Research on log canonical divisors on higher dimensional algebraic varieties

设X是正规完备代数簇，L是其上的可逆丛。我考虑了如下有效的非零猜想：“假设X上存在一个R-因子B，使得(X，B)是Kit的，L是nef的，L-(K_X+B)是nef且大的，则存在L的非零全纯整体截口。”在L的数值Kodaira维至多为2，或者X是极小3重或Fano 4重的情况下证明了这一点。在证明过程中，我得到了代数纤维空间的半正性定理的一个对数版本。结合我前面证明的加法定理，我们可以把这个结果应用于Fano簇上的梯子的存在问题。我考虑了Fujita Free猜想的以下相对版本，它可能导致Fujita在任意维的原始猜想的解：“设f是从一个光滑射影簇Y到另一个光滑射影簇X的满射，使得f…更多的是在X上的正规交叉因子的补上光滑的，L是X上的一个充分线丛.设F是Y的正则层f的直象层.如果m至少是n+1，则F与L的m次方的张量积是由整体截集生成的.得到的结果表明，当n至多为4时，相对猜想归结为关于局部存在某个对数正则因子的猜想，从而当n至多为4时，相对猜想被证实。为了证明这一结果，我用F上的抛物结构推广了直象叶F的q除法定理，其中抛物结构是用由Hodge结构的变异的单数决定的Hodge丛的滤子来定义的，我从代数簇上凝聚导叶的有界导出范畴的观点考虑了一种新的方法来解决翻转的存在性问题。作为准备，我证明了翻转的存在性问题归结为翻转的存在性问题。然后，我举例说明，对于具有商奇性的簇，通常的有界相干层派生范畴在FLOPS下不一定不变。然后，我展示了，如果我们考虑奥比福斯滑轮而不是通常的滑轮，在某些特殊情况下，一切都工作得很好。较少

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