Tensor differential forms on algebraic varieties

Tensor (differential) forms on projective varieties are defined and studied in connection with certain birational invariants.
A sufficiently complete picture of the set of all tensor forms of the first kind on smooth projective hypersurfaces is given.
Restrictions of tensor forms, given on any smooth projective variety, to its intersection with a hypersurface are investigated. In particular, an analog of Lefschetz's theorem involving the usual skew-symmetric form is proved.