Godeaux surface

Short description: One of the surfaces of general type introduced by Lucien Godeaux in 1931

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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.

## Contents

* 1 Construction
* 2 Invariants
* 3 See also
* 4 References

## 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 q=0,pg=0 like rational surfaces do, though it is not rational. The square of the first Chern class c12=1 (and moreover the canonical class is ample).

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## 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

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