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

TL;DR

This paper proves the existence of complete scalar-flat Kähler metrics on the complement of a hypersurface in a polarized manifold, given specific geometric and algebraic conditions, extending the understanding of scalar-flat metrics in complex geometry.

Contribution

It establishes new conditions under which the complement of a hypersurface admits a complete scalar-flat Kähler metric, linking scalar curvature properties with algebraic ampleness conditions.

Findings

Existence of scalar-flat Kähler metrics on $X \setminus D$ under specified conditions.

Conditions involve scalar curvature bounds and algebraic ampleness.

Results apply to manifolds with no holomorphic vector fields vanishing on $D$.

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.