  The cohomology of a compact Kaehler (resp. hyperKaehler) manifold admits the
action of the Lie algebra so(2,1) (resp. so(4,1)). In this paper we show,
following an idea of Witten, how this action follows from supersymmetry, in
particular from the symmetries of certain supersymmetric sigma models. In
addition, many of the fundamental identities in Hodge-Lefschetz theory are also
naturally derived from supersymmetry.
