Toward a construction of 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 study the existence of a complete scalar-flat K\"{a}hler metric on $X \setminus D$ on the assumption that $D$ has a constant positive scalar curvature K\"{a}hler metric.