  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. 
Surfaces of general type with $p_g=1$, $q=0$, $K^2=6$ and Grassmannians