  The theorem of Barth-Lefschetz is a statement about the cohomology of a
submanifold X of some projective space, in a range depending on the codimension
of the embedding. Here this is generalized to the case of a submanifold X of a
smooth projective toric variety Y, under some natural conditions on X. In
particular, in the Barth-Lefschetz range a formula for the Hodge numbers of X
in terms of the toric, i.e. combinatorial data of Y is given.
