Holomorphic geometric structures on Hopf manifolds
Matthieu Madera
TL;DR
This work develops a coherent geometric framework to study complex Hopf manifolds by constructing holomorphic G-structures of order 1 and extending them to higher orders, yielding flat Cartan geometries with model $(\widetilde{G^r_\beta},G^r_\beta)$. The approach hinges on sub-resonant polynomial transformations determined by the contraction’s eigenvalues $\beta$ and their resonances, together with Mall-bundle cohomology to control obstructions. In dimension $n\ge 3$, the authors show existence and uniqueness of extensions from order $k$ to $k+1$ for all $k$, while Hopf surfaces require refined analysis, including Mall’s theorem and Ornea-Verbitsky results, due to potential nontrivial cohomology. The core achievement is a Frobenius-type integrability result that yields a flat Cartan geometry, which in turn provides special charts and a global, geometric proof of the Poincaré-Dulac theorem in this setting, clarifying the link between normal forms and holomorphic geometric structures.
Abstract
We construct integrable holomorphic G-structures and flat holomorphic Cartan geometries on every complex Hopf manifold, without using the normal forms given by the Poincaré-Dulac Theorem. We provide a new proof of the latter using charts adapted with the geometric structures.