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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 thatare zero section of some special homogeneous vector bundle on Grassmannians. Welink as well our construction to a classical Campedelli surface, using thePfaffian-Grassmannian correspondence.

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