\urladdr

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Weighted cscK metrics on Kähler varieties

Chung-Ming Pan Centre interuniversitaire de recherches en géométrie et topologies (CIRGET); Université du Québec à Montréal; Case postale 8888, Succursale centre-ville, Montréal, Québec, H3C 3P8, Canada pan.chung_ming@uqam.ca and Tat Dat Tô Institut de Mathématiques de Jussieu-Paris Rive Gauche; Sorbonne Université - Campus Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05, France tat-dat.to@imj-prg.fr

(September 6, 2025)

Abstract.

We study the weighted constant scalar curvature Kähler equations on mildly singular Kähler varieties. Assuming the existence of a suitable resolution of singularities, we establish the existence of singular weighted cscK metrics when the weighted Mabuchi functional is coercive for an extremal weight. This extends the works of Chen-Cheng and He to the singular weighted setting. Moreover, we provide a method for constructing examples of singular cscK metrics inspired by the work of Arezzo–Pacard. In contrast to the usual gluing techniques, our approach does not require a precise understanding about of the metric behavior near the singular locus.

Key words and phrases:

Weighted cscK metric, Log terminal singularities, A priori estimates

1991 Mathematics Subject Classification:

53C55, 32J27, 32Q20, 32W20, 35A23

Introduction

The constant scalar curvature Kähler (cscK) metric problem has become one of the central focus in Kähler geometry during the last decades. The Yau–Tian–Donaldson conjecture asserts that given a compact Kähler manifold with a fixed Kähler class, the existence of cscK metrics in the Kähler class is equivalent to an algebro-geometric notion called ”K-stability”.

Several progresses in the literature [DR17, BDL20, CC21a, CC21b] have shown that the existence of a unique cscK metric in a Kähler class is equivalent to the coercivity of the Mabuchi functional, whose minimizers are cscK metrics. Boucksom–Hisamoto–Jonsson [BHJ19] demonstrated that the coercivity of the Mabuchi functional implies uniform K-stability (see [DR17, Der18, SD18, SD20] for a transcendental setup). Conversely, C. Li [Li22b] (and the recent transcendental version by Mesquita-Piccione [MP24]) showed that the strong uniform K-stability implies the coercivity of Mabuchi functional. The remaining challenge in proving the uniform Yau–Tian–Donaldson conjecture lies in establishing strong uniform K-stability from uniform K-stability.

Typical examples of cscK metrics are Kähler–Einstein metrics. Motivated by Minimal Model Program and moduli theory, Kähler–Einstein metrics have been well studied on smooth and mildly singular Kähler varieties [Aub78, Yau78, EGZ09, CDS15, BBE⁺19, BBJ21, LTW21, LTW22, Li22a] and their families [Koi83, RZ11a, RZ11b, SSY16, LWX19, DGG23, PT25] etc.

However, there are very few results regarding cscK metrics in the singular setting. We shall focus on the analytic part of the Yau–Tian–Donaldson conjecture on mildly singular varieties, particularly the relation between the existence of singular cscK metrics and the coercivity of the Mabuchi functional and explore under the weighted formalism. In a recent joint work with Trusiani [PTT23], when the Mabuchi functional is coercive, we establish the existence of singular cscK metrics on $\mathbb{Q}$ -Gorenstein smoothable Kähler varieties with log terminal singularities. One of the key ingredients is the stability of the coercivity of the Mabuchi functional [PTT23, Thm. A]. A similar strategy for establishing openness of coercivity has been applied to the resolution setting by Boucksom–Jonsson–Trusiani [BJT24] under an appropriate condition on the resolution.

This article aims to remove the additional smoothable assumption and to investigate existence results in a more general weighted setting. The weighted framework introduced by Lahdili [Lah19] (see also [Ino22]) includes various notions of canonical Kähler metrics, for example, extremal Kähler metrics and Kähler–Ricci solitons. For further results on the existence of weighted cscK metrics, we refer to [AJL23, Lah23, DJL24, DJL25, HL25] and the references therein.

We quickly review the basic setup and notations for the weighted cscK metrics below (see Section 1 for more details).

Setting (GS).

Let $X$ be an $n$ -dimensional compact Kähler variety with log terminal singularities, let $T\subset\operatorname{Aut}_{\operatorname{red}}(X)$ be a maximal real torus in the reduced automorphism group, and let $\omega$ be a $T$ -invariant Kähler metric on $X$ . Denote by $T_{\mathbb{C}}$ the complexification of $T$ , $\mathfrak{t}$ the Lie algebra of $T$ and $m_{\omega}:X\to\mathfrak{t}^{\vee}$ a moment map associated to $\omega$ . Consider functions $v,w\in\mathcal{C}^{\infty}(\mathfrak{t}^{\vee},\mathbb{R})$ with $v>0$ on $m_{\omega}(X)$ .

A metric $\omega^{\prime}$ is a singular $(v,w)$ -cscK metric if it has locally bounded potentials on $X$ and it solves the weighted cscK equation $S_{v}(\omega^{\prime})=w(m_{\omega^{\prime}})$ on $X^{\mathrm{reg}}$ , where $S_{v}$ is the weighted scalar curvature. Such an equation is the Euler–Lagrange equation of the $(v,w)$ -weighted Mabuchi functional $\mathbf{M}_{v,w}$ . When $w=\ell w_{0}$ , where $w_{0}>0$ on $m_{\omega}(X)$ and $\ell$ is an affine function such that the Mabuchi functional $\mathbf{M}_{v,w}$ is $T_{\mathbb{C}}$ -invariant, the corresponding $(v,w)$ -cscK metrics are called $(v,w_{0})$ -extremal metrics. In this context, $w$ is referred to as an extremal weight. The cscK metrics and extremal Kähler metrics correspond to the special case of $(1,1)$ -extremal metrics.

Before presenting our results, we introduce the following additional condition:

Condition (A).

There exists a $T$ -equivariant resolution of singularities $\pi:Y\to X$ such that $Y$ is Kähler and $\pi$ is an isomorphism over $\pi^{-1}(X^{\mathrm{reg}})$ . Also, there exist a Kähler metric $\omega_{Y}$ on $Y$ , a positive constant $K_{1}>0$ , and a function $\rho_{1}\in\operatorname{QPSH}(Y)$ such that

\operatorname{Ric}(\omega_{Y})\geq-K_{1}(\pi^{\ast}\omega+dd^{c}\rho_{1})\quad\text{and}\quad\int_{Y}e^{-K_{1}\rho_{1}}\omega^{n}_{Y}<+\infty.

In the above notation, $\operatorname{QPSH}(Y)$ is the set of all quasi-plurisubharmonic functions on $Y$ . A resolution of singularities described in Condition (A) is referred to as a resolution of Fano type in [BJT24] (see Section 3.3.1 and [BJT24, Sec. 4.1] for further discussions and examples).

Under Condition (A), we establish the following existence theorem for singular weighted cscK metrics:

Theorem A.

Let $(X,\omega)$ be a compact Kähler variety with log terminal singularities that satisfies Setting (GS). Assume that $(X,\omega)$ satisfies Condition (A), $v$ is $\log$ -concave and $w$ is an extremal weight. If the weighted Mabuchi functional $\mathbf{M}_{v,w}$ is $T_{\mathbb{C}}$ -coercive, then $X$ admits a singular $(v,w)$ -cscK metric in $\{\omega\}$ , which also minimizes $\mathbf{M}_{v,w}$ .

For cscK and extremal metrics, the weight $v\equiv 1$ on $\mathfrak{t}^{\vee}$ (hence $\log$ -concave). In the smooth setup, existence results under the coercivity of the Mabuchi functional are obtained by [CC21b] and [He19] for cscK and extremal metrics, respectively. Our Theorem A, in particular, extends their results to singular settings. We state a direct corollary for the case of cscK metrics, corresponding $v=1$ and $w=\frac{4\pi nc_{1}(X)\cdot[\omega]^{n-1}}{[\omega]^{n}}$ in the following.

Corollary B.

Let $X$ be a compact Kähler variety with log terminal singularities and let $T\subset\operatorname{Aut}_{\operatorname{red}}(X)$ be a maximal real torus. Assume that $\omega$ is a $T$ -invariant Kähler metric and that $(X,\omega)$ satisfies Condition (A). If the Mabuchi functional $\mathbf{M}_{\omega}$ on $X$ is $T_{\mathbb{C}}$ -coercive, then $X$ admits a singular cscK metric in $\{\omega\}$ which also minimizes $\mathbf{M}_{\omega}$ . In particular, if $\operatorname{Aut}(X)$ is discrete, the coercivity of the Mabuchi functional implies the existence of singular cscK metrics in $\{\omega\}$ .

The strategy for proving Theorem A is to establish uniform a priori estimates for the solutions to a family of weighted cscK equations on the resolution of singularities $\pi:Y\to X$ . Denote by $E$ the exceptional set of $\pi$ . Consider a family of perturbed Kähler metrics $\omega_{\varepsilon}:=\pi^{\ast}\omega+\varepsilon\omega_{Y}$ on the resolution for $\varepsilon\in(0,1]$ . Under Condition (A), and assuming $w$ is an extremal weight, [BJT24, Thm. A] has proved the openness of uniform coercivity on $Y$ . By [DJL25, HL25], there exists a Kähler metric $\omega_{\varepsilon,\varphi_{\varepsilon}}:=\omega_{\varepsilon}+dd^{c}\varphi_{\varepsilon}$ solving the weighted cscK equation on $Y$ . Then the uniform coercivity of weighted Mabuchi functional yields a uniform control on the entropy $\mathbf{H}_{\varepsilon}(\varphi_{\varepsilon})$ . Under Condition (A) and the bound on the entropy we establish priori estimates for the weighted cscK equation extending the estimates of Chen and Cheng to the degenerate weighted setting. Consequently, one can then extract a subsequence $\varphi_{\varepsilon}$ converging in $\mathcal{C}^{\infty}_{\mathrm{\ell oc}}(Y\setminus E)$ to a bounded $\pi^{\ast}\omega$ -psh function $\varphi_{0}$ , which is smooth on $Y\setminus E\simeq X^{\mathrm{reg}}$ and solves the weighted cscK equation there.

In particular, within the context of Corollary B, these estimates shows that singular Kähler–Einstein metrics on a Kähler varieties can be approximated by extremal metrics, provided Condition (A) holds, and this generalizes a recent result by Székelyhidi [Szé24, Thm. 3] for non-discrete automorphism group.

We highlight below the main difficulties and contributions made in the article:

•

$L^{\infty}$ -estimate and Condition (A). The main difficulty to obtain uniform $L^{\infty}$ -estimate is the lack of uniform bounds for the Ricci curvature of $\omega_{Y}$ with respect to the reference metrics $(\omega_{\varepsilon})_{\varepsilon}$ on the resolution of singularities. That is the reason why the estimates of Chen–Cheng [CC21a] and Guo–Phong [GP24] cannot be applied directly. To overcome this difficulty, we follow and generalize the Guo-Phong’s approach by incorporating Condition (A), which provides a weak version of lower bound for $\operatorname{Ric}(\omega_{Y})$ , together with the strong openness [Ber13, GZ15], and Demailly–Kollar’s theorem [DK01]. Moreover, our $L^{\infty}$ -estimate, Theorem 2.1, does not require that $v$ be log-concave and $w$ be an extremal weight.
•

Constructing examples. In Section 3.3, we present a method for constructing singular cscK metrics inspired by Arezzo–Pacard [AP06] under Condition (A). Additionally, we propose a mixed construction that integrates the result on the smoothable setting [PTT23]. An important ingredient in the construction is the stability of the coercivity of the Mabuchi functional for the blow-up along compact submanifolds in the smooth locus of singular varieties (cf. Lemma 3.6) instead of desingularizations as in [BJT24]. Comparing to the usual gluing technique (cf. [AP06]), our method does not require a very precise understanding of how the metric behaves near the singular locus.

We provide further detailed comments on the proof and the technical assumption on the $\log$ -concavity of $v$ . For higher order estimates, we adapt the strategy of Chen–Cheng [CC21b] and its generalization for the weighted setting [HL25, DJL24]. Due to the degeneration of $\omega_{\epsilon}$ , we modify the test function for the Laplacian by adding a strictly $\pi^{*}\omega$ -psh function $\rho$ with analytic singularities, ensuring $\omega_{\varepsilon}+dd^{c}\rho\geq C\omega_{Y}$ to absorb problematic terms. We obtain an integral Laplacian estimate with respect to $\mu=e^{C^{\prime}\rho}\omega_{Y}^{n}$ , giving a local integral Laplacian estimate away from the exceptional locus. The $\log$ -concavity of $v$ is used in the uniform integral Laplacian estimate in order to eliminate a problematic term in the weighted Aubin–Yau inequality. Local Laplacian and higher order estimates then follow from a generalization of Chen–Cheng’s local estimates for the weighted setting (cf. Appendix B). We remark that our estimates apply to general $w$ without requiring it to be an extremal weight.

Acknowledgements.

The authors are grateful to C. Arezzo, S. Boucksom, E. Di Nezza, S. Jubert, A. Lahdili, Y. Odaka, S. Sun, G. Székelyhidi, and A. Trusiani for helpful and inspiring discussions. The authors would like to thank S. Boucksom, M. Jonsson, A. Trusiani, and G. Székelyhidi for kindly sharing their articles. The authors would also like to thank V. Guedj and H. Guenancia for their suggestions that helped improve the exposition.

Part of this article is based upon work supported by the National Science Foundation under Grant No. DMS-1928930, while the first named author was in residence at the Simons Laufer Mathematical Sciences Institute (formerly MSRI) in Berkeley, California, during the Fall 2024 semester. The second named author is partially supported by ANR-21-CE40-0011-01 (research project MARGE), PEPS-JCJC-2024 (CRNS) and Tremplins-2024 (Sorbonne University).

1. Preliminaries

In this section, we recall several basic notions of pluripotential theory on singular spaces and weighted cscK metrics. We define $d^{c}:=\mathrm{i}(\bar{\partial}-\partial)$ and then we have $dd^{c}=2\mathrm{i}\partial\bar{\partial}$ . By variety, we always mean an irreducible reduced complex analytic space.

1.1. Pluripotential theory on normal Kähler varieties

Let $X$ be an $n$ -dimensional compact normal complex variety. A smooth Kähler metric on $X$ is defined by a Kähler metric $\omega$ on $X_{\mathrm{reg}}$ , and it is locally a restriction of a Kähler metric defined near the image of a local embedding $j:X\underset{\mathrm{\ell oc}.}{\hookrightarrow}\mathbb{C}^{N}$ . By Kähler variety, we mean a complex variety $X$ equipped with a smooth Kähler metric $\omega$ . More generally, a smooth form $\alpha$ on $X$ is defined as a smooth form on $X^{\mathrm{reg}}$ such that $\alpha$ extends smoothly under any local embedding $X\underset{\mathrm{\ell oc}.}{\hookrightarrow}\mathbb{C}^{N}$ .

Definition 1.1.

A function $\phi:X\to\mathbb{R}\cup\{-\infty\}$ is $\omega$ -plurisubharmonic ( $\omega$ -psh for short) if $\phi+u$ is plurisubharmonic where $u$ is a local potential of $\omega$ ; i.e. $\phi+u$ is the restriction of an psh function defined near an open neighborhood of $\operatorname{im}(j)$ . Denote by $\operatorname{PSH}(X,\omega)$ the set of all integrable $\omega$ -psh functions.

By Bedford–Taylor’s theory [BT82], the complex Monge–Ampère operator can be extended to bounded $\omega$ -psh functions on smooth complex manifolds. In the singular setting, the complex Monge–Ampère operator of locally bounded psh functions can also be defined by taking zero through singular locus (cf. [Dem85] for more details).

1.1.1. Finite energy class

Set $V:=\int_{X}\omega^{n}$ . For all $\varphi\in\operatorname{PSH}(X,\omega)\cap L^{\infty}(X)$ , the Monge–Ampère energy is defined by

\mathbf{E}(\varphi):=\frac{1}{(n+1)}\sum_{j=0}^{n}\int_{X}\varphi\,\omega_{\varphi}^{j}\wedge\omega^{n-j},

where $\omega_{\varphi}:=\omega+dd^{c}\varphi$ . The energy satisfies $\mathbf{E}(\varphi+c)=\mathbf{E}(\varphi)+cV$ for all $c\in\mathbb{R}$ and for $\varphi,\psi\in\operatorname{PSH}(X,\omega)\cap L^{\infty}(X)$ , if $\varphi\leq\psi$ then $\mathbf{E}(\varphi)\leq\mathbf{E}(\psi)$ with equality iff $\varphi=\psi$ . By the later property, $\mathbf{E}$ extends uniquely to $\operatorname{PSH}(X,\omega)$ by

\mathbf{E}(\varphi):=\inf\left\{{\mathbf{E}(\psi)}\,\middle|\,{\varphi\leq\psi\in\operatorname{PSH}(X,\omega)\cap L^{\infty}(X)}\right\}.

The finite energy class is then given as

\mathcal{E}^{1}(X,\omega)=\left\{{\varphi\in\operatorname{PSH}(X,\omega)}\,\middle|\,{\mathbf{E}(\varphi)>-\infty}\right\}.

The space of finite energy potentials $\mathcal{E}^{1}(X,\omega)$ admits a metric topology induced by the so-called $d_{1}$ -distance, which can be expressed as follows (cf. [Dar17, Thm. 2.1], [DG18, Thm. B])

d_{1}(u,v)=\mathbf{E}(u)+\mathbf{E}(v)-2\mathbf{E}(P_{\omega}(u,v)),

where $P_{\omega}(u,v):=\left(\sup\left\{{w\in\operatorname{PSH}(X,\omega)}\,\middle|\,{w\leq\min(u,v)}\right\}\right)^{*}$ .

1.2. Weighted cscK metrics

In this section, we review some basic concepts related to weighted cscK metrics.

1.2.1. Automorphism group and holomorphic vector fields

We recall here certain well-known properties of the Lie algebra of $\operatorname{Aut}(X)$ and some of its subgroups (cf. [LS94, Gau20]).

Let $(X,\omega)$ be a compact Kähler manifold and denote $J$ to be its complex structure. The automorphism group $\operatorname{Aut}(X)$ is a complex Lie group, whose Lie algebra $\mathfrak{h}=\left\{{\xi\in\Gamma(TX)}\,\middle|\,{\mathcal{L}_{\xi}J=0}\right\}$ consists of real holomorphic vector fields on $X$ . A vector field $\xi$ is real holomorphic if and only if $\xi^{\prime}:=\frac{1}{2}(\xi-iJ\xi)$ is a holomorphic, so $\mathfrak{h}$ can be identified with $H^{0}(X,T_{X})$ the space of holomorphic vector fields. Denote by $\operatorname{Aut}_{0}(X)$ the identity component of $\operatorname{Aut}(X)$ .

The Albanese torus is defined by (cf. [Uen75, Thm. 9.7])

\text{Alb}(X)=H^{0}(X,\Omega_{X}^{1})^{\vee}/H_{1}(X,\mathbb{Z}),

where the inclusion $H_{1}(X,\mathbb{Z})\xhookrightarrow{}H^{0}(X,\Omega_{X}^{1})^{\vee}$ is induced via the map $\alpha\mapsto\int_{c}\alpha$ for any loop $c$ and holomorphic $1$ -form $\alpha$ . Then one can define a Lie group homomorphism $\tau:\operatorname{Aut}_{0}(X)\rightarrow\text{Alb}(X)$ as follows. Fix a $x_{0}\in X$ , for any $\sigma\in\operatorname{Aut}_{0}(X)$ , we define $\tau(\sigma):\alpha\mapsto\int_{x_{0}}^{\sigma\cdot x_{0}}\alpha$ , which does not depend on $x_{0}$ as $\alpha$ is harmonic. The homomorphism $\tau$ induces an action of $\operatorname{Aut}_{0}(X)$ on $\text{Alb}(X)$ by translation. Moreover, the derivative of $\tau$ at the identity is $\tau^{\prime}:\mathfrak{h}\rightarrow H^{0}(X,\Omega_{X}^{1})^{\vee}$ , $\xi\mapsto(\tau^{\prime}(\xi):\alpha\mapsto\alpha(\xi))$ . Define $\mathfrak{h}_{\operatorname{red}}:=\ker\tau^{\prime}$ , which consists of all holomorphic vector field $\xi\in H^{0}(X,T_{X})$ such that $\alpha(\xi)=0$ for all $\alpha\in H^{0}(X,\Omega_{X}^{1})$ . By [LS94, Thm. 1], we have

\mathfrak{h}_{\operatorname{red}}=\{\xi\in H^{0}(X,T_{X})\mid\xi=\nabla^{1,0}f=g^{j\bar{k}}\partial_{\bar{k}}f\frac{\partial}{\partial z^{j}},\,f\in\mathcal{C}^{\infty}(X,\mathbb{C})\}.

Then $\mathfrak{h}_{\operatorname{red}}$ generates a Lie subgroup $\operatorname{Aut}_{\operatorname{red}}(X)\subset\operatorname{Aut}_{0}(X)$ which acts trivially on the Albanese torus. We also identify $\mathfrak{h}_{\operatorname{red}}\subset\mathfrak{h}$ with the Lie algebra of real holomorphic vector fields with zeros (cf. [LS94, Thm. 1]).

1.2.2. Weighted setting

Fix a real torus $T\subset\operatorname{Aut}_{\operatorname{red}}(X)$ with Lie algebra $\mathfrak{t}\subset\mathfrak{h}_{\operatorname{red}}$ . Any closed, real, $T$ -invariant $(1,1)$ -form $\omega$ admits a moment map $m_{\omega}:X\rightarrow\mathfrak{t}^{\vee}$ , that is a unique (up to additive constant) $T$ -invariant smooth map such that for each $\xi\in\mathfrak{t}$ , $m_{\omega}^{\xi}:=\langle m_{\omega},\xi\rangle:X\rightarrow\mathbb{R}$ satisfies $-dm_{\omega}^{\xi}=i_{\xi}\omega=\omega(\xi,\cdot)$ ; in other words, $m^{\xi}_{\omega}$ is a Hamiltonian function of $\xi$ with respect to $\omega$ .

