9.10.17 Extensions of the results in the smooth settingCE1C14C908CB

Let $X$ be an irreducible unibranch compact Kähler space of dimension $n$. Let $\theta $ be a closed real smooth $(1,1)$-form on $X$. We say the cohomology class $\{ \theta \} $ is big if for any proper bimeromorphic morphism $\pi \colon Y\rightarrow X$ from a compact Kähler manifold $Y$, $\{ \pi ^*\theta \} $ is big. This definition is independent of the choice of $\pi $.

The non-pluripolar products can be defined exactly as in section 13.1 and the results in that chapter hold mutatis mutandis.

section 17.10 and section 16.7 can be extended without big changes. The only exception is theorem 16.9.59, where we do not have the notion of multiplier ideal sheaves. So we do not know how to extend this theorem.

section 18.1 can be extended except for subsection 18.1.44 for the same reason as above.

If $Y\subseteq X$ is an irreducible analytic subspace not contained in $X^{\operatorname{Sing}}$, then the trace operator of section 20.2 can be extended to this setting by using an embedded resolution. In general, due to the lack of Demailly regularization on singular spaces, we do not know how to define the trace operator.

section 17.11 can be easily extended.

section 19.1 is easy to extend since the partial Okounkov bodies are bimeromorphically invariant in the sense of theorem 19.5.26.

section 20.1 is unchanged, since we always take projective limits with respect to all models in that section.

section 21.13 can be extended using the toric resolution [Theorem 11.1.9, CLS11].

section 22.1 can be extended except for the parts involving the trace operator.

section 18.28 can be easily extended by considering a resolution.