Normal forms and geometric structures on Hopf manifolds

TL;DR

This paper demonstrates that Hopf manifolds can support holomorphic geometric structures, extending known results to higher dimensions by revisiting classical normal form theories and group actions.

Contribution

It extends the existence of holomorphic $(G,X)$-structures on Hopf manifolds to all dimensions, building on prior work and classical normal form approaches.

Findings

Hopf manifolds admit holomorphic $(G,X)$-structures in any dimension.

Revisits and applies Guysinsky-Katok's group of invertible sub-resonant polynomials.

Utilizes Poincaré-Dulac normal form theory in the context of complex geometry.

Abstract

We prove that Hopf manifolds admit holomorphic $(G,X)$-structures, extending to any dimension a result of McKay and Pokrovskiy. For this, we revisit Guysinsky-Katok's group of invertible sub-resonant polynomials, and Bertheloot's approach of Poincar\'e-Dulac normal form theory.