1.1 Metrics on singular spaces

Let $X$ be a reduced complex analytic space of pure dimension $n \geq 1$. We denote by $X^{\mathrm{reg}}$ the complex manifold of regular points of $X$ and $X^{\mathrm{sing}} := X \setminus X^{\mathrm{reg}}$ the singular set of $X$. Now, we give the definition of hermitian metrics on reduced complex analytic space $X$.

Note that in Definition 1.1 a hermitian metric on $X$ is more than just a metric on $X^{\mathrm{reg}}$. Now, we define Gauduchon metrics in the following two different concepts.

We define the complex Laplacian and the norm of gradients with respect to $\omega$ by

\begin{align*} \Delta_\omega f &:= \operatorname{tr}_\omega (\mathrm{dd}^{\mathrm{c}} f) = \frac{n \mathrm{dd}^{\mathrm{c}} f \wedge \omega^{n-1}}{\omega^{n}}\\ \big\lvert {\mathrm{d} f} \big\rvert_\omega^{2} &:= \operatorname{tr}_\omega(\mathrm{d} f \wedge \mathrm{d}^{\mathrm{c}} f) = \frac{n \mathrm{d} f \wedge \mathrm{d}^{\mathrm{c}} f \wedge \omega^{n-1}}{\omega^{n}}. \end{align*}

1.2 Currents on singular spaces

Recall that smooth forms on $X$ are defined as restriction of smooth forms in local embeddings. We denote by

1. (i) $\mathcal {D}_{p,q}(X)$ the space of compactly supported smooth forms of bidegree $(p,q)$;

2. (ii) $\mathcal {D}_{p,p}(X)_\mathbb {R}$ the space of real smooth $(p,p)$-forms with compact support.

The notion of currents on $X$, $\mathcal {D}'_{p,q}(X)$ and $\mathcal {D}'_{p,p}(X)_\mathbb {R}$, is defined by acting on compactly supported smooth forms on $X$. The operators $\mathrm {d}$, $\mathrm {d}^{\mathrm {c}}$ and $\mathrm {dd}^{\mathrm {c}}$ are well-defined by duality (see [Reference DemaillyDem85] for detail arguments).

1.3 Example

We give an example of a non-Kähler variety satisfying Setup (S) extracted from [Reference Lu and TianLT94]. The manifold $M = (\Gamma _1 \times \Gamma _2) \cap H \subset \mathbb {CP}^{3} \times \mathbb {CP}^{3}$ is a simply connected Calabi–Yau threefold with $b_2 = 14$, where

\begin{align*} \Gamma_1 &= \{x_0^{3} + x_1^{3} + x_2^{3} + x_3^{3} = 0\} \subset \mathbb{CP}^{3},\\ \Gamma_2 &= \{y_0^{3} + y_1^{3} + y_2^{3} + y_3^{3} = 0\} \subset \mathbb{CP}^{3},\\ H &= \{x_0 y_0 + x_1 y_1 + x_2 y_2 + x_3 y_3 = 0\} \subset \mathbb{CP}^{3} \times \mathbb{CP}^{3}. \end{align*}

There exists 15 disjoint rational curves $\ell _1, \ldots, \ell _{15}$ with normal bundles $\mathcal {O}_{\mathbb {CP}^{1}}(-1)^{\oplus 2}$, $\ell _1, \ldots, \ell _{14}$ generate $H_2(M,\mathbb {Z})$, and there exists $a_j \neq 0$ for $j \in \{1,\ldots,15\}$ such that $\sum _{j=1}^{15} a_j [\ell _j] = 0$. Then one can contract these curves and obtain a singular space $X_0$ with $15$ ordinary double points. Moreover, there is a smoothing $\pi : \mathcal {X} \rightarrow \mathbb {D}$ of $X_0$ such that for all $t\neq 0$, $X_t$ is diffeomorphic to a connected sum of $\mathbb {S}^{3} \times \mathbb {S}^{3}$. This implies that $X_t$ does not admit a Kähler metric since $b_2(X_t) = 0$. One can check that $X_0$ is not Kähler.

1.4 Remarks on the family setting

Under Setup (S), the family of metrics $(\omega _t)_{t}$ satisfies the following properties: For all $t \in \overline {\mathbb {D}}_{1/2}$, there is a constant $B \geq 0$ independent of $t$ such that

(1.1)\begin{equation} -B \omega_t^{n} \leq \mathrm{dd}^{\mathrm{c}}_t \omega_t^{n-1} \leq B \omega_t^{n}, \end{equation}

where $\mathrm {d}^{\mathrm {c}}_t$ is the twisted exterior derivative with respect to the complex structure of $X_t$. Indeed, in a local embedding, we have $-B\omega ^{n} \leq \mathrm {dd}^{\mathrm {c}} \omega ^{n-1} \leq B \omega ^{n}$ on $\mathcal {X}$; hence (1.1) is just the restriction on each fibre $X_t$. On the other hand, the volume of $(X_t, \omega _t)$ is comparable to the volume of $(X_0, \omega _0)$ for all $t \in \overline {\mathbb {D}}_{1/2}$. Namely, we have a uniform constant $C_V \geq 1$ such that

