Locally conformal Kähler structures on compact solvmanifolds
Consider a simply connected solvable Lie group \(G\) with a left-invariant complex structure \(J\), and with a lattice, or discrete co-compact subgroup, \(\Gamma\); then \(\Gamma\backslash G\) is called a solvmanifold. Now, if \(g\) is some Hermitian metric on \(\Gamma\backslash G\), it is said to be locally conformally Kähler if, defining a 2-form \(\Omega\) as \(g(\cdot,J,\cdot)\), there exists a closed one-form \(\omega\) such that \(d\Omega=\omega\wedge \Omega\); \(\Omega\) is then called the fundamental 2-form, and \(\omega\), the Lee form. The main result of this article is the following: consider a locally conformally Kähler solvmanifold \((\Gamma\backslash G,J,\Omega)\) of Lee form \(\omega\). Assume moreover that there exists a left-invariant 1-form \(\omega_0\) cohomologous to \(\omega\), and that \(\Omega= -\omega\wedge\eta+d\eta\) with \(\eta=-\omega\circ J\). Then \(\Gamma\backslash G\) admits a left-invariant locally conformally Kähler structure \((J,g_0,\Omega_0)\) of Lee form \(\omega_0\), for which \(\omega_0\) is furthermore parallel; in other words, \((\Gamma\backslash G,J,g_0)\) is a generalized Hopf surface. As a corollary, by a previous result of \textit{I. Vaisman} [Rend. Mat., VI. Ser. 12, 263--264 (1979; Zbl 0447.53032)], any example satisfying these conditions, such as Kodaira-Thurston and secondary Kodaira-Thurston manifolds, have a structure of principal \(S^1\)-bundle over a compact Sasakian manifold (this was already known for other examples such as Hopf manifolds and generalized Kodaira-Thurston manifolds). Explicit examples are also dealt with. The author's construction is as follows: he introduces a natural candidate for \(g_0\) through its fundamental 2-form \(\Omega_0\) defined as \[ \Omega_0(X,Y) = \int_{\Gamma\backslash G} (e^f\Omega)(X,Y) d\mu, \] where \(df=\omega_0-\omega\), and \(d\mu\) is induced by a bi-invariant volume element on \(G\). Most of the work is thus devoted to proving that \(\omega_0\) is indeed parallel for such a \(g_0\). This is done by working extensively on the Lie algebra \(\mathfrak{g}\) of \(G\), using in a crucial its solvability -- and in particular, that \(G\) has the ``strong unimodular'' property, as raised in [\textit{C. Benson} and \textit{C. S. Gordon}, Proc. Am. Math. Soc. 108, No. 4, 971--980 (1990; Zbl 0689.53036)].
scientific article; zbMATH DE number 6166425
scientific article