On the structure theorem for Vaisman solvmanifolds
Introduction
Let be a simply-connected solvable Lie group. A discrete co-compact subgroup of is said to be a lattice in . We call the compact manifold a solvmanifold. If is a nilpotent Lie group, we call the compact manifold 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 be a 2-dimensional compact Hermitian manifold. We denote by the fundamental 2-form, that is, the 2-form defined by . A Hermitian manifold is said to be locally conformal Kähler (LCK) if there exists a closed 1-form such that . The closed 1-form is called Lee form. Note that if , then is Kähler. A LCK manifold is said to be a Vaisman manifold if Lee form is parallel with respect to Levi-Civita connection of the metric .
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 be a solvmanifold with a left-invariant complex structure . In the case of a nilmanifold , has a LCK structure if and only if , where 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 except for , then has no Vaisman structures [14].
In this paper, we consider LCK structures on a solvmanifold. Let be a LCK solvmanifold with Lee form such that is left-invariant, and be the Lie algebra of . We assume that there exists a left-invariant closed 1-form such that Lee form is cohomologous to . Then, a LCK structure induces the left-invariant LCK structure with Lee form [4]. We call a locally conformal Kähler (LCK) solvable Lie algebra. Moreover, if is a Vaisman structure, then is also a Vaisman structure [17]. We call a Vaisman solvable Lie algebra.
In Section 2, we prove that
Main Theorem 1 Let be a Vaisman solvable Lie algebra as above, and be the nilradical of , that is, the maximal nilpotent ideal of . Then, there exists such that , where is a Heisenberg Lie algebra.
Since admits a lattice , 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 , where 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 has a Vaisman structure, then 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 , and they are modifications of , where is a Heisenberg Lie algebra.
Section snippets
Preliminary
In this section, we obtain a left-invariant LCK metric provided one begins with a left-invariant complex structure.
Let be a manifold and be the de Rham complex of with the exterior differential operator . For a closed -form on , we define the new differential operator from to by A -form is called -closed if . It is called -exact if there exists a -form such that . Since is closed, we see that . Similarly, we can define a
Proof of Main Theorem 1
We use same notation introduced in Section 1. In this section, we prove Main Theorem 1.
We take an orthogonal decomposition and put . Note that and . Let be the natural projection from to . Since , is homomorphic.
We consider the structure on . Let be a linear map from to induced by Thus, we can define a linear map from to by
Proof of Main Theorem 2
We use same notation introduced in Sections 1 Preliminary, 2 Proof of. In this section, we prove Main Theorem 2.
We define
Definition 3.1 A Lie algebra is called type I if all eigenvalues of the linear map from to are in , for any .
We see that a nilpotent Lie algebra is type I, and a completely solvable Lie algebra of type I is nilpotent. Moreover, from Theorem 2.2, a unimodular Kähler solvable Lie algebra is type I.
Let be a Vaisman solvable Lie algebra. We have an orthogonal
Examples
In this section, we give examples of Vaisman solvmanifolds.
Let be a -dimensional Heisenberg Lie group given by where the product of the group is defined by The Lie group admits a lattice : Let be the Lie algebra corresponding to . The nilpotent Lie algebra is given by
We define a left-invariant metric on such that is an
Acknowledgments
The author would like to express his deep appreciation to Professor Yusuke Sakane and Professor Takumi Yamada for their thoughtful guidance and encouragement during the completion of this paper. This work was supported by JSPS KAKEN Grant number JP17K05235, Japan.
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