The size of $PSH(X,\omega)$ is therefore related to that of ${\mathcal{T}}_{[\omega]}(X)$ hence only depends on the positivity of the cohomology class $[\omega]$. The more positive $[\omega]$, the bigger $PSH(X,\omega)$.

When $[\omega]$ is Kähler then $PSH(X,\omega)$ is large: if e.g. $\chi$ is any ${\mathcal{C}}^{2}$-function on $X$ then $\varepsilon\chi\in PSH(X,\omega)$ for $\varepsilon>0$ small enough. We will see (theorem 5.2) that $PSH(X,\omega)$ characterizes locally pluripolar sets when $[\omega]$ is Kähler. It follows from proposition 1.3 that $PSH(X,\omega)$ and $PSH(X,\omega^{\prime})$ have the same ”size” if $\omega$ and $\omega^{\prime}$ are both Kähler.

Note on the other hand that $PSH(X,\omega)\simeq\mathbb{R}$ when $\omega$ is cohomologous to $[E]$, the current of integration along the exceptional divisor of a smooth blow up. Indeed let $\pi:X\rightarrow\tilde{X}$ be a blow up with smooth center $Y$, $codim_{\mathbb{C}}Y\geq 2$ (see e.g. chapter 2 of [15] for the definition of blow-ups). Let $E=\pi^{-1}(Y)$ denote the exceptional divisor and $\omega=[E]$ be the current of integration along $E$. If $\varphi\in PSH(X,[E])$ then $dd^{c}(\varphi\circ\pi^{-1})\geq 0$ in $\tilde{X}\setminus Y$. Since $codim_{\mathbb{C}}Y\geq 2$, $\varphi\circ\pi^{-1}$ extends trivially through $Y$ has a global psh function on $\tilde{X}$. By the maximum principle $\varphi\circ\pi^{-1}$ is constant hence so is $\varphi$. Alternatively there is no positive closed current of bidegree $(1,1)$ on $X$ which is cohomologous to $[E]$ except $[E]$ itself.