  The rational homology groups of the packing complexes are important in
algebraic geometry since they control the syzygies of line bundles on
projective embeddings of products of projective spaces (Segre--Veronese
varieties). These complexes are a common generalization of the multidimensional
chessboard complexes and of the matching complexes of complete uniform
hypergraphs, whose study has been a topic of interest in combinatorial
topology. We prove that the multivariate version of representation stability, a
notion recently introduced and studied by Church and Farb, holds for the
homology groups of packing complexes. This allows us to deduce stability
properties for the syzygies of line bundles on Segre--Veronese varieties. We
provide bounds for when stabilization occurs and show that these bounds are
sometimes sharp by describing the linear syzygies for a family of line bundles
on Segre varieties.
  As a motivation for our investigation, we show in an appendix that Ein and
Lazarsfeld's conjecture on the asymptotic vanishing of syzygies of coherent
sheaves on arbitrary projective varieties reduces to the case of line bundles
on a product of (at most three) projective spaces.