(1.2)\begin{equation} C_V^{-1} \leq \operatorname{Vol}_{\omega_t}(X_t) \leq C_V, \quad \forall t \in \overline{\mathbb{D}}_{1/2}. \end{equation}

The lower bound is obvious. One can prove the upper bound by the continuity of the total mass of currents $(\omega ^{n} \wedge [X_t])_{t \in \overline {\mathbb {D}}_{1/2}}$. The proof goes as follows: the current of integration $[X_t]$ can be written as $\mathrm {dd}^{\mathrm {c}} \log |\pi - t|$ by the Poincaré–Lelong formula. As $|\pi - t|$ converges to $|\pi |$ uniformly when $t \rightarrow 0$, $\log |\pi - t|$ converges to $\log |\pi |$ almost everywhere and, thus, $\log |\pi - t| \rightarrow \log |\pi |$ in $L^{1}$ when $t \rightarrow 0$ by Hartogs’ lemma. Therefore, $\omega ^{n} \wedge [X_t] \xrightarrow [t \rightarrow 0]{} \omega ^{n} \wedge [X_0]$ in the sense of currents and this implies $\operatorname {Vol}_{\omega _t}(X_t) \xrightarrow [t \rightarrow 0]{} \operatorname {Vol}_{\omega _0}(X_0)$. Thus, using the compactness of $X_0$, we obtain a uniform upper bound $C_V$ of $\operatorname {Vol}_{\omega _t}(X_t)$ for all $t \in \overline {\mathbb {D}}_{1/2}$.

Finally, we give a remark on non-smoothable singularities: The first non-smoothable example was given by Thom and reproduced by Hartshorne [Reference HartshorneHar74]. They considered a cone in $\mathbb {C}^{6}$ over the Segre embedding of $\mathbb {CP}^{1} \times \mathbb {CP}^{2}$ into $\mathbb {CP}^{5}$ and they proved that the cone does not admit a smoothing because of a topological constraint.

2.1 Proof of Theorem 2.1

In this subsection, we establish two gradient estimates ((2.1) and (2.3)) and then apply Moser's iteration argument to obtain an upper bound of $\rho$. In order to check the dependence on each data, we formulate the dependence of given constants. For convenience, we write $V := \operatorname {Vol}_\omega (X)$. We start the proof with a useful formula.

Lemma 2.4 Suppose that $\rho \in \ker \Delta ^{\ast }_\omega$. Then we have

\[ \int_X F'(\rho) \mathrm{d}\rho \wedge \mathrm{d}^{\mathrm{c}}\rho \wedge \omega^{n-1} = \int_X G(\rho) \mathrm{dd}^{\mathrm{c}}\omega^{n-1}, \]

where $F$ and $G$ are two real $\mathcal {C}^{1}$-functions defined on $\mathbb {R}_{>0}$ which satisfy $x F'(x) = G'(x)$.

Proof. As $\rho \in \ker \Delta _\omega ^{\ast }$, we have $\mathrm {dd}^{\mathrm {c}}(\rho \omega ^{n-1}) = 0$ equivalently. From Stokes’ theorem, it follows that

\begin{align*} 0 &= \int_X F(\rho) \mathrm{dd}^{\mathrm{c}}(\rho \omega^{n-1}) = - \int_X F'(\rho)\mathrm{d}\rho \wedge \mathrm{d}^{\mathrm{c}}(\rho \omega^{n-1})\\ &= - \int_X F'(\rho)\mathrm{d}\rho \wedge \mathrm{d}^{\mathrm{c}}\rho \wedge \omega^{n-1} - \int_X F'(\rho)\rho \mathrm{d}\rho \wedge \mathrm{d}^{\mathrm{c}}\omega^{n-1}. \end{align*}

By the assumption $xF'(x) = G'(x)$, we obtain the desired formula

\begin{align*} \int_X F'(\rho) \mathrm{d}\rho \wedge \mathrm{d}^{\mathrm{c}}\rho \wedge \omega^{n-1} &= -\int_X \rho F'(\rho)\mathrm{d}\rho \wedge \mathrm{d}^{\mathrm{c}}\omega^{n-1}\\ &= -\int_X \mathrm{d} G(\rho) \wedge \mathrm{d}^{\mathrm{c}} \omega^{n-1} = \int_X G(\rho) \mathrm{dd}^{\mathrm{c}} \omega^{n-1}. \end{align*}

Consider $F'(x) = ( {np^{2}}/{4}) x^{-p-2}$ and $G(x) = - ({np}/{4})x^{-p}$ for $p \geq 1$. By Lemma 2.4, we have the following inequality

(2.1)\begin{align} \int_X \big\lvert {\mathrm{d}\big(\rho^{-{p}/{2}}\big)} \big\rvert_\omega^{2} \omega^{n} &= \int_X n \mathrm{d} \big(\rho^{-{p}/{2}}\big) \wedge \mathrm{d}^{\mathrm{c}} \big(\rho^{-{p}/{2}}\big) \wedge \omega^{n-1}\nonumber\\ &= \int_X F'(\rho) \mathrm{d}\rho \wedge \mathrm{d}^{\mathrm{c}}\rho \wedge \omega^{n-1} = \int_X G(\rho) \mathrm{dd}^{\mathrm{c}}\omega^{n-1}\nonumber\\ &= \frac{np}{4} \int_X \rho^{-p} \big(-\mathrm{dd}^{\mathrm{c}}\omega^{n-1}\big)\nonumber\\ &\leq p\frac{nB}{4} \int_X \rho^{-p} \omega^{n}. \end{align}