Denote by $P_{\omega}:=\operatorname{im}(m_{\omega})$ a compact set in $\mathfrak{t}^{\vee}$ and $\mathcal{H}^{T}_{\omega}:=\mathcal{H}^{T}(X,\omega)$ the set of $T$ -invariant, strictly positive, smooth, $\omega$ -psh functions. For any $\phi\in\mathcal{H}^{T}_{\omega}$ , we normalize $m_{\phi}:=m_{\omega_{\phi}}$ by $m_{\phi}^{\xi}=m_{\omega}^{\xi}+d^{c}\phi(\xi)$ for all $\xi\in\mathfrak{t}$ . Then from [Lah19, Lem. 1], under this normalization, for any $\phi\in\mathcal{H}^{T}_{\omega}$ , one has $P_{\omega_{\phi}}=P_{\omega}$ and

\int_{X}m^{\xi}_{\phi}\omega^{n}_{\phi}=\int_{X}m^{\xi}_{\omega}\omega^{n},\quad\forall\xi\in\mathfrak{t}.

(1.1)

Let $v,w\in\mathcal{C}^{\infty}(\mathfrak{t}^{\vee},\mathbb{R})$ with $v>0$ on $P:=P_{\omega}$ . We recall the following notations and definitions from [BJT24, Sec. 3],

Definition 1.2.

Let $(\theta,m_{\theta})$ be a $T$ -invariant pair and $f$ be a $T$ distribution.

(1)

A moment map $m_{dd^{c}f}$ for the $T$ -invariant $(1,1)$ -current $dd^{c}f$ is defined by

$m_{dd^{c}f}^{\xi}:=d^{c}f(\xi)=i_{\xi}d^{c}f;$

(2)

For a smooth function $g:\mathbb{R}_{>0}\to\mathbb{R}$ , and another $T$ -invariant pair $(\omega,m_{\omega})$ ,

\langle(g(v))^{\prime}(m_{\omega}),m_{\theta}\rangle:=g^{\prime}(v(m_{\omega}))\sum_{\alpha}v_{\alpha}(m_{\omega})m_{\theta}^{\xi_{\alpha}}

where $(\xi_{\alpha})_{\alpha}$ is a basis of $\mathfrak{t}$ , and $(v_{\alpha})_{\alpha}$ are the partial derivatives of $v$ with respect to the dual basis $(\xi^{\alpha})_{\alpha}$ in $\mathfrak{t}^{\vee}$ ;

(3)

The $v$ -weighted trace is given by

$\operatorname{tr}_{\omega,v}\theta:=\operatorname{tr}_{\omega}\theta+\langle(\log v)^{\prime}(m_{\omega}),m_{\theta}\rangle;$

(4)

The $v$ -weighted Laplacian is defined as

\Delta_{\omega,v}f:=\operatorname{tr}_{\omega,v}(dd^{c}f)=\Delta_{\omega}f+\langle(\log v)^{\prime}(m_{\omega}),m_{dd^{c}f}\rangle;

(5)

The $v$ -weighted Ricci curvature is

\operatorname{Ric}_{v}(\omega):=\operatorname{Ric}(v(m_{\omega})\omega^{n})=-dd^{c}\log(v(m_{\omega})\omega^{n})=\operatorname{Ric}(\omega)-dd^{c}\log v(m_{\omega});

(6)

The $v$ -weighted scalar curvature is

S_{v}(\omega):=\operatorname{tr}_{\omega,v}\operatorname{Ric}_{v}(\omega)=S(\omega)-\frac{\langle v^{\prime}(m_{\omega}),\Delta_{\omega}m_{\omega}\rangle+\Delta_{\omega}v(m_{\omega})}{v(m_{\omega})}.

As [BJT24, (3.4)], by applying the interior product $i_{\xi}$ to the trivial relation $d^{c}f\wedge\omega^{n}=0$ , one can obtain $m_{dd^{c}f}^{\xi}\omega^{n}=nd^{c}f\wedge i_{\xi}\omega\wedge\omega^{n-1}$ . Combining this formula with

dv(m_{\omega})=\sum_{\alpha}v_{\alpha}(m_{\omega})dm_{\omega}^{\xi_{\alpha}}=-\sum_{\alpha}v_{\alpha}(m_{\omega})i_{\xi_{\alpha}}\omega,\quad\text{and}\quad\langle v^{\prime}(m_{\omega}),m_{dd^{c}f}\rangle:=\sum_{\alpha}v_{\alpha}(m_{\omega})m_{dd^{c}f}^{\xi_{\alpha}},

one can derive that

\langle v^{\prime}(m_{\omega}),m_{dd^{c}f}\rangle=n\frac{dv(m_{\omega})\wedge d^{c}f\wedge\omega^{n-1}}{\omega^{n}}=\operatorname{tr}_{\omega}(dv(m_{\omega})\wedge d^{c}f).

Therefore, we have the following formula for the weighted Laplacian

\Delta_{\omega,v}f=\Delta_{\omega}f+\operatorname{tr}_{\omega}(d\log v(m_{\omega})\wedge d^{c}f).

(1.2)

The $v$ -weighted Monge–Ampère operator is defined as

\operatorname{MA}_{v}(\phi):=\operatorname{MA}_{\omega,v}(\phi):=v(m_{\phi})(\omega+dd^{c}\phi)^{n}

for any $\phi\in\mathcal{H}^{T}_{\omega}$ . From [BJT24, (3.32)], one has the following integration-by-parts formula

\int_{X}g(\Delta_{\omega_{\phi},v}f)\operatorname{MA}_{v}(\phi)=\int_{X}f(\Delta_{\omega_{\phi},v}g)\operatorname{MA}_{v}(\phi)=-\int_{X}\langle df,dg\rangle_{\omega_{\phi}}\operatorname{MA}_{v}(\phi),

(1.3)

for all $T$ -invariant distributions $f,g$ such that at least one of which is smooth, where

\langle df,dg\rangle_{\omega}:=\operatorname{tr}_{\omega}(df\wedge d^{c}g).

1.2.3. Weighted cscK equations

Consider the weighted cscK problem

S_{v}(\omega_{\phi})=w(m_{\phi}).

The above equation can be rewritten as

\begin{cases}v(m_{\phi})(\omega+dd^{c}\phi)^{n}=e^{F}\omega^{n},\quad\sup_{X}\phi=0,\\ \Delta_{\phi,v}F=-w(m_{\phi})+\operatorname{tr}_{\phi,v}(\operatorname{Ric}(\omega)),\end{cases}

where $\operatorname{tr}_{\phi,v}:=\operatorname{tr}_{\omega_{\phi},v}$ and $\Delta_{\phi,v}:=\Delta_{\omega_{\phi},v}$ . For any volume form $\mu_{X}$ on $X$ , taking $\widetilde{F}:=F+\log\frac{\omega^{n}}{\mu_{X}}$ , the original coupled equations are equivalent to the following coupled equations

\begin{cases}v(m_{\phi})(\omega+dd^{c}\phi)^{n}=e^{\widetilde{F}}\mu_{X},\quad\sup_{X}\phi=0,\\ \Delta_{\phi,v}\widetilde{F}=-w(m_{\phi})+\operatorname{tr}_{\phi,v}(\operatorname{Ric}(\mu_{X})).\end{cases}

Since $P$ is compact, there are positive constants $C_{v}$ and $C_{w}$ such that, on $P$ ,

C_{v}^{-1}\leq v+\sum_{\alpha}|v_{\alpha}|+\sum_{\alpha,\beta}|v_{\alpha\beta}|\leq C_{v},\text{ and }\,C_{w}^{-1}\leq|w|+\sum_{\alpha}|w_{\alpha}|+\sum_{\alpha,\beta}|w_{\alpha\beta}|\leq C_{w}.

This yields that

|\operatorname{tr}_{\omega_{\phi},v}(\eta)-\operatorname{tr}_{\omega_{\phi}}(\eta))|=|\langle(\log v)^{\prime}(m_{\phi}),m_{\eta}\rangle|\leq C_{v,\eta},

where $C_{v,\eta}$ depending only on $C_{v}$ and $\eta$ . In particular, for any $\varphi,\psi\in\mathcal{H}^{T}(X,\omega)$ and $\eta:=\omega_{\psi}$ ,

|\operatorname{tr}_{\omega_{\varphi},v}(\omega_{\psi})-\operatorname{tr}_{\omega_{\varphi}}(\omega_{\psi})|=|\langle(\log v)^{\prime}(m_{\varphi}),m_{\psi}\rangle|\leq C_{v},

(1.4)

since $m_{\varphi}(X)=m_{\omega}(X)=P$ is compact.

1.2.4. Weighted Mabuchi functional

Let $\mu_{X}$ be a fixed $T$ -invariant volume form on $X$ . The $v$ -weighted relative entropy is defined by

\mathbf{H}_{v}(\phi):=\mathbf{H}_{v,\mu_{X}}(\phi):=\int_{X}\log\left(\frac{\operatorname{MA}_{v}(\phi)}{\mu_{X}}\right)\operatorname{MA}_{v}(\phi).

The $w$ -weighted Monge–Ampère energy $\mathbf{E}_{w}:=\mathbf{E}_{\omega,w}:\mathcal{H}_{\omega}^{T}\rightarrow\mathbb{R}$ is given by

(d\mathbf{E}_{w})_{\phi}(f):=\int_{X}fw(m_{\omega_{\phi}})\omega^{n}_{\phi},\quad\mathbf{E}_{w}(0)=0

for any $f\in\mathcal{C}^{\infty}(X)^{T}$ (cf. [Lah19, Lem. 3]).

Fix $\theta$ a $T$ -invariant closed $(1,1)$ -form and $m_{\theta}$ is its moment map. Then the twisted weighted Monge–Ampère energy $\mathbf{E}_{v}^{\theta}:=\mathbf{E}_{\omega,v}^{\theta}:\mathcal{H}^{T}_{\omega}\rightarrow\mathbb{R}$ is the primitive of $\left(v(m_{\phi})n\theta\wedge\omega_{\phi}^{n}+\langle v^{\prime}(m_{\phi}),m_{\theta}\rangle\omega_{\phi}^{n}\right)$ with $\mathbf{E}_{\omega,v}^{\theta}(0)=0$ ; in other words,

(d\mathbf{E}_{\omega,v}^{\theta})_{\phi}(f)=\int_{X}f\left(v(m_{\phi})n\theta\wedge\omega_{\phi}^{n}+\langle v^{\prime}(m_{\phi}),m_{\theta}\rangle\omega_{\phi}^{n}\right),\quad\mathbf{E}_{\omega,v}^{\theta}(0)=0.

for any $f\in\mathcal{C}^{\infty}(X)^{T}$ (see [Lah19, Lem. 4] for the well-definedness).

Definition 1.3.

The weighted Mabuchi energy $\mathbf{M}_{v,w}$ is an Euler–Lagrange functional of $\mathcal{H}^{T}_{\omega}\ni\phi\mapsto\mathbf{M}^{\prime}_{v,w}(\phi):=(w(m_{\phi})-S_{v}(\omega_{\phi}))\operatorname{MA}_{v}(\phi)$ .

From [Lah19, BJT24], we have the following lemma.

Lemma 1.4.

The following formula holds on $\mathcal{H}_{\omega}^{T}$

\mathbf{M}_{v,w}:=\mathbf{M}_{v,w,\mu_{X}}=\mathbf{H}_{v,\mu_{X}}+\mathbf{E}_{v}^{-\operatorname{Ric}(\mu_{X})}+\mathbf{E}_{vw}.

The above lemma follows from the fact that the functional $\mathcal{H}^{T}_{\omega}\ni\phi\mapsto\mathbf{H}_{v}+\mathbf{E}_{v}^{-\operatorname{Ric}(\mu_{X})}$ is an Euler–Lagrange functional for $\phi\mapsto-S_{v}(\omega_{\phi})\operatorname{MA}_{v}(\phi)$ (cf. [BJT24, Lem. 3.29]).

Lemma 1.5.

Any $(v,w)$ -cscK metric in $\{\omega\}$ is a minimizer of $\mathbf{M}_{v,w,\mu_{X}}$ for any choice of $\mu_{X}$ .

Proof.

It follows from [Lah23, Cor. 1] that any $(v,w)$ -cscK metric in $\{\omega\}$ is a minimizer of $\mathbf{M}_{v,w,\omega^{n}}$ . On the other hand, by [BJT24, Lem. 3.48] there is a constant $C$ such that $\mathbf{M}_{v,w,\mu_{X}}=\mathbf{M}_{v,w,\omega^{n}}+C$ , hence any $(v,w)$ -cscK metric in $\{\omega\}$ is also a minimizer of $\mathbf{M}_{v,w,\mu_{X}}$ . ∎

1.2.5. Weighted extremal metrics

We recall here the definition of weighted extremal metrics and relative weighted Mabuchi energy, which is a modification of Mabuchi energy such that it is $T_{\mathbb{C}}$ -invariant. We refer to [BJT24, Sec. 3.6] for more details.

Lemma 1.6 ([BJT24, Lem. 3.34]).

The weighted Mabuchi functional $\mathbf{M}_{v,w}$ is translation invariant and $T_{\mathbb{C}}$ -invariant iff for all affine function $\ell=\xi+c\in\mathfrak{t}\bigoplus\mathbb{R}$ on $\mathfrak{t}^{\vee}$ we have $\int_{X}\ell(m_{\omega})\mathbf{M}^{\prime}_{v,w}(0)=0$ , i.e.

\int_{X}\ell(m_{\omega})w(m_{\omega})\operatorname{MA}_{v}(0)=\int_{X}\ell(m_{\omega})S_{v}(\omega)\operatorname{MA}_{v}(0).

Now, let $w_{0}\in\mathcal{C}^{\infty}(P,\mathbb{R}_{>0})$ . The weighted Futaki–Mabuchi pairing on the space $\mathfrak{t}\bigoplus\mathbb{R}$ of affine functions on $\mathfrak{t}^{\vee}$ is defined by

\langle\ell,\ell^{\prime}\rangle:=\int_{X}\ell(m_{\omega})\ell^{\prime}(m_{\omega})w_{0}(m_{\omega})\operatorname{MA}_{v}(0)=\int_{X}\ell(m_{\omega})\ell^{\prime}(m_{\omega})\operatorname{MA}_{vw_{0}}(0).

Then this is positive definite, and there exists the unique affine function $\ell^{\operatorname{ext}}=\ell^{\operatorname{ext}}_{\omega,v,w_{0}}$ on $\mathfrak{t}^{\vee}$ such that

\langle\ell,\ell^{\operatorname{ext}}\rangle=\int_{X}\ell(m_{\omega})S_{v}(\omega)\operatorname{MA}_{v}(0).

(1.5)

Then the weighted Mabuchi energy $\mathbf{M}_{v,w}$ with $w=w_{0}\ell^{\operatorname{ext}}$ is $T_{\mathbb{C}}$ -invariant. We define $\mathbf{M}^{\rm re\ell}_{v,w_{0}}:=\mathbf{M}_{v,w_{0}\ell^{\operatorname{ext}}}$ is the relative weighted Mabuchi energy.

Definition 1.7.

Let $v,w_{0}\in\mathcal{C}^{\infty}(P,\mathbb{R}_{>0})$ . A metric $\omega$ is called a $(v,w_{0})$ -extremal metric of it is $(v,w)$ -cscK with $w=w_{0}\ell^{\operatorname{ext}}$ . In this case, $w$ is called an extremal weight.

1.2.6. Extremal Kähler metrics

We explain here how the problem of finding extremal Kähler metrics is a special case of the one for weighted setting. This corresponds to the $(1,1)$ -extremal metric, i.e. $v=1$ and $w_{0}=1$ .

Let $(X,\omega)$ be a compact Kähler manifold, and let $g$ be the Riemannian metric defined by $\omega$ . The metric $\omega$ is said to be extremal if the Hamiltonian vector field $\xi_{\omega}=J\nabla\operatorname{S}(\omega)$ is a Killing vector field for $g$ , i.e $\mathcal{L}_{\xi_{\omega}}g=0$ .

Let $T$ be a maximal compact torus of $\operatorname{Aut}_{\operatorname{red}}(X)$ and $\alpha$ be a Kähler class. From [FM95] (see also [Lah19, Sec. 3.1]), the projection of $\operatorname{S}(\omega)$ with respect to $L^{2}(\omega^{n})$ -inner product, to the sub-space $\{m_{\omega}^{\xi}+c:\xi\in\mathfrak{t},c\in\mathbb{R}\}$ is written as $\Pi_{\omega}^{T}(\operatorname{S}(\omega))=m_{\omega}^{\xi_{\operatorname{ext}}}+c_{\operatorname{ext}}$ for some $\xi_{\operatorname{ext}}\in\mathfrak{t}$ where $\xi_{\operatorname{ext}}$ only depends on $T$ and $\{\omega\}$ . In particular, it implies that $c_{\operatorname{ext}}$ also depend only on $T$ and $\{\omega\}$ , since

nc_{1}(X)\cdot\{\omega\}^{n-1}=\int_{X}\operatorname{S}(\omega)\omega^{n}=\int_{X}\Pi^{T}_{\omega}(\operatorname{S}(\omega))\omega^{n}=\int_{X}m_{\omega}^{\xi_{\operatorname{ext}}}\omega^{n}+c_{\operatorname{ext}}\{\omega\}^{n},

where the last integral only depends $\{\omega\}$ and $\xi_{\operatorname{ext}}$ by (1.1). One can also normalize $\int_{X}m_{\omega}^{\xi_{\operatorname{ext}}}\omega^{n}=0$ to get $c_{\operatorname{ext}}=\overline{s}:=n\frac{c_{1}(X)\cdot\{\omega\}^{n-1}}{\{\omega\}^{n}}$ . Therefore, we obtain the unique affine function $\ell^{\operatorname{ext}}(p)=\langle\xi_{\operatorname{ext}},p\rangle+c_{\operatorname{ext}}$ defined in (1.5) and the problem of finding extremal metric is equivalent to the one for $(1,\ell^{\operatorname{ext}})$ -cscK metric.

1.2.7. Extension on $\mathcal{E}^{1,T}$ and coercivity

Denote by $\mathcal{E}^{1,T}_{\omega}:=\mathcal{E}^{1}(X,\omega)^{T}\subset\mathcal{E}^{1}(X,\omega)$ the space of $T$ -invariant finite energy potentials and $\mathcal{E}^{1,T}_{\rm norm}(X,\omega):=\{u\in\mathcal{E}^{1,T}_{\omega}\mid\mathbf{E}_{\omega}(u)=0\}$ . From [BJT24, Prop. 3.41], one can extend all functionals above on $\mathcal{E}^{1,T}_{\omega}$ .

Since $\operatorname{Aut}_{0}(X)$ acts trivially on cohomologies, for each $\sigma\in\operatorname{Aut}_{0}(X)$ , one can find a unique function $\tau_{\sigma}\in\operatorname{PSH}(X,\omega)\cap\mathcal{C}^{\infty}(X)$ such that

\sigma^{\ast}\omega=\omega+dd^{c}\tau_{\sigma},\quad\mathbf{E}(\tau_{\sigma})=0.

For $u\in\operatorname{PSH}(X,\omega)$ and $\sigma\in\operatorname{Aut}_{0}(X)$ , define $\sigma\cdot u:=\sigma^{\ast}u+\tau_{\sigma}$ . Set

d_{1,T}(u,0):=\inf_{\sigma\in T_{\mathbb{C}}}d_{1}(\sigma\cdot u,0).

(1.6)

Definition 1.8.

The weighted Mabuchi functional $\mathbf{M}_{v,w}$ is $T_{\mathbb{C}}$ -coercive if there exist constants $A>0$ and $B>0$ such that

\mathbf{M}_{v,w}(\phi)\geq Ad_{1,T}(\phi,0)-B

for any $\phi\in\mathcal{E}^{1,T}_{\rm norm}(X,\omega)$ .

By definition, if $\mathbf{M}_{v,w}$ is $T_{\mathbb{C}}$ -coercive, then it is $T_{\mathbb{C}}$ -invariant since it is bounded from below (cf. [BJT24, Section 1.1]).

1.3. Weighted variational formalism in the singular setting

We recall here the weighted formalism for the variational problem of singular weighted cscK metrics on Kähler varieties with log terminal singularities as introduced in [BJT24, Sec. 3.8, 4.1].

1.3.1. Reduced automorphism group and moment maps

Let $(X,\omega)$ be a normal compact Kähler variety. The automorphism group $\operatorname{Aut}(X)$ is a complex Lie group, whose Lie algebra is $H^{0}(X,T_{X})\simeq H^{0}(X^{\mathrm{reg}},T_{X^{\mathrm{reg}}})$ the space of holomorphic vector fields, that are global section of the tangent sheaf $T_{X}:=\operatorname{Hom}(\Omega_{X}^{1},\mathcal{O}_{X})$ . There exists an $\operatorname{Aut}_{0}(X)$ -equivariant resolution of singularities $\pi:\tilde{X}\to X$ , i.e. $\pi$ is an isomorphism over $X^{\mathrm{reg}}$ and any $\sigma\in\operatorname{Aut}_{0}(X)$ can be extended to a unique $\sigma^{\prime}\in\operatorname{Aut}_{0}(\tilde{X})$ . Hence, we get the inclusion $\operatorname{Aut}_{0}(X)\subset\operatorname{Aut}_{0}(\tilde{X})$ . Moreover, any holomorphic vector field on $\tilde{X}$ descends to an element in $H^{0}(X^{\mathrm{reg}},T_{X^{\mathrm{reg}}})$ , so we get $\operatorname{Aut}_{0}(X)\simeq\operatorname{Aut}_{0}(\tilde{X})$ .

