In toric geometry, it is known that the geometric properties of a toric line bundle are closely related to a convex polytope, known as the Newton polytope. Based on the work of Okounkov, Lazarsfeld--Musta\c{t}\u{a} and Kaveh—Khovanskii extended the Newton polytope to big line bundles on general projective manifolds.
In the thesis of Y. Deng, the construction was extended to general transcendental big cohomology classes on compact Kähler manifolds as well. It remains unclear if the transcendental Okounkov bodies have the expected volume. In this talk, we will confirm this and hence answering a conjecture of Lazarsfeld--Musta\c{t}\u{a}, Demailly and Deng. This part is a joint work with Kewei Zhang, Tamás Darvas, David Witt Nyström and Remi Reboulet.
During the study of transcendental Okounkov bodies, it turns out that it is very helpful to understand the flag valuations of general closed positive (1,1)-currents with arbitrary singularities. As I will explain in the talk, there is a natural way to restrict such currents to subvarieties along which the generic Lelong numbers vanish, which allows a recursive definition of their valuations. We call this restriction operator the trace operator, in analogue with the trace operator in the theory of Sobolev spaces.
