Chern class can be any integer from 3 to 11. Surfaces with all these values are known; a few of the many examples that have been studied are: c2 = 3: Fake projective plane (Mumford surface). The first example was found by Mumford using p-adic geometry, and there are 50 examples altogether. They have the same Betti numbers as the projective plane, but are not homeomorphic to it as their fundamental groups are infinite. c2 = 4: Beauville surfaces are named for Arnaud Beauville and have infinite fundamental group. c2 ≥ 4: Burniat surfaces c2 = 10: Campedelli surfaces. Surfaces with the same Hodge numbers are called numerical Campedelli surfaces. c2 = 10: Catanese surfaces are simply connected. c2 = 11: Godeaux surfaces. The cyclic group of order 5 acts freely on the Fermat surface of points in P3 satisfying by mapping to where ρ is a fifth root of 1. The quotient by this action is the original Godeaux surface. Other surfaces constructed in a similar way with the same Hodge numbers are also sometimes called Godeaux surfaces. Surfaces with the same Hodge numbers (such as Barlow surfaces) are called numerical Godeaux surfaces. The fundamental group (of the original Godeaux surface) is cyclic of order 5. c2 = 11: Barlow surfaces are simply connected. Together with the Craighero-Gattazzo surface, these are the only known examples of simply connected surfaces of general type with pg = 0. Todorov surfaces give counterexamples to the conclusion of the Torelli theorem Other Examples Castelnuovo surfaces: Another extremal case, Castelnuovo proved that if the canonical bundle is very ample for a surface of general type then Castelnuovo surface are surfaces of general type such that the canonical bundle is very ample and that Complete intersections: A smooth complete intersection of hypersurfaces of degrees in Pn is a surface of general type unless the degrees are (2), (3), (2, 2) (rational), (4), (3, 2), (2, 2, 2) (Kodaira dimension 0). Complete intersections are all simply connected. A special case are hypersurfaces: for example, in P3, non-singular surfaces of degree at least 5 are of general type (Non-singular hypersurfaces of degree 4 are K3 surfaces, and those of degree less than 4 are rational). Fano surfaces of lines on a cubic 3-fold. Hilbert modular surfaces are mostly of general type. Horikawa surfaces are surfaces with q = 0 and or (which implies that they are more