Given an $\operatorname{Aut}_{0}(X)$ -equivariant resolution of singularities $\pi:{\widetilde{X}}\to X$ , the reduced automorphism group $\operatorname{Aut}_{\operatorname{red}}(X)\subset\operatorname{Aut}_{0}(X)$ is defined as the subgroup of $\operatorname{Aut}_{0}(X)\simeq\operatorname{Aut}_{0}({\widetilde{X}})$ acting trivially on the Albanese torus of ${\widetilde{X}}$ . This definition is independent of the choice of $\operatorname{Aut}_{0}(X)$ -equivariant resolution of singularities $\pi:\tilde{X}\rightarrow X$ , by the bimeromorphic invariance of Albanese torus (cf. [Uen75, Prop. 9.12]). Since $\operatorname{Aut}_{0}(X)\simeq\operatorname{Aut}_{0}(\tilde{X})$ , we get $\operatorname{Aut}_{\operatorname{red}}(X)\simeq\operatorname{Aut}_{\operatorname{red}}(\tilde{X})$ .

Denote by $\mathcal{Z}$ (reps. $\mathcal{Z}^{\prime}$ ) the space of locally $dd^{c}$ -exact real $(1,1)$ -form (resp. currents) on $X$ . In particular, any current $\theta\in\mathcal{Z}^{\prime}$ can be written as $\theta=\omega+dd^{c}u$ for some $\omega\in\mathcal{Z}$ and $u$ is a distribution [BG13, Sec. 4.6.1]. Then the group $\operatorname{Aut}(X)$ acts on $\mathcal{Z}$ and $\mathcal{Z}^{\prime}$ . In particular, $\operatorname{Aut}_{0}(X)$ acts trivially on the the classes of $\mathcal{Z}$ (cf. [BJT24, Lem. 3.54]): for any $\sigma\in\operatorname{Aut}_{0}(X)$ , there exist $f\in\mathcal{C}^{\infty}(X)$ such that $\sigma^{*}\omega=\omega+dd^{c}f$ .

Fix a compact torus $T\subset\operatorname{Aut}_{\operatorname{red}}(X)$ with Lie algebra $\mathfrak{t}$ . Any $T$ -invariant $(1,1)$ -form (resp. current) $\omega\in\mathcal{Z}$ admits a moment map $m_{\omega}:X\rightarrow\mathfrak{t}^{\vee}$ which is a $T$ -invariant smooth map (resp. a distribution), unique up to an additive constant, such that for each $\xi\in\mathfrak{t}$ , $m_{\omega}^{\xi}:=\langle m_{\omega},\xi\rangle:X\rightarrow\mathbb{R}$ satisfies $-dm_{\omega}^{\xi}=i_{\xi}\omega=\omega(\xi,\cdot)$ . Denote by $P:=\operatorname{im}(m_{\omega})$ which is a compact subset in $\mathfrak{t}^{\vee}$ and normalize such that $P=\operatorname{im}(m_{\omega_{\phi}})$ for all $T$ -invariant smooth $\omega$ -psh function $\phi$ . Take $v,w\in\mathcal{C}^{\infty}(t^{\vee},\mathbb{R})$ with $\inf_{P}v>0$ .

1.3.2. Singular weighted cscK metrics

Suppose that $X$ is $\mathbb{Q}$ -Gorenstein, meaning that $X$ is normal and $K_{X}$ is $m$ -Cartier for some $m\in\mathbb{N}^{\ast}$ . Below, we recall the definitions of adapted measures and log terminal singularities from [EGZ09, Sec. 5-6].

Definition 1.9.

Let $h^{m}$ be a smooth hermitian metric on $mK_{X}$ . Taking $\Omega$ a local generator of $mK_{X}$ , the adapted measure associated with $h^{m}$ is defined by

\mu_{h}:=\mathrm{i}^{n^{2}}\left(\frac{\Omega\wedge\overline{\Omega}}{\left\lvert{\Omega}\right\rvert_{h^{m}}^{2}}\right)^{1/m}.

This definition does not depend on the choice of $\Omega$ , and two adapted measures differ by a smooth positive density. The Ricci form of the adapted measure $\mu_{h}$ is given as

\operatorname{Ric}(\mu_{h}):=dd^{c}\log|\Omega|^{2/m}_{h^{m}}

which belongs to $\mathcal{Z}$ . The $\mathbb{Q}$ -Gorenstein variety $X$ has log terminal singularities if the measure $\mu_{h}$ has finite masses near $X^{\mathrm{sing}}$ .

We now further assume $X$ is log terminal. As in [EGZ09, Sec. 5], $\operatorname{Ric}(\mu_{h})$ is canonically attached to an element in $H^{0}(X,\mathcal{C}^{\infty}_{X}/\operatorname{PH}_{X})$ where $\mathcal{C}^{\infty}_{X}$ (resp. $\operatorname{PH}_{X}$ ) is the subsheaf of continuous functions on $X$ that are local restrictions of smooth functions (resp. pluriharmonic functions) under local embeddings. The first Chern class of $\operatorname{Ric}(\mu_{h})$ , denoted $\{\operatorname{Ric}(\mu_{h})\}$ , is the image of $\operatorname{Ric}(\mu_{h})$ in $H^{1}(X,\operatorname{PH}_{X})$ via the connecting homomorphism $\{\bullet\}$ in the following exact sequence

H^{0}(X,\mathcal{C}^{\infty}_{X})\to H^{0}(X,\mathcal{C}^{\infty}_{X}/\operatorname{PH}_{X})\xrightarrow[]{\{\bullet\}}H^{1}(X,\operatorname{PH}_{X})\to 0

which is induced by the short exact sequence $0\to\operatorname{PH}_{X}\to\mathcal{C}^{\infty}_{X}\to\mathcal{C}^{\infty}_{X}/\operatorname{PH}_{X}\to 0$ .

Assume that $h^{m}$ is a $T$ -invariant metric on $mK_{X}$ so that $\operatorname{Ric}(\mu_{h})$ is an equivariant curvature form. Fix a compact torus $T\subset\operatorname{Aut}_{\operatorname{red}}(X)$ with Lie algebra $\mathfrak{t}$ . Set $P:=P_{\omega}:=m_{\omega}(X)\subset\mathfrak{t}^{\vee}$ and take $v\in\mathcal{C}^{\infty}(P,\mathbb{R}_{>0})$ . For any $\phi\in\mathcal{H}^{T}_{\omega}$ , the weighted Ricci current of $\omega_{\phi}:=\omega+dd^{c}\phi$ is defined as

\operatorname{Ric}_{v}^{T}(\omega_{\phi})=\operatorname{Ric}^{T}(\operatorname{MA}_{v}(\phi)):=-dd^{c}\log\left(\frac{v(m_{\phi})\omega_{\phi}^{n}}{\mu_{h}}\right)+\operatorname{Ric}(\mu_{h}).

and the $v$ -weighted scalar curvature is the distribution expressed by

S_{v}({\omega_{\phi}}):=\operatorname{tr}_{\omega_{\phi}}(\operatorname{Ric}^{T}(\omega_{\phi})).

One can extend the weighted energy and the weighted Ricci energy on $\mathcal{E}^{1,T}_{\omega}$ (cf. [PTT23, Sec. 4.1.2], [BJT24, Sec. 3.8]). Fix a $T$ -invariant adapted measure $\mu_{X}$ , for any $w\in\mathcal{C}^{\infty}(P,\mathbb{R})$ , the weighted Mabuchi energy $\mathbf{M}_{v,w}:\mathcal{E}^{1,T}_{\omega}\rightarrow\mathbb{R}\cup\{+\infty\}$ can be expressed as

\mathbf{M}_{v,w}:=\mathbf{M}_{v,w,\mu_{X}}:=\mathbf{H}_{v}+\mathbf{E}^{-\operatorname{Ric}(\mu_{X})}_{v}+\mathbf{E}_{vw}.

Definition 1.10.

Let $v\in\mathcal{C}^{\infty}(P,\mathbb{R}_{>0})$ and $w\in\mathcal{C}^{\infty}(P,\mathbb{R})$ . Then $\omega_{\phi}\in\{\omega\}$ is a singular $(v,w)$ -cscK metric if $\phi\in\operatorname{PSH}(X,\omega)\cap L^{\infty}(X)$ which is also smooth on $X^{\mathrm{reg}}$ , and

S_{v}(\omega_{\phi})=w(m_{\phi})\quad\text{on }X^{\mathrm{reg}}.

Let $w_{0}\in\mathcal{C}^{\infty}(P,\mathbb{R}_{>0})$ and $\ell^{\operatorname{ext}}$ be the unique affine function $\ell^{\operatorname{ext}}=\ell^{\operatorname{ext}}_{\omega,v,w_{0}}$ on $\mathfrak{t}^{\vee}$ such that

\langle\ell,\ell^{\operatorname{ext}}\rangle=\int_{X}\ell(m_{\omega})S_{v}(\omega)\operatorname{MA}_{v}(0).

Then $\omega_{\phi}$ is called a singular $(v,w_{0})$ -extremal metric if it is a singular $(v,w)$ -cscK metric with $w=w_{0}\ell^{\operatorname{ext}}$ .

Let $\pi:Y\rightarrow X$ be an $T$ -equivariant resolution of singularities. We can also define $\mathbf{M}^{\rm re\ell}_{\pi^{*}\omega}:\pi^{*}\mathcal{H}^{T}_{\omega}\rightarrow\mathbb{R}$ by

\mathbf{M}^{\rm re\ell}_{\pi^{*}\omega}(\pi^{*}u):=\mathbf{H}_{\omega_{Y}^{n}}(\pi^{*}u)+\mathbf{E}_{\pi^{*}\omega}^{-\operatorname{Ric}(\omega_{Y})}(\pi^{*}u)+\mathbf{E}_{\pi^{\ast}\omega,vw\ell^{\operatorname{ext}}}(\pi^{\ast}u).

Under Condition (A), following [BJT24, Lem. 4.21], one can extend the weighted energy, the weighted Ricci energy, and $\mathbf{M}^{\rm re\ell}_{\pi^{*}\omega}$ to $\mathcal{E}^{1,T}_{\pi^{*}\omega}$ .

2. A priori estimates

In this section, we shall establish a priori estimates for weighted cscK equations on compact Kähler manifolds when the reference metrics are degenerating. These estimates are crucial for obtaining the existence of singular weighted cscK metrics in the next section.

Once and for all, we fix $X$ to be an $n$ -dimensional compact Kähler manifold, $T\subset\operatorname{Aut}_{\operatorname{red}}(X)$ a compact torus with Lie algebra $\mathfrak{t}$ , and $\omega$ a $T$ -invariant Kähler metric on $X$ with the moment polytope $P$ of $\omega$ and total volume $V$ . Take $v\in\mathcal{C}^{\infty}(\mathfrak{t}^{\vee},\mathbb{R})$ such that $v>0$ on $P$ .

2.1. $L^{\infty}$ -estimates

In the following, we shall establish a uniform $L^{\infty}$ -estimate following and generalizing the approach of Guo and Phong [GP24, Thm. 3] with certain modifications (cf. [PTT23, Thm. 5.4]).

Theorem 2.1.

Let $\mu$ be a $T$ -invariant smooth volume form such that $\omega^{n}=g\mu$ with $\|g^{1/n}\|_{L^{\infty}}\leq K_{0}$ for some constant $K_{0}>0$ . Suppose that $(\varphi,F)\in\mathcal{H}^{T}(X,\omega)\times\mathcal{C}^{\infty}(X)^{T}$ is a solution to the coupled equations

\begin{cases}v(m_{\varphi})(\omega+dd^{c}\varphi)^{n}=e^{F}\mu,\quad\sup_{X}\varphi=0,\\ \Delta_{\varphi,v}F=-S+\operatorname{tr}_{\varphi,v}(\operatorname{Ric}(\mu)),\end{cases}

for some $S\in\mathcal{C}^{\infty}(X)^{T}$ . In addition, assume that there are positive constants $K_{1},K_{3},K_{4}$ such that

(1)

$-K_{1}(\omega+dd^{c}\rho_{1})\leq{\rm Ric}(\mu)$ , where $\rho_{1}$ is a $T$ -invariant quasi-psh function, $\sup_{X}\rho_{1}=0$ and $\int_{X}e^{-K_{1}\rho_{1}}d\mu<+\infty$ ;
(2)

${\bf H}_{\mu}(\varphi)=\int_{X}\log\left(\frac{\omega_{\varphi}^{n}}{\mu}\right)\omega_{\varphi}^{n}\leqslant K_{3}$ ;
(3)

there exists $\alpha>0$ such that $\int_{X}e^{-\alpha(\phi-\sup_{X}\phi)}\mu\leqslant K_{4}$ for all $\phi\in{\rm PSH}(X,\omega)$ .

Then there is a uniform constant $C>0$ depending only on $n,\max_{X}S,V,\alpha,K_{0},K_{1},K_{3},K_{4},C_{v},\mu,\rho_{1}$ such that

\|\varphi\|_{L^{\infty}}\leq C\,\text{ and }\,F\leq C.

Furthermore, if we have that

\operatorname{Ric}(\mu)\leq K_{2}(\omega+dd^{c}\rho_{2}),

for a $T$ -invariant $\omega$ -psh function $\rho_{2}\in\mathcal{C}^{\infty}(X\setminus Z)$ , with $Z:=\{\rho_{2}=-\infty\}$ and $\sup_{X}\rho_{2}=0$ , there is a constant $C_{2}>0$ depending only on $n,\max_{X}|S|,V,\alpha,K_{0},K_{1},K_{2},K_{3},K_{4},C_{v},\mu,\rho_{1}$ such that

F\geq K_{2}\rho_{2}-C_{2}.

Proof.

Consider $\tau_{k}:\mathbb{R}\to\mathbb{R}_{>0}$ a sequence of positive smooth functions decreasing towards the function $x\mapsto x\cdot\mathds{1}_{\mathbb{R}_{>0}}(x).$ Let $\phi_{k}$ be a solution to the following auxiliary complex Monge–Ampère equation

V^{-1}(\omega+dd^{c}\phi_{k})^{n}=\frac{\tau_{k}(-\varphi+\lambda F)+1}{A_{k}}e^{F}\mu,\quad\sup_{X}\phi_{k}=0,

(2.1)

where

A_{k}=\int_{X}(\tau_{k}(-\varphi+\lambda F)+1)e^{F}\mu\xrightarrow[k\to+\infty]{}\int_{X}(-\varphi+\lambda F)_{+}e^{F}\mu+\int e^{F}\mu=A_{\infty}.

We remark that $\phi_{k}$ is also $T$ -invariant since the RHS of (2.1) is $T$ -invariant. Applying Young’s inequality with $\chi(s)=(s+1)\log(s+1)-s$ and $\chi^{\ast}(s)=e^{s}-s-1$ ,

\int_{X}(-\varphi)e^{F}\omega^{n}\leq\int_{X}\chi(\alpha^{-1}e^{F})\omega^{n}+\int_{X}\chi^{*}(-\alpha\varphi)\omega^{n}

where $\alpha>0$ is the constant in (3). It follows from (2) and (3) that $C_{v}^{-1}V\leq A_{\infty}\leq C(K_{3},K_{4},V)$ . Thus $C_{v}^{-1}V\leq A_{k}\leq C_{1}=C(K_{3},K_{4},V)$ for $k$ sufficiently large.

By strong openness [Ber13, GZ15], we have a constant $\delta>0$ such that $I:=\int_{X}e^{-(K_{1}+\delta)\rho_{1}}<+\infty$ . Using Demailly’s approximation theorem [Dem92] (see also [Dem12, Sec. 14B]) and Demailly–Kollár’s convergence result [DK01, Main Thm. 0.2 (2)], there exists a sequence quasi-plurisubharmonic functions $(\rho_{1,k})_{k}$ with analytic singularities and smooth away from their singular locus, such that $-K_{1}(\omega+dd^{c}\rho_{1,k})-\frac{1}{k}\omega\leq\operatorname{Ric}(\mu)$ , $\rho_{1,k}\to\rho_{1}$ in $L^{1}$ as $k\to+\infty$ , and $\int_{X}e^{-(K_{1}+\delta)\rho_{1,k}}\leq 2I$ for all $k\gg 1$ . Hence, $-K_{1}^{\prime}(\omega+dd^{c}\rho^{\prime}_{1,k})\leq\operatorname{Ric}(\mu)$ where $K_{1}^{\prime}=K_{1}+\frac{1}{k}$ with $k>1/\delta$ and $\rho^{\prime}_{1,k}=\frac{K_{1}}{K_{1}+1/k}\rho_{1,k}$ . Replacing $\rho_{1}$ by $\rho^{\prime}_{1,k}$ and $K_{1}$ by $K_{1}^{\prime}$ , one can assume $\rho_{1}$ has analytics singularities.

Consider the function

\Phi=-\epsilon(-\phi_{k}+\Lambda)^{\frac{n}{n+1}}-\varphi+\lambda(F+K_{1}\rho_{1})

with $\Lambda=\left(\frac{2n}{n+1}\epsilon\right)^{n+1}$ and $\epsilon=\left(\frac{(n+1)(n+\lambda\bar{s}+L)}{n^{2}}\right)^{\frac{n}{n+1}}A_{k}^{\frac{1}{n+1}}$ , where $\bar{s}:=\max_{X}S$ , $L:=\lambda(C_{v,\mu}+K_{1}C_{v})+2C_{v}$ with $C_{v,\mu}>0$ a constant independent of $\varphi$ so that $\langle(\log v)^{\prime}(m_{\varphi}),m_{\operatorname{Ric}(\mu)}\rangle\geq-C_{v,\mu}$ , and $\lambda>0$ is a constant such that $n+\lambda\bar{s}>0$ and $\lambda K_{1}<1/2$ . Since $\rho_{1}(x)\to-\infty$ as $x\to Z_{1}:=\{\rho_{1}=-\infty\}$ , the maximal points of $\Phi$ only occur in $X\setminus Z_{1}$ . Fix $x_{0}\in X\setminus Z_{1}$ a maximal point of $\Phi$ . At $x_{0}$ , we have

	$\displaystyle 0$	$\displaystyle\geq\Delta_{\varphi,v}\Phi=\Delta_{\omega_{\varphi}}\Phi+\langle(\log v)^{\prime}(m_{\varphi}),m_{dd^{c}\Phi}\rangle$
		$\displaystyle\geq\frac{\epsilon n}{n+1}(-\phi_{k}+\Lambda)^{-\frac{1}{n+1}}\Delta_{\omega_{\varphi}}\phi_{k}-\Delta_{\omega_{\varphi}}\varphi+\lambda\Delta_{\varphi,v}(F+K_{1}\rho_{1})$
		$\displaystyle\quad+\frac{\epsilon n}{n+1}(-\phi_{k}+\Lambda)^{\frac{-1}{n+1}}\langle(\log v)^{\prime}(m_{\varphi}),m_{dd^{c}\phi_{k}}\rangle-\langle(\log v)^{\prime}(m_{\varphi}),m_{dd^{c}\varphi}\rangle.$

Since $-C_{v}\leq\langle(\log v)^{\prime}(m_{\phi}),m_{dd^{c}\psi}\rangle\leq C_{v}$ for any $\phi,\psi\in\operatorname{PSH}(X,\omega)^{T}$ (see (1.4)), we infer that

	$\displaystyle 0$	$\displaystyle\geq\frac{\epsilon n}{n+1}(-\phi_{k}+\Lambda)^{-\frac{1}{n+1}}\Delta_{\omega_{\varphi}}\phi_{k}-\Delta_{\omega_{\varphi}}\varphi+\lambda\Delta_{\varphi,v}(F+K_{1}\rho_{1})-C_{v}\left(\frac{\epsilon n}{n+1}\Lambda^{\frac{-1}{n+1}}+1\right)$
		$\displaystyle=\frac{\epsilon n}{n+1}(-\phi_{k}+\Lambda)^{-\frac{1}{n+1}}(\operatorname{tr}_{\omega_{\varphi}}\omega_{\phi_{k}}-\operatorname{tr}_{\omega_{\varphi}}\omega)-\operatorname{tr}_{\omega_{\varphi}}(\omega_{\varphi}-\omega)+\lambda(-S+\operatorname{tr}_{\omega_{\varphi}}({\rm Ric}(\mu)+K_{1}dd^{c}\rho_{1}))$
		$\displaystyle\quad+\lambda\left(\langle(\log v)^{\prime}(m_{\varphi}),m_{\operatorname{Ric}(\mu)}\rangle+K_{1}\langle(\log v)^{\prime}(m_{\varphi}),m_{dd^{c}\rho_{1}}\rangle\right)-2C_{v}.$

Then

	$\displaystyle 0$	$\displaystyle\geq\frac{n^{2}\epsilon}{n+1}(-\phi_{k}+\Lambda)^{-\frac{1}{n+1}}\left(\frac{\tau_{k}(-\varphi+\lambda F)+1}{A_{k}}\right)^{1/n}-n-\lambda\bar{s}-L+\left(1-\frac{n\epsilon}{n+1}\Lambda^{-\frac{1}{n+1}}-\lambda K_{1}\ \right)\operatorname{tr}_{\omega_{\varphi}}\omega$
		$\displaystyle\geq\frac{n^{2}\epsilon}{n+1}(-\phi_{k}+\Lambda)^{-\frac{1}{n+1}}\left(\frac{\tau_{k}(-\varphi+\lambda F)+1}{A_{k}}\right)^{1/n}-n-\lambda\bar{s}-L.$

Since $n+\lambda\bar{s}>0$ and $\lambda K_{1}<1/2$ , at $x_{0}$ , we obtain

-\varphi+\lambda(F+K_{1}\rho_{1})\leq-\varphi+\lambda F\leq\left(\frac{(n+\lambda\bar{s}+L)(n+1)}{n^{2}\epsilon}\right)^{n}A_{k}(-\phi_{k}+\Lambda)^{\frac{n}{n+1}};

(2.2)

therefore, $\Phi(x_{0})\leq 0$ and $\Phi\leq 0$ on $X$ . By the choice of $\epsilon,\Lambda$ and $V\leq A_{k}\leq C(K_{3},K_{4},V)$ , and Young’s inequality, we derive that for any $\delta>0$

\lambda(F+K_{1}\rho_{1})\leq-\varphi+\lambda(F+K_{1}\rho_{1})\leq C(V,K_{1},K_{3},K_{4})(-\phi_{k}+\Lambda)^{\frac{n}{n+1}}\leq-\delta\phi_{k}+C_{2},

(2.3)

with $C_{2}=C(\delta,V,K_{2},K_{3},K_{4})$ .

