Complete 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 $L_{X}$. We prove that if $D$ has a constant positive scalar curvature K\"{a}hler metric, $X \setminus D$ admits a complete scalar-flat K\"{a}hler metric, under the following three conditions: (i) $n \geq 6$ and there is no nonzero holomorphic vector field on $X$ vanishing on $D$, (ii) an average of a scalar curvature on $D$ denoted by $\hat{S}_{D}$ satisfies the inequality $0 < 3 \hat{S}_{D} < n(n-1)$, (iii) 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.