  In this paper we show some Lefschetz-type theorems for the effective cone of
Hyperk\"ahler varieties. In particular we are able to show that the inclusion
of any smooth ample divisor induces an isomorphism of effective cones. Moreover
we deduce a similar statement for some effective exceptional divisors, which
yields the computation of the effective cone of e.g. projectivized cotangent
bundles and some projectivized Lazarsfeld--Mukai bundles.
