Non-classical Godeaux surfaces

A non-classical Godeaux surface is a minimal surface of general type with χ = K ² = 1 but with h ⁰¹ ≠ 0. We prove that such surfaces fulfill h ⁰¹ = 1 and they can exist only over fields of positive characteristic at most 5. Like non-classical Enriques surfaces they fall into two classes: the singular and the supersingular ones. We give a complete classification in characteristic 5 and compute their Hodge-, Hodge–Witt- and crystalline cohomology (including torsion). Finally, we give an example of a supersingular Godeaux surface in characteristic 5.