Test configurations and Okounkov bodies

By David Witt Nyström. Compositio Mathematica, 2010

60 citations recorded by the source corpus.

Abstract We associate to a test configuration for a polarized variety a filtration of the section ring of the line bundle. Using the recent work of Boucksom and Chen we get a concave function on the Okounkov body whose law with respect to Lebesgue measure determines the asymptotic distribution of the weights of the test configuration. We show that this is a generalization of a well-known result in toric geometry. As an application, we prove that the pushforward of the Lebesgue measure on the Okounkov body is equal to a Duistermaat–Heckman measure of a certain deformation of the manifold. Via the Duisteraat–Heckman formula, we get as a corollary that in the special case of an effective ℂ×-action on the manifold lifting to the line bundle, the pushforward of the Lebesgue measure on the Okounkov body is piecewise polynomial.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/55325e7f3698909b438883c7a9950df17c0a5cb7

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Citations in the corpus

The Kähler–Ricci flow and optimal degenerations (2016, methodology): “Let us recall here that as shown by Witt Nyström [39] (see also [7, Proposition 2.”
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].”
Complex optimal transport and the pluripotential theory of K\"ahler-Ricci solitons (2014, methodology): “For L ample and T = S(1) this was shown in [53] (but it also follows from general results on torus actions in symplectic geometry).”
Vanishing sequences and Okounkov bodies (2013, result): “More general filtrations are in turn closely related to the socalled ‘test configurations’ in Donaldson’s definition of K -stability [8,20,21].”
Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, II (2015, background): “Notice as in [35], [32], a test configuration forW can also be viewed as a filtration on the co-ordinate ring of W .”
Kähler–Ricci flow, Kähler–Einstein metric, and K–stability (2015, methodology): “It is interesting to understand the coordinate ring of X in terms of the language of filtrations introduced in [42].”
Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs (2015, methodology): “See also [9, 83, 45] for an analytic approach to Duistermaat–Heckman measures via geodesic rays.”
K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics (2012, methodology): “As pointed out above Lemma 3.7 is a special case of a general result of Hisamoto [24], saying that the measure (dφ t dt )∗MA(φ t) on R only depends on the test configuration (X ,L) and moreover is equal to the limiting normalized weight measures for the C∗−action, as conjectured by Witt-Nyström [36]”