Let $(X,L_{ X})$ be an $n$-dimensional polarized manifold. Let $D$ be a smooth hypersurface defined by a holomorphic section of $L_{ X}$. In this paper, we show the existence of a complete K\"{ a}hler metric on $X \setminus D$ whose scalar curvature is flat away from some divisor if there are positive integers $l(>n),m$ such that the line bundle $K_{ X}^{ -l} \otimes L_{ X}^{ m}$ is very ample and the ratio $m/l$ is sufficiently small.
Categories: math.DG math.AP math.CV