From Condition (A), we have $\int_{Y}e^{-K_{1}\rho_{1}}\mu<+\infty$ . The strong openness [Ber13, GZ15] yields a constant $0<a\ll 1$ such that $\int_{X}e^{-(1+a)K_{1}\rho_{1}}\mu\leq C_{a}$ for some constant $C_{a}>0$ . By (2.3), Hölder inequality and (3), for $\beta=\frac{1+a/2}{\lambda}$ and $\delta>0$ such that $\delta\beta<\alpha/\gamma^{*}$ , with $\gamma=\frac{1+a}{1+a/2}$ and $\frac{1}{\gamma}+\frac{1}{\gamma^{\ast}}=1$ , we obtain

\int_{X}e^{\beta\lambda F}\mu\leq e^{\beta C_{2}}\int_{X}e^{-\frac{\alpha}{\gamma^{*}}\phi_{k}-(1+\frac{a}{2})K_{1}\rho_{1}}\mu=e^{\beta C_{2}}\int_{X}e^{-\frac{\alpha}{\gamma^{*}}\phi_{k}-(1+a)K_{1}\rho_{1}/\gamma}\mu\leq C(a,C_{a},\alpha).

(2.4)

By a refined version of Kołodziej’s $L^{\infty}$ -estimate [Koł98] (see [DGG23, Thm. A] for the version we referred), a uniform control $C_{v}^{-1}\leq v(m_{\varphi})\leq C_{v}$ and (2.4), we obtain $\|\varphi\|_{L^{\infty}}\leq C(n,V,\alpha,K_{1},K_{3},K_{4},C_{v},\mu,\rho_{1})$ . Also, combining the $L^{\infty}$ -estimate of $\varphi$ with (2.4), we infer that

\|(\tau_{k}(-\varphi+\lambda F)+1)e^{F}\|_{L^{p^{\prime}}(X,\omega^{n})}\leq C(n,\alpha,V,K_{1},K_{3},K_{4},C_{v},\mu,\rho_{1})

for some $p^{\prime}>1$ and for all $k>0$ sufficiently large. Again, Kołodziej’s $L^{\infty}$ -estimate yields a uniform bound $\|\phi_{k}\|_{L^{\infty}}\leq C(n,V,\alpha,K_{1},K_{3},K_{4},C_{v},\rho_{1})$ . Then the inequality (2.2) provides a uniform upper bound for $F$ .

In the second part, we consider the function $H:=F+(K_{2}+1)\varphi-K_{2}\rho_{2}$ . Since $\rho_{2}=-\infty$ along $Z$ , one can assume that $H$ admits a minimum at $x_{0}\in X\setminus Z$ . At $x_{0}$ , we have

	$\displaystyle 0\leq\Delta_{\varphi,v}H$	$\displaystyle=-S+\operatorname{tr}_{\omega_{\varphi}}(\operatorname{Ric}(\mu))+(K_{2}+1)n-(K_{2}+1)\operatorname{tr}_{\omega_{\varphi}}\omega-K_{2}\operatorname{tr}_{\omega_{\varphi}}(\omega+dd^{c}\rho_{2})+K_{2}\operatorname{tr}_{\omega_{\varphi}}\omega$
		$\displaystyle\quad+\langle(\log v)^{\prime}(m_{\varphi}),m_{\operatorname{Ric}(\mu)}\rangle+(K_{2}+1)\langle(\log v)^{\prime}(m_{\varphi}),m_{dd^{c}\phi}\rangle-K_{2}\langle(\log v)^{\prime}(m_{\varphi}),m_{dd^{c}\rho_{2}}\rangle$
		$\displaystyle\leq-S+\operatorname{tr}_{\omega_{\varphi}}(\operatorname{Ric}(\mu)-K_{2}(\omega+dd^{c}\rho_{2}))+(K_{2}+1)n-\operatorname{tr}_{\omega_{\varphi}}\omega+C_{v,\mu}+2(K_{2}+1)C_{v}$
		$\displaystyle\leq(K_{2}+1)n-S-\operatorname{tr}_{\omega_{\varphi}}\omega+C_{v,\mu}+2(K_{2}+1)C_{v}$
		$\displaystyle\leq(K_{2}+1)n-S-n\left(\frac{\omega^{n}}{\omega^{n}_{\varphi}}\right)^{1/n}+C_{v,\mu}+2(K_{2}+1)C_{v}$
		$\displaystyle\leq(K_{2}+1)n-S-n\left(\frac{gv(m_{\varphi})}{e^{F}}\right)^{1/n}+C_{v,\mu}+2(K_{2}+1)C_{v}$
		$\displaystyle\leq(K_{2}+1)n-\min_{X}S-nK_{0}C_{v}^{-1/n}e^{-F/n}+C_{v,\mu}+2(K_{2}+1)C_{v}.$

Therefore, at $x_{0}$ , $F(x_{0})\geq-C(n,K_{0},K_{2},\min_{X}S,C_{v,\mu},C_{v})$ . We obtain

	$\displaystyle F$	$\displaystyle\geq K_{2}(\rho_{2}-\rho_{2}(x_{0}))-C(n,K_{0},K_{2},\min_{X}S,C_{v,\mu},C_{v})-(K_{2}+1)\\|\varphi\\|_{L^{\infty}}$
		$\displaystyle\geq K_{2}\rho_{2}-C(n,K_{0},\ldots,K_{4},\max_{X}\|S\|,C_{v},\mu,\rho_{1})$

as required. ∎

2.2. Local $L^{p}$ -estimate for Laplacian

Let $\omega_{X}$ be another $T$ -invariant Kähler metric on $X$ . Since $P_{\omega_{X}}=\operatorname{im}(m_{\omega_{X}})$ is compact, one can further assume that

C_{v}^{-1}\leq|v|+\sum_{\alpha}|v_{\alpha}|+\sum_{\alpha,\beta}|v_{\alpha\beta}|\leq C_{v}\quad\text{on }P_{\omega_{X}}.

Consider the following weighted cscK equations

\begin{cases}v(m_{\varphi})(\omega+dd^{c}\varphi)^{n}=e^{F}\omega_{X}^{n},\quad\sup_{X}\varphi=0,\\ \Delta_{\varphi,v}F=-S+\operatorname{tr}_{\varphi,v}(\operatorname{Ric}(\omega_{X})).\end{cases}

(2.5)

In this section, we shall further assume $\mathfrak{t}^{\vee}\ni\zeta\mapsto\log v(\zeta)\in\mathbb{R}$ to be concave. With the concavity condition on $\log v$ , from Lemma A.1, we have the following weighted Aubin–Yau type inequality:

\Delta_{\varphi,v}\log\operatorname{tr}_{\omega_{X}}\omega_{\varphi}\geq\frac{\Delta_{\omega_{X}}F}{\operatorname{tr}_{\omega_{X}}\omega_{\varphi}}-\mathcal{B}\operatorname{tr}_{\omega_{\varphi}}\omega_{X}-\mathcal{C}

for some uniform constant $\mathcal{B},\mathcal{C}>0$ .

Proposition 2.2.

Suppose that $(\varphi,F)\in\mathcal{H}^{T}(X,\omega)\times\mathcal{C}^{\infty}(X)^{T}$ is a solution to (2.5). Fix $p>1$ . Assume that $\omega\leq C_{\omega}\omega_{X}$ , $\operatorname{Ric}(\omega_{X})\leq A\omega_{X}$ , $\operatorname{Bisec}(\omega_{X})\geq-B$ , $\omega+dd^{c}\rho\geq C_{\rho}\omega_{X}$ with $C_{\rho}>0$ , and

\|\varphi\|_{L^{\infty}}\leq C_{0},\quad F\leq C_{0},\quad\text{and}\quad F\geq K_{2}\rho-C_{0},

where $\rho\in\mathcal{C}^{\infty}(X\setminus\{\rho=-\infty\})$ is a $T$ -invariant $\omega$ -psh function. Then for any $\mathcal{K}$ compact set of $X\setminus\{\rho=-\infty\}$ , one has the following estimate

\|\operatorname{tr}_{\omega_{X}}\omega_{\varphi}\|_{L^{2p+2}(\mathcal{K},\omega_{X}^{n})}\leq C_{1},

where $C_{1}$ only depends on $\mathcal{K},n,p,A,B,C_{0},C_{\rho},\min_{X}S$ .

Proof.

We shall adapt the approach in the smooth setting of Chen and Cheng [CC21a] for cscK metrics and also its generalization by [DJL24] and [HL25] for weighted cscK metrics. We highlight some differences in the following:

•

We shall work with the trace taken with respect to the reference metric $\omega_{X}$ instead of $\omega$ as $\omega$ is moving when we are going to apply the result. However, the metric $\omega$ still plays a role and must be carefully merged during the computations, especially since we only have the upper bound $\omega\leq C_{\omega}\omega_{X}$ .
•

In the last step, we need to use bounds on $\varphi$ and $F$ . Special attention is required for $F$ since its lower bound is uniform only up to a term involving $K_{2}\rho$ , which is not bounded from below.

Take $u:=e^{-a(F+b\varphi-b\rho)}\operatorname{tr}_{\omega_{X}}\omega_{\varphi}\geq 0$ , where $a,b$ are constants to be determined later. Recall that from (1.2),

\Delta_{\omega,v}f=\Delta_{\omega}f+\operatorname{tr}_{\omega}(d\log v(m_{\omega})\wedge d^{c}f);

hence, we have

\begin{split}\Delta_{\varphi,v}u&=\Delta_{\varphi,v}e^{\log u}=\frac{|du|^{2}_{\omega_{\varphi}}}{u}+u\Delta_{\omega_{\varphi}}\log u+\operatorname{tr}_{\omega_{\varphi}}(d\log v(m_{\omega_{\varphi}})\wedge d^{c}u)\\ &=\frac{|du|^{2}_{\omega_{\varphi}}}{u}+u\Delta_{\omega_{\varphi}}\log u+u\operatorname{tr}_{\omega_{\varphi}}(d\log v(m_{\omega_{\varphi}})\wedge d^{c}\log u)=\frac{|du|^{2}_{\omega_{\varphi}}}{u}+u\Delta_{\varphi,v}\log u\\ &\geq u\Delta_{\varphi,v}\log u=-au\Delta_{\varphi,v}(F+b\varphi-c\rho)+u\Delta_{\varphi,v}\log(\operatorname{tr}_{\omega_{X}}\omega_{\varphi}).\end{split}

(2.6)

Using $\operatorname{Ric}(\omega_{X})\leq A\omega_{X}$ , $|\langle(\log v)^{\prime}(m_{\varphi}),m_{\operatorname{Ric}(\omega_{X})}\rangle|\leq C_{v,\operatorname{Ric}(\omega_{X})}$ , $\operatorname{tr}_{\varphi,v}\omega_{\varphi}\leq n+C_{v}$ , and $\operatorname{tr}_{\omega_{\varphi}}{\omega_{\rho}}\geq C_{\rho}\operatorname{tr}_{\varphi,v}\omega_{X}-C_{v}$ , we derive

\begin{split}\Delta_{\varphi,v}(F+b\varphi-b\rho)&=-S+\operatorname{tr}_{\varphi,v}(\operatorname{Ric}(\omega_{X}))+b\operatorname{tr}_{\varphi,v}(dd^{c}\varphi-dd^{c}\rho)\\ &\leq-\min_{X}S+A\operatorname{tr}_{\omega_{\varphi}}\omega_{X}+\langle(\log v)^{\prime}(m_{\varphi}),m_{\operatorname{Ric}(\omega_{X})}\rangle-b\operatorname{tr}_{\varphi,v}\omega_{\rho}+b\operatorname{tr}_{\varphi,v}\omega_{\varphi}\\ &\leq-\min_{X}S+(A-bC_{\rho})\operatorname{tr}_{\omega_{\varphi}}\omega_{X}+b(2C_{v}+n)+C_{v,\operatorname{Ric}(\omega_{X})}\\ &=C_{1}+(A-bC_{\rho})\operatorname{tr}_{\omega_{\varphi}}\omega_{X}\end{split}

(2.7)

where $C_{1}>0$ only depends on $-\min_{X}S,b,C_{v},C_{v,\operatorname{Ric}(\omega_{X})}$ .

By a weighted Aubin–Yau’s inequality for weighted Monge–Ampère equation obtained in [DJL24] (see also Lemma A.1 for the version we apply), we have

\Delta_{\varphi,v}\log\operatorname{tr}_{\omega_{X}}\omega_{\varphi}\geq\frac{1}{\operatorname{tr}_{\omega_{X}}\omega_{\varphi}}\Delta_{\omega_{X}}F-\mathcal{B}\operatorname{tr}_{\omega_{\varphi}}\omega_{X}-\mathcal{C}

(2.8)

where $\mathcal{B},\mathcal{C}$ depend on $\omega_{X},C_{v}$ . Combining (2.6), (2.7), and (2.8), we get

\Delta_{\varphi,v}u\geq e^{-a(F+b\varphi-b\rho)}\left\{-(aC_{1}+\mathcal{C})\operatorname{tr}_{\omega_{X}}\omega_{\varphi}+\Delta_{\omega_{X}}F+(abC_{\rho}-Aa-\mathcal{B})\operatorname{tr}_{\omega_{X}}\omega_{\varphi}\operatorname{tr}_{\omega_{\varphi}}\omega_{X}\right\}.

Using the fact that

\operatorname{tr}_{\omega_{X}}\omega_{\varphi}\operatorname{tr}_{\omega_{\varphi}}\omega_{X}\geq(\operatorname{tr}_{\omega_{X}}\omega_{\varphi})^{\frac{n}{n-1}}\left(\frac{\omega_{X}^{n}}{\omega_{\varphi}^{n}}\right)^{\frac{1}{n-1}}=(\operatorname{tr}_{\omega_{X}}\omega_{\varphi})^{\frac{n}{n-1}}\left(ve^{-F}\right)^{\frac{1}{n-1}}\geq(\operatorname{tr}_{\omega_{X}}\omega_{\varphi})^{\frac{n}{n-1}}\left(C_{v}e^{F}\right)^{\frac{-1}{n-1}},

and choosing $b$ large enough such that $abC_{\rho}-Aa-\mathcal{B}\geq abC_{\rho}/2$ , we get

\displaystyle\Delta_{\varphi,v}u\geq-(aC_{1}+\mathcal{C})u+\frac{abC_{\rho}}{2}C^{-\frac{1}{n-1}}e^{\frac{-F}{n-1}}(\operatorname{tr}_{\omega_{X}}\omega_{\varphi})^{\frac{1}{n-1}}u+e^{-a(F+b\varphi-b\rho)}\Delta_{\omega_{X}}F.

(2.9)

Since $|\nabla u|_{\omega_{\varphi}}^{2}\operatorname{tr}_{\omega_{X}}\omega_{\varphi}\geq|\nabla u|^{2}_{\omega_{X}}$ , we infer that

\frac{1}{2p+1}\Delta_{\varphi,v}u^{2p+1}=u^{2p}\Delta_{\varphi,v}u+2pu^{2p-1}|\nabla u|^{2}_{\omega_{\varphi}}\geq u^{2p}\Delta_{\varphi,v}u+2pu^{2p-2}e^{-a(F+b\varphi-b\rho)}|\nabla u|^{2}_{\omega_{X}}

(2.10)

By (2.9), (2.10), and the weighted integration-by-parts formula (1.3), we obtain

\begin{split}0&=\int_{X}\frac{1}{2p+1}\Delta_{\varphi,v}u^{2p+1}v(m_{\varphi})\omega_{\varphi}^{n}\\ &\geq\int_{X}2pu^{2p-2}e^{-a(F+b\varphi-b\rho)+F}|\nabla u|^{2}_{\omega_{X}}\omega_{X}^{n}-(aC_{1}+\mathcal{C})\int_{X}u^{2p+1}e^{F}\omega_{X}^{n}\\ &\quad\quad+\frac{abC_{\rho}}{2}C_{v}^{-\frac{1}{n-1}}\int_{X}u^{2p+1}e^{\frac{n-2}{n-1}F}(\operatorname{tr}_{\omega_{X}}\omega_{\varphi})^{\frac{1}{n-1}}\omega_{X}^{n}+\underset{=:I}{\underbrace{\int_{X}u^{2p}e^{-a(F+b\varphi-b\rho)+F}\Delta_{\omega_{X}}F\omega_{X}^{n}}}.\end{split}

(2.11)

Set $G=(1-a)F-ab(\varphi-\rho)$ and consider $a>1$ . The last term $I$ can be written as the following two parts

\displaystyle I=\int_{X}u^{2p}\Delta_{\omega_{X}}Fe^{G}\omega_{X}^{n}=\underset{=:I_{1}}{\underbrace{\frac{1}{1-a}\int_{X}u^{2p}\Delta_{\omega_{X}}(G)e^{G}\omega_{X}^{n}}}+\underset{=:I_{2}}{\underbrace{\frac{ab}{1-a}\int_{X}u^{2p}\Delta_{\omega_{X}}(\varphi-\rho)e^{G}\omega_{X}^{n}}}.

For $I_{1}$ , the weighted integration-by-parts formula (1.3) implies

	$\displaystyle I_{1}$	$\displaystyle=\frac{1}{1-a}\int_{X}u^{2p}\Delta_{\omega_{X}}(G)e^{G}\omega_{X}^{n}$
		$\displaystyle=\frac{1}{a-1}\int_{X}u^{2p}\|\nabla G\|^{2}_{\omega_{X}}e^{G}\omega_{X}^{n}+\frac{1}{a-1}\int_{X}n2pu^{2p-1}e^{G}du\wedge d^{c}G\wedge\omega_{X}^{n-1}.$

By Cauchy–Schwarz inequality,

\left|2pu^{2p-1}\frac{du\wedge d^{c}G\wedge\omega_{X}^{n-1}}{\omega_{X}^{n}}\right|\leq\frac{4p^{2}}{2}u^{2p-2}|\nabla u|^{2}_{\omega_{X}}+\frac{1}{2}u^{2p}|\nabla G|^{2}_{\omega_{X}}.

The above two inequalities yield

\displaystyle I_{1}\geq-\frac{2p^{2}}{a-1}\int_{X}u^{2p-2}|\nabla u|^{2}_{\omega_{X}}e^{G}\omega_{X}^{n}.

(2.12)

We next control the term $I_{2}$ . We compute

	$\displaystyle\Delta_{\omega_{X},v}(\varphi-\rho)=\operatorname{tr}_{\omega_{X},v}(\omega_{\varphi}-\omega_{\rho})$	$\displaystyle=\operatorname{tr}_{\omega_{X}}(\omega_{\varphi}-\omega_{\rho})+\langle(\log v)^{\prime}(m_{\omega_{X}}),(m_{\varphi}-m_{\rho})\rangle$
		$\displaystyle\leq\operatorname{tr}_{\omega_{X}}\omega_{\varphi}+C_{3}$

where $C_{3}$ only depends on $P,C_{v}$ . Since $a>1$ , we get

\displaystyle I_{2}\geq\frac{ab}{1-a}\int_{X}u^{2p}\operatorname{tr}_{\omega_{X}}\omega_{\varphi}e^{G}\omega_{X}^{n}+\frac{abC_{3}}{1-a}\int_{X}u^{2p}e^{G}\omega_{X}^{n}.

