Research on log canonical divisors on higher dimensional algebraic varieties

高维代数簇的对数正则因数研究

基本信息

批准号：
11440002
负责人：
KAWAMATA Yujiro
金额：
$ 5.25万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

项目摘要

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

令X为正常的完整代数品种，并在其上进行可逆捆。我考虑了以下有效的非变化概念：“假设x上存在一个R-divisor b，使得（x，b）为套件，l是nef。，l-（k_x +b）是nef and big。然后存在非零的holomorphic holomorphic lomorphic lomorphic lomorphic lomorphic lomerphic los l''。我证明了这是在L的数值Kodaira尺寸最多为2的情况下，或者X是最小的3倍或Fano 4倍。在证明过程中，我获得了代数纤维空间的半积极性理论的对数版本。结合我之前证明的调整理论的结合，可以将结果应用于fano品种上的存在问题。我考虑了以下相对版本，这可能会导致福吉塔在任意维度中的原始概念的解决方案，从而使f从平稳的变种中脱落到平稳的变化中……在X上，在X上有足够的线束。让F为Y的直接图像捆，然后F。获得的结果指出，相对猜想被简化为对某些对数规范除数的本地存在的猜想，因此，当n最多4最多4时，确认了相对缔合。霍奇结构变化的单构型。我从代数品种上相干滑轮的有限派生类别的观点的角度考虑了一种新的方法。作为准备工作，我规定，翻转的存在问题减少到了失败的存在问题。然后，我以举例说明，对于引号奇异性的变化，相干滑轮的通常有限的派生类别不一定在拖鞋下不变。然后，我表明，如果我们考虑Orbifold滑轮而不是通常的束带，那么在某些特殊情况下，一切都很好。较少的