  We generalize the notion of coisotropic hypersurfaces to subvarieties of
Grassmannians having arbitrary codimension. To every projective variety X,
Gel'fand, Kapranov and Zelevinsky associate a series of coisotropic
hypersurfaces in different Grassmannians. These include the Chow form and the
Hurwitz form of X. Gel'fand, Kapranov and Zelevinsky characterized coisotropic
hypersurfaces by a rank one condition on conormal spaces, which we use as the
starting point for our generalization. We also study the dual notion of
isotropic varieties by imposing rank one conditions on tangent spaces instead
of conormal spaces.