(2.13)

Combining (2.11), (2.12) and (2.13), one can derive

	$\displaystyle 0$	$\displaystyle\geq 2\left(p-\frac{p^{2}}{a-1}\right)\int_{X}u^{2p-2}e^{G}\|\nabla u\|^{2}_{\omega_{X}}\omega_{X}^{n}-(aC_{1}+\mathcal{C})\int_{X}u^{2p+1}e^{F}\omega_{X}^{n}$
		$\displaystyle\qquad+\frac{abC_{\rho}}{2}C_{v}^{-\frac{1}{n-1}}\int_{X}u^{2p+1}e^{\frac{n-2}{n-1}F}(\operatorname{tr}_{\omega_{X}}\omega_{\varphi})^{\frac{1}{n-1}}\omega_{X}^{n}+\frac{ab}{1-a}\int_{X}u^{2p}\operatorname{tr}_{\omega_{X}}\omega_{\varphi}e^{G}\omega_{X}^{n}+\frac{abC_{3}}{1-a}\int_{X}u^{2p}e^{G}\omega_{X}^{n}.$

Taking $a>1$ sufficiently large so that $p>p^{2}/(a-1)$ , and using the upper bound $F\leq C_{0}$ , one gains $\frac{n-2}{n-1}F\geq F-C_{0}/(n-1)$ and thus,

\displaystyle 0

\displaystyle\geq-C_{4}\int_{X}(\operatorname{tr}_{\omega_{X}}\omega_{\varphi})^{2p+1}e^{(2p+1)(G-F)+F}\omega_{X}^{n}-C_{5}\int_{X}u^{2p}e^{G}\omega_{X}^{n}+C_{6}\int_{X}(\operatorname{tr}_{\omega_{X}}\omega_{\varphi})^{2p+1+\frac{1}{n-1}}e^{(2p+1)(G-F)+F}\omega_{X}^{n},

where $C_{4},C_{5},C_{6}$ only depend on $a,b,p,C_{0},C_{\rho},C_{v},n$ and $|\nabla(v(m_{\omega_{X}}))|^{2}_{\omega_{X}}$ . Set $\mu:=e^{(2p+1)(G-F)+F}\omega_{X}^{n}=e^{-[a(2p+1)-1]F-ab(2p+1)\varphi+ab(2p+1)\rho)}\omega_{X}^{n}$ . Choose $b\gg 1$ such that $ab(2p+1)>K_{2}(a(2p+1)-1)$ . By $F\geq K_{2}\rho-C_{0}$ in the assumption,

C_{7}^{-1}e^{ab(2p+1)\rho}\leq\frac{\mu}{\omega_{X}^{n}}\leq C_{7},

and it implies that $C_{8}^{-1}\leq\int_{X}\mu\leq C_{8}$ . Therefore, we obtain

\displaystyle 0

\displaystyle\geq-C_{4}\int_{X}(\operatorname{tr}_{\omega_{X}}\omega_{\varphi})^{2p+1}d\mu-C_{5}\int_{X}(\operatorname{tr}_{\omega_{X}}\omega_{\varphi})^{2p}d\mu+C_{6}\int_{X}(\operatorname{tr}_{\omega_{X}}\omega_{\varphi})^{2p+1+\frac{1}{n-1}}d\mu,

and Hölder’s inequality shows $\|\operatorname{tr}_{\omega_{X}}\omega_{\varphi}\|_{L^{2p+1}(X,\mu)}\leq C$ . This concludes the proof. ∎

2.3. Local higher-order estimates

We next provide higher-order estimates that are away from the degenerating locus. We shall use the following local estimate, which generalizes [CC21a] for cscK equations (see also [DJL24, HL25]). Its proof follows a similar argument in [CC21a] with further analysis of the weighted terms. For full details, the reader is referred to Appendix B.

Theorem 2.3.

Assume that $\mathfrak{t}^{\vee}\ni\zeta\mapsto\log v(\zeta)\in\mathbb{R}$ is concave and $w\in\mathcal{C}^{\infty}(\mathfrak{t}^{\vee},\mathbb{R})$ . Let $\phi$ be a smooth solution of

\begin{cases}v(m_{dd^{c}\phi})\det{(\phi_{k\bar{j}})}=e^{G}\\ \Delta_{\phi,v}G=-w(m_{dd^{c}\phi})\end{cases}

in $B_{1}(0)\subset\mathbb{C}^{n}$ , where $\Delta_{\phi,v}f:=\phi^{k\bar{j}}\partial_{k}\partial_{\bar{j}}f+\operatorname{tr}_{dd^{c}\phi}(d\log v(m_{dd^{c}\phi})\wedge d^{c}f)$ , such that $\Delta\phi\in L^{p}(B_{1}(0))$ and $\operatorname{tr}_{\phi}\omega_{\mathrm{euc\ell}}\in L^{p}(B_{1}(0))$ for some $p>3n$ . Then there exists a constant $A>0$ , depending only on $C_{v}$ , $C_{w}$ , $p$ , $\|\Delta\phi\|_{L^{p}(B_{1}(0))}$ , $\|\operatorname{tr}_{\phi}\omega_{\mathrm{euc\ell}}\|_{L^{p}(B_{1}(0))}$ , such that $|\phi_{k\bar{j}}|\leq A$ , $|\nabla G|\leq A$ in $B_{\frac{1}{2}}(0))$ and for any $k\geq 2$

\|D^{k}\phi\|_{L^{\infty}(B_{1/2}(0))}\leq C(k,A,\|\phi\|_{L^{\infty}(B_{1}(0))}).

Combining Theorem 2.1, Proposition 2.2, and Theorem 2.3, we obtain the following result:

Theorem 2.4.

Under the same assumption of Theorem 2.1 and assuming that $\log v$ is concave and $S=w(m_{\phi}),w\in\mathcal{C}^{\infty}(\mathfrak{t}^{\vee},\mathbb{R})$ , there is a uniform constant $C>0$ depending only on $n,V,\alpha,K_{0},K_{1},K_{2},K_{3},K_{4},C_{v},C_{w},\mu,\rho_{1}$ such that

\|\varphi\|_{L^{\infty}}\leq C\,\text{ and }\,F\leq C\,\text{ and }F\geq K_{2}\rho_{2}-C,

where $\rho_{2}$ is a quasi-plurisubharmonic function with analytic singularities, such that $\omega+dd^{c}\rho_{2}$ is a Kähler current with $\operatorname{Ric}(\mu)\leq K_{2}(\omega+dd^{c}\rho_{2})$ .

Moreover, for any compact set $K\subset X\setminus\{\rho_{2}=-\infty\}$ and $\ell\in\mathbb{N}$ , we have $\|\varphi_{\varepsilon}\|_{\mathcal{C}^{\ell}(K,\omega_{X})}\leq C_{K,\ell}$ for some uniform constant $C_{K,\ell}>0$ , where $\omega_{X}$ is a fixed Kähler metric on $X$ .

3. Existence of singular weighted cscK metrics

We shall combine the a priori estimates obtained in the previous section and the uniform coercivity established in [BJT24] to construct singular weighted cscK metrics.

3.1. Setup

In the sequel, we always assume the following setting:

Setting 3.1.

Let $X$ be an $n$ -dimensional compact Kähler variety with log terminal singularities. Fix a compact torus $T\subset\operatorname{Aut}_{\operatorname{red}}(X)$ and denote by $\mathfrak{t}=\operatorname{Lie}(T)$ . Assume that $X$ admits a $T$ -equivariant resolution of singularities $\pi:Y\to X$ with $K_{Y}=\pi^{*}K_{X}+\sum_{i}a_{i}E_{i}$ , $a_{i}>-1$ , $Y$ is Kähler, and $\pi$ is an isomorphism over $X^{\mathrm{reg}}$ . Let $\omega_{Y}$ be a $T$ -invariant Kähler metric on $Y$ . Given a $T$ -invariant Kähler metric $\omega$ on $X$ , by [Bou04, Thm. 3.17], there exists a $T$ -invariant quasi-psh function $\rho$ that is smooth on $Y\setminus E$ and has analytic singularities along $E$ such that $\pi^{\ast}\omega+dd^{c}\rho$ is a Kähler current, i.e. $\pi^{\ast}\omega+dd^{c}\rho\geq C_{\rho}\omega_{Y}$ for some $C_{\rho}>0$ . Fix a $T$ -invariant adapted measure $\mu_{X}$ on $X$ , which is normalized by $\int_{Y}\log\left(\frac{\pi^{*}\mu_{X}}{\omega_{Y}^{n}}\right)\pi^{*}\omega^{n}=0$ .

For $\varepsilon\in(0,1]$ , we denote by $\omega_{\varepsilon}:=\pi^{\ast}\omega+\varepsilon\omega_{Y}$ and $P_{\varepsilon}:=\operatorname{im}(m_{\omega_{\varepsilon}})=\operatorname{im}(m_{\pi^{*}\omega})+\varepsilon\operatorname{im}(m_{\omega_{Y}})$ which contained in a compact set of $\mathfrak{t}^{\vee}$ for all $\varepsilon$ small. Let $v,w\in\mathcal{C}^{\infty}(\mathfrak{t}^{\vee},\mathbb{R})$ with $v,w>0$ on $P_{\omega}$ . Since $P_{\omega}$ is compact, there are constants $C_{v},C_{w}\geq 1$ such that for any $\varepsilon$ sufficiently small, on $P_{\varepsilon}$ ,

C_{v}^{-1}\leq v+\sum_{\alpha}|v_{\alpha}|+\sum_{\alpha,\beta}|v_{\alpha\beta}|\leq C_{v},\quad\text{and}\quad C_{w}^{-1}\leq|w|+\sum_{\alpha}|w_{\alpha}|+\sum_{\alpha,\beta}|w_{\alpha\beta}|\leq C_{w}.

Denote by $\ell^{\operatorname{ext}}_{\varepsilon}$ (reps. $\ell^{\operatorname{ext}}$ ) the extremal affine function on $\mathfrak{t}^{\vee}$ associated to $\omega_{\varepsilon}$ (resp. $\omega$ ). Since $\omega_{\varepsilon}$ converges smoothly to $\pi^{\ast}\omega$ , the moment map $m_{\omega_{\varepsilon}}$ converges smoothly to $m_{\pi^{\ast}\omega}$ and $\ell^{\operatorname{ext}}_{\varepsilon}\rightarrow\ell^{\operatorname{ext}}$ in $\mathfrak{t}\oplus\mathbb{R}$ (cf. [BJT24, Lem. 4.18]).

Assume that a smooth pair $(\varphi_{\varepsilon},F_{\varepsilon})$ solves

\begin{cases}v(m_{\varphi_{\varepsilon}})(\omega_{\varepsilon}+dd^{c}\varphi_{\varepsilon})^{n}=e^{F_{\varepsilon}}\omega_{Y}^{n},\quad\sup_{Y}\varphi_{\varepsilon}=0,\\ \Delta_{\varphi_{\varepsilon},v}F_{\varepsilon}=-(w\ell^{\operatorname{ext}}_{\varepsilon})(m_{\varphi_{\varepsilon}})+\operatorname{tr}_{\varphi_{\varepsilon},v}(\operatorname{Ric}(\omega_{Y})).\end{cases}

(3.1)

Under Setting 3.1, one can conclude the following Theorem 3.2 by applying Theorem 2.4.

Theorem 3.2.

Under Setting 3.1, suppose that Condition (A) holds and $\log v$ is concave. Let $(\varphi_{\varepsilon},F_{\varepsilon})$ be a solution to (3.1). If there is a constant $C>0$ independent of sufficient small $\varepsilon>0$ such that

\mathbf{H}_{\varepsilon}(\varphi_{\varepsilon})\leq\int_{Y}\log\left(\frac{\omega_{\varepsilon,\varphi_{\varepsilon}}^{n}}{\omega_{Y}^{n}}\right)\omega_{\varepsilon,\varphi_{\varepsilon}}^{n}\leq C,

then there exists a uniform constant $C_{0}>0$ such that

\|\varphi_{\varepsilon}\|_{L^{\infty}}\leq C_{0},\quad C_{0}\geq F_{\varepsilon}\geq K_{2}\rho-C_{0},

where $K_{2}$ is a positive constant such that $\operatorname{Ric}(\omega_{Y})\leq K_{2}(\pi^{\ast}\omega+dd^{c}\rho)$ . Moreover, for any compact subset $K\subset Y\setminus\operatorname{Exc}(\pi)$ and $\ell\in\mathbb{N}$ , we have $\|\varphi_{\varepsilon}\|_{\mathcal{C}^{\ell}(K,\omega_{Y})}\leq C_{K,\ell}$ for some uniform constant $C_{K,\ell}>0$ .

3.2. Proof of Theorem A

We now prove the following result on the existence of singular weighted cscK metrics.

Theorem 3.3.

Under Setting 3.1, suppose that Condition (A) is fulfilled and $\log v$ is concave. If the weighted Mabuchi functional $\mathbf{M}_{\omega,v,w\ell^{\operatorname{ext}}}$ is $T_{\mathbb{C}}$ -coercive, then $X$ admits a singular $(v,w)$ -extremal metric (i.e. $(v,w\ell^{\operatorname{ext}})$ -cscK metric) in $\{\omega\}$ , and it is also a minimizer of $\mathbf{M}_{\omega,v,w\ell^{\operatorname{ext}}}$ .

Proof.

Part 1: existence. Since $\mathbf{M}^{\rm re\ell}_{X}:=\mathbf{M}_{\omega,v,w\ell^{\operatorname{ext}}}$ is $T_{\mathbb{C}}$ -coercive, there are positive constants $A_{0}$ and $B_{0}$ such that

\mathbf{M}^{\rm re\ell}_{X}\geq A_{0}d_{1,T}-B_{0}\quad\text{on }\mathcal{E}^{1}_{{\mathrm{norm}}}(X,\omega)^{T}.

By [BJT24, Thm. A], for any $A\in(0,A_{0})$ , there exists $B>0$ such that for all sufficiently small $\varepsilon>0$ ,

\mathbf{M}_{\varepsilon}^{\rm re\ell}\geq Ad_{1,\omega_{\varepsilon},T}-B\quad\text{on}\ \,\mathcal{E}^{1}_{\mathrm{norm}}(Y,\omega_{\varepsilon})^{T},

(3.2)

where $d_{1,\omega_{\varepsilon},T}$ is $d_{1,T}$ with respect to $\omega_{\varepsilon}$ in (1.6) and

\mathbf{M}_{\varepsilon}^{\rm re\ell}:=\mathbf{M}_{\omega_{\varepsilon},v,w\ell^{\operatorname{ext}}_{\varepsilon}}=\mathbf{H}_{\varepsilon,v,\omega_{Y}^{n}}+\mathbf{E}_{\varepsilon,v}^{-\operatorname{Ric}(\omega_{Y})}+\mathbf{E}_{\varepsilon,vw\ell^{\operatorname{ext}}_{\varepsilon}}

is the weighted Mabuchi functional on $(Y,\omega_{\varepsilon})$ . By the existence result obtained in [DJL25, HL25], there is a smooth pair $(\varphi_{\varepsilon},F_{\varepsilon})$ solving

\begin{cases}v(m_{\varphi_{\varepsilon}})(\omega_{\varepsilon}+dd^{c}\varphi_{\varepsilon})^{n}=e^{F_{\varepsilon}}\omega_{Y}^{n},\quad\mathbf{E}_{\omega_{\varepsilon}}(\varphi_{\varepsilon})=0\\ \Delta_{\varphi_{\varepsilon},v}F_{\varepsilon}=-(w\ell^{\operatorname{ext}}_{\varepsilon})(m_{\varphi_{\varepsilon}})+\operatorname{tr}_{\varphi_{\varepsilon},v}(\operatorname{Ric}(\omega_{Y}))\end{cases}

(3.3)

Namely, $\omega_{\varepsilon,\varphi_{\varepsilon}}$ is a $(v,w\ell^{\operatorname{ext}}_{\varepsilon})$ -cscK metric and $\varphi_{\varepsilon}$ minimizes $\mathbf{M}_{\varepsilon}^{\rm re\ell}$ .

From Lemma 1.5, we know that $\varphi_{\varepsilon}$ minimizes the relative Mabuchi functional $\mathbf{M}_{\varepsilon}^{\rm re\ell}$ , so $\mathbf{M}_{\varepsilon}^{\rm re\ell}(\varphi_{\varepsilon})\leq\mathbf{M}_{\varepsilon}^{\rm re\ell}(0)\leq C_{1}$ , which gives

d_{1,\omega_{\varepsilon},T}(\varphi_{\varepsilon},0)\leq\frac{C_{1}+B}{A}

using the coercivity condition. By the definition of $d_{1,\omega_{\varepsilon},T}$ , for each $\varepsilon$ there exists $\sigma_{\varepsilon}\in T_{\mathbb{C}}$ such that $d_{1,\omega_{\varepsilon}}(\sigma_{\varepsilon}\cdot\varphi_{\varepsilon},0)\leq\frac{C_{1}+B}{A}$ . Set $\tilde{\varphi}_{\varepsilon}:=\sigma_{\varepsilon}\cdot\varphi_{\varepsilon}$ . By the $T_{\mathbb{C}}$ -action, we have

\omega_{\varepsilon,\tilde{\varphi}_{\varepsilon}}=\sigma_{\varepsilon}^{*}\omega_{\varepsilon,\varphi_{\varepsilon}},\quad m_{\tilde{\varphi}_{\varepsilon}}=\sigma_{\varepsilon}^{*}m_{\varphi_{\varepsilon}}

(see e.g. [BJT24, Lem. 3.19]). Pulling back the first equation in (3.3) by $\sigma_{\varepsilon}$ , we get

v(m_{\tilde{\varphi}_{\varepsilon}})(\omega_{\varepsilon}+dd^{c}\tilde{\varphi}_{\varepsilon})^{n}=\sigma_{\varepsilon}^{\ast}\left(v(m_{\varphi_{\varepsilon}})(\omega_{\varepsilon}+dd^{c}\varphi_{\varepsilon})^{n}\right)=e^{\sigma^{*}_{\varepsilon}F_{\varepsilon}}(\sigma^{*}\omega_{Y})^{n}=e^{\tilde{F}_{\varepsilon}}\omega_{Y}^{n}

where $\tilde{F}_{\varepsilon}=\sigma_{\varepsilon}^{*}F_{\varepsilon}+\log\frac{(\sigma_{\varepsilon}^{*}\omega_{Y})^{n}}{\omega_{Y}^{n}}$ . For any $T$ -invariant Kähler metric $\omega$ on $Y$ and $\sigma\in T_{\mathbb{C}}$ , we have

\sigma^{\ast}\left(\operatorname{tr}_{\omega,v}\bullet\right)=\operatorname{tr}_{\sigma^{\ast}\omega,v}(\sigma^{\ast}\bullet)\quad\text{and}\quad\sigma^{\ast}\left(\Delta_{\omega,v}\bullet\right)=\Delta_{\sigma^{\ast}\omega,v}(\sigma^{\ast}\bullet).

Combining this with the second equation in (3.3) yields

	$\displaystyle\Delta_{\tilde{\varphi}_{\varepsilon},v}\tilde{F}_{\varepsilon}$	$\displaystyle=\Delta_{\sigma_{\varepsilon}^{\ast}\omega_{\varepsilon,\varphi_{\varepsilon}},v}\sigma_{\varepsilon}^{\ast}F_{\varepsilon}+\Delta_{\tilde{\varphi}_{\varepsilon},v}\log\frac{(\sigma_{\varepsilon}^{}\omega_{Y})^{n}}{\omega_{Y}^{n}}=\sigma_{\varepsilon}^{}\Delta_{\varphi_{\varepsilon},v}F_{\varepsilon}+\Delta_{\tilde{\varphi}_{\varepsilon},v}\log\frac{(\sigma_{\varepsilon}^{*}\omega_{Y})^{n}}{\omega_{Y}^{n}}$
		$\displaystyle=-(w\ell^{\operatorname{ext}}_{\varepsilon})(m_{\tilde{\varphi}_{\varepsilon}})+\operatorname{tr}_{\tilde{\varphi}_{\varepsilon},v}(\operatorname{Ric}(\sigma_{\varepsilon}^{}\omega_{Y}))+\operatorname{tr}_{\tilde{\varphi}_{\varepsilon},v}(\operatorname{Ric}(\omega_{Y})-\operatorname{Ric}(\sigma_{\varepsilon}^{}\omega_{Y}))$
		$\displaystyle=-(w\ell^{\operatorname{ext}}_{\varepsilon})(m_{\tilde{\varphi}_{\varepsilon}})+\operatorname{tr}_{\tilde{\varphi}_{\varepsilon},v}(\operatorname{Ric}(\omega_{Y})).$

Therefore, $(\sigma_{\varepsilon}\cdot\varphi_{\varepsilon},\tilde{F}_{\varepsilon})$ also solves (3.3). Replacing $\varphi_{\varepsilon}$ by $\sigma_{\varepsilon}\cdot\varphi_{\varepsilon}$ and $F_{\varepsilon}$ by $\tilde{F}_{\varepsilon}$ , we thus obtain

d_{1,\omega_{\varepsilon}}(\varphi_{\varepsilon},0)\leq\frac{C_{1}+B}{A}.

(3.4)

Since $\mathbf{M}_{\varepsilon}^{\rm re\ell}$ is $T_{\mathbb{C}}$ -invariant, this still minimize $\mathbf{M}_{\varepsilon}^{\rm re\ell}$ .

Recall that $\pi^{\ast}\omega+dd^{c}\rho\geq C_{\rho}\omega_{Y}$ . We shall verify the assumptions in Theorem 2.1 with $\rho_{2}=\rho$ and $\mu=\omega_{Y}^{n}$ . It is obvious that the condition (3) holds. We now show that the uniform control on the entropies $\mathbf{H}_{\varepsilon}(\varphi_{\varepsilon})$ , which will confirm the condition (2).

If a closed $(1,1)$ -current $\Theta\geq-A\pi^{\ast}\omega$ for some constant $A>0$ , then

\mathbf{E}_{\varepsilon,v}^{\Theta}(\psi)\leq C(A,v,\Theta)(d_{1,\varepsilon}(\psi,0)+1).

(3.5)

for any $\varepsilon\in(0,1]$ and for all $\psi\in\mathcal{E}^{1}(Y,\omega_{\varepsilon})$ (c.f [BJT24, Lem. 4.22]). By Condition (A) and the strong openness [Ber13, GZ15], there is a constant $a>1$ such that $\int_{Y}e^{-aK_{1}\rho_{1}}\omega_{Y}^{n}<+\infty$ . Set $\mu_{Y}:=e^{-aK_{1}\rho_{1}}\omega_{Y}^{n}$ . Similar to [BJT24, Sec. 4.4], we consider

\widehat{\mathbf{H}}_{\varepsilon,v}(\psi):=\mathbf{H}_{\mu_{Y}}(\operatorname{MA}_{\varepsilon,v}(\psi))\quad\text{and}\quad\widehat{\operatorname{Ric}}:=\operatorname{Ric}(\omega_{Y})+K_{1}dd^{c}\rho_{1},

then $\widehat{\operatorname{Ric}}\geq-K_{1}\pi^{\ast}\omega\geq-K_{1}\omega_{\varepsilon}$ by assumption. It follows from [BJT24, Lem. A.2] and [BJT24, Prop. 3.44] that

\frac{1}{a}\widehat{\mathbf{H}}_{\varepsilon,v}(\psi)=\frac{1}{a}\mathbf{H}_{\varepsilon,v}(\psi)+\int_{Y}K_{1}\rho_{1}\operatorname{MA}_{\varepsilon,v}(\psi)

(3.6)

and

\mathbf{E}^{-\widehat{\operatorname{Ric}}}_{\varepsilon,v}(\psi)=\mathbf{E}^{-{\operatorname{Ric}(\omega_{Y})}}_{\varepsilon,v}(\psi)-\int_{Y}K_{1}\rho_{1}\operatorname{MA}_{\varepsilon,v}(\psi)+\int_{Y}K_{1}\rho_{1}\operatorname{MA}_{\varepsilon,v}(0).

