Levi-flat hypersurfaces and their complement in complex surfaces

Abstract

In this work we study analytic Levi-flat hypersurfaces in complex algebraic surfaces. First, we show that if this foliation admits chaotic dynamics (i.e. if it does not admit a transverse invariant measure), then the connected components of the complement of the hypersurface are modifications of Stein domains. This allows us to extend the CR foliation to a singular algebraic foliation on the ambient complex surface. We apply this result to prove, by contradiction, that analytic Levi-flat hypersurfaces admitting a transverse affine structure in a complex algebraic surface have a transverse invariant measure. This leads us to conjecture that Levi-flat hypersurfaces in complex algebraic surfaces that are diffeomorphic to a hyperbolic torus bundle over the circle are fibrations by algebraic curves.

Presque toute surface K3 contient une infinité d’hypersurfaces Levi-plates linéaires

“…Lins Neto [20] a montré l'inexistence d'hypersurfaces Levi-plates analytiques réelles dans l'espace projectif complexe P n (C) (n ⩾ 3). Citons aussi par exemple l'article de survol [24], et les articles [4], [16], [22], [11], [9] et [1] pour diverses constructions et résultats d'inexistence. Notons en particulier [23] qui construit des exemples différents de ceux que nous allons étudier dans les surfaces de Kummer.…”