Let
X
X
be a smooth projective variety over
C
\mathbb {C}
and
L
L
a nef-big (resp. ample) divisor on
X
X
. Then
(
X
,
L
)
(X,L)
is called a quasi-polarized (resp. polarized) manifold. Then we conjecture that
g
(
L
)
≥
q
(
X
)
g(L)\geq q(X)
, where
g
(
L
)
g(L)
is the sectional genus of
L
L
and
q
(
X
)
=
dim
H
1
(
O
X
)
q(X)=\operatorname {dim}H^{1}(\mathcal {O}_{X})
is the irregularity of
X
X
. In general it is unknown whether this conjecture is true or not, even in the case of
dim
X
=
2
\operatorname {dim}X=2
. For example, this conjecture is true if
dim
X
=
2
\operatorname {dim}X=2
and
dim
H
0
(
L
)
>
0
\operatorname {dim}H^{0}(L)>0
. But it is unknown if
dim
X
≥
3
\operatorname {dim}X\geq 3
and
dim
H
0
(
L
)
>
0
\operatorname {dim}H^{0}(L)>0
. In this paper, we prove
g
(
L
)
≥
q
(
X
)
g(L)\geq q(X)
if
dim
X
=
3
\operatorname {dim}X=3
and
dim
H
0
(
L
)
≥
2
\operatorname {dim}H^{0}(L)\geq 2
. Furthermore we classify polarized manifolds
(
X
,
L
)
(X,L)
with
dim
X
=
3
\operatorname {dim}X=3
,
dim
H
0
(
L
)
≥
3
\operatorname {dim}H^{0}(L)\geq 3
, and
g
(
L
)
=
q
(
X
)
g(L)=q(X)
.