(3.7)

Therefore, we get

\frac{1}{a}\mathbf{H}_{\varepsilon,v,\omega_{Y}^{n}}(\psi)+\mathbf{E}_{\varepsilon,v}^{-\operatorname{Ric}(\omega_{Y})}(\psi)=\frac{1}{a}\widehat{\mathbf{H}}_{\varepsilon,v}(\psi)+\mathbf{E}_{\varepsilon,v}^{-\widehat{\operatorname{Ric}}}(\psi)-\int_{Y}K_{1}\rho_{1}\operatorname{MA}_{\varepsilon,v}(0),

(3.8)

with $\int_{Y}K_{1}\rho_{1}\operatorname{MA}_{\varepsilon,v}(0)\rightarrow\int_{Y}K_{1}\rho_{1}v(m_{\pi^{*}\omega})(\pi^{*}\omega)^{n}$ as $\varepsilon\to 0$ . Here without loss of generality, we assume that $\int_{Y}\rho_{1}v(m_{\pi^{*}\omega})(\pi^{*}\omega)^{n}=0$ . Then one can derive

\begin{split}\mathbf{M}_{\varepsilon}^{\rm re\ell}(\psi)&=\mathbf{H}_{\varepsilon,v,\omega_{Y}^{n}}(\psi)+\mathbf{E}_{\varepsilon,v}^{-\operatorname{Ric}(\omega_{Y})}(\psi)+\mathbf{E}_{\varepsilon,vw\ell^{\operatorname{ext}}}(\psi)\\ &\geq\left(1-\frac{1}{a}\right)\mathbf{H}_{\varepsilon,v,\omega_{Y}^{n}}(\psi)+\mathbf{E}_{\varepsilon,v}^{-\widehat{\operatorname{Ric}}}(\psi)+\mathbf{E}_{\varepsilon,vw\ell^{\operatorname{ext}}}(\psi)-C.\end{split}

(3.9)

By [BJT24, Lem. 4.20], $|\mathbf{E}_{\varepsilon}(\psi)|\leq C_{2}d_{1,\omega_{\varepsilon}}(\psi,0)$ for all $\psi\in\mathcal{E}^{1}(Y,\omega_{\varepsilon})$ , hence (3.4) implies that

|\mathbf{E}_{\varepsilon,vw\ell^{\operatorname{ext}}}(\varphi_{\varepsilon})|\leq C_{3}=C(A,B,C_{1},C_{2},C_{v},C_{w}),

for all $\varepsilon$ small, and then (3.5) and (3.9) imply that

\left(1-\frac{1}{a}\right)\mathbf{H}_{\varepsilon,v,\omega_{Y}^{n}}(\varphi_{\varepsilon})\leq C-\mathbf{E}_{\varepsilon,vw\ell^{\operatorname{ext}}}(\varphi_{\varepsilon})+\mathbf{E}^{\widehat{\operatorname{Ric}}}_{\varepsilon,v}(\varphi_{\varepsilon})\leq C_{1}+C_{3}+C(K_{1},v,\widehat{\operatorname{Ric}})C_{4}.

Therefore, $\mathbf{H}_{\varepsilon,v,\omega_{Y}^{n}}(\varphi_{\varepsilon})\leq C^{\prime}$ for all sufficiently small $\varepsilon>0$ and for some uniform constant $C^{\prime}>0$ . We have

	$\displaystyle\|\mathbf{H}_{\varepsilon,v,\omega_{Y}^{n}}(\varphi_{\varepsilon})-\mathbf{H}_{\varepsilon,\omega_{Y}^{n}}(\varphi_{\varepsilon})\|=\left\|\int_{Y}\log\left(\frac{v(m_{\varphi_{\varepsilon}})\omega_{\varepsilon,\varphi_{\varepsilon}}^{n}}{\omega_{Y}^{n}}\right)v(m_{\varphi_{\varepsilon}})\omega_{\varepsilon,\varphi_{\varepsilon}}^{n}-\int_{Y}\log\left(\frac{\omega_{\varepsilon,\varphi_{\varepsilon}}^{n}}{\omega_{Y}^{n}}\right)\omega_{\varepsilon,\varphi_{\varepsilon}}^{n}\right\|$
	$\displaystyle=\left\|\int_{Y}\left(1-\frac{1}{v(m_{\varphi_{\varepsilon}})}\right)\log\left(\frac{v(m_{\varphi_{\varepsilon}})\omega_{\varepsilon,\varphi_{\varepsilon}}^{n}}{\omega_{Y}^{n}}\right)v(m_{\varphi_{\varepsilon}})\omega_{\varepsilon,\varphi_{\varepsilon}}^{n}+\int_{Y}\log(v(m_{\varphi_{\varepsilon}}))\omega_{\varepsilon,\varphi_{\varepsilon}}^{n}\right\|\leq C_{v}^{\prime}C^{\prime}+C_{v}^{\prime},$

where $C^{\prime}_{v}>0$ depend only on $C_{v}$ ; hence, we obtain a uniform upper bound for $\mathbf{H}_{\varepsilon,\omega_{Y}^{n}}(\varphi_{\varepsilon})$ as required.

By Theorem 2.4, we obtained uniform $L^{\infty}$ and local $\mathcal{C}^{\ell}$ -estimates for $\varphi_{\varepsilon}-\sup_{Y}\varphi_{\varepsilon}$ . The Arzelà–Ascoli theorem shows that there is a subsequence $\varphi_{\varepsilon}-\sup_{Y}\varphi_{\varepsilon}$ converging in $\mathcal{C}_{\mathrm{\ell oc}}^{\infty}(Y\setminus E)$ to $\varphi_{0}\in\operatorname{PSH}(Y,\pi^{*}\omega)\cap L^{\infty}(Y)\cap\mathcal{C}^{\infty}(Y\setminus E)$ which satisfies $(v,w\ell^{\operatorname{ext}})$ -cscK equations on $Y\setminus E$ . This deduces the existence of a singular weighted cscK metric $\omega+dd^{c}\psi_{0}$ in $\{\omega\}$ on $X$ , where $\psi_{0}\in\operatorname{PSH}(X,\omega)\cap L^{\infty}(X)\cap\mathcal{C}^{\infty}(X^{\mathrm{reg}})$ and $\varphi_{0}=\pi^{*}\psi_{0}$ .

Part 2: Minimizer. It remains to show that $\psi_{0}$ is a minimizer of $\mathbf{M}_{X}^{\rm re\ell}$ on $\mathcal{E}^{1,T}_{\omega}$ . Note that $\varphi_{\varepsilon}$ are uniformly bounded and converging locally smoothly to $\pi^{\ast}\psi_{0}$ on $Y\setminus E$ . Once can derive $\mathbf{E}_{\varepsilon,vw\ell^{\operatorname{ext}}}(\varphi_{\varepsilon})\to\mathbf{E}_{vw\ell^{\operatorname{ext}}}(\pi^{\ast}\psi_{0})$ as $\varepsilon\to 0$ . Thus, by (3.8) with $a=1$ and by the semi-continuity with respect to strong convergence of $\widehat{\mathbf{H}}_{\varepsilon,v}$ and $\mathbf{E}_{\varepsilon,v}^{-\widehat{\operatorname{Ric}}}$ (cf. [BJT24, Lem. 4.22]) , one gets

\liminf_{\varepsilon\to 0}\mathbf{M}_{\varepsilon}^{\rm re\ell}(\varphi_{\varepsilon})\geq\widehat{\mathbf{H}}_{v}(\pi^{*}\psi_{0})+\mathbf{E}^{-\widehat{\operatorname{Ric}}}_{v}(\pi^{*}\psi_{0})+\mathbf{E}_{vw\ell^{\operatorname{ext}}}(\pi^{*}\psi_{0}),

where we used the fact that $\int_{Y}K_{1}\rho_{1}\operatorname{MA}_{\varepsilon,v}(0)\rightarrow\int_{Y}K_{1}\rho_{1}v(m_{\pi^{*}\omega})(\pi^{*}\omega)^{n}=0$ . It follows from [BJT24, Lem. 4.26] that

\widehat{\mathbf{H}}_{v}(\pi^{*}\psi_{0})+\mathbf{E}^{-\widehat{\operatorname{Ric}}}_{v}(\pi^{*}\psi_{0})\geq\mathbf{H}_{\mu_{X}}(\psi_{0})+\mathbf{E}^{-{\operatorname{Ric}(\mu_{X})}}_{v}(\psi_{0}).

This shows that

\liminf_{\varepsilon\to 0}\mathbf{M}_{\varepsilon}^{\rm re\ell}(\varphi_{\varepsilon})\geq\mathbf{M}_{X}^{\rm re\ell}(\psi_{0}).

(3.10)

Fix an arbitrary $u\in\mathcal{E}^{1,T}_{\omega}$ . Without loss of generality, assume that $\mathbf{H}_{v,\mu_{X}}(u)<+\infty$ and denote by $f=\omega^{n}_{u}/\omega^{n}$ which is $T$ -invariant. By [PTT23, Lem. 3.4], there exists $0\leq f^{j}\in\mathcal{C}^{\infty}(X)$ converging to $f$ in $L^{\chi}$ as $j\rightarrow 0$ with $\chi(s):=(s+1)\log(s+1)-s$ . Consider $f^{T,j}(x):=\int_{T}f^{j}(\sigma\cdot x)d\mu(\sigma)$ , where $\mu$ is the Haar measure on $T$ normalized by $\int_{T}d\mu(\sigma)=1$ . We claim that $f^{T,j}\in\mathcal{C}^{\infty}(X)$ also converges to $f$ in $L^{\chi}$ . Indeed, since $\chi$ is convex,

\chi(|f^{T,j}-f|(x))=\chi\left(\left|\int_{T}(f^{T,j}-f)(\sigma\cdot x)d\mu(\sigma)\right|\right)\leq\int_{T}\chi(|f^{j}-f|(\sigma\cdot x))d\mu(\sigma).

By Fubini’s theorem and $T$ -invariance of $\omega$ , we obtain that

	$\displaystyle\int_{x\in X}\chi(\|f^{j,T}-f\|(x))\omega^{n}(x)\leq\int_{x\in X}\left(\int_{\sigma\in T}\chi(\|f^{j}-f\|(\sigma\cdot x))d\mu(\sigma)\right)\omega^{n}(x)$
	$\displaystyle=\int_{\sigma\in T}\left(\int_{x\in X}\chi(\|f^{j}-f\|(\sigma\cdot x))\omega^{n}(x)\right)d\mu(\sigma)=\int_{\sigma\in T}\left(\int_{x\in X}\chi(\|f^{j}-f\|(\sigma\cdot x))\omega^{n}(\sigma\cdot x)\right)d\mu(\sigma).$

Hence, $f^{T,j}$ converges to $f$ in $L^{\chi}$ . Consider $u_{j}\in\mathcal{C}^{\infty}(X^{\mathrm{reg}})\cap L^{\infty}(X)$ solving

(\omega+dd^{c}u_{j})^{n}=c_{j}f^{T,j}\omega^{n}\quad\text{with}\quad\sup_{X}u_{j}=0

where $c_{j}>0$ is a normalizing constant. Moreover, it follows from [CC24, Thm. 1.3] that $u_{j}$ is continuous on $X$ . By [PTT23, Lem. 3.4], $u_{j}$ converges strongly to $u$ , and $\mathbf{H}_{\mu_{X}}(u_{j})\rightarrow\mathbf{H}_{\mu_{X}}(u)$ ; thus, $\mathbf{M}_{v,w\ell^{\operatorname{ext}}}(u_{j})\to\mathbf{M}_{v,w\ell^{\operatorname{ext}}}(u)$ .

Then for each $j$ fixed, one can find a family of functions $u_{j,\varepsilon}\in\operatorname{PSH}(Y,\omega_{\varepsilon})$ such that

•

$(u_{j,\varepsilon})_{\varepsilon\in(0,1)}$ are uniformly bounded and continuous;
•

$u_{j,\varepsilon}$ converges locally smoothly on $Y\setminus E$ as $\varepsilon\to 0$ ;
•

$u_{j,\varepsilon}$ decreases to $\pi^{\ast}u_{j}$ as $\varepsilon\to 0$ .

Since $\pi^{*}u_{j}$ is continuous on $Y$ , Dini’s theorem implies $u_{j,\varepsilon}$ converges uniformly to $\pi^{*}u_{j}$ as $\varepsilon\rightarrow 0$ . Indeed, for a fixed $j$ , consider $u_{j,\varepsilon}$ the unique solution to the following equation

(\omega_{\varepsilon}+dd^{c}u_{j,\varepsilon})^{n}=e^{u_{j,\varepsilon}-\pi^{\ast}u_{j}}\pi^{*}(c_{j}f^{T,j}\omega^{n}).

Then by [EGZ09], one has a uniform $\mathcal{C}^{0}$ -estimate for $u_{j,\varepsilon}$ on $Y$ , local $\mathcal{C}^{2}$ -estimate for $u_{j,\varepsilon}$ away from $E$ , and $u_{j,\varepsilon}$ converges locally smoothly to $\pi^{*}u_{j}$ in $Y\setminus E$ . We now check that $u_{j,\varepsilon}$ is decreasing as $\varepsilon\to 0$ . For $0<\varepsilon_{1}<\varepsilon_{2}$ , we have

(\omega_{\varepsilon_{2}}+dd^{c}u_{j,\varepsilon_{1}})^{n}\geq(\omega_{\varepsilon_{1}}+dd^{c}u_{j,\varepsilon_{1}})^{n}=e^{u_{j,\varepsilon_{1}}-\pi^{\ast}u_{j}}\pi^{*}(c_{j}f^{T,j}\omega^{n}),

so $u_{j,\varepsilon_{1}}$ is a subsolution to the equation

(\omega_{\varepsilon_{2}}+dd^{c}\varphi)=e^{\varphi}e^{-\pi^{\ast}u_{j}}\pi^{*}(c_{j}f^{T,j}\omega^{n}).

Thus, we obtain that $u_{j,\varepsilon_{1}}\leq u_{j,\varepsilon_{2}}$ for any $0<\varepsilon_{1}<\varepsilon_{2}$ .

Combining with (3.10), we get

\mathbf{M}_{X}^{\rm re\ell}(\psi_{0})\leq\liminf_{\varepsilon\to 0}\mathbf{M}_{\varepsilon}^{\rm re\ell}(\varphi_{\varepsilon})\leq\liminf_{\varepsilon\to 0}\mathbf{M}_{\varepsilon}^{\rm re\ell}(u_{\varepsilon,j})=\mathbf{M}_{X}^{\rm re\ell}(u_{j}),

where the second inequality follows from the fact that $\varphi_{\varepsilon}$ is a minimizer for $\mathbf{M}_{\varepsilon,v,w\ell^{\operatorname{ext}}}$ and the last equality follows from Lemma 3.4 below. Letting $j\rightarrow+\infty$ , we obtain that $\mathbf{M}_{v,w\ell^{\operatorname{ext}}}(\psi_{0})\leq\mathbf{M}_{v,w\ell^{\operatorname{ext}}}(u)$ for arbitrary $u\in\mathcal{E}^{1,T}_{\omega}$ and this finishes the proof. ∎

Lemma 3.4.

Consider $u\in\operatorname{PSH}(X,\omega)^{T}\cap\mathcal{C}^{0}(X)$ such that $u$ is smooth on $X^{\mathrm{reg}}$ and $\mathbf{H}_{\mu_{X}}(u)<+\infty$ . Assume that $u_{\varepsilon}\in\operatorname{PSH}(Y,\omega_{\varepsilon})^{T}\cap\mathcal{C}^{0}(Y)\cap\mathcal{C}^{\infty}(Y\setminus E)$ converges smoothly to $\pi^{*}u$ in $Y\setminus E$ , uniformly on $Y$ and $\mathbf{H}_{\varepsilon,\omega^{n}_{Y}}(u_{\varepsilon})\rightarrow\mathbf{H}_{\omega^{n}_{Y}}(\pi^{*}u)$ as $\varepsilon\to 0$ . Then $\mathbf{M}_{\varepsilon}^{\rm re\ell}(u_{\varepsilon})\rightarrow\mathbf{M}^{\rm re\ell}_{X}(u)$ as $\varepsilon\rightarrow 0$ .

Proof.

The proof follows a similar approach to that of [BJT24, Prop. 4.16] where the authors assume $u$ to be smooth on $X$ instead of $X^{\mathrm{reg}}$ . Here, we only provide a brief outline and indicate the necessary modifications.

Set $D=K_{Y}-\pi^{*}K_{X}=\sum_{i}a_{i}E_{i}$ and define $\rho_{Y/X}=\log\frac{\pi^{\ast}\mu_{X}}{\omega_{Y}^{n}}$ . Then we have

[D]-dd^{c}\rho_{Y/X}=\pi^{*}\operatorname{Ric}(\mu_{X})-\operatorname{Ric}(\omega_{Y}^{n}).

For $u\in\operatorname{PSH}(X,\omega)\cap L^{\infty}(X)$ and any smooth $(1,1)$ -form $\theta$ (or positive closed $(1,1)$ -current),

\mathbf{E}^{\theta}_{\pi^{*}\omega}(\pi^{*}u):=\sum_{j=0}^{n-1}\int_{Y}(\pi^{*}u)\theta\wedge(\pi^{*}\omega+dd^{c}\pi^{*}u)^{j}\wedge(\pi^{*}\omega)^{n-1-j}.

Note that $\mathbf{E}^{[D]}_{\pi^{*}\omega}(\pi^{*}u)=0$ for any $u\in\operatorname{PSH}(X,\omega)\cap L^{\infty}(X)$ . Following the same argument in [BJT24, Lem. 4.15], one can infer that for any $u\in\operatorname{PSH}(X,\omega)\cap L^{\infty}(X)$ and smooth on $X^{\mathrm{reg}}$ we have

(\mathbf{H}_{\omega_{Y}^{n},v}(\pi^{*}u)+\mathbf{E}^{-\operatorname{Ric}(\omega_{Y})}_{\pi^{*}\omega}(\pi^{*}u))-(\mathbf{H}_{\mu_{X},v}(u)+\mathbf{E}^{-\operatorname{Ric}(\mu_{X})}_{\omega}(u))=\mathbf{E}^{[D]}_{\pi^{*}\omega}(\pi^{*}u)=0;

therefore,

\mathbf{H}_{\omega_{Y}^{n},v}(\pi^{*}u)+\mathbf{E}^{-\operatorname{Ric}(\omega_{Y})}_{\pi^{*}\omega}(\pi^{*}u)=\mathbf{H}_{\mu_{X},v}(u)+\mathbf{E}^{-\operatorname{Ric}(\mu_{X})}_{\omega}(u).

(3.11)

Since $u_{\varepsilon}$ converges uniformly to $\pi^{*}u$ , we have $\mathbf{E}_{\omega_{\varepsilon}}^{-\operatorname{Ric}(\omega_{Y})}(u_{\varepsilon})\to\mathbf{E}_{\pi^{*}\omega}^{-\operatorname{Ric}(\omega_{Y})}(\pi^{*}u)$ as $\varepsilon\to 0$ . Moreover, since $\ell^{\operatorname{ext}}_{\varepsilon}\rightarrow\ell^{\operatorname{ext}}$ in $\mathfrak{t}\oplus\mathbb{R}$ , we have $\mathbf{E}_{\varepsilon,vw\ell^{\operatorname{ext}}}(u_{\varepsilon})\to\mathbf{E}_{\omega,vw\ell^{\operatorname{ext}}}(u)$ . The hypothesis $\mathbf{H}_{\varepsilon,\omega_{Y}^{n}}(u_{\varepsilon})\rightarrow\mathbf{H}_{\omega_{Y}^{n}}(\pi^{*}u)$ implies $\mathbf{H}_{\varepsilon,\omega_{Y}^{n},v}(u_{\varepsilon})\rightarrow\mathbf{H}_{\omega_{Y}^{n},v}(\pi^{*}u)$ by generalized dominated convergent theorem and $u_{\varepsilon}\to\pi^{\ast}u$ locally smoothly on $Y\setminus E$ . All in all, these yield

	$\displaystyle\lim_{\varepsilon\to 0}\mathbf{M}_{\varepsilon}^{\rm re\ell}(u_{\varepsilon})$	$\displaystyle=\mathbf{H}_{\omega_{Y}^{n},v}(\pi^{}u)+\mathbf{E}_{\pi^{}\omega}^{-\operatorname{Ric}(\omega_{Y})}(\pi^{*}u)+\mathbf{E}_{\omega,vw\ell^{\operatorname{ext}}}(u)$
		$\displaystyle=\mathbf{H}_{\mu_{X},v}(u)+\mathbf{E}^{-\operatorname{Ric}(\mu_{X})}_{\omega}(u)+\mathbf{E}_{\omega,vw\ell^{\operatorname{ext}}}(u)=\mathbf{M}_{X}^{\rm re\ell}(u),$

as required, where the second equality follows from (3.11). ∎

3.2.1. Singular cscK and extremal metrics

We begin by considering the problem of finding a singular cscK metric.

Proof of Corollary B.

