Study of Vector Bundles on Manifolds

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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.