Lattices in almost abelian Lie groups with locally conformal Kähler or symplectic structures
The authors study left-invariant locally conformally Kähler and locally conformally symplectic structures on almost abelian Lie groups and on their compact quotients. In particular, they focus on the existence of lattices for such Lie groups endowed with such structures. Locally conformally Kähler, respectively symplectic, structures are structures that locally admit a conformal change that is Kähler, respectively symplectic. For some classes of solvable Lie groups, criteria are known that assure the existence of lattices: in this case, left-invariant lcK and lcs structures move to the solvmanifold defined as the compact quotient. For nilpotent Lie groups, a criterion for the existence of lattices is given in [\textit{A. I. Mal'tsev}, Izv. Akad. Nauk SSSR, Ser. Mat. 9, 329--356 (1945; Zbl 0061.05303)], left-invariant lcK structures are studied in [\textit{H. Sawai}, Geom. Dedicata 125, 93--101 (2007; Zbl 1121.53052)], and left-invariant lcs structures are studied in [\textit{G. Bazzoni} and \textit{J. C. Marrero}, Bull. Sci. Math. 143, 1--57 (2018; Zbl 1383.22004)]. For almost abelian Lie groups, (that is, Lie groups whose Lie algebra has a codimension-one abelian ideal) a criterion for the existence of lattices is given in [\textit{C. Bock}, Asian J. Math. 20, No. 2, 199--262 (2016; Zbl 1346.53052)], and left-invariant lcK and lcs structures are the object of the paper under review. Section 3 deals with lcK structures: Theorem 3.3 characterizes the Lie algebras of almost abelian Lie groups admitting left-invariant lcK structures; Theorems 3.7 and 3.9 show that lattices for such Lie groups exist only in dimension \(4\), where Inoue surfaces of type \(S^0\) appear. Section 4 deals with lcs structures: Theorems 4.1 and 4.5 characterize the Lie algebras of almost abelian Lie groups admitting left-invariant lcs structures; Corollary 4.3 and Table 1 specify whether there exist such structures of first kind or second kind (this meaning that the Lee morphism is surjective, respectively zero); finally, Theorem 4.8 determines which of these Lie groups in dimension \(4\) admit lattices, and Section 4.2.2 constructs families of almost abelian solvmanifolds with lcs structures of the second kind and with no left-invariant complex structure in any higher dimension (their de Rham and Lichnerowicz cohomologies are also studied for low dimensions).
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