Put $\beta = {n}/({n-1}) > 1$. Combining Sobolev inequality and (2.1), we obtain

\begin{align*} \bigg(\int_X (\rho^{-1})^{p\beta} \omega^{n}\bigg)^{{1}/{\beta}} &\leq C_S \bigg( \int_X \big\lvert {\mathrm{d} \big(\rho^{-{p}/{2}}\big)} \big\rvert^{2}_\omega \omega^{n} + \int_X \big\lvert {\rho^{-{p}/{2}}} \big\rvert^{2} \omega^{n}\bigg)\\ &\leq C_S \bigg( p \frac{n B}{4} \int_X (\rho^{-1})^{p} \omega^{n} + \int_X (\rho^{-1})^{p} \omega^{n}\bigg)\\ &\leq p \bigg( \frac{nB}{4} + 1 \bigg) C_S \int_X \rho^{-p} \omega^{n}. \end{align*}

For all $p \geq 1$, we have

\[ \left\lVert {\rho^{-1}} \right\rVert_{L^{p\beta}(X,\omega)} \leq p^{{1}/{p}} C_1^{{1}/{p}} \left\lVert {\rho^{-1}} \right\rVert_{L^{p}(X,\omega)}, \]

where $C_1$ is a constant depending only on $n, B, C_S$. Inductively, we obtain

\begin{align*} \left\lVert {\rho^{-1}} \right\rVert_{L^{p\beta^{k}}(X,\omega)} &\leq p^{({1}/{p}) \sum_{j=0}^{k-1} {1}/{\beta^{j}}} \beta^{({1}/{p}) \sum_{j=0}^{k-1} {j}/{\beta^{j}}} C_1^{({1}/{p}) \sum_{j=0}^{k-1} {1}/{\beta^{j}}} \left\lVert {\rho^{-1}} \right\rVert_{L^{p}(X,\omega)}\\ &\leq p^{{n}/{p}} \beta^{{n(n-1)}/{p}} C_1^{{n}/{p}} \left\lVert {\rho^{-1}} \right\rVert_{L^{p}(X,\omega)}. \end{align*}

Let $p=1$ and $k \rightarrow \infty$ and, thus, we obtain

\[ 1 = \sup_X \rho^{-1} \leq \beta^{n(n-1)} C_1^{n} \int_X \rho^{-1} \omega^{n}. \]

Therefore, the $L^{1}$-norm of $\rho ^{-1}$ is bounded away from zero by a constant $\delta$ which depends only on $n, B, C_S$:

\[ \int_X \rho^{-1} \omega^{n} \geq \frac{1}{\beta^{n(n-1)} C_1^{n}} =:2\delta. \]

Choosing sufficiently small $\delta$, we may assume $A := V/\delta \geq 1$. Then we have

\[ 2\delta \leq \int_{\{\rho< A\}} \rho^{-1} \omega^{n} + \int_{\{\rho \geq A\}} \rho^{-1} \omega^{n} \leq \int_{\{\rho< A\}} \omega^{n} + \frac{1}{A} \int_X \omega^{n}. \]

Hence, the volume of $\{\rho < A\}$ is bounded away from zero:

(2.2)\begin{equation} \int_{\{\rho < A\}} \omega^{n} \geq \delta. \end{equation}

Now, we consider

\[ F'(x) = \frac{n(p+1)^{2}}{4} \frac{(\log x)^{p-1}}{x^{2}}\quad {\rm and}\quad G(x) = \frac{n(p+1)^{2}}{4p} (\log x)^{p} \quad {\rm for}\ p \geq 1. \]

From Lemma 2.4, we find the following estimate

(2.3)\begin{align} \int_X \big\lvert {\mathrm{d} (\log \rho)^{({p+1})/{2}}} \big\rvert_\omega^{2} \omega^{n} &= \frac{n(p+1)^{2}}{4} \int_X \frac{(\log \rho)^{p-1}}{\rho^{2}}\mathrm{d}\rho \wedge \mathrm{d}^{\mathrm{c}}\rho \wedge \omega^{n-1}\nonumber\\ &= \int_X F'(\rho) \mathrm{d}\rho \wedge \mathrm{d}^{\mathrm{c}}\rho \wedge \omega^{n-1} = \int_X G(\rho) \mathrm{dd}^{\mathrm{c}}\omega^{n-1}\nonumber\\ &= \frac{n(p+1)^{2}}{4p} \int_X (\log \rho)^{p} \mathrm{dd}^{\mathrm{c}} \omega^{n-1}\nonumber\\ &\leq p n B \int_X (\log\rho)^{p} \omega^{n}. \end{align}

