Complete scalar-flat Kähler 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 Kähler metric, $X\setminus D$ admits a complete scalar-flat Kähler metric, under the following three conditions: (i) $n\ge 6$ 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.

arXiv comment18 pages, no figures, This paper is merged with arXiv:1907.09780 and arXiv:1908.05583 and appear in Math. Z./

References in corpus (1)

• Regularity properties of the degenerate Monge--Ampère equations compact Kähler manifolds