Special birational structures on non-K\"ahler complex surfaces

TL;DR

This paper explores the existence of birational structures on non-Kähler complex surfaces, particularly class VII_0^+ surfaces, and introduces new normal forms for contracting germs in the Cremona group.

Contribution

It demonstrates that certain Kato surfaces with specific configurations admit a special birational structure via new normal forms, advancing understanding of non-Kähler surface structures.

Findings

Kato surfaces with a cycle and one branch of rational curves have a birational structure.

Surfaces with GSS and 0<b_2(S)<4 admit a birational structure.

Existence of meromorphic maps from universal covers to P^2 that blow down rational curves.

Abstract

We investigate the following conjecture: all compact non-K\"ahler complex surfaces admit birational structures. After Inoue-Kobayashi-Ochiai, the remaining cases to study are essentially surfaces in class VII_0^+. In case of Kato surfaces with a cycle and one branch of rational curves we show that they have a special birational structure given by new normal forms of contracting germs in Cremona group Bir(P^2(C)). In particular all surfaces S with GSS and 0<b_2(S)<4 admit a birational structure. From the existence of a special birational structure we deduce meromorphic mappings from the universal cover of S to the projective plane which blow down an infinite number of rational curves.