Overview
This study investigates the biholomorphic equivalence between simply connected nilpotent Lie groups, equipped with specific complex structures, and complex Euclidean space, $\mathbb{C}^n$. The core finding establishes that such Lie groups are biholomorphic to $\mathbb{C}^n$, with the biholomorphism explicitly constructed through polynomial maps in exponential coordinates. A direct consequence of this construction involves the action of lattices within these Lie groups on $\mathbb{C}^n$ through holomorphic polynomial automorphisms.
Research Context
The broader area of inquiry involves the geometric and algebraic properties of Lie groups and their associated complex structures. Specifically, the research considers simply connected nilpotent Lie groups endowed with left-invariant nilpotent complex structures. This context sets the stage for exploring how these complex manifolds relate to standard complex spaces like $\mathbb{C}^n$. The concept of a biholomorphism, a bijective holomorphic map with a holomorphic inverse, is central to establishing equivalence between complex manifolds.
Approach
The methodology employed in this research centers on demonstrating the existence of a biholomorphism and subsequently providing its explicit construction. The demonstration proceeds from the foundational properties of simply connected nilpotent Lie groups and their left-invariant nilpotent complex structures. The explicit construction of the biholomorphism leverages polynomial maps. These maps are formulated within the framework of exponential coordinates, a common and effective coordinate system for Lie groups, particularly nilpotent ones. This explicit construction is crucial for understanding the nature of the equivalence established.
Findings
- Every simply connected nilpotent Lie group equipped with a left-invariant nilpotent complex structure is biholomorphic to $\mathbb{C}^n$.
- The biholomorphism from such a Lie group to $\mathbb{C}^n$ can be explicitly constructed using polynomial maps.
- This explicit construction utilizes exponential coordinates for the polynomial maps.
- A direct consequence is that every lattice within such a Lie group admits a free, properly discontinuous, and cocompact action on $\mathbb{C}^n$.
- This action is performed by holomorphic polynomial automorphisms.
Why This Matters
The study provides a holomorphic analogue of polynomial crystallographic actions, as originally introduced by Dekimpe, Igodt, and Lee. This connection extends the understanding of crystallographic actions from real polynomial settings to a holomorphic context, potentially offering new perspectives on group actions on complex spaces.