Project/Area Number 
16540027

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear 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)

CoInvestigator(Kenkyūbuntansha) 
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

Project Status 
Completed (Fiscal Year 2006)

Budget Amount *help 
¥3,800,000 (Direct Cost: ¥3,800,000)
Fiscal Year 2006: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2005: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2004: ¥1,800,000 (Direct Cost: ¥1,800,000)

Keywords  Algebra / Vector Bundles / Hartshorne Conjecture / Determinantal Varieties / Hilbert Schemes / Frobenius morphisms / Stable Vector Bundles / Hartshome予想 / フロべニウス写像 / 射影空間上のベクトル束 / ベクトル束の分解 
Research Abstract 
We have studied splitting problem of rank two vector bundles on ndimensional 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_1^24c_2 ≧ 0 ( c_1 being the ith 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((2qr)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 semistable with respect to H and K_xH > 0. Then for any line bundle L on X, the direct image F_*(L) is semistable with respect H. In particular, if X is a nonsingular minimal surface of general type whose Ω_x^1 is semistable with respect to K_x, then for any line bundle L, the direct image F_*(L) is semistable with respect K_x. Further let X be a nonsingular projective surface such that K_x is numerically trivial and Ω_x^1 is semistable with respect to a numerically positive line bundle H on X. Then for any line bundle L, we see that F_*(L) is semistable with respect H.
