$\partial \overline{\partial}$-lemma for Irreducible, Normal Projective Varieties

Question

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.

Answer 1

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 $\mathbb{P}^{N_1}$.