| Project/Area Number |
16540027
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Hiroshima University |
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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)
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| 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)
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| Project Period (FY) |
2004 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| 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)
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| 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 |
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| 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.
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