Solution to a non-Archimedean Monge-Ampère equation

Abstract

Let X be a smooth projective Berkovich space over a complete discrete valuation field K of residue characteristic zero, and assume that X is defined over a function field admitting K as a completion. Let further m be a positive measure on X and L be an ample line bundle such that the mass of m is equal to the degree of L. Then we show the existence a continuous semipositive metric whose associated measure is equal to m in the sense of Zhang and Chambert-Loir. This we do under a technical assumption on the support of m, which is, for instance, fulfilled if the support is a finite set of divisorial points. Our method draws on analogues of the variational approach developed to solve complex Monge-Ampère equations on compact Kähler manifolds by Berman, Guedj, Zeriahi and the first named author, and of Kołodziej's continuity estimates. It relies in a crucial way on the compactness properties of singular semipositive metrics, as defined and studied in a companion article.

47 pages

References in corpus (6)

• The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension
• Singular Kahler-Einstein metrics
• Monge-Ampère equations in big cohomology classes
• A variational approach to complex Monge-Ampere equations
• Intrinsic capacities on compact Kähler manifolds
• A non-archimedean analogue of Calabi-Yau theorem for totally degenerate abelian varieties

Cited by in corpus (3)

• Singular semipositive metrics in non-Archimedean geometry
• SYZ conjecture for Calabi-Yau hypersurfaces in the Fermat family
• Invariants of Varieties and Singularities inspired by Kahler-Einstein problems