Godeaux surface

In mathematics, a Godeaux surface is one of the surfaces of general type introduced by Lucien Godeaux in 1931. 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.

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. It has invariants [math]\displaystyle{ q = 0, p_g = 0 }[/math] like rational surfaces do, though it is not rational. The square of the first Chern class [math]\displaystyle{ c_1^2 = 1 }[/math] (and moreover the canonical class is ample).

See also

• Hodge theory

References

• Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3