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

设 X 为正规完备代数簇，L 为其可逆束，我考虑了以下有效的非零猜想：“假设 X 上存在 R 约数 B，使得 (X, B) 为 KIT， L 是 nef 且 L - (K_X +B) 是 nef 且很大。则存在 L 的非零全纯全局部分。我在数值 Kodaira 维数的情况下证明了这一点。 L 至多为 2，或 X 为最小 3 倍或 Fano 4 倍在证明过程中，我结合我的附加定理得到了代数纤维空间的半正定理的对数版本。前面证明了，我们可以将结果应用于Fano品种的梯子存在问题。我考虑了藤田自由度猜想的以下相对版本，它可能导致藤田原始猜想的解决方案任意维猜想：“设 f 是从一个平滑射影簇 Y 到另一个平滑射影簇 X 的满射态射，使得 f … 更多在 X 上的正常交叉除数的补集上是平滑的，并且 L 是一个充足的线束X. 设 F 为 Y 的规范束的直像束，如果 m 至少为 n+1，则 F 与 L 的 m 次方的张量乘积由全局截面生成。” 得到的结果。指出相对猜想被简化为关于某个对数正则除数局部存在的猜想，因此当n至多4时相对猜想得到证实。为了证明这个结果，我扩展了消失定理的Q除数版本对于直接图像层 F 而言，F 上的抛物线结构，其中抛物线结构是使用霍奇丛的过滤来定义的，该霍奇丛由霍奇结构变分的单向性确定。从代数簇相干滑轮有界派生范畴理论的角度考虑了翻转的存在性问题的一种新方法。作为准备，我证明了翻转的存在性问题被简化为翻转的存在性问题。然后我通过例子证明，对于具有商奇点的簇，相干滑轮的通常有界派生类别在翻牌下不一定是不变的。然后我表明，如果我们考虑轨道滑轮而不是通常的滑轮，那么在某些情况下一切都会很好。特殊情况较少