Almost scalar-flat K\"{a}hler metrics on affine algebraic manifolds

Abstract

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.