Let $(M, \omega)$ be a compact Kähler manifold. An $\omega$-quasi-plurisubharmonic function on $M$ is an upper semi-continuous function $\varphi : M \to \mathbb{R} \cup \{ - \infty \}$ such that $\omega + dd^c \varphi \geq 0$ in the sense of currents. The prototype in the non-compact or quasi-projective setting is $\log | f |^2$, where $f$ is a holomorphic function (e.g., the local defining section for a divisor $D \subset M$).
Are there examples of quasi-plurisubharmonic functions with polynomial decay?
Here is a non-example: Suppose we have a smooth divisor $D$ in $M$ with local defining section $s_D$. For $\varepsilon >0$, we may consider $| s_D |^{-2\varepsilon}$, where the norm is taken with respect to a Hermitian metric on the line bundle $\mathscr{O}_D$ associated to $D$. A standard calculation gives $$\sqrt{-1} \partial \overline{\partial} | s_D |^{-2\varepsilon} \geq \varepsilon | s_D |^{-2\varepsilon} R_D,$$ where $R_D$ is the curvature form of the Hermitian metric on $\mathscr{O}_D$. This example blows up at a polynomial rate, but in the wrong direction. I want a quasi-plurisubharmonic function which decays to $-\infty$ polynomially. Of course, this is not quasi-plurisubharmonic either.
