Complete scalar-flat Kähler metrics on affine algebraic manifolds

Abstract

AbstractLet $$(X,L_{X})$$ ( X , L X ) be an n-dimensional polarized manifold. Let D be a smooth hypersurface defined by a holomorphic section of $$L_{X}$$ L X . We prove that if D has a constant positive scalar curvature Kähler metric, $$X {\setminus } D$$ X \ D admits a complete scalar-flat Kähler metric, under the following three conditions: (i) $$n \ge 6$$ n ≥ 6 and there is no nonzero holomorphic vector field on X vanishing on D, (ii) the average of a scalar curvature on D denoted by $${\hat{S}}_{D}$$ S ^ D satisfies the inequality $$0< 3 {\hat{S}}_{D} < n(n-1)$$ 0 < 3 S ^ D < n ( n - 1 ) , (iii) there are positive integers $$l(>n),m$$ l ( > n ) , m such that the line bundle $$K_{X}^{-l} \otimes L_{X}^{m}$$ K X - l ⊗ L X m is very ample and the ratio m/l is sufficiently small.