In mathematics, a Godeaux surface is one of the surfaces of general type introduced by Lucien Godeaux

Lucien Godeaux

Lucien Godeaux was a prolific Belgian mathematician. His total of more than 1000 papers and books, 669 of which are found in Mathematical Reviews, made him one of the most published mathematicians. He was the sole author of all but one of his papers, and his Erdős number is undefined.He is best...

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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 surface

Barlow surface

In mathematics, a Barlow surface is one of the complex surfaces introduced by .They are simply connected surfaces of general type with pg = 0. They are homeomorphic but not diffeomorphic to a projective plane blown up in 8 points....

s) are called numerical Godeaux surfaces.

Construction

The cyclic group of order 5 acts freely on the Fermat surface of points (w : x : y : z)
in P³ satisfying w⁵ + x⁵ + y⁵ + z⁵ = 0 by mapping (w : x : y : z) to (w:ρx:ρ²y:ρ³z) where ρ is a fifth root of 1. The quotient by this action is the original Godeaux surface.

Invariants

The fundamental group (of the original Godeaux surface) is cyclic of order 5.

Hodge diamond:

		1
	0		0
0		9		0
	0		0
		1