Let $X$ be an algebraic variety over $\mathbb{C}$ and $X^{{\text{\rm an}}}$ its associated complex analytic space. We recall the definition of differential forms on $X^{{\text{\rm an}}}$ introduced by Bloom and Herrera [BH69]. The space $X^{{\text{\rm an}}}$ can be covered by a family of open subsets $\{U_{i}\}_{i}$ such that each $U_{i}$ can be identified with a closed analytic subset of an open ball in $\mathbb{C}^{r}$ for some $r$. On each $U_{i}$, the differential forms are defined as the restriction to this subset of smooth complex-valued differential forms defined on an open neighbourhood of $U_{i}$ in $\mathbb{C}^{r}$. Two differential forms on $U_{i}$ are identified if they coincide on the non-singular locus of $U_{i}$. We denote by $\mathscr{A^{\ast}}(U_{i})$ the complex of differential forms of $U_{i}$, which is independent of the chosen embedding. In particular, if $U_{i}$ is non-singular, we recover the usual complex of differential forms. These complexes glue together to define a sheaf $\mathscr{A}^{\ast}_{X^{{\text{\rm an}}}}$. This sheaf is equipped with differential operators  d, $\operatorname{d^{c}}$, $\partial$, $\bar{\partial}$, an external product and inverse images with respect to analytic morphisms: these operations are defined locally on each $\mathscr{A^{\ast}}(U_{i})$ by extending the differential forms to a neighbourhood of $U_{i}$ in $\mathbb{C}^{r}$ as above and applying the corresponding operations for $\mathbb{C}^{r}$. We write $\mathcal{O}_{X^{{\text{\rm an}}}}$ and $C^{\infty}_{X^{{\text{\rm an}}}}=\mathscr{A}^{0}_{X^{{\text{\rm an}}}}$ for the sheaves of analytic functions and of smooth functions of $X^{{\text{\rm an}}}$, respectively.

Let $L$ be an algebraic line bundle on $X$ and $L^{{\text{\rm an}}}$ its analytification.

We remark that what we call “metric” in this text is called “continuous metric” in other contexts.

Let ${\overline{L}}=(L,\|\cdot\|)$ be a smooth metrized line bundle. Given a local section $s$ of $L^{{\text{\rm an}}}$ on an open subset $U$, the first Chern form of ${\overline{L}}$ is the $(1,1)$-form defined on $U$ as

It does not depend on the choice of local section and can be extended to a global closed $(1,1)$-form. Observe that we are using the algebro-geometric convention, and so $\operatorname{c}_{1}({\overline{L}})$ determines a class in $H^{2}(X^{{\text{\rm an}}},2\pi i\,\mathbb{Z})$.

Many smooth metrics can be obtained as the inverse image of the Fubini-Study metric. Let $X$ be a variety over $\mathbb{C}$ and $L$ a line bundle on $X$, and assume that there is an integer $e\geq 1$ such that $L^{\otimes e}$ is generated by global sections. Choose a basis of the space of global sections $\Gamma(X,L^{\otimes e})$ and let $\varphi\colon X\to\mathbb{P}^{M}_{\mathbb{C}}$ be the induced morphism. Given a local section $s$ of $L$, let $s^{\prime}$ be a local section of ${\mathcal{O}}(1)$ such that $s^{\otimes e}=\varphi^{\ast}s^{\prime}$. Then, the smooth metric on $L^{{\text{\rm an}}}$ obtained from the Fubini-Study metric by inverse image is given by

for any $p\in X^{{\text{\rm an}}}$ which is not a pole of $s$.

A family of smooth metrized line bundles ${\overline{L}}_{0},\dots,{\overline{L}}_{d-1}$ on $X$ and a $d$-dimensional cycle $Y$ of $X$ define a signed measure on $X^{{\text{\rm an}}}$ as follows. First suppose that $Y$ is a subvariety of $X$ and let $\delta_{Y}$ denote the current of integration along the analytic subvariety $Y^{{\text{\rm an}}}$, defined as $\delta_{Y}(\omega)=\frac{1}{(2\pi i)^{d}}\int_{Y^{{\text{\rm an}}}}\omega$ for $\omega\in\mathscr{A}^{2d}_{X^{{\text{\rm an}}}}$. Then the current

is a signed measure on $X^{{\text{\rm an}}}$. This notion extends by linearity to $Y\in Z_{d}(X)$. If ${\overline{L}}_{i}$, $i=0,\dots,d-1$, are semipositive and $Y$ is effective, this signed measure is a measure.