The Kazdan-Warner problem on compact Kähler surfaces

In this paper, we investigate a Kazdan-Warner problem on compact Kähler surfaces, which corresponds to prescribing sign-changing Chern scalar curvatures, and establish a Chen-Li type existence theorem on compact Kähler surfaces when the candidate curvature function is of negative average. Moreover, we give an alternative proof of Ding-Liu’s theorem [Trans. Amer. Math. Soc. 347(1995) 1059-1066] on prescribing sign-changing Gaussian curvatures.