Calabi-Yau locally conformally Kähler manifolds

We study compact locally conformally Kähler (lcK) manifolds which are Calabi–Yau, in the sense that $c_1^{BC}(X)=0$. First of all, we prove that all the known lcK manifolds which are Calabi–Yau are Vaisman. Then we prove that an lcK Chern–Ricci flat metric that is Gauduchon is necessarily Vaisman. Finally, 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.