Lefschetz Hyperplane Theorem for Stacks

Details Lefschetz Hyperplane Theorem for Stacks Lefschetz Hyperplane Theorem for Stacks

2010-08-04

arxiv.org/abs/1008.0891v1

Mathematics

Differential Geometry

16 pages

Scientific paper

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 {\L}ojasiewicz which allows us to prove the theorem without any genericity or nondegeneracy hypotheses on $Z$.

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