Reference: W. Ding, G. Tian -- Kähler--Einstein metrics and the Generalised Futaki Invariant, *Inventiones mathematicae*, (1992). 


Let $X$ be a normal projective variety which is irreducible. Given an ample line bundle $L \to X$ over $X$, there exists an $m \in \mathbb{N}$ such that the global sections of $L^{\otimes m}$ furnish a Kodaira embedding $\phi\_m : X \hookrightarrow \mathbb{CP}^N$ for some $N \in \mathbb{N}$. Let $\omega\_{\text{FS}}$ denote the Fubini--Study metric on $\mathbb{CP}^N$, which may be pulled back via $\phi\_m$ to obtain a metric $$\omega = \frac{\alpha}{m} \phi\_m^{\ast} \left( \omega\_{\text{FS}} + \frac{\sqrt{-1}}{2\pi} \partial \overline{\partial} \psi \right),$$ on $X$, where $\alpha >0$ and $\psi \in \mathscr{C}^{\infty}(\mathbb{CP}^N, \mathbb{R})$. We call a metric $\omega$ of this form *admissible*. 


One may show that since $\dim\_{\mathbb{C}}\text{Sing}(X) \leq \dim\_{\mathbb{C}} X - 1$, $\omega$ defines a cohomology class of type $(1,1)$ on $X$. In other words, $[\omega] = \alpha c\_1(L)$, where $c\_1(L)$ denotes the first Chern class of $L$. 


It is claimed in the above reference that it follows from the definition that, given two admissible metrics $\omega\_1, \omega\_2 \in [\omega]$, there is a bounded, continuous functon $\varphi \in \mathscr{C}(X) \cap \mathscr{C}^{\infty}(\text{Reg}(X))$, such that over $\text{Reg}(X)$, we have $$\omega\_2 - \omega\_1 = \frac{\sqrt{-1}}{2\pi} \partial \overline{\partial} \varphi,$$
where $\text{Reg}(X)$ denotes the regular part of $X$.


**Q:** Does anyone have a reference for where I may find this *generalised $\partial \overline{\partial}$--lemma*? It is likely to be in Griffith's and Harris' *Principles of Algebraic Geometry*, but I cannot seem to find the result that ensures that $\varphi$ is bounded and continuous on $\text{Sing}(X)$.


Thanks in advance. 



$\partial \overline{\partial}$-lemma for Irreducible, Normal Projective Varieties
This just follows from the definition of admissible. Indeed $\omega\_1 = c(\omega\_{FS} + i\partial\bar\partial \psi\_1),$ and $\omega\_2 = c(\omega\_{FS} + i\partial\bar\partial \psi\_2$). Thus $\omega\_1 - \omega\_2 = i\partial\bar\partial c(\psi\_1 -\psi\_2)$, with $\psi\_1 - \psi\_2$ the restriction of a smooth function on $\mathbb{P}^N$. $\phi\_m$ is an embedding of $X$, hence an embedding of the smooth locus of $X$, so $\psi\_1 -\psi\_2$ is smooth on the smooth locus of $X$. Similarly the restriction of a continuous function is continuous, so $\psi\_1 - \psi\_2$ is continuous on all of $X$.


If you want to allow the $m$ used for $\omega\_1$ and $\omega\_2$ to vary, say $m\_1
$ and $m\_2$ with $m\_1<m\_2$, one could first embed $\mathbb{P}^{N\_1}$ into $\mathbb{P}^{N\_2}$ and argue in the same way, using the $\partial\bar\partial$-lemma on $\mathbb{P}^{N\_1}$.


