试验构型和Okounkov体
Test configurations and Okounkov bodies
摘要：
我们将一个充分的线丛的测试构形与线丛的截面环的一个过滤联系起来。利用Boucksom-Chen最近的工作，我们得到了Okounkov体上的一个凹函数，它关于勒贝格测度的定律决定了试验构形权的渐近分布.我们表明，这是一个众所周知的结果在复曲面几何的推广。作为应用，我们证明了Okounkov体上的勒贝格测度的前推等于流形的某一变形的Duistermaat-Heckman测度.通过Duisteraat-Heckman公式，我们得到了在流形上的有效$\mathbb{C^{\times}}$作用提升到线丛的特殊情况下，Okounkov体上的勒贝格测度的前推是分段多项式的推论.
英文摘要：
We associate to a test configuration of an ample line bundle a filtration of the section ring of the line bundle. Using the recent work of Boucksom-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 $\mathbb{C^{\times}}$-action on the manifold lifting to the line bundle, the pushforward of the Lebesgue measure on the Okounkov body is piecewise polynomial.
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