Again, Sobolev inequality, (2.3) and Hölder inequality yield the following inequalities:

\begin{align*} \bigg(\int_X (\log\rho)^{(p+1)\beta} \omega^{n} \bigg)^{{1}/{\beta}} &\leq C_S \bigg( \int_X \big\lvert {\mathrm{d} (\log \rho)^{({p+1})/{2}}} \big\rvert^{2}_\omega \omega^{n} + \int_X (\log\rho)^{p+1} \omega^{n} \bigg)\\ &\leq C_S\bigg( p n B \int_X (\log\rho)^{p} \omega^{n} + \int_X (\log\rho)^{p+1} \omega^{n}\bigg)\\ &\leq C_S\bigg( p n B V^{{1}/({p+1})} \bigg(\int_X (\log\rho)^{p+1} \omega^{n}\bigg)^{{p}/({p+1})} + \int_X (\log\rho)^{p+1} \omega^{n} \bigg)\\ &\leq (p+1) 2 n B C_S \max\{V,1\} \max\bigg\{\int_X (\log\rho)^{p+1} \omega^{n},1 \bigg\}. \end{align*}

Write $q = p+1 \geq 2$. We have

\[ \left\lVert {\log \rho} \right\rVert_{L^{q\beta}(X,\omega)} \leq q^{{1}/{q}} C_2^{{1}/{q}} \max\big\{\left\lVert {\log\rho} \right\rVert_{L^{q}(X,\omega)}, 1 \big\}, \]

where $C_2 > 0$ is a constant depending only on $n, B, C_S, V$. Using the similar strategy of Moser's iteration again, we derive

\begin{align*} \sup_X (\log\rho)&\leq 2^{{n}/{2}} \beta^{{n(n-1)}/{2}} C_2^{{n}/{2}} \max\{\left\lVert {\log\rho} \right\rVert_{L^{2}(X,\omega)},1\}\\ &\leq 2^{{n}/{2}} \beta^{{n(n-1)}/{2}}C_2^{{n}/{2}} \max \bigg\{\Big(\sup_X \log\rho\Big)^{{1}/{2}} \bigg(\int_X \log\rho \omega^{n}\bigg)^{{1}/{2}},1 \bigg\} \end{align*}

and, thus,

\[ \sup_X (\log\rho) \leq C_3 \max\bigg\{\int_X \log\rho \omega^{n}, 1\bigg\} \]

for some constant $C_3 = C_3(n,B,C_S,V)$.

Now, everything comes down to bounding $\int _X \log \rho \omega ^{n}$ from above. Using Poincaré inequality and (2.3) with $p=1$, we obtain

(2.4)\begin{equation} \int_X \big\lvert {\log \rho - \underline{\log\rho}} \big\rvert^{2} \omega^{n} \leq C_P \int_X \big\lvert {\mathrm{d} \log\rho} \big\rvert^{2}_\omega \omega^{n} \leq C_4 \int_X \log\rho \omega^{n}, \end{equation}

where $\underline {\log \rho } = ({1}/{V}) \int _X \log \rho \omega ^{n}$ is the average of $\log \rho$ and $C_4 = C_4(n, B, C_P)$. Then by (2.2), we can infer that

(2.5)\begin{align} \delta \int_X \log\rho \omega^{n} = V \delta \underline{\log\rho} &\leq V \int_{\{\rho < A\}} \underline{\log\rho} \omega^{n}\nonumber\\ &\leq V \int_{\{\rho < A\}} \big( \log A - \log\rho + \underline{\log\rho}\big) \omega^{n}\nonumber\\ &\leq V \int_X \big( \big\lvert {\log\rho - \underline{\log\rho}} \big\rvert + \log A\big) \omega^{n}. \end{align}

We use (2.4) and (2.5) to obtain

\begin{align*} \int_X \log\rho \omega^{n} &\leq \frac{V}{\delta} \bigg( \int_X \big\lvert {\log\rho - \underline{\log\rho}} \big\rvert \omega^{n} + V \log A \bigg)\\ &\leq \frac{V}{\delta} \bigg( V^{{1}/{2}} \bigg(\int_X \big\lvert {\log\rho - \underline{\log\rho}} \big\rvert^{2} \omega^{n}\bigg)^{{1}/{2}} + V \log A \bigg)\\ &\leq \frac{V}{\delta} \bigg(V^{{1}/{2}} C_4^{{1}/{2}} \bigg(\int_X \log\rho \omega^{n}\bigg)^{{1}/{2}} + V \log A\bigg). \end{align*}

Note that if $x^{2} \leq ax +b$ for $a, b >0$, then $x \leq {a}/{2}+(b+ {a^{2}}/{4})^{ {1}/{2}}$. Eventually, we obtain

\[ \int_X \log\rho \omega^{n} \leq C_5(n,B,C_S,C_P,V) \]

and this completes the proof of Theorem 2.1.

3.1 Gauduchon metric on the central fibre

Proof. We shall apply standard elliptic theory on some relatively compact subsets of $X_0^{\mathrm{reg}}$ to get a smooth function $\rho$. This $\rho$ is the limit of $(\rho _{t_j})_{j \in \mathbb {N}}$ defined on the fibre $X_{t_j}$ for some sequence $t_j \rightarrow 0$ when $j \rightarrow + \infty$.

