  We construct examples of surfaces of general type with $p_g=1$, $q=0$ and
$K^2=6$. We use as key varieties Fano fourfolds and Calabi-Yau threefolds that
are zero section of some special homogeneous vector bundle on Grassmannians. We
link as well our construction to a classical Campedelli surface, using the
Pfaffian-Grassmannian correspondence.
