Study of Vector Bundles on Manifolds

研究了定义在代数闭域k （P = chark > 0）上的n维射影空间P^n （n≧4）上的两个秩向量束的分裂问题，得到了如下结果。1) Bogomolov分解的变形研究：设E为P^4上满足c_1^2-4c_2≧0 （c_1为E的第i个陈氏数）的二阶向量束。让X是一个行列式表面相关E和Z, Z ^ *因子X相关E .此外,让我们表示由E ^ < (q) > | X矢量的逆象束E | X F是学位的弗罗贝尼乌斯射q = p ^ n X然后我们看到E ^ < (q) > | X∈H ^ 1 (X, O (q (Z + Z ^ *))和任何变形G∈H ^ 1 (X, O (q (Z + Z ^ *))的E ^ < (q) > | X Bogomolov感是不稳定的。G有如下的Bogomolov分解：0→O（qC+rZ）→G→I叉乘O((2q-r)Z))→0。定理：当且仅当r≥q时，E是线束的直接和。2)向量束的Frobenius态射直接像的稳定性研究：设X是定义在代数闭域k （p = chark > 0）上的非奇异投影曲面，F是X的Frobenius态射。定理：设X为非奇异投影曲面，H为X上的数值正直线束，设Ω_x^1对H和K_xH >是半稳定的。则对于X上的任何线束L，其直接像F_*(L)对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对K_x是半稳定的，则对于任何线束L，其直接像F_*(L)对K_x是半稳定的。进一步设X是一个非奇异投影曲面，使得K_x在数值上是平凡的，并且Ω_x^1对于X上的数值正的线束H是半稳定的，那么对于任何线束L，我们看到F_*(L)对于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.