Recall that $\pi : \mathcal {X} \rightarrow \mathbb {D}$ is submersive on $X_0^{\mathrm{reg}}$. By the tubular neighborhood theorem, there is an open neighborhood $\mathcal {M}$ of $X^{\mathrm{reg}}_0$ in $\mathcal {X}^{\mathrm{reg}}$ such that the following statements hold.

1. (i) For all $U \Subset X_0^{\mathrm{reg}}$, there exists an open set $\mathcal {M}_U \subset \mathcal {X}^{\mathrm{reg}}$, a constant $\delta _U > 0$, and a diffeomorphism $\psi _U: U \times \mathbb {D}_{\delta _U} \xrightarrow {\sim } \mathcal {M}_U$ such that the diagram

   
   commutes. In particular, for all $t \in \mathbb {D}_{\delta _U}$, $\psi _U(\cdot, t): U \rightarrow \mathcal {M}_U$ is a diffeomorphism onto its image $M_{U,t} = \psi _U(U,t) = \mathcal {M}_U \cap X_t$.

2. (ii) If $U \Subset V \Subset X_0^{\mathrm{reg}}$, we have $\delta _U \geq \delta _V > 0$ and $\psi _U(x, t) = \psi _V(x, t)$ for all $x \in U$ and $t \in \mathbb {D}_{\delta _V}$.

Now, we denote by $P_t = \Delta _{\omega _t}^{\ast }$ and fix $U_1 \Subset U_2 \Subset X_0^{\mathrm{reg}}$ which are connected open subsets. Note that $U_i$ can be identified with $M_{U_i,t} = \psi _{U_2}(U_i,t)$ for $i \in \{1,2\}$ and for all $t \in \mathbb {D}_{\delta _{U_2}}$ and, hence, $P_t$ can act on smooth functions defined on $U_2$. In addition, on $U_2$, the Riemannian metric $g_0$ induced by $\omega _0$ is quasi-isometric to $g_t$ induced by $\omega _t$, and the volume form $\omega _0^{n}$ is comparable with $\omega _t^{n}$. In other words, we have a uniform constant $C_{U_2} > 0$ such that

(3.1)\begin{equation} C_{U_2}^{-1} \langle {\cdot}, {\cdot}\rangle_{\omega_0} \leq \langle {\cdot}, {\cdot}\rangle_{\omega_t} \leq C_{U_2} \langle {\cdot}, {\cdot}\rangle_{\omega_0} \quad\text{and}\quad C_{U_2}^{-1} \omega_0^{n} \leq \omega_t^{n} \leq C_{U_2} \omega_0^{n}. \end{equation}

By Gårding inequality, we have

\[ \left\lVert {u} \right\rVert_{L^{2}_2(U_1,\omega_0)} \leq C_{U_1, U_2}\big(\left\lVert {P_t u} \right\rVert_{L^{2}(U_2,\omega_0)} + \left\lVert {u} \right\rVert_{L^{2}(U_2,\omega_0)} \big) \]

for all $u \in \mathcal {C}^{\infty }_c(U_2)$. The constant $C_{U_1,U_2}$ can be chosen independent of $t$ because the coefficients of $P_t$ move smoothly in $t$. Choose a cutoff function $\chi$ such that $\operatorname {supp}(\chi ) \subset U_2$ and $\chi \equiv 1$ on $U_1$. From Gårding inequality, we obtain

(3.2)\begin{equation} \left\lVert {\rho_t} \right\rVert_{L^{2}_2(U_1,\omega_0)} \leq C_{U_1,U_2} \big(\left\lVert {P_t(\chi\rho_t)} \right\rVert_{L^{2}(U_2,\omega_0)} + \left\lVert {\chi\rho_t} \right\rVert_{L^{2}(U_2,\omega_0)} \big). \end{equation}

In (3.2), the second term $\left \lVert {\chi \rho _t} \right \rVert _{L^{2}(U_2,\omega _0)}$ is uniformly bounded because of Corollary 2.3. Hence, we only need to estimate $\left \lVert {P_t(\chi \rho _t)} \right \rVert _{L^{2}(U_2,\omega _0)}$. Note that

(3.3)\begin{align} P_t(\chi \rho_t) &= \frac{n}{\omega_t^{n}} \mathrm{dd}^{\mathrm{c}}_t(\chi\rho_t \omega_t^{n-1})\nonumber\\ &= \frac{n}{\omega_t^{n}} \big( 2\mathrm{d}\rho_t\wedge\mathrm{d}^{\mathrm{c}}_t\chi\wedge\omega_t^{n-1} + \chi\mathrm{dd}^{\mathrm{c}}_t(\rho_t \omega_t^{n-1}) + \rho_t \mathrm{dd}^{\mathrm{c}}(\chi\omega_t^{n-1}) - \chi\rho_t\mathrm{dd}^{\mathrm{c}}_t\omega_t^{n-1}\big)\nonumber\\ &= 2 \langle {\mathrm{d}\rho_t}, {\mathrm{d}\chi}\rangle_{\omega_t} + \rho_t P_t(\chi) - \rho_t\chi\frac{\mathrm{dd}^{\mathrm{c}}_t\omega_t^{n}}{\omega_t^{n}}. \end{align}

