On the structure theorem for Vaisman solvmanifolds

Abstract

The purpose in this paper is to give the structure theorem for Vaisman solvmanifolds. Moreover, we prove that a Vaisman solvmanifold has no non-Vaisman LCK structures.

Introduction

Let $G$ be a simply-connected solvable Lie group. A discrete co-compact subgroup $\Gamma$ of $G$ is said to be a lattice in $G$. We call the compact manifold $\Gamma \setminus G$ a solvmanifold. If $G$ is a nilpotent Lie group, we call the compact manifold $\Gamma \setminus G$ a nilmanifold. It is well known that a Kähler solvmanifold is a finite quotient of a complex torus which has a structure of a complex torus bundle over a complex torus [2], [9]. In this paper, we consider a structure generalized a Kähler structure on solvmanifolds.

Let $(M,g,J)$ be a 2$n$-dimensional compact Hermitian manifold. We denote by $\Omega$ the fundamental 2-form, that is, the 2-form defined by $\Omega (X,Y)=g(X,JY)$. A Hermitian manifold $(M,g,J)$ is said to be locally conformal Kähler (LCK) if there exists a closed 1-form $\omega$ such that $d\Omega =\omega \wedge \Omega$. The closed 1-form $\omega$ is called Lee form. Note that if $\omega =df$, then $(M,e^{-f}g,J)$ is Kähler. A LCK manifold $(M,g,J)$ is said to be a Vaisman manifold if Lee form $\omega$ is parallel with respect to Levi-Civita connection $\nabla$ of the metric $g$.

The main non-Kähler examples of LCK manifolds are Hopf manifold [21], Inoue surfaces [20], Kodaira–Thurston manifold [5] and Oeljeklaus–Toma manifold [12] (cf. [7]). Note that Hopf manifold and Kodaira–Thurston manifold are Vaisman manifolds, and Inoue surfaces and Oeljeklaus–Toma manifold are not Vaisman manifolds. Moreover, Inoue surfaces, Kodaira–Thurston manifold and Oeljeklaus–Toma manifold have a structure of a solvmanifold.

Let $\Gamma \setminus G$ be a solvmanifold with a left-invariant complex structure $J$. In the case of a nilmanifold $\Gamma \setminus G$, $(\Gamma \setminus G,J)$ has a LCK structure if and only if $G=\mathbb{R}\times H$, where $H$ is a Heisenberg Lie group [15]. In the case of a solvmanifold, if a Vaisman completely solvable solvmanifold has the same structure [18]. On the other hand, if $G={\mathbb{R}}^n⋉{\mathbb{R}}^m$ except for $\mathbb{R}\times H$, then $(\Gamma \setminus G,J)$ has no Vaisman structures [14].

In this paper, we consider LCK structures on a solvmanifold. Let $(\Gamma \setminus G,g,J)$ be a LCK solvmanifold with Lee form $\omega$ such that $J$ is left-invariant, and $\mathfrak{g}$ be the Lie algebra of $G$. We assume that there exists a left-invariant closed 1-form ${\omega }_0$ such that Lee form $\omega$ is cohomologous to ${\omega }_0$. Then, a LCK structure $(g,J)$ induces the left-invariant LCK structure $(〈,〉,J)$ with Lee form ${\omega }_0$ [4]. We call $(\mathfrak{g},〈,〉,J)$ a locally conformal Kähler (LCK) solvable Lie algebra. Moreover, if $(g,J)$ is a Vaisman structure, then $(〈,〉,J)$ is also a Vaisman structure [17]. We call $(\mathfrak{g},〈,〉,J)$ a Vaisman solvable Lie algebra.

In Section 2, we prove that

Main Theorem 1

Let $(\mathfrak{g},〈,〉,J)$ be a Vaisman solvable Lie algebra as above, and $\mathfrak{n}$ be the nilradical of $\mathfrak{g}$, that is, the maximal nilpotent ideal of $\mathfrak{g}$. Then, there exists $k\in \mathbb{N}\cup \left\{0\right\}$ such that $\mathfrak{n}={\mathbb{R}}^k\times \mathfrak{h}$, where $\mathfrak{h}$ is a Heisenberg Lie algebra.

Since $G$ admits a lattice $\Gamma$, $\mathfrak{g}$ is unimodular. Alekseevsky–Hasegawa–Kamishima [1] define a modification with respect to a Hermitian structure on a unimodular Lie algebra (For detail, see Remark 2.8), and prove that a Vaisman solvable Lie algebra is a modification of $\mathbb{R}\times \mathfrak{h}$, where $\mathfrak{h}$ is a Heisenberg Lie algebra. In Main Theorem 1, we determine the nilradical of a Vaisman solvable Lie algebra.

Moreover, in Section 3, we prove that

Main Theorem 2

If a solvable Lie algebra $\mathfrak{g}$ has a Vaisman structure, then $\mathfrak{g}$ has no non-Vaisman LCK structures.

As a corollary, we see that a non-Vaisman LCK solvable Lie algebra has no Vaisman structures. Thus, Inoue surfaces and Oeljeklaus–Toma manifold have no Vaisman structures with a left-invariant complex structure (cf. [4], [10], [14], [19]).

In Section 4, we give examples of a Vaisman solvmanifold. Moreover, we see that the nilradical is given by ${\mathbb{R}}^k\times \mathfrak{h}$, and they are modifications of $\mathbb{R}\times \mathfrak{h}$, where $\mathfrak{h}$ is a Heisenberg Lie algebra.