权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

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是正规完备代数簇，L是其上的可逆层.我考虑了以下有效的非零猜想：“假设X上存在一个R-因子B，使得对（X，B）是KIT，L是nef。而L -（K_X +B）是nef和big。则存在L”的非零全纯整体截面。我证明了它的情况下，数值科代拉维数的L是最多2，或X是一个最小的3倍或法诺4倍。在证明过程中，我得到了代数纤维空间的半正性定理的对数形式。结合我前面证明的附加定理，可以将结果应用于Fano簇上梯子的存在问题。我考虑了以下相对版本的Fujita自由猜想，这可能会导致Fujita原始猜想在任意维度上的解：“令f是从光滑投射簇Y到另一光滑投射簇X的满射态射，使得f ...更多信息在X上的正规交叉因子的补上是光滑的，L是X上的样本线丛。设F是Y的标准层被f的直像层。则F和L的m次幂的张量积由整体截面生成，如果m至少为n+1。“所得结果表明，相对猜想被归结为关于局部存在某个对数典型因子的猜想，从而当n不大于4时，相对猜想被证实.为了证明这一结果，我利用F上的抛物结构推广了直象层F的Q-除灭定理，其中抛物结构是利用由Hodge结构的变分单值性所决定的Hodge丛的滤子来定义的，我从有界导范畴理论的观点考虑了一种新的方法来解决翻转的存在性问题代数簇上的凝聚层作为准备，我证明了翻转的存在性问题归结为触发器的存在性问题。然后，我通过例子表明，对于具有商奇点的簇，通常的凝聚层的有界导出范畴在触发器下不一定是不变的。然后我证明了，如果我们考虑orbifold层而不是通常的层，在某些特殊情况下一切都很好。少