Obviously, $\rho _t P_t(\chi )$ and $\rho _t \chi ({\mathrm {dd}^{\mathrm {c}}_t\omega _t^{n-1}}/{\omega _t^{n}})$ are uniformly bounded, so we only need to control the $L^{2}$-norm of the first term $\langle {\mathrm {d}\rho _t}, {\mathrm {d}\chi }\rangle _{\omega _t}$:

\begin{align*} \int_{U_2} \big\lvert {\langle {\mathrm{d}\rho_t}, {\mathrm{d}\chi}\rangle_{\omega_t}} \big\rvert^{2} \omega_0^{n} &\leq C_{U_2}^{2} \Big(\sup_{U_2} \big\lvert {\mathrm{d}\chi} \big\rvert^{2}_{\omega_0}\Big) \int_{U_2} \big\lvert {\mathrm{d}\rho_t} \big\rvert_{\omega_t}^{2} \omega_t^{n}\\ &= C_{U_2}^{2} \Big(\sup_{U_2} \big\lvert {\mathrm{d}\chi} \big\rvert^{2}_{\omega_0}\Big) \int_{M_{U_2,t}} \big\lvert {\mathrm{d}\rho_t} \big\rvert_{\omega_t}^{2} \omega_t^{n}\\ &\leq C_{U_2}^{2} \Big(\sup_{U_2} \big\lvert {\mathrm{d}\chi} \big\rvert^{2}_{\omega_0}\Big) \int_{X_t} \big\lvert {\mathrm{d}\rho_t} \big\rvert^{2}_{\omega_t} \omega_t^{n}\\ &\leq C_{U_2}^{2} \Big(\sup_{U_2} \big\lvert {\mathrm{d}\chi} \big\rvert^{2}_{\omega_0}\Big) \frac{nB}{2} \int_{X_t} \rho_t^{2} \omega_t^{n}. \end{align*}

Here the first line is by Cauchy–Schwarz inequality and (3.1). The fourth line follows from an argument similar to that used in (2.1) (just replace $-p$ by $p$). As $1 \leq \rho _t \leq C_G$ and $\operatorname {Vol}_{\omega _t}(X_t) \leq C_V$, we find a uniform bound of $\left \lVert {P_t(\chi \rho _t)} \right \rVert _{L^{2}(U_2,\omega _0)}$. Hence, $\left \lVert {\rho _t} \right \rVert _{L^{2}_2(U_1,\omega _0)}$ is uniformly bounded by some uniform constant $C(U_1,U_2)$.

For higher-order estimates, we apply higher-order Gårding inequalities on the fixed domains $U_1 \Subset U_2 \Subset X_0^{\mathrm{reg}}$:

\[ \left\lVert {u} \right\rVert_{L^{2}_{s+2}(U_1,\omega_0)} \leq C_{s, U_1, U_2}\big(\left\lVert {P_t u} \right\rVert_{L^{2}_s(U_2,\omega_0)} + \left\lVert {u} \right\rVert_{L^{2}(U_2,\omega_0)} \big) \]

for all $u \in \mathcal {C}^{\infty }_c(U_2)$. Let $\mathcal {U} = (U_i)_{i\in \mathbb {N}}$ be a relatively compact exhaustion of $X_0^{\mathrm{reg}}$. Differentiating (3.3) on both sides and using a bootstrapping argument, we obtain $\left \lVert {\rho _t} \right \rVert _{L^{2}_s(U_1,\omega _0)} < C(s,\mathcal {U})$ where $C(s,\mathcal {U})$ does not depend on $t$. By Rellich's theorem, there exists a subsequence $(\rho _{t_j})_{j \in \mathbb {N}}$ such that $\rho _{t_j}$ converges to $\rho$ in $\mathcal {C}^{k}(\overline {U_1})$ for all $k\in \mathbb {N}$ when $t_j \rightarrow 0$. Therefore, $\mathrm {dd}^{\mathrm {c}}_0(\rho \omega _0^{n-1}) = \lim _{j \rightarrow + \infty } \mathrm {dd}^{\mathrm {c}}_{t_j} (\rho _{t_j} \omega _{t_j}^{n-1}) = 0$ on $U_1$. Using a diagonal argument, we can infer that there is a smooth function $\rho$ on $X_0^{\mathrm{reg}}$ which is bounded between $1$ and $C_G$, and satisfies $\mathrm {dd}^{\mathrm {c}}_0(\rho \omega _0^{n-1}) = 0$ on $X_0^{\mathrm{reg}}$.

3.2 Proof of Theorem B

In this subsection, we always assume that $X$ is an irreducible reduced compact complex space and $\omega _\mathrm {G}$ is a bounded Gauduchon metric on $X$. Before proving the uniqueness result and extension property, we recall the existence of cutoff functions with small $L^{2}$-gradients. From a classical property in Riemannian geometry, these cutoff functions do exist on so-called parabolic manifolds (see, e.g., [Reference GlasnerGla83, Reference Evans and GariepyEG92]). In our case, one can easily construct explicit cutoff functions by Hironaka's desingularization and $\log \log$-potentials (cf. [Reference BerndtssonBer12, Lemma 2.2] and [Reference Campana, Guenancia and PăunCGP13, § 9]):

These functions allow us to perform integration by parts as on a compact manifold. Then we argue that the desired results hold when $\varepsilon$ tends to zero. Now, we give a proof of Theorem B:

Proof of Theorem B We divide the proof in two parts.

