Cumrun Vafa with Greene-Shapere-Yau introduced semi-Ricci flat metric here
B. Greene, A. Shapere, C. Vafa, and S.-T. Yau. Stringy cosmic strings and noncompact Calabi-Yau manifolds. Nuclear Physics B, 337(1):1–36, 1990
I know that such metrics are Kahler in fiber direction. Is there any counter example to show that such metrics are not Kahler in horizontal direction?