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

## TL;DR

This paper constructs complete Kähler metrics with scalar curvature flatness away from a divisor on affine algebraic manifolds, under specific ampleness and ratio conditions of line bundles.

## Contribution

It demonstrates the existence of almost scalar-flat Kähler metrics on affine algebraic manifolds using line bundle ampleness and ratio conditions.

## Key findings

- Existence of complete Kähler metrics with flat scalar curvature away from divisors.
- Conditions involving very ample line bundles and small ratios ensure metric construction.
- Provides new examples of affine algebraic manifolds with special Kähler metrics.

## 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.