A complex manifold
X
X
of dimension
n
n
together with an ample vector bundle
E
E
on it will be called a generalized polarized variety. The adjoint bundle of the pair
(
X
,
E
)
(X,E)
is the line bundle
K
X
+
d
e
t
(
E
)
K_X + det(E)
. We study the positivity (the nefness or ampleness) of the adjoint bundle in the case
r
:=
r
a
n
k
(
E
)
=
(
n
−
2
)
r := rank (E) = (n-2)
. If
r
≥
(
n
−
1
)
r\geq (n-1)
this was previously done in a series of papers by Ye and Zhang, by Fujita, and by Andreatta, Ballico and Wisniewski.
If
K
X
+
d
e
t
E
K_X+detE
is nef then, by the Kawamata-Shokurov base point free theorem, it supports a contraction; i.e. a map
π
:
X
⟶
W
\pi :X \longrightarrow W
from
X
X
onto a normal projective variety
W
W
with connected fiber and such that
K
X
+
d
e
t
(
E
)
=
π
∗
H
K_X + det(E) = \pi ^*H
, for some ample line bundle
H
H
on
W
W
. We describe those contractions for which
d
i
m
F
≤
(
r
−
1
)
dimF \leq (r-1)
. We extend this result to the case in which
X
X
has log terminal singularities. In particular this gives Mukai’s conjecture 1 for singular varieties. We consider also the case in which
d
i
m
F
=
r
dimF = r
for every fiber and
π
\pi
is birational.