It is known that the hard Lefschetz action, together with K¨ahler identities
for K¨ahler (resp. hyperk¨ahler) manifolds, determines a su(1, 1)_sup (resp. sp(1, 1)_sup)
Lie superalgebra action on differential forms. In this paper, we explain the geometric
origin of this action, and we also generalize it to manifolds with other holonomy groups.
For semi-flat Calabi-Yau (resp. hyperk¨ahler) manifolds, these symmetries can be
enlarged to a so(2, 2)_sup (resp. su(2, 2)_sup) action.