Let
S
S
be a nonsingular complex algebraic variety and
V
\mathcal {V}
a polarized variation of Hodge structure of weight
2
p
2p
with polarization form
Q
Q
. Given an integer
K
K
, let
S
(
K
)
{S^{(K)}}
be the space of pairs
(
s
,
u
)
(s,u)
with
s
∈
S
s \in S
,
u
∈
V
s
u \in {\mathcal {V}_s}
integral of type
(
p
,
p
)
(p,p)
, and
Q
(
u
,
u
)
≤
K
Q(u,u) \leq K
. We show in Theorem 1.1 that
S
(
K
)
{S^{(K)}}
is an algebraic variety, finite over
S
S
. When
V
\mathcal {V}
is the local system
H
2
p
(
X
s
,
Z
)
{H^{2p}}({X_s},\mathbb {Z})
/torsion associated with a family of nonsingular projective varieties parametrized by
S
S
, the result implies that the locus where a given integral class of type
(
p
,
p
)
(p,p)
remains of type
(
p
,
p
)
(p,p)
is algebraic.