# Toward a construction of scalar-flat K\"{a}hler metrics on affine algebraic manifolds

## TL;DR

This paper investigates conditions under which complete scalar-flat Kähler metrics can be constructed on the complement of a hypersurface in a polarized manifold, assuming the hypersurface admits a constant positive scalar curvature Kähler metric.

## Contribution

It provides a new approach to constructing scalar-flat Kähler metrics on affine algebraic manifolds using hypersurfaces with constant positive scalar curvature.

## Key findings

- Established existence conditions for scalar-flat Kähler metrics on $X \setminus D$
- Connected hypersurface scalar curvature to the construction of affine metrics
- Extended previous methods to a broader class of 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.