[Paper Review] Small deformations and non-left-invariant complex structures on a compact solvmanifold

Keizo Hasegawa|arXiv (Cornell University)|Mar 26, 2007

Geometry and complex manifolds8 references3 citations

TL;DR

This paper constructs a six-dimensional compact solvmanifold that admits a continuous family of non-left-invariant complex structures, demonstrating that small deformations of left-invariant complex structures on solvmanifolds need not preserve left-invariance. It completes the classification of three-dimensional complex solvmanifolds and determines which admit pseudo-Kähler structures, showing that all left-invariant complex structures on three-dimensional simply connected complex solvable Lie groups are biholomorphic to $\mathbb{C}^3$. The key result is the existence of non-left-invariant complex structures via deformation theory on a type (3b) solvmanifold.

ABSTRACT

We observed in our previous paper that all the complex structures on four-dimensional compact solvmanifolds, including tori, are left-invariant. In this paper we will give an example of a six-dimensional compact solvmanifold which admits a continuous family of non-left-invariant complex structures. Furthermore, we will make a complete classification of three-dimensional compact homogeneous complex solvmanifolds; and determine which of them admit pseudo-Kaehler structures.

Motivation & Objective

• To investigate whether non-left-invariant complex structures can exist on higher-dimensional solvmanifolds, extending previous results that all complex structures on 4D solvmanifolds are left-invariant.
• To classify all three-dimensional compact homogeneous complex solvmanifolds by analyzing lattices in unimodular complex solvable Lie groups.
• To determine which three-dimensional complex solvmanifolds admit pseudo-Kähler structures, identifying the necessary and sufficient conditions.

Proposed method

• Using deformation theory of complex structures on compact solvmanifolds, particularly leveraging small deformations of complex structures on three-dimensional solvmanifolds.
• Applying the Nijenhuis tensor condition to verify integrability of almost complex structures on Lie algebras.
• Analyzing the complex subalgebra structure of the complexified Lie algebra to characterize left-invariant complex structures.
• Constructing a complex automorphism of the complexified Lie algebra to show that all left-invariant complex structures on 3D complex solvable Lie groups are biholomorphic to $\mathbb{C}^3$.
• Using Kodaira's deformation theory results to identify a continuous family of non-Stein universal coverings in type (3b) solvmanifolds.
• Establishing equivalence of short exact sequences of Lie algebras to lift automorphisms from the Lie algebra level to the group level.

Experimental results

Research questions

• RQ1 Does there exist a compact solvmanifold of dimension greater than four that admits non-left-invariant complex structures?
• RQ2 Are all left-invariant complex structures on three-dimensional simply connected complex solvable Lie groups biholomorphic to $\mathbb{C}^3$?
• RQ3 Which three-dimensional complex solvmanifolds admit a pseudo-Kähler structure?
• RQ4 Is the property of being left-invariant preserved under small deformations of complex structures on solvmanifolds?
• RQ5 Can a continuous family of non-left-invariant complex structures arise from small deformations of a left-invariant structure on a solvmanifold?

Key findings

• A six-dimensional compact solvmanifold exists that admits a continuous family of non-left-invariant complex structures, constructed via small deformations of a type (3b) three-dimensional complex solvmanifold.
• All left-invariant complex structures on three-dimensional simply connected complex solvable Lie groups are biholomorphic to $\mathbb{C}^3$, regardless of type (abelian, nilpotent, or non-nilpotent).
• The classification of three-dimensional compact homogeneous complex solvmanifolds is completed by determining all lattices in unimodular complex solvable Lie groups.
• A three-dimensional complex solvmanifold admits a pseudo-Kähler structure if and only if it is of type (1) (abelian) or type (3b) (non-nilpotent with $h^1 = 3$).
• Small deformations of left-invariant complex structures on solvmanifolds need not be left-invariant, as demonstrated by the existence of non-left-invariant structures in the deformation space of a type (3b) solvmanifold.
• The canonical map from the solvmanifold to its universal covering space remains biholomorphic under deformation, but the universal coverings of the deformed structures are not Stein, indicating non-left-invariance.