On Duistermaat-Heckman measure for filtered linear series

We revisit work of S. Boucksom, C. Favre, and M. Jonsson (J. Algebraic Geom. \textbf{18} (2009), no. 2, 279--308); Boucksom and H. Chen (Compos. Math. \textbf{147} (2011), no. 4, 1205--1229); and S. Boucksom, A. K\"{u}ronya, C. Maclean, and T. Szemberg (Math. Ann. \textbf{361} (2015), no.~3--4, 811--834). The key point is to associate a Duistermaat-Heckman measure to a filtered big linear series on a given projective variety. The expectation of the measure admits a description via the theory of Newton-Okounkov bodies. Such considerations have origins in symplectic geometry. They have applications for $\mathrm{K}$-stability and Diophantine arithmetic geometry of projective varieties.