Pluricanonical systems on minimal algebraic varieties
Pluricanonical systems on minimal algebraic varieties
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DOI：
10.1007/bf01388524
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发表时间：
1985-10
影响因子：
3.1
通讯作者：
Y. Kawamata
中科院分区：
文献类型：
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作者：
Y. Kawamata
一个极小代数簇X是一个正规投射簇，它只有标准奇点，其标准因子Kx是nef。如果它的科代拉维数等于数值科代拉维数，则称之为好的（精确定义见第1节）。本文的主要结果是：对于定义在特征为零的代数闭域上的好的极小代数簇X，m-标准系统ImKxl对某个正整数m是基点自由的（定理1.（一）。如果Kx是大的，那么这已经在Benveniste [1]和Kawamata [12]中在dim X= 3的情况下以及在Shokuroy [19]中得到了证明（也参见Kawamata [13]）。我们利用J. Koll~ r的消失定理[15]证明了我们的结果，而不是像[1，12，13]和[19]中那样使用科代拉消失定理的修改版本.这个证明与[13，定理2.6]的证明几乎是平行的。本文的结构如下。在第1节中，我们修正了我们的符号，并陈述了主要结果。第二节证明了nef和好因子本质上是nef和大因子的拉回。第三节给出了稍微推广形式的科尔格尔消失定理。第四节得到了广义正规交叉簇的一些基本性质。在证明了第5节中的一个关键引理H~的非零性之后，我们给出了第6节中主要结果的一个推广证明，最后的第7节是为了进一步的发展而增加的。
A minimal algebraic variety X is a normal projective variety having only canonical singularities whose canonical divisor K x is nef. It is called good if its Kodaira dimension is equal to the numerical Kodaira dimension (for the precise definition see Sect. 1). The main result of this paper is the following: For a good minimal algebraic variety X defined over an algebraically closed field of characteristic zero, m-canonical system ImKxl is base point free for some positive integer m (Theorem 1. I). If K x is big, then this was already proved in Benveniste [1] and Kawamata [12] in case dim X= 3 and in Shokuroy [19] generally (see also Kawamata [13]). Instead of using a modified version of the Kodaira vanishing theorem as in [1, 12, 13] and [19], we make use of J. Koll~ r's vanishing theorem [15] to prove our result. The proof is almost parallel to that of [13, Theorem 2.6]. The construction of this paper is as follows. In Sect. 1 we fix our notation and state the main result. It is shown in Sect. 2 that nef and good divisors are essentially the pull-backs of nef and big divisors, We give J. Kollgr's vanishing theorem in a slightly generalized form in Sect. 3. Section 4 obtains some elementary properties of generalized normal crossing varieties. After proving a technically key lemma, the non-vanishing of H~ in Sect. 5, we give a proof of the main result in Sect. 6 in a generalized version, The last Sect, 7 is added for further development, In this paper we always assume that the ground field k is algebraically closed and of characteristic zero,