Surfaces of general type with pg=1, q=0, K2=6 and grassmannians

We construct examples of surfaces of general type with 𝑝𝑔 = 1, 𝑞 = 0 and 𝐾2 = 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.