Take $T\subset\operatorname{Aut}_{\operatorname{red}}(X)$ to be a maximal torus and $\pi:Y\rightarrow X$ to be the $T$ -equivariant resolution of singularities in Condition (A). For cscK metrics, we have $v=1$ , $w=1$ , and $\ell^{\operatorname{ext}}=\bar{s}$ where $\bar{s}:=\frac{n}{V_{\omega}}c_{1}(X)\cdot\{\omega\}^{n-1}$ , so $\mathbf{M}_{\omega}=\mathbf{M}_{v,\ell^{\operatorname{ext}}}$ . By the hypothesis, $\mathbf{M}_{v,\ell^{\operatorname{ext}}}$ is $T_{\mathbb{C}}$ -coercive, the existence of a singular cscK metric on $X$ now follows from Theorem 3.3. ∎

Let $X$ be a normal compact Kähler variety with log terminal singularities. Fix a Kähler class $\alpha$ and take a maximal torus $T\subset\operatorname{Aut}_{\operatorname{red}}(X)$ . Let $\pi:Y\rightarrow X$ be a $T$ -equivariant resolution of singularities. We now consider the problem of the existence of singular extremal metrics in the Kähler class $\alpha$ .

Let $\xi_{\operatorname{ext}}$ be the extremal vector field defined by $T$ and $\alpha$ (cf. Section 1.2.6) and let $\omega\in\alpha$ be a $T$ -equivariant Kähler metric. A singular extremal metric in this setting is defined as a positive current of the form $\pi^{*}\omega+dd^{c}\varphi$ where $\varphi\in\operatorname{PSH}(Y,\pi^{\ast}\omega)\cap L^{\infty}(Y)$ and $\varphi$ is smooth away from $\operatorname{Exc}(\pi)$ . Additionally, $\pi^{*}\omega+dd^{c}\varphi$ is a genuine $(1,\ell^{\operatorname{ext}})$ -cscK metric on $Y\setminus\operatorname{Exc}(\pi)$ where $\ell^{\operatorname{ext}}(p)=\langle\xi_{\operatorname{ext}},p\rangle+\bar{s}$ , where $\bar{s}:=\frac{n}{V_{\omega}}c_{1}(X)\cdot\{\omega\}^{n-1}$ .

Under Condition (A), Theorem 3.3 implies the following existence result of singular extremal metrics:

Theorem 3.5.

Under the above setting, moreover, assume that $X$ satisfies Condition (A). If the weighted Mabuchi functional $\mathbf{M}_{\omega,v,\ell^{\operatorname{ext}}}$ on $X$ is $T_{\mathbb{C}}$ -coercive, then $X$ admits a singular extremal metric in $\alpha$ .

3.3. Constructing examples of singular cscK metrics

We shall give a way to construct examples of singular cscK metric in the spirit of Arezzo–Pacard [AP06] (see also [AP09, APS11, Szé15]) and use variational argument and our existence result. Before illustrating the process, we need the following lemma:

Lemma 3.6.

Let $(X,\omega)$ be a compact Kähler variety with log terminal singularities and let $f:Y\to X$ be a blowup along a compact submanifold $S\subset X^{\mathrm{reg}}$ of codimension $\geq 2$ . Consider $\omega_{Y}$ a Kähler metric on $Y$ and $\omega_{\varepsilon}=f^{\ast}\omega+\varepsilon\omega_{Y}$ for $\varepsilon\in[0,1]$ . If $\mathbf{M}_{\omega}$ is coercive, then $\mathbf{M}_{\omega_{\varepsilon}}$ is coercive for all sufficiently small $\varepsilon>0$ .

Proof.

The proof follows the same strategy in [PTT23, Thm. 4.11] and [BJT24, Thm. A]. Since $f$ is a blowup of $S$ , for $m\in\mathbb{N}^{\ast}$ so that $mK_{Y}$ and $mK_{X}$ are both $\mathbb{Q}$ -Cartier, there is a smooth hermitain metric $h_{Y}$ of $mK_{Y}$ such that

-\frac{1}{m}\Theta(mK_{Y},h_{Y})\geq-Df^{\ast}\omega

(3.12)

for some constant $D\geq 0$ . Denote by $\Theta_{Y}=\frac{1}{m}\Theta(mK_{Y},h_{Y})$ and $\Theta_{X}=\frac{1}{m}\Theta(mK_{X},h_{X})$ for some hermitian metric $h_{X}$ on $mK_{X}$ . We also set $\mu_{Y}$ (resp. $\mu_{X}$ ) to be the corresponding probability measure of $h_{Y}$ (resp. $h_{X}$ ).

Recall that from [PTT23, Sec. 4],

\mathbf{M}_{\omega_{\varepsilon}}=\mathbf{H}_{\mu_{Y},\omega_{\varepsilon}}+\bar{s}_{\varepsilon}\mathbf{E}_{\omega_{\varepsilon}}-n\mathbf{E}_{\Theta_{Y},\omega_{\varepsilon}}-C_{\varepsilon}\quad\text{on }\mathcal{E}^{1}(Y,\omega_{\varepsilon})

and

\mathbf{M}_{\omega}=\mathbf{H}_{\mu_{X},\omega}+\bar{s}\mathbf{E}_{\omega}-n\mathbf{E}_{\Theta_{X},\omega}-C\quad\text{on }\mathcal{E}^{1}(X,\omega)

where $\bar{s}_{\varepsilon}=\frac{nc_{1}(Y)\cdot[\omega_{\varepsilon}]^{n-1}}{[\omega_{\varepsilon}]^{n}}$ , $\bar{s}=\frac{nc_{1}(X)\cdot[\omega]^{n-1}}{[\omega]^{n}}$ , $C_{\varepsilon}=\mathbf{H}_{\mu_{Y},\omega_{\varepsilon}}(0)=\int_{Y}\log\left(\frac{\omega_{\varepsilon}^{n}}{\mu_{Y}}\right)\mu_{Y}$ and $C=\mathbf{H}_{\mu_{X},\omega_{\varepsilon}}(0)=\int_{X}\log\left(\frac{\omega^{n}}{\mu_{X}}\right)\mu_{X}$ . Let $A_{0},B_{0}>0$ be two constant such that $\mathbf{M}_{\omega}\geq A_{0}(-\mathbf{E}_{\omega})-B_{0}$ on $\mathcal{E}^{1}_{{\mathrm{norm}}}(X,\omega)$ . We claim that for all $0<A<A_{0}$ there are $\varepsilon_{0}>0$ and $B>0$ such that for all $\varepsilon\in(0,\varepsilon_{0})$ ,

\mathbf{M}_{\omega_{\varepsilon}}\geq A(-\mathbf{E}_{\omega_{\varepsilon}})-B

s on $\mathcal{E}^{1}_{{\mathrm{norm}}}(Y,\omega_{\varepsilon})$ .

Suppose otherwise, for an $A\in(0,A_{0})$ , there are $\varepsilon_{k}\to 0$ , $B_{k}\to+\infty$ , as $k\to+\infty$ , and $u_{k}\in\mathcal{E}^{1}_{{\mathrm{norm}}}(Y,\omega_{\varepsilon_{k}})$ such that

\mathbf{M}_{\omega_{\varepsilon_{k}}}(u_{k})<A(-\mathbf{E}_{\omega_{\varepsilon_{k}}}(u_{k}))-B_{k}.

Without loss of generality, one can assume that $u_{k}$ is bounded. Otherwise, by [PTT23, Lem. 3.4], one can find a sequence bounded $\omega_{k}$ -psh functions $(v_{k,j})_{j}$ converging strongly to $u_{k}$ and their entropies also converge to $\mathbf{H}_{\mu_{Y},\omega_{\varepsilon_{k}}}(u_{k})$ . Then for sufficiently large $j$ , we have $\mathbf{M}_{\omega_{\varepsilon_{k}}}(v_{k,j})<A(-\mathbf{E}_{\omega_{\varepsilon_{k}}}(v_{k,j}))-B_{k}$ .

Note that from (3.12), for all $\psi\in\mathcal{E}_{{\mathrm{norm}}}^{1}(Y,\omega_{\varepsilon})$ ,

	$\displaystyle-n\mathbf{E}_{\Theta_{Y},\omega_{\varepsilon}}(\psi)$	$\displaystyle=-\frac{1}{V_{\varepsilon}}\sum_{j=0}^{n-1}\int_{Y}\psi\Theta_{Y}\wedge\omega_{\varepsilon,\psi}^{j}\wedge\omega_{\varepsilon}^{n-1-j}$
		$\displaystyle\geq\frac{1}{V_{\varepsilon}}\sum_{j=0}^{n-1}\int_{Y}\psi Df^{\ast}\omega\wedge\omega_{\varepsilon,\psi}^{j}\wedge\omega_{\varepsilon}^{n-1-j}\geq(n+1)D\mathbf{E}_{\omega_{\varepsilon}}(\psi).$

Hence,

\mathbf{H}_{\mu_{Y},\omega_{\varepsilon}}(u_{k})+(\bar{s}_{\varepsilon_{k}}+(n+1)D)\mathbf{E}_{\omega_{\varepsilon_{k}}}(u_{k})\leq\mathbf{M}_{\omega_{\varepsilon_{k}}}(u_{k})<A(-\mathbf{E}_{\omega_{\varepsilon_{k}}}(u_{k}))-B_{k}.

After enlarging $D$ , one may assume that for all $\varepsilon\in[0,1]$ , $(n+1)D+A-\bar{s}_{\varepsilon}>\delta$ for a uniform $\delta>0$ . Then we have $B_{k}<((n+1)D+A-\bar{s}_{\varepsilon})(-\mathbf{E}_{\omega_{\varepsilon_{k}}}(u_{k}))$ and this implies that $d_{k}:=-\mathbf{E}_{\omega_{\varepsilon_{k}}}(u_{k})\to+\infty$ as $k\to+\infty$ . Let $g_{k}(s)$ be the $d_{1}$ -geodesic connecting $0$ and $u_{k}$ in $\mathcal{E}^{1}_{{\mathrm{norm}}}(Y,\omega_{\varepsilon_{k}})$ . Fix a constant $R>0$ and define $v_{k}:=g_{k}(R)$ . By the convexity of Mabuchi functional [PTT23, Prop. 4.7],

\mathbf{M}_{\omega_{\varepsilon_{k}}}(v_{k})\leq\frac{d_{k}-R}{d_{k}}\mathbf{M}_{\omega_{\varepsilon_{k}}}(0)+\frac{R}{d_{k}}\mathbf{M}_{\omega_{\varepsilon_{k}}}(u_{k})\leq\frac{R}{d_{k}}(Ad_{k}-B_{k})\leq AR.

From the expression of Mabuchi functional, we have $\mathbf{H}_{\mu_{Y},\omega_{\varepsilon_{k}}}(v_{k})\leq((n+1)D+A-\bar{s}_{\varepsilon_{k}})R$ .

In the argument below, although $Y$ is singular, corresponding proofs in [BJT24] proceed exactly the same. By the strong compactness [BJT24, Thm. 2.10], up to a subsequence, $v_{k}$ converges in $v_{0}\in\mathcal{E}^{1}_{{\mathrm{norm}}}(Y,f^{\ast}\omega)$ and $\mathbf{E}_{\omega_{\varepsilon_{k}}}(v_{k})\to\mathbf{E}_{f^{\ast}\omega}(v_{0})$ as $k\to+\infty$ . Note that $v$ can descend to a function in $\mathcal{E}^{1}_{{\mathrm{norm}}}(X,\omega)$ , which we still denote by $v$ . By [BJT24, Lem. 4.16], $\mathbf{H}_{\mu_{X},\omega}(v)-n\mathbf{E}_{\Theta_{X},\omega}(v)\leq\mathbf{H}_{\mu_{Y},f^{\ast}\omega}(v)-n\mathbf{E}_{\Theta_{Y},f^{\ast}\omega}(v)$ and [BJT24, Lem. 4.6 and Lem. 4.9] shows that $\mathbf{M}_{\omega_{0}}(v_{0})\leq\liminf_{k\to+\infty}\mathbf{M}_{\omega_{\varepsilon_{k}}}(v_{k})$ . All in all, we obtain

A_{0}R-B_{0}\leq\mathbf{M}_{\omega}(v_{0})\leq\liminf_{k\to+\infty}\mathbf{M}_{\omega_{\varepsilon_{k}}}(v_{k})\leq AR.

Letting $R=\frac{B_{0}}{A_{0}-A}+1$ , this yields a contradiction. ∎

3.3.1. Construct singular cscK on blowups of singular KEs

Let $(X,\omega)$ be a compact Kähler variety with log-terminal singularity. Suppose that $K_{X}$ is $m$ -Cartier for some $m\in\mathbb{N}^{\ast}$ , and pick a hermitian metric $h$ on $mK_{X}$ . Assume that either

•

$K_{X}$ is ample and $\omega=\frac{\mathrm{i}}{m}\Theta(mK_{X},h)$ ; or
•

$K_{X}$ numerically trivial and $\frac{\mathrm{i}}{m}\Theta(mK_{X},h)=0$ ; or else
•

$K_{X}$ is anti-ample, $\omega=-\frac{\mathrm{i}}{m}\Theta(mK_{X},h)$ , and $(X,K_{X})$ is K-stable.

Note that in the above cases, $\{\omega\}$ contains a unique singular Kähler–Einstein metric and the Mabuchi function with respect to $\omega$ is coercive. We further assume that $X$ admits a resolution of Fano type $\pi:\widehat{X}\to X$ .

Let $f:Y\to X$ be a blowup of $N$ distinct points in the smooth locus of $X$ . Denote the irreducible components of the exceptional divisor of $f$ by $(D_{i})_{i}$ . Since $f$ is an isomorphism near the singularities of $Y$ and $X$ , one can verify that $Y$ also admits a resolution of Fano type.

For any $a:=(a_{1},\cdots,a_{N})\in\mathbb{R}_{>0}^{n}$ , there exists a constant $\delta_{a}>0$ , such that for all $\delta\in(0,\delta_{a})$ , the class $\{f^{\ast}\omega\}-\delta\sum_{i=1}^{N}a_{i}c_{1}(\mathcal{O}(D_{i}))$ contains a Kähler metric $\omega_{Y,a,\delta}$ . By Lemma 3.6, the Mabuchi functional with respect to $f^{\ast}\omega+\varepsilon\omega_{Y,a,\delta}$ is coercive for all sufficiently small $\varepsilon$ . Corollary B then ensures the existence of a singular cscK metric in the class $(1+\varepsilon)\{f^{\ast}\omega\}-\varepsilon\delta\sum_{i=1}^{N}a_{i}c_{1}(\mathcal{O}(D_{i}))$ on $Y$ .

If $X$ is a Kähler variety with log terminal singularities, $\operatorname{Aut}(X)$ is discrete and it admits a crepant resolution $\pi:Y\to X$ , it is not difficult to check that $\pi:Y\to X$ satisfies Condition (A). In the case of Kähler–Einstein varieties with log terminal singularities and discrete automorphism groups that admit crepant resolutions, the above construction provides a method to produce numerous singular cscK metrics on their blowups at points within the smooth locus.

In dimension two, all surfaces with canonical singularities admit crepant resolutions. In dimension three, the famous result of [BKR01, Thm. 1.2] establishes that singular varieties locally modeled on $\mathbb{C}^{3}/G$ , where $G<\mathrm{SL}(3,\mathbb{C})$ is finite, also admit crepant resolutions. Three-dimensional ODP singularity also admits a crepant resolution, and it is not a quotient singularity.

We also extract the following example from [Szé24, Rmk. 34] and [BJT24, Example 4.10] that is not a crepant resolution.

Example 3.7 (Isolated cone singularities).

Let $V$ be a smooth projective variety and let $L$ be an ample line bundle on $V$ . Set $C_{a}(V,L):=\operatorname{Spec}\sum_{m\geq 0}H^{0}(V,L^{m})$ the corresponding affine cone. Assuming $K_{V}\sim_{\mathbb{Q}}r\cdot L$ for some $r\in\mathbb{Q}$ , by [Kol13, Lem. 3.1], $C_{a}(V,L)$ is klt if and only if $r<0$ . Moreover, $C_{a}(V,L)$ is canonical if and only if $r\leq-1$ . Therefore, one can choose $r\in(-1,0)$ to get a klt isolated singularity which is not canonical.

Assume that $X$ has klt isolated singularities, and each singular point is locally isomorphic to an affine cone $C_{a}(V,L)$ where $V$ is a Fano manifold. Then blowing up the singularities yields a resolution of singularities such that $\pi:Y\rightarrow X$ is projective. In this case, $-K_{Y}=-\pi^{*}K_{X}-\sum_{i}a_{i}E_{i}$ is $\pi$ -nef if and only if $X$ has canonical singularities, i.e. $a_{i}\geq 0$ . When $-K_{Y}$ is $\pi$ -nef, Condition (A) holds by choosing $\rho_{1}=0$ .

We now consider a slightly more general situation that $-K_{Y}$ is not $\pi$ -nef and explain Condition (A) in this setting. Let $\mu_{X}$ and $\mu_{Y}$ be two smooth $T$ -invariant volume forms on $X$ and $Y$ , respectively. Then define a $T$ -invariant current $\operatorname{Ric}(\mu_{Y})+dd^{c}\psi:=\pi^{\ast}\operatorname{Ric}(\mu_{X})-\sum_{i}a_{i}[E_{i}],$ where $\psi=-\sum_{i}a_{i}\log|s_{i}|^{2}_{h_{i}}$ for some smooth Hermitian metric $h_{i}$ on $\mathcal{O}_{X}(E_{i})$ . Since $E_{i}$ are disjoint in our construction, one has $\{\pi^{\ast}\omega\}-\sum_{a_{i}>0}a_{i}c_{1}(\mathcal{O}_{X}(E_{i}))>0$ . Therefore, there exists a constant $K>0$ such that $-\sum_{a_{i}>0}a_{i}[E_{i}]\geq-K\{\pi^{\ast}\omega\}-\sum_{a_{i}>0}dd^{c}a_{i}\log|s_{i}|^{2}_{h^{\prime}_{i}}$ for some smooth hermitian metrics $h^{\prime}_{i}$ on $\mathcal{O}_{X}(E_{i})$ . We then have $\operatorname{Ric}(\mu_{Y})+dd^{c}(\psi+\sum_{a_{i}>0}a_{i}\log|s_{i}|^{2}_{h^{\prime}_{i}})\geq-K\pi^{*}\omega$ . Set $\phi:=\psi+\sum_{a_{i}>0}a_{i}\log|s_{i}|^{2}_{h_{i}^{\prime}}=u-\sum_{a_{j}<0}a_{j}\log|s_{j}|^{2}_{h_{j}}$ for some $u\in\mathcal{C}^{\infty}(Y)$ . One can obtain $\int_{X}e^{-\phi}\mu_{Y}^{n}<+\infty$ as $a_{j}>-1$ . Hence, Condition (A) holds by taking $\rho_{1}=\phi/K$ .

3.3.2. Mixing construction with smoothing

Let us stress that this construction also works on the $\mathbb{Q}$ -Gorenstein smoothable setting. Consider a $\mathbb{Q}$ -Gorenstein smoothing $f:(\mathcal{X},\omega_{\mathcal{X}})\to\mathbb{D}$ of $(X_{0},{\omega_{\mathcal{X}}}_{|X_{0}})$ where $X_{0}$ is a Kähler–Einstein variety with log terminal singularities and $\operatorname{Aut}(X)$ is discrete, and $\{{\omega_{\mathcal{X}}}_{|X_{0}}\}$ is a Kähler–Einstein class. Denote by $\mathcal{Z}$ the singular set of $\mathcal{Z}$ . Take a finite set of points $\{p_{1},\cdots,p_{N}\}$ in $X_{0}^{\mathrm{reg}}$ . There exists smooth curves $C_{1},\cdots,C_{N}$ in $\mathcal{X}$ such that these curves are disjoint, each $C_{i}$ intersects $X_{0}$ transversely at the single point $p_{i}$ , $C_{i}\subset\mathcal{X}\setminus\mathcal{Z}$ , and the restriction $f_{|C_{i}}:C_{i}\to\mathbb{D}$ is an isomorphism. Now consider the blowup map $\mu:\mathcal{Y}\to\mathcal{X}$ along all the curves $(C_{i})_{i}$ and let $Y_{0}=\operatorname{B\ell}_{\{p_{1},\cdots,p_{N}\}}X$ . Then $\pi=f\circ\mu:\mathcal{Y}\to\mathbb{D}$ is a $\mathbb{Q}$ -Gorenstein smoothing of $(Y_{0},{\omega_{\mathcal{Y},\varepsilon}}_{|Y_{0}})$ where $\omega_{\mathcal{Y},\varepsilon}=\mu^{\ast}\omega_{\mathcal{X}}+\varepsilon\omega_{\mathcal{Y}}$ . Here $\omega_{\mathcal{Y}}$ is a hermitian metric on $\mathcal{Y}$ and relatively Kähler that defined by

\omega_{\mathcal{Y}}=\mu^{\ast}\omega_{\mathcal{X}}-\sum_{i=1}^{N}a_{i}\Theta_{h_{i}}(\mathcal{O}(E_{i}))

where $E_{i}$ is the exceptional divisor over $C_{i}$ , $a_{i}>0$ and $h_{i}$ is some hermitian metric on $\mathcal{O}(E_{i})$ such that $\omega$ . Since the Mabuchi functional $\mathbf{M}_{{\omega_{\mathcal{X}}}_{|X_{0}}}$ on $X_{0}$ is coercive, it follows from Lemma 3.6 $\mathbf{M}_{{\omega_{\mathcal{Y},\varepsilon}}_{|Y_{0}}}$ on $Y_{0}$ as well. By [PTT23, Thm. C], this implies the existence of a cscK metric in the class $\{{\omega_{\mathcal{Y},\varepsilon}}|Y_{0}\}$ .

