Let $L$ be an ample line bundle on an smooth projective $K$-analytic variety $X$. Fix a model metric $\|\cdot\|$ on $L$ with curvature form $\theta$. A $\theta$-plurisubharmonic function on $X$ is then a function $\varphi:X\to[-\infty,+\infty)$ such that:

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  $\varphi$ is upper semicontinuous (usc).

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  $\varphi\leq\varphi\circ p_{\mathcal{X}}$ for each SNC model $\mathcal{X}$.

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  $\varphi$ is a uniform limit on each dual complex $\Delta_{\mathcal{X}}$ of $\theta$-psh model functions.

A singular semipositive metric is a metric $\|\cdot\|e^{-\varphi}$ with $\varphi$ a $\theta$-psh function.