In order to study unitals in the projective plane
$PG\left(2,{q}^{2}\right)$, F. Buekenhout gave a
representation in
$PG\left(4,q\right)$ of the
unitary polar space
$H\left(2,{q}^{2}\right)$ as points
of a quadratic cone on a
${Q}^{}\left(3,q\right)$.
G. Lunardon used the BarlottiCofman representation of
$PG\left(3,{q}^{2}\right)$ to
represent
$H\left(3,{q}^{2}\right)$
in
$PG\left(6,q\right)$ as a
cone on a
${Q}^{+}\left(5,q\right)$.
He also proved that to any locally hermitian ovoid of
$H\left(3,{q}^{2}\right)$ corresponds
an ovoid of
${Q}^{+}\left(5,q\right)$
and conversely.
In this paper, we study the BarlottiCofman representation of the unitary polar
space
$H\left(n,{q}^{2}\right)$
for all
$n$
and we prove that to any locally hermitian partial ovoid of such spaces corresponds a
partial ovoid of an orthogonal polar space, and conversely. Further the locally hermitian
partial ovoid is maximal if and only if the corresponding partial ovoid of the orthogonal
polar space is maximal. As a consequence of the previous connection and a result of
A. Klein we obtain a geometric proof to derive that the orthogonal polar space
${Q}^{+}\left(4n+1,q\right)$ has no
ovoid when
$n>{q}^{3}$.
