A non-algebraizable foliation
Paulo Sad
TL;DR
The article constructs a pair of holomorphic foliations on complex surfaces that are topologically equivalent but differ in holomorphic/algebraic type. It uses a Neeman-type construction and a twisted gluing to produce a non-algebraizable germ, leveraging holomorphic convexity, Stein/1-convex techniques, and a generalized Hartogs extension to prove non-embeddability. The main result shows that topological equivalence of foliations does not imply holomorphic or algebraic equivalence of the ambient surfaces, highlighting a nontrivial moduli phenomenon for foliations on non-algebraic backgrounds. This provides a concrete example where a foliation topologically mirrors an algebraic one while residing on a non-algebraizable surface, clarifying limitations of holomorphic classification in foliations.
Abstract
It is presented an example of a holomorphic foliation of a non-algebraizable surface which is topologically equivalent to an algebraic foliation.