Definition 6.14.36A70B026D8EF

Definition 6.14.3.6A70B026D8EF Let $X$ be a complex manifold. A function $\varphi \colon X\to [-\infty ,\infty )$ is quasi-plurisubharmonic if it is quasi-plurisubharmonic near every point of $X$ in the sense of corollary 6.9.3, that is, if every point of $X$ has a neighborhood $V$ on which $\varphi =u+f$ pointwise, with $u\in \mathrm{PSH}(V)$ and $f$ a smooth real function on $V$. For a smooth closed real $(1,1)$-form $\theta $ on $X$ we put

and call its elements $\theta $-plurisubharmonic ($\theta $-psh) functions.