Part 1: uniqueness of singular Gauduchon metrics. Assume that $\omega _\mathrm {G}$ and $\omega _\mathrm {G}'$ are both bounded Gauduchon metrics in the same conformal class. We write $\rho$ to be the bounded Gauduchon factor satisfying $\rho ^{ {1}/({n-1})} \omega _\mathrm {G} = \omega _\mathrm {G}'$. Let $(\chi _\varepsilon )_{\varepsilon >0}$ be cutoff functions given in Lemma 3.2. From Stokes formula and direct computations, we derive

(3.4)\begin{equation} \int_{X^{\mathrm{reg}}} \chi_\varepsilon |\mathrm{d} \rho|^{2}_{\omega_\mathrm{G}} \omega_\mathrm{G}^{n} = \frac{1}{2} \int_{X^{\mathrm{reg}}} \rho^{2} \mathrm{dd}^{\mathrm{c}}\chi_\varepsilon \wedge \omega_\mathrm{G}^{n-1}. \end{equation}

Indeed,

\begin{align*} \int_{X^{\mathrm{reg}}} \chi_\varepsilon |\mathrm{d} \rho|_{\omega_\mathrm{G}}^{2} \omega_\mathrm{G}^{n-1} &= \int_{X^{\mathrm{reg}}} \chi_\varepsilon \mathrm{d} \rho \wedge \mathrm{d}^{\mathrm{c}} \rho \wedge \omega_\mathrm{G}^{n-1} = - \int_{X^{\mathrm{reg}}} \rho \mathrm{d} \big(\chi_\varepsilon \mathrm{d}^{\mathrm{c}} \rho \wedge \omega_\mathrm{G}^{n-1} \big)\\ &= - \int_{X^{\mathrm{reg}}} \rho \big(\mathrm{d} \chi_\varepsilon \wedge \mathrm{d}^{\mathrm{c}} \rho \wedge \omega_\mathrm{G}^{n-1} + \chi_\varepsilon \mathrm{dd}^{\mathrm{c}} \rho \wedge \omega_\mathrm{G}^{n-1} - \chi_\varepsilon \mathrm{d}^{\mathrm{c}} \rho \wedge \mathrm{d} \omega_\mathrm{G}^{n-1}\big)\\ &= -\frac{1}{2} \int_{X^{\mathrm{reg}}} \mathrm{d} \rho^{2} \wedge \mathrm{d}^{\mathrm{c}} \chi_\varepsilon \wedge \omega_\mathrm{G}^{n-1} + \int_{X^{\mathrm{reg}}} \rho \chi_\varepsilon \mathrm{d} \rho \wedge \mathrm{d}^{\mathrm{c}} \omega_\mathrm{G}^{n-1}\\ &= \frac{1}{2} \int_{X^{\mathrm{reg}}} \rho^{2} \mathrm{d} \big( \mathrm{d}^{\mathrm{c}} \chi_\varepsilon \wedge \omega_\mathrm{G}^{n-1}\big) - \frac{1}{2} \int_{X^{\mathrm{reg}}} \rho^{2} \mathrm{d}\big(\chi_\varepsilon \mathrm{d}^{\mathrm{c}}\omega_\mathrm{G}^{n-1}\big)\\ &= \frac{1}{2} \int_{X^{\mathrm{reg}}} \rho^{2} \mathrm{dd}^{\mathrm{c}} \chi_\varepsilon \wedge \omega_\mathrm{G}^{n-1} - \frac{1}{2} \int_{X^{\mathrm{reg}}} \rho^{2} \mathrm{d}^{\mathrm{c}} \chi_\varepsilon \wedge \mathrm{d} \omega_\mathrm{G}^{n-1}\\ &\quad - \frac{1}{2} \int_{X^{\mathrm{reg}}} \rho^{2} \mathrm{d} \chi_\varepsilon \wedge \mathrm{d}^{\mathrm{c}} \omega_\mathrm{G}^{n-1}\\ &= \frac{1}{2} \int_{X^{\mathrm{reg}}} \rho^{2} \mathrm{dd}^{\mathrm{c}} \chi_\varepsilon \wedge \omega_\mathrm{G}^{n-1}. \end{align*}

By Lemma 3.2 and the boundedness of $\omega _\mathrm {G}$, we can see that $\int _{X} |\mathrm {dd}^{\mathrm {c}} \chi _\varepsilon \wedge \omega _G^{n-1}|$ converges to zero when $\varepsilon$ tends to zero. As $\rho$ is bounded, the right-hand side of (3.4) goes to zero and, hence,

\[ \int_{X^{\mathrm{reg}}} |\mathrm{d} \rho|^{2}_{\omega_\mathrm{G}} \omega_\mathrm{G}^{n} = 0. \]

Because $X^{\mathrm{reg}}$ is connected by the irreducible assumption, $\rho$ is a constant.

