Calabi-Yau locally conformally Kähler manifolds

Abstract: We study compact locally conformally K\"ahler (lcK) manifolds which are Calabi-Yau, in the sense that $c_1^{BC}(X)=0$. We prove that an lcK Gauduchon metric that is Chern-Ricci flat is necessarily Vaisman. Specializing to Calabi--Yau solvmanifolds with left-invariant complex structure, we prove that a left-invariant metric is lcK if and only if it is Vaisman. Therefore, they are finite quotients of the Kodaira manifold.