SEMISTABLE REDUCTION FOR ABELIAN VARIETIES
SEMISTABLE REDUCTION FOR ABELIAN VARIETIES
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阿贝尔簇的半稳态约简
DOI：
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发表时间：
2011
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通讯作者：
B. Conrad
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作者：
B. Conrad
在Christian的笔记中讨论了曲线的半稳定约化定理。在这些注记中，我们将利用这一结果来证明一个关于交换簇的类似定理。在介绍了半阿贝尔变种(使我们相信这个概念是稳健的)之后，我们将回顾半阿贝尔方案的概念(在克里斯蒂安的演讲中介绍)，回顾半稳定约化定理的陈述，并解释为什么在雅可比的特殊情况下证明这一结果就足够了。关于这种情况下的证明的讨论，指出Artin和Raynaud关于Picard函子的结果的作用，将在注释的最后部分给出。这些注解的主要内容是通过使用半稳定约化定理作为“黑箱”来证明关于交换簇的许多有用的事实(我们将其证明推迟到注解末尾的另一个原因)。
The semistable reduction theorem for curves was discussed in Christian’s notes. In these notes, we will use that result to prove an analogous theorem for abelian varieties. After some preliminaries on semi-abelian varieties (to convince us that the notion is a robust one), we will review the notion of a semi-abelian scheme (introduced in Christian’s lecture), recall the statement of the semistable reduction theorem, and explain why it suffices to prove that result in the special case of Jacobians. A discussion of the proof in that case, indicating the role of results of Artin and Raynaud concerning Picard functors, will be given in the final section of the notes. The bulk of these notes are concerned with proving many useful facts about abelian varieties by using the semistable reduction theorem as a “black box” (another reason we postpone its proof until the end of the notes).