov decomposition : 0→O( qC+rZ )→G→I cross product O((2q-r)Z)) →0.Theorem : E is a direct sum of line bundles if and only if r ≧ q for large q.2) Study of stability of direct images of vector bundles by Frobenius morphisms : Let X be a nonsingular projective surface defined over an algebraically closed field k ( p = chark > 0) and F the Frobenius morphism of X. As for stability of direct images of vector bundles by Frobenius morphisms, we obtained the following.Theorem : Let X be a nonsingular projective surface and H a numerically positive line bundle on X. Assume that Ω_x^1 is semi-stable with respect to H and K_xH > 0. Then for any line bundle L on X, the direct image F_*(L) is semi-stable with respect H. In particular, if X is a nonsingular minimal surface of general type whose Ω_x^1 is semi-stable with respect to K_x, then for any line bundle L, the direct image F_*(L) is semi-stable with respect K_x. Further let X be a nonsingular projective surface such that K_x is numerically trivial and Ω_x^1 is semi-stable with respect to a numerically positive line bundle H on X. Then for any line bundle L, we see that F_*(L) is semi-stable with respect H.