Singular semipositive metrics in non-Archimedean geometry

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

135 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, endowed with an ample line bundle L. We introduce a general notion of (possibly singular) semipositive (or plurisubharmonic) metrics on L, and prove the analogue of the following two basic results in the complex case: the set of semipositive metrics is compact modulo constants, and each semipositive metric is a decreasing limit of smooth semipositive ones. In particular, for continuous metrics our definition agrees with the one by S.-W. Zhang. The proofs use multiplier ideals and the construction of suitable models of X over the valuation ring of K, using toroidal techniques.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4c0fc4be978510167c41e0afa81d492466bdb2e9

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In Geometry and Topology

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

In Mathematical Physics

1. Chi Li
2. Mattias Jonsson
3. S. Boucksom
4. Robert J. Berman
5. C. Favre
6. Y. Soibelman
7. M. Kontsevich
8. M. Abouzaid

Citations in the corpus

• A non-Archimedean approach to K-stability (2018, background): “This class has many good properties, as studied in [BoJ18], see also [BFJ16a, BoJ18] for the discretely valued case.”
• Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and Mirror Symmetry (2013, background): “er metrics on a given manifold. There is an analog of this notion in the non-archimedean geometry. More precisely, there a sheaf of monoids Psh⊂Conton Xanof continuous plurisubharmonic functions (see [BouFaJo1]). The sheaf of abelian groups Phis defined as Psh∩(−Psh). The quotient Psh/Phis by definiti”
• Solution to a non-Archimedean Monge-Ampère equation (2011, background): “ve (1,1)-form on X, such that {ω}∈ N1(X) is furthermore ample. This situation arises for instance when ωis the curvature form of a semipositive model metric on an ample line bundle L. As was shown in [BFJ11], one may then define a class PSH(X,ω) of ω-psh functions with the following properties: •Each”
• Geodesic rays and stability in the cscK problem (2020, methodology): “Non-Archimedean psh metrics were originally characterized as limits of decreasing nets of metrics from HNA (see [15]).”
• A variational approach to the Yau–Tian–Donaldson conjecture (2015, methodology): “It is endowed with the weak topology of pointwise convergence on X div , and it is proved in [BoJ18a], as a consequence of [BFJ16], that the space of sup-normalized functions Lemma 6.1.”