  Given a morphism between complex projective varieties, we make several
conjectures on the relations between the set of pseudo-effective (co)homology
classes which are annihilated by pushforward and the set of classes of
varieties contracted by the morphism. We prove these conjectures for classes of
curves or divisors. We also prove that one of these conjectures implies
Grothendieck's generalized Hodge conjecture for varieties with Hodge coniveau
at least 1.
