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arXiv · 2501.10346

Normal forms and geometric structures on Hopf manifolds

Paul Boureau
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Abstract

We prove that every Hopf manifold of dimension $n\geq2$, primary or secondary, admits a holomorphic $(G,X)$-structure compatible with its complex structure, where $X=\mathbb{C}^n$ and $G$ is generated by the translations and the Guysinsky--Katok group of invertible sub-resonant polynomials. This extends to any dimension a result of B.~McKay and A.~Pokrovskiy, and rests on a self-contained treatment of Berteloot's approach to the Poincar\'e--Dulac normal form. We then study, in the body of the paper, to what extent the structure is unique: marked uniqueness fails, since diagonal Hopf manifolds already carry $n!$ pairwise inequivalent structures, but any two compatible structures differ by a global automorphism of $\mathbb{C}^n$, exactly one structure is aligned, and the compatible structures are classified by a coset space of the sub-resonant group.

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Paul Boureau. 2025-01-17. Normal forms and geometric structures on Hopf manifolds. https://arxiv.org/abs/2501.10346

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