Algebraic varieties are homeomorphic to varieties defined over number fields

Adam Parusinski (1) , Guillaume Rond (2)

1 LJAD - Laboratoire Jean Alexandre Dieudonné
2 I2M - Institut de Mathématiques de Marseille

We show that every affine or projective algebraic variety defined over the field of real or complex numbers is homeomorphic to a variety defined over the field of algebraic numbers. We construct such a homeomorphism by choosing a small deformation of the coefficients of the original equations. This method is based on the properties of Zariski equisingular families of varieties. Moreover we construct an algorithm, that, given a system of equations defining a variety $V$, produces a system of equations with algebraic coefficients of a variety homeomorphic to $V$

Domaines: Géométrie algébrique [math.AG]

Adam Parusinski, Guillaume Rond. Algebraic varieties are homeomorphic to varieties defined over number fields. Commentarii Mathematici Helvetici, 2020, 95, pp.339--359. ⟨10.4171/CMH/490⟩. ⟨hal-02536240⟩