A note on the abundance conjecture
A note on the abundance conjecture
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DOI：
10.14231/ag-2015-020
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发表时间：
2014-06
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通讯作者：
Tobias Dorsch;Vladimir Lazi'c
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作者：
Tobias Dorsch;Vladimir Lazi'c
在假设低维最小模型程序的条件下，我们证明了n维非无界Kawamata对数终端对的丰度猜想蕴含着n维无界Kawamata对数终端对的丰度猜想.对于KX+，伪集有一个极小模型(Y；Y)使得KY+Y是半充分的。这样的最小模型被称为好模型。众所周知，好模型的存在蕴含着丰度猜想，该猜想预言每个极小模型都是好的。我们说，如果潜在的簇是这样的，那么一个对是无规则的，对于一个非无规则的对也是如此。在这篇文章中，我们证明了上述猜想对于非uniruled对是成功的。更准确地说，以下是我们的主要结果。定理1.1。假设最多存在n1维的KLT对的好模型。如果丰富度猜想对n维的非无序KLT对成立，那么跳跃猜想对n维的非无序KLT对也成立。定理1.2。假设n维KLT对存在好模型，则n维非无界KLT对好模型的存在蕴含着n维非无界KLT对好模型的存在。通过传递到终端调制(见定理2.1和定理2.2)，并利用(BDPP13)的主要结果，定理1.1和1.2证明了好模型的存在性和终端对(X；)的充分性猜想。
We prove that the abundance conjecture for non-uniruled Kawamata log terminal pairs in dimension n implies the abundance conjecture for uniruled Kawamata log terminal pairs in dimension n, assuming the Minimal Model Program in lower dimensions. with KX + pseudo- eective has a minimal model ( Y; Y ) such that KY + Y is semiample. Such a minimal model is called a good model. It is well known that the existence of good models implies the abundance conjecture, which predicts that every minimal model is good. We say that a pair is uniruled if the underlying variety is so, and similarly for a non-uniruled pair. In this paper, we show that it suces to prove the aforementioned conjectures for non- uniruled pairs. More precisely, the following are our main results. Theorem 1.1. Assume the existence of good models for klt pairs in dimensions at most n 1. If the abundance conjecture holds for non-uniruled klt pairs in dimension n, then the abun- dance conjecture holds for uniruled klt pairs in dimension n. Theorem 1.2. Assume the existence of good models for klt pairs in dimensions at most n 1. Then the existence of good models for non-uniruled klt pairs in dimension n implies the existence of good models for uniruled klt pairs in dimension n. Observe that by passing to a terminal modication (see Denition 2.1 and Theorem 2.2), and by using the main result of (BDPP13), Theorems 1.1 and 1.2 show that it suces to prove the existence of good models and the abundance conjecture for terminal pairs (X; )