Study of Vector Bundles on Manifolds

2006 Fiscal Year Final Research Report Summary

• Project/Area Number: 16540027
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Hiroshima University
• Principal Investigator: SUMIHIRO Hideyasu Hiroshima University, Graduate School of Science, Honorary Professor, 大学院理学研究科, 名誉教授 (60068129)
• Co-Investigator(Kenkyū-buntansha): ISHII Akira Hiroshima University, Graduate School of Science, Associated Professor, 大学院理学研究科, 助教授 (10252420)
• Project Period (FY): 2004 – 2006
• Keywords: Algebra / Vector Bundles / Hartshorne Conjecture / Determinantal Varieties / Hilbert Schemes / Frobenius morphisms / Stable Vector Bundles

Research Abstract

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.