Part 2: extension to a pluriclosed current. We already have a smooth pluriclosed bounded positive $(n-1,n-1)$-form $\omega _\mathrm {G}^{n-1}$ on $X^{\mathrm{reg}}$. Note that $\omega _\mathrm {G}^{n-1}$ can extend trivially as a bounded positive $(n-1,n-1)$-current $T$ on $X$. If $\omega _G^{n-1}$ is closed, so would be $T$ by the Skoda–El Mir extension result [Reference SkodaSko82, Reference El MirEM84]. Some results for plurisubharmonic currents do exist in the literature (see, e.g., [Reference Alessandrini and BassanelliAB93, Reference Dabbek, Elkhadhra and El MirDEEM03]), but we could not find the one of interest for us here. Therefore, we provide a quick proof.

Again, we are going to use good cutoff functions $(\chi _\varepsilon )_{\varepsilon > 0}$ in Lemma 3.2 to prove that $T$ is pluriclosed in the sense of currents. Fix a smooth function $f$ on $X$. We need to show that

\[ \langle {f}, {\mathrm{dd}^{\mathrm{c}} T}\rangle := \int_{X} \mathrm{dd}^{\mathrm{c}} f \wedge T = 0. \]

Using the cutoff functions constructed in Lemma 3.2, we can write

\[ \int_{X} \mathrm{dd}^{\mathrm{c}} f \wedge T = \underbrace{\int_{R_\varepsilon} \chi_\varepsilon \mathrm{dd}^{\mathrm{c}} f \wedge T}_{:= {\rm I}_\varepsilon} + \underbrace{\int_{S_\varepsilon} (1-\chi_\varepsilon) \mathrm{dd}^{\mathrm{c}} f \wedge T}_{:={\rm II}_\varepsilon}, \]

where $R_\varepsilon$ is a small open neighborhood of the closure of $\{\chi _\varepsilon > 0\}$ contained in $X^{\mathrm{reg}}$ and similarly $S_\varepsilon$ is a small neighborhood of the closure of $\{\chi _\varepsilon < 1\}$. According to Lemma 3.2, as $\varepsilon \rightarrow 0$, $R_\varepsilon$ tends to $X^{\mathrm{reg}}$ and $S_\varepsilon$ shrinks to $X^{\mathrm{sing}}$. Therefore, ${\rm II}_\varepsilon$ converges to zero when $\varepsilon$ goes to zero. We compute the term ${\rm I}_\varepsilon$ by the Stokes formula:

\begin{align*} {\rm I}_\varepsilon&= \int_{R_\varepsilon} f \mathrm{dd}^{\mathrm{c}}\big( \chi_\varepsilon \omega_\mathrm{G}^{n-1}\big) = \int_{R_\varepsilon} f \big(\mathrm{dd}^{\mathrm{c}} \chi_\varepsilon \wedge \omega_\mathrm{G}^{n-1} + 2 \mathrm{d} \omega_\mathrm{G}^{n-1} \wedge \mathrm{d}^{\mathrm{c}} \chi_\varepsilon\big)\\ &= -\underbrace{\int_{R_\varepsilon} f \mathrm{dd}^{\mathrm{c}} \chi_\varepsilon \wedge \omega_\mathrm{G}^{n-1}}_{:= {\rm III}_\varepsilon} - 2 \underbrace{\int_{R_\varepsilon} \mathrm{d} f \wedge \mathrm{d}^{\mathrm{c}} \chi_\varepsilon \wedge \omega_\mathrm{G}^{n-1}}_{:= {\rm IV}_\varepsilon}. \end{align*}

By Cauchy–Schwarz inequality, the term ${\rm IV}_\varepsilon$ can be bounded by

\[ |{\rm IV}_\varepsilon| \leq \underbrace{\int_{R_\varepsilon} \mathbb{1}_{\{\mathrm{d} \chi_\varepsilon \neq 0\}} \mathrm{d} f \wedge \mathrm{d}^{\mathrm{c}} f \wedge \omega_\mathrm{G}^{n-1}}_{:= {\rm V}_\varepsilon} + \underbrace{\int_{R_\varepsilon} \mathrm{d}\chi_\varepsilon \wedge \mathrm{d}^{\mathrm{c}}\chi_\varepsilon \wedge \omega_\mathrm{G}^{n-1}}_{:= {\rm VI}_\varepsilon}. \]

Using the dominated convergence theorem, ${\rm V}_\varepsilon$ converges to zero when $\varepsilon$ goes to zero, because the set $\{\mathrm {d}\chi _\varepsilon \neq 0\}$ is contained in $S_\varepsilon$ and $S_\varepsilon$ shrinks to $X^{\mathrm{sing}}$. Applying Lemma 3.2, ${\rm III}_\varepsilon$ and ${\rm VI}_\varepsilon$ converge to zero as $\varepsilon$ tending to zero. These yield that ${\rm I}_\varepsilon \rightarrow 0$ as $\varepsilon \rightarrow 0$. All in all, we have $\int _{X} \mathrm {dd}^{\mathrm {c}} f \wedge T = 0$ for all $f \in \mathcal {C}^{\infty }(X)$; hence, $\mathrm {dd}^{\mathrm {c}} T = 0$ in the sense of currents.