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
ZβX 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.Comment: 16 page