Solution to a non-Archimedean Monge-Ampère equation

By S. Boucksom, C. Favre, Mattias Jonsson. 2011

96 citations recorded by the source corpus.

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-Ampere equations on compact Kahler manifolds by Berman, Guedj, Zeriahi and the first named author, and of Ko{\l}odziej'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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4553b1315ed8f10881a5a7ba0798214194cec854

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In Geometry and Topology

1. Chi Li
2. Mattias Jonsson
3. S. Boucksom
4. Robert J. Berman
5. C. Favre
6. Yuchen Liu
7. H. Blum
8. Xinyi Yuan

In Mathematical Physics

1. Chi Li
2. Mattias Jonsson
3. S. Boucksom
4. Robert J. Berman
5. C. Favre
6. Xinyi Yuan
7. Shou-wu Zhang
8. H. Blum

Citations in the corpus

• A variational approach to the Yau–Tian–Donaldson conjecture (2015, methodology): “ such functions. To any ’2E1;NA is attached a non-Archimedean Monge-Ampere measure MA(’), a Radon probability measure on XNA. By the non-Archimedean Calabi-Yau theorem proved in [BoJ18a] building on [BFJ15], the nonArchimedean Monge-Ampere operator sets up a one-to-one correspondence between E1;NA=R”
• DEGENERATION OF ENDOMORPHISMS OF THE COMPLEX PROJECTIVE SPACE IN THE HYBRID SPACE (2016, methodology): “ghborhood of X¯r; • there exists a positive closed (1,1) current Ton Xr such that T|Xr = ω+ddcϕ; • the function gT extends continuously to XC((t)),r. The forward implication is easy. It was proved in [BFJ15] (see also [BGJ+16]) that one can solve the Monge-Amp`ere equation MAL(g) = µfor a suitable c”
• Geodesic rays and stability in the cscK problem (2020, methodology): “On the other hand, the proof given combines our analysis with the fundamental works on non-Archimedean Monge-Ampère equations in [14, 19], in addition to [25, 26].”
• The arithmetic Hodge index theorem for adelic line bundles (2013, result): “Some partial results have been obtained by Liu [26] and Boucksom–Favre– Jonsson [5].”