On line bundles arising from the LCK structure over locally conformal Kähler solvmanifolds

Abstract

We can construct a real line bundle arising from the locally conformal Kähler (LCK) structure over an LCK manifold. We study the properties of this line bundle over an LCK solvmanifold whose complex structure is left-invariant. Mainly, we prove that this line bundle LL over an LCK solvmanifold Γ\G\Gamma \backslash G with left-invariant complex structure is flat and GG has a global closed 2-form, which induces an Hermitian structure on the holomorphic tangent bundle twisted by the line bundle LC=L⊗C{L}^{{\mathbb{C}}}=L\otimes {\mathbb{C}} if the Lee form is cohomologous to a left-invariant 1-form on GG.

Keywords

• locally conformal kähler manifold
• solvmanifold
• primary 53c55
• secondary 22e25