In mathematics, a Godeaux surface is one of the
surfaces of general type In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. Because of Algebraic geometry and analytic geometry#Chow.27s theorem, Chow's theorem any compact complex manifold of dimension 2 and with Kodaira ...
introduced by
Lucien Godeaux
Lucien Godeaux (1887–1975) 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 o ...
in 1931.
Other surfaces constructed in a similar way with the same
Hodge number
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohom ...
s are also sometimes called Godeaux surfaces. Surfaces with the same Hodge numbers (such as
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. T ...
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 ''P''
3 satisfying ''w''
5 + ''x''
5 + ''y''
5 + ''z''
5 = 0 by mapping (''w'' : ''x'' : ''y'' : ''z'') to (''w:ρx:ρ
2y:ρ
3z'') 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
like rational surfaces do, though it is not rational. The square of the first Chern class
(and moreover the canonical class is ample).
See also
*
Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohom ...
References
*{{Citation , last1=Barth , first1=Wolf P. , last2=Hulek , first2=Klaus , last3=Peters , first3=Chris A.M. , last4=Van de Ven , first4=Antonius , title=Compact Complex Surfaces , publisher= Springer-Verlag, Berlin , series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. , isbn=978-3-540-00832-3 , mr=2030225 , year=2004 , volume=4
Algebraic surfaces
Complex surfaces