Okounkov bodies of filtered linear series

By S. Boucksom, Huayi Chen. Compositio Mathematica, 2009

132 citations recorded by the source corpus.

Abstract We associate to certain filtrations of a graded linear series of a big line bundle a concave function on its Okounkov body, whose law with respect to the Lebesgue measure describes the asymptotic distribution of the jumps of the filtration. As a consequence, we obtain a Fujita-type approximation theorem in this general filtered setting. We then specialize these results to the filtrations by minima in the usual context of Arakelov geometry (and for more general adelically normed graded linear series), thereby obtaining in a simple way a natural construction of an arithmetic Okounkov body, the existence of the arithmetic volume as a limit and an arithmetic Fujita approximation theorem for adelically normed graded linear series. We also obtain an easy proof of the existence of the sectional capacity previously obtained by Lau, Rumely and Varley.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7e985d4e273683624a32a0ec71b50ab90e5127b7

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Extremal K\"ahler metrics (2014, background): “In the proof of Theorem 9 the main additional ingredient is the use of the Okounkov body [37, 8, 29, 58].”
Asymptotic Multiplicities of Graded Families of Ideals and Linear Series (2014, methodology): “This observation is exploited by Boucksom and Chen in [1] where some limits on geometrically integral arithmetic varieties are computed.”
Kähler–Einstein metrics and volume minimization (2016, methodology): “We recall the relevant definitions about filtrations after [9] (see also [12, Section 1] and [26, Section 4.”
K-stability of Fano varieties via admissible flags (2020, unknown): “We define S(L; v) to be the limit limm→∞ Sm (L; v), which exists by [2, 8].”
Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs (2015, background): “Let us we review and complement the study in [11] of a natural measure on R associated to a general R-filtration F •R on R.”
On the Yau‐Tian‐Donaldson Conjecture for Generalized Kähler‐Ricci Soliton Equations (2020, background): “Let F = {FRm} be a T -invariant filtration as considered in [15] (see [54, 2.”
Filtrations and test-configurations (2011, methodology): “A key new ingredient in the proof is the Okounkov body [21], and the concave (in our case convex) transform of a filtration introduced by Boucksom–Chen [4], which was also used in the context of test-configurations by Witt Nyström [31].”