2001 Fiscal Year Final Research Report Summary
Research on log canonical divisors on higher dimensional algebraic varieties
| Project/Area Number |
11440002
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | The University of Tokyo |
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Principal Investigator |
KAWAMATA Yujiro Graduate School of Mathematical Sciences, The University of Tokyo, Professor, 大学院・数理科学研究科, 教授 (90126037)
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| Co-Investigator(Kenkyū-buntansha) |
TERASOMA Tomohide Graduate School of Mathematical Sciences, The University of Tokyo, Associate Professor, 大学院・数理科学研究科, 助教授 (50192654)
ODA Takayuki Graduate School of Mathematical Sciences, The University of Tokyo, Professor, 大学院・数理科学研究科, 教授 (10109415)
KATSURA Toshiyuki Graduate School of Mathematical Sciences, The University of Tokyo, Professor, 大学院・数理科学研究科, 教授 (40108444)
OGUISO Keiji Graduate School of Mathematical Sciences, The University of Tokyo, Associate Professor, 大学院・数理科学研究科, 助教授 (40224133)
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| Project Period (FY) |
1999 – 2001
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| Keywords | non-vanishing theorem / algebraic variety / minimal variet / semipositivity theorem / Fujita conjecture / log canonical divisor / vanishing theorem / flip |
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| Research Abstract |
Let X be a normal complete algebraic variety and L an invertible sheaf on it . I considered the following effective non-vanishing conjecture : "Assume that there exists an R-divisor B on X such that the pair (X, B) is KIT, L is nef. and L - (K_X +B) is nef and big. Then there exists a non-zero holomorphic global sections of L". I proved it in the case where the numerical Kodaira dimension of L is at most 2, or X is a minimal 3-fold or a Fano 4-fold. In the course of the proof, I obtained a logarithmic version of the semipositivity theorem for algebraic fiber spaces. Combining with the adjunction theorem which I proved earlier, one can apply the result for the existence problem of ladders on Fano varieties.
I considered the following relative version of the Fujita freeness conjecture which may lead to the solution of Fujita's original conjecture in arbitrary dimension : "Let f be a surjective morphism from a smooth projective variety Y to an other smooth projective variety X such that f
… More
is smooth over the complement of a normal crossing divisor on X, and L an ample line bundle on X. Let F be the direct image sheaf of the canonical sheaf of Y by f. Then the tensor product of F and the m-th power of L is generated by global sections if m is at least n+1." The result obtained states that the relative conjecture is reduced to the conjecture on the local existence of certain log canonical divisor, and thus the relative conjecture is confirmed when n is at most 4. In order to prove the result, I extended the Q-divisorial version of the vanishing theorem for the direct image sheaf F in terms of the parabolic structure on F, where the parabolic structure is defined using the filtration of the Hodge bundle determined by the monodromy of the variation of Hodge structures.
I considered a new approach toward the existence problem of the flip from the view point of the theory of bounded derived categories of coherent sheaves on algebraic varieties. As a preparation, I proved that the existence problem of the flip is reduced to the existence problem of the flop. Then I showed by example that for varieties with quotient singularities, the usual bounded derived categories of coherent sheaves are not necessarily invariant under the flops. Then I showed that if we consider orbifold sheaves instead of usual sheaves, everything works well in some special cases. Less
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