  I use local differential geometric techniques to prove that the algebraic
cycles in certain extremal homology classes in Hermitian symmetric spaces are
either rigid (i.e., deformable only by ambient motions) or quasi-rigid (roughly
speaking, foliated by rigid subvarieties in a nontrivial way).
  These rigidity results have a number of applications: First, they prove that
many subvarieties in Grassmannians and other Hermitian symmetric spaces cannot
be smoothed (i.e., are not homologous to a smooth subvariety). Second, they
provide characterizations of holomorphic bundles over compact Kahler manifolds
that are generated by their global sections but that have certain polynomials
in their Chern classes vanish (for example, c_2 = 0, c_1c_2 - c_3 = 0, c_3 = 0,
etc.).
