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Normal forms and geometric structures on Hopf manifolds

Paul Boureau

TL;DR

The work addresses the existence of holomorphic $(G,X)$-structures on Hopf manifolds in all dimensions, extending the known Hopf-surface result. It combines the $SR^*(L)$ sub-resonant polynomial framework with a constructive $Poincaré-Dulac$ normal-form method and incorporates translations to enlarge the structure group to $G=\langle SR^*(L), \mathbb{C}^n\rangle$. The main contributions show that $SR^*(L)$ is a finite-dimensional algebraic group, provide a convergent conjugacy of any contracting germ to a polynomial in $SR^*(L)$, and realize Hopf manifolds as quotients by a cyclic action generated by a normal form $h\in SR^*(L)$, yielding a holomorphic $(G,X)$-structure. This generalizes the surface case and supplies explicit algebraic models with potential links to Cartan-geometric frameworks for complex manifolds.

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 Poincaré-Dulac normal form theory.

Normal forms and geometric structures on Hopf manifolds

TL;DR

The work addresses the existence of holomorphic -structures on Hopf manifolds in all dimensions, extending the known Hopf-surface result. It combines the sub-resonant polynomial framework with a constructive normal-form method and incorporates translations to enlarge the structure group to . The main contributions show that is a finite-dimensional algebraic group, provide a convergent conjugacy of any contracting germ to a polynomial in , and realize Hopf manifolds as quotients by a cyclic action generated by a normal form , yielding a holomorphic -structure. This generalizes the surface case and supplies explicit algebraic models with potential links to Cartan-geometric frameworks for complex manifolds.

Abstract

We prove that Hopf manifolds admit holomorphic -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 Poincaré-Dulac normal form theory.
Paper Structure (4 sections, 12 theorems, 34 equations)

This paper contains 4 sections, 12 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Sub-resonance type polynomials
  3. A theorem of Poincaré and Dulac
  4. Geometric structures on Hopf manifolds

Key Result

Lemma 2.3

Let $P_i: \mathbb{C}^n \to E_{\lambda_i}(L)$ be a homogeneous term of a sub-resonant polynomial relative to $L$ of type $s = (s_1, \dots, s_l)$, then:

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Theorem 2.4: M. Guysinsky and A. Katok
  • proof
  • Theorem 3.1
  • Proposition 3.2
  • Definition 3.3
  • Definition 3.4
  • Proposition 3.5
  • ...and 14 more