关于栈的Lefschetz超平面定理
Lefschetz Hyperplane Theorem for Stacks
摘要：
利用莫尔斯理论证明了Lefschetz超平面定理对复流形上的紧致光滑Deligne-Mumford栈成立。对于超平面截面$Z \subset X$，$X$可由$Z$通过一系列形变收缩核和附加的高维有限圆盘变形获得。我们用它来推导更熟悉的关于$(X,Z)$对的相对同伦、同调和上同调群的陈述。我们还证明了一些初步的结果表明，莱夫谢茨超平面定理成立的Artin堆栈以及。一个技术上的创新是重新引入一个不等式，它允许我们证明定理没有任何一般性或非退化假设$Z$。
英文摘要：
We use Morse theory to prove that the Lefschetz Hyperplane Theorem holds for compact smooth Deligne-Mumford stacks over the site of complex manifolds. For $Z \subset X$ a hyperplane section, $X$ can be obtained from $Z$ by a sequence of deformation retracts and attachments of high-dimensional finite disc quotients. We use this to derive more familiar statements about the relative homotopy, homology, and cohomology groups of the pair $(X,Z)$. We also prove some preliminary results suggesting that the Lefschetz Hyperplane Theorem holds for Artin stacks as well. One technical innovation is to reintroduce an inequality of Łojasiewicz which allows us to prove the theorem without any genericity or nondegeneracy hypotheses on $Z$.
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