For examples of smoothable Calabi–Yau varieties, we refer the reader to [DG18, Sec. 8]. For the smoothable Fano case, [LX19, Liu22] prove that mildly singular cubic varieties in dimensions three and four are K-stable. Additionally, explicit examples of K-stable singular cubic threefolds can be found, for instance, in [CTZ25, Sec. 3, 4] and [CMTZ24, Sec. 5].

Appendix A Weighted Aubin–Yau inequality

In this section, for the reader’s convenience, we provide detailed proof of a Laplacian inequality (see also [DJL24, Lem. 5.6]), which generalizes [Siu87, p. 98–99] to the weighted setting.

Lemma A.1.

Let $\omega$ and $\omega_{X}$ be two $T$ -invariant Kähler metrics. Assume that $\varphi\in\mathcal{H}^{T}_{\omega}$ satisfies

v(m_{\varphi})(\omega+dd^{c}\varphi)^{n}=e^{F}\omega_{X}^{n}.

Then there exist positive constants $\mathcal{B}_{0},\mathcal{C}_{0}$ such that

\displaystyle\Delta_{\omega_{\varphi},v}\log\operatorname{tr}_{\omega_{X}}\omega_{\varphi}

\displaystyle\geq\frac{\Delta_{\omega_{X}}F}{\operatorname{tr}_{\omega_{X}}\omega_{\varphi}}-\mathcal{B}_{0}\operatorname{tr}_{\omega_{\varphi}}\omega_{X}-\mathcal{C}_{0}-\frac{1}{\operatorname{tr}_{\omega_{X}}\omega_{\varphi}}\sum_{\alpha,\beta}(\log v)_{\alpha\beta}(m_{\varphi})\langle dm^{\xi_{\alpha}}_{\varphi},dm^{\xi_{\beta}}_{\varphi}\rangle_{\omega_{X}}.

In particular, if $\mathfrak{t}^{\vee}\ni p\mapsto\log v(p)\in\mathbb{R}$ is concave, then

\Delta_{\omega_{\varphi},v}\log\operatorname{tr}_{\omega_{X}}\omega_{\varphi}\geq\frac{\Delta_{\omega_{X}}F}{\operatorname{tr}_{\omega_{X}}\omega_{\varphi}}-\mathcal{B}_{0}\operatorname{tr}_{\omega_{\varphi}}\omega_{X}-\mathcal{C}_{0}.

We note that the constants $\mathcal{B}_{0}=B+2C_{v}^{2}C_{\operatorname{Ric}}$ and $\mathcal{C}_{0}=C_{v}^{2}(C_{2}+C_{v}^{2}C_{\xi})$ where

•

$\operatorname{Bisec}(\omega_{X})\geq-B$ is a negative lower bound for the bisectional curvature of $\omega_{X}$ ;

•

$C_{v}>0$ is a constant such that

\begin{split}C_{v}^{-1}&\leq v+\sum_{\alpha}|v_{\alpha}|+\sum_{\alpha,\beta}|v_{\alpha\beta}|\leq C_{v}\quad{\text{on $P=\operatorname{im}(m_{\omega})$}},\\ C_{v}^{-1}&\leq|v|+\sum_{\alpha}|v_{\alpha}|+\sum_{\alpha,\beta}|v_{\alpha\beta}|\leq C_{v}\quad{\text{on $P_{\omega_{X}}=\operatorname{im}(m_{\omega_{X}})$}};\end{split}

(A.1)

•

$C_{2}$ depending only on $n$ and $C_{1}$ where $C_{1}>0$ is a constant so that for any $\alpha$ ,

$-C_{1}\omega_{X}\leq\mathcal{L}_{J\xi_{\alpha}}\omega_{X}\leq C_{1}\omega_{X};$ (A.2)
•

$C_{\operatorname{Ric}}>0$ and $C_{\xi}>0$ are constants so that $\|m_{\operatorname{Ric}(\omega_{X})}^{\xi_{\alpha}}(X)\|\leq C_{\operatorname{Ric}}$ and $|\xi_{\alpha}|_{\omega_{X}}^{2}\leq C_{\xi}$ , for all $\alpha$ , respectively, where $\|\cdot\|$ is a norm on the dual Lie algebra of $T$ .

Remark A.2.

The concavity condition on $\log v(p)$ holds in many interested cases. Here we extract some examples from [Lah19, Sec. 3]:

•

cscK and extremal metrics: $v(p)=1$ ,
•

Kähler–Ricci solitons: $v(p)=e^{\langle\xi,p\rangle}$ for some fixed $\xi\in\mathfrak{t}$ ,
•

Kähler metrics given by the generalized Calabi construction: $v(p)=\Pi_{j}(\langle\xi_{j},p\rangle+a_{j})^{d_{j}}$ with $d_{j}>0$ .

Proof.

Before entering the proof, we recall a basic equality of Lie derivative about the quotient of volume forms. Suppose that $\alpha$ is an $(n,n)$ -form and $\beta$ is a volume form on $X$ . For all vector fields $V$ , one can derive the following

\mathcal{L}_{V}\left(\frac{\alpha}{\beta}\right)=\frac{\mathcal{L}_{V}\alpha}{\beta}-\frac{\alpha}{\beta}\cdot\frac{\mathcal{L}_{V}\beta}{\beta}.

As a consequence, we have

\mathcal{L}_{V}\left(\frac{\omega_{\varphi}^{n}}{\omega_{X}^{n}}\right)=\frac{n(\mathcal{L}_{V}\omega_{\varphi})\wedge\omega_{\varphi}^{n-1}}{\omega_{\varphi}^{n}}-\frac{n(\mathcal{L}_{V}\omega_{X})\wedge\omega_{X}^{n-1}}{\omega_{X}^{n}}

(A.3)

and

\mathcal{L}_{V}\left(\operatorname{tr}_{\omega_{X}}\omega_{\varphi}\right)=\frac{n(\mathcal{L}_{V}\omega_{\varphi})\wedge\omega_{X}^{n-1}+n(n-1)(\mathcal{L}_{V}\omega_{X})\wedge\omega_{\varphi}\wedge\omega_{X}^{n-2}}{\omega_{X}^{n}}-(\operatorname{tr}_{\omega_{X}}\omega_{\varphi})\frac{n(\mathcal{L}_{V}\omega_{X})\wedge\omega_{X}^{n-1}}{\omega_{X}^{n}}

(A.4)

By the standard Aubin–Yau’s inequality, we have

\Delta_{\omega_{\varphi}}\log\operatorname{tr}_{\omega_{X}}\omega_{\varphi}\geq\frac{\Delta_{\omega_{X}}(F-\log v(m_{\varphi}))}{\operatorname{tr}_{\omega_{X}}\omega_{\varphi}}-B\operatorname{tr}_{\omega_{\varphi}}\omega_{X},

and thus,

\begin{split}&\Delta_{\omega_{\varphi},v}\log\operatorname{tr}_{\omega_{X}}\omega_{\varphi}\\ &\geq\frac{\Delta_{\omega_{X}}(F-\log v(m_{\varphi}))}{\operatorname{tr}_{\omega_{X}}\omega_{\varphi}}-B\operatorname{tr}_{\omega_{\varphi}}\omega_{X}+\operatorname{tr}_{\omega_{\varphi}}[d\log v(m_{\varphi})\wedge d^{c}\log\operatorname{tr}_{\omega_{X}}\omega_{\varphi}]\\ &=\frac{\Delta_{\omega_{X}}F}{\operatorname{tr}_{\omega_{X}}\omega_{\varphi}}-B\operatorname{tr}_{\omega_{\varphi}}\omega_{X}+\frac{1}{\operatorname{tr}_{\omega_{X}}\omega_{\varphi}}\bigg{\{}\operatorname{tr}_{\omega_{\varphi}}[d\log v(m_{\varphi})\wedge d^{c}\operatorname{tr}_{\omega_{X}}\omega_{\varphi}]-\Delta_{\omega_{X}}\log v(m_{\varphi})\bigg{\}}.\end{split}

(A.5)

It suffices to establish a suitable lower bound for the terms in $\{\cdots\}$ in (A.5). Note that $P$ does not depend on $\varphi$ and $d^{c}=J^{-1}\circ d\circ J$ . By Cartan’s formula, $i_{V}d^{c}f=-i_{JV}df=-\mathcal{L}_{JV}f$ . Using (A.4), (A.2) and (A.1), we obtain the following:

\begin{split}&\operatorname{tr}_{\omega_{\varphi}}(d\log v(m_{\varphi})\wedge d^{c}\operatorname{tr}_{\omega_{X}}\omega_{\varphi})=\sum_{\alpha}(\log v)_{\alpha}(m_{\varphi})i_{\xi_{\alpha}}d^{c}\operatorname{tr}_{\omega_{X}}\omega_{\varphi}=-\sum_{\alpha}(\log v)_{\alpha}(m_{\varphi})(\mathcal{L}_{J\xi_{\alpha}}\operatorname{tr}_{\omega_{X}}\omega_{\varphi})\\ &=-\sum_{\alpha}(\log v)_{\alpha}(m_{\varphi})\left(\frac{n(\mathcal{L}_{J\xi_{\alpha}}\omega_{\varphi})\wedge\omega_{X}^{n-1}+n(n-1)(\mathcal{L}_{J\xi_{\alpha}}\omega_{X})\wedge\omega_{\varphi}\wedge\omega_{X}^{n-2}}{\omega_{X}^{n}}-\operatorname{tr}_{\omega_{X}}\omega_{\varphi}\cdot\frac{n(\mathcal{L}_{J\xi_{\alpha}}\omega_{X})\wedge\omega_{X}^{n-1}}{\omega_{X}^{n}}\right)\\ &\geq-\sum_{\alpha}(\log v)_{\alpha}(m_{\varphi})\left(\frac{n(\mathcal{L}_{J\xi_{\alpha}}\omega_{\varphi})\wedge\omega_{X}^{n-1}}{\omega_{X}^{n}}\right)-C_{2}C_{v}^{2}\operatorname{tr}_{\omega_{X}}\omega_{\varphi}\\ &=\sum_{\alpha}(\log v)_{\alpha}(m_{\varphi})\Delta_{\omega_{X}}m_{\varphi}^{\xi_{\alpha}}-C_{2}C_{v}^{2}\operatorname{tr}_{\omega_{X}}\omega_{\varphi}\\ &=\frac{1}{v(m_{\varphi})}\left(\Delta_{\omega_{X}}v(m_{\varphi})-\sum_{\alpha,\beta}v_{\alpha\beta}(m_{\varphi})\langle dm_{\varphi}^{\xi_{\alpha}},dm_{\varphi}^{\xi_{\beta}}\rangle_{\omega_{X}}\right)-C_{2}C_{v}^{2}\operatorname{tr}_{\omega_{X}}\omega_{\varphi}\\ &=\Delta_{\omega_{X}}\log v(m_{\varphi})-\sum_{\alpha,\beta}(\log v)_{\alpha\beta}(m_{\varphi})\langle dm_{\varphi}^{\xi_{\alpha}},dm_{\varphi}^{\xi_{\beta}}\rangle_{\omega_{X}}-C_{2}C_{v}^{2}\operatorname{tr}_{\omega_{X}}\omega_{\varphi}\end{split}

(A.6)

where $C_{2}>0$ is a constant depending only on $C_{1}$ and $n$ . To conclude, combining (A.5) and (A.6), we finally obtain

\displaystyle\Delta_{\omega_{\varphi},v}\log\operatorname{tr}_{\omega_{X}}\omega_{\varphi}\geq\frac{\Delta_{\omega_{X}}F}{\operatorname{tr}_{\omega_{X}}\omega_{\varphi}}-B\operatorname{tr}_{\omega_{\varphi}}\omega_{X}-C_{2}C_{v}^{2}-\frac{1}{\operatorname{tr}_{\omega_{X}}\omega_{\varphi}}\sum_{\alpha,\beta}(\log v)_{\alpha\beta}(m_{\varphi})\langle dm^{\xi_{\alpha}}_{\varphi},dm^{\xi_{\beta}}_{\varphi}\rangle_{\omega_{X}}

as required. ∎

Appendix B Weighted local Chen–Cheng’s $\mathcal{C}^{2}$ -estimate

This section aims to give details for the $\mathcal{C}^{2}$ -estimate in Theorem 2.3. The proof follows a similar argument in [CC21a, Prop. 4.1] with further analysis of the weighted terms. Recall that we have two local equations on $B_{1}(0)\subset\mathbb{C}^{n}$ ,

v(m_{dd^{c}\phi})\det(\phi_{i\bar{j}})=e^{G}\quad\text{and}\quad\Delta_{\phi,v}G=-S=-w(m_{dd^{c}\phi}).

Here we denote $\Delta f:=\sum_{k=1}^{n}\partial_{k}\partial_{\bar{k}}f$ and $\Delta_{\phi,v}f:=\sum_{i,j=1}^{n}\phi^{i\bar{j}}\partial_{i}\partial_{\bar{j}}f+\frac{1}{2}\langle d\log v(m_{dd^{c}\phi}),df\rangle_{dd^{c}\phi}$ which are different from the scaling we used before. We first prove the following lemma:

Lemma B.1.

There exists positive constants $K,C$ depending on $v,w,\|G\|_{L^{\infty}(B_{1}(0))}$ and $\|\phi\|_{L^{\infty}(B_{1}(0))}$ such that

\Delta_{\phi,v}u\geq-C\cdot(\Delta\phi)\cdot u\quad\text{on $B_{1}(0)$}

where $u=e^{\frac{G}{2}}|dG|_{\phi}^{2}+K\Delta\phi$ .

Proof.

We first remark that by arithmetic and geometric means inequality, one has

\Delta\phi\geq n\det(\phi_{i\bar{j}})^{1/n}=ne^{G/n}v(m_{dd^{c}\phi})^{-1/n}\geq c\quad\text{and}\quad\operatorname{tr}_{\phi}\omega\geq n\det(\phi_{i\bar{j}})^{-1/n}=ne^{-G/n}v(m_{dd^{c}\phi})^{1/n}\geq c.

where $\omega$ is the Euclidean metric and $c>0$ is a constant depending only on $n$ , $\|G\|_{L^{\infty}(B_{1}(0))}$ and $C_{v}$ . We next review the following estimate by Chen and Cheng [CC21a, p. 16, (4.3)]. Under the normal coordinates with respect to $\phi_{i\bar{j}}$ , we may assume that $\omega_{i\bar{j}}(x_{0})=\delta_{ij}$ , $\phi_{i\bar{j}}(x_{0})=\lambda_{i}\delta_{ij}$ and $\partial_{k}\phi_{i\bar{j}}(x_{0})=0$ at a fixed point $x_{0}$ , then, at $x_{0}$ , we have

\begin{split}e^{-\frac{G}{2}}\Delta_{\phi}(e^{\frac{G}{2}}|dG|_{\phi}^{2})&=\Delta_{\phi}|\nabla^{\phi}G|_{\phi}^{2}+\frac{1}{2}\phi^{i\bar{i}}(\partial_{i}G\cdot\partial_{\bar{i}}|\nabla^{\phi}G|_{\phi}^{2}+\partial_{\bar{i}}G\cdot\partial_{i}|\nabla^{\phi}G|_{\phi}^{2})+\frac{1}{4}|\nabla^{\phi}G|_{\phi}^{4}+\frac{1}{2}\Delta_{\phi}G\cdot|\nabla^{\phi}G|_{\phi}^{2}\\ &=\phi^{i\bar{i}}(\partial_{i}\Delta_{\phi}G\cdot\partial_{\bar{i}}G+\partial_{\bar{i}}\Delta_{\phi}G\cdot\partial_{i}G)+\frac{1}{2}\operatorname{Ric}_{\phi}(\nabla^{\phi}G,\nabla^{\phi}G)+|\nabla^{\phi}\nabla^{\phi}G|_{\phi}^{2}+|\nabla^{\phi}\overline{\nabla^{\phi}}G|_{\phi}^{2}\\ &\quad+\frac{1}{2}\phi^{i\bar{i}}\phi^{j\bar{j}}(G_{i}G_{j}G_{\bar{i}\bar{j}}+G_{i}G_{\bar{j}}G_{j\bar{i}}+G_{\bar{i}}G_{\bar{j}}G_{ij}+G_{\bar{i}}G_{j}G_{i\bar{j}})+\frac{1}{4}|\nabla^{\phi}G|_{\phi}^{4}+\frac{1}{2}\Delta_{\phi}G\cdot|\nabla^{\phi}G|_{\phi}^{2}\\ &\geq\phi^{i\bar{i}}(\partial_{i}\Delta_{\phi}G\cdot\partial_{\bar{i}}G+\partial_{\bar{i}}\Delta_{\phi}G\cdot\partial_{i}G)+\frac{1}{2}\operatorname{Ric}_{\phi}(\nabla^{\phi}G,\nabla^{\phi}G)+|\nabla^{\phi}\overline{\nabla^{\phi}}G|_{\phi}^{2}\\ &\quad+\frac{1}{2}\phi^{i\bar{i}}\phi^{j\bar{j}}(G_{i}G_{\bar{j}}G_{j\bar{i}}+G_{\bar{i}}G_{j}G_{i\bar{j}})+\frac{1}{2}\Delta_{\phi}G\cdot|\nabla^{\phi}G|_{\phi}^{2}\end{split}

(B.1)

where $\operatorname{Ric}_{\phi}:=\operatorname{Ric}(dd^{c}\phi)=-dd^{c}\log\det(\phi_{i\bar{j}})$ . The last inequality comes from the fact that

\frac{1}{4}|\nabla^{\phi}G|_{\phi}^{4}+\frac{1}{2}\phi^{i\bar{i}}\phi^{j\bar{j}}(G_{i}G_{j}G_{\bar{i}\bar{j}}+G_{\bar{i}}G_{\bar{j}}G_{ij})+|\nabla^{\phi}\nabla^{\phi}G|_{\phi}^{2}=\left|G_{ij}+\frac{1}{2}\sqrt{\lambda_{i}^{-1}\lambda_{j}^{-1}}G_{i}G_{j}\right|_{\phi}^{2}\geq 0.

From the first equation, we have $\operatorname{Ric}(dd^{c}\phi)=-dd^{c}(G-\log v(m_{dd^{c}\phi}))$ . Rewrite (B.1) as follows

	$\displaystyle e^{-\frac{G}{2}}\Delta_{\phi}(e^{\frac{G}{2}}\|dG\|_{\phi}^{2})$	$\displaystyle\geq\phi^{i\bar{i}}(\partial_{i}\Delta_{\phi}G\cdot\partial_{\bar{i}}G+\partial_{\bar{i}}\Delta_{\phi}G\cdot\partial_{i}G)-\phi^{i\bar{i}}\phi^{j\bar{j}}G_{i\bar{j}}G_{i}G_{\bar{j}}$
		$\displaystyle\quad+\phi^{i\bar{i}}\phi^{j\bar{j}}(\log v(m_{dd^{c}\phi}))_{i\bar{j}}G_{i}G_{\bar{j}}+\phi^{i\bar{i}}\phi^{j\bar{j}}G_{i\bar{j}}G_{i}G_{\bar{j}}+\|\nabla^{\phi}\overline{\nabla^{\phi}}G\|_{\phi}^{2}+\frac{1}{2}\Delta_{\phi}G\cdot\|dG\|_{\phi}^{2}$
		$\displaystyle=2\operatorname{tr}_{\phi}(d\Delta_{\phi}G\wedge d^{c}G)+\phi^{i\bar{i}}\phi^{j\bar{j}}(\log v(m_{dd^{c}\phi}))_{i\bar{j}}G_{i}G_{\bar{j}}+\|\nabla^{\phi}\overline{\nabla^{\phi}}G\|_{\phi}^{2}+\frac{1}{2}\Delta_{\phi}G\cdot\|dG\|_{\phi}^{2}.$

Considering the weighted version of the above inequality, one can infer

\begin{split}e^{-\frac{G}{2}}\Delta_{\phi,v}(e^{\frac{G}{2}}|dG|_{\phi}^{2})&\geq 2\operatorname{tr}_{\phi}(d\Delta_{\phi}G\wedge d^{c}G)+\phi^{i\bar{i}}\phi^{j\bar{j}}(\log v(m_{dd^{c}\phi}))_{i\bar{j}}G_{i}G_{\bar{j}}+|\nabla^{\phi}\overline{\nabla^{\phi}}G|_{\phi}^{2}+\frac{1}{2}\Delta_{\phi}G\cdot|dG|_{\phi}^{2}\\ &\quad+\frac{1}{2}\operatorname{tr}_{\phi}(d\log v(m_{dd^{c}\phi})\wedge d^{c}G)|dG|_{\phi}^{2}+\operatorname{tr}_{\phi}(d\log v(m_{dd^{c}\phi})\wedge d^{c}|dG|_{\phi}^{2})\\ &=2\operatorname{tr}_{\phi}(d\Delta_{\phi}G\wedge d^{c}G)+\phi^{i\bar{i}}\phi^{j\bar{j}}(\log v(m_{dd^{c}\phi}))_{i\bar{j}}G_{i}G_{\bar{j}}+|\nabla^{\phi}\overline{\nabla^{\phi}}G|_{\phi}^{2}+\frac{1}{2}\Delta_{\phi,v}G\cdot|dG|_{\phi}^{2}\\ &\quad+\operatorname{tr}_{\phi}(d\log v(m_{dd^{c}\phi})\wedge d^{c}|dG|_{\phi}^{2})\end{split}

(B.2)

Under normal coordinates with respect to $\phi_{i\bar{j}}$ at $x_{0}$ , we obtain

	$\displaystyle\operatorname{tr}_{\phi}(d\log v(m_{dd^{c}\phi})\wedge d^{c}\|dG\|_{\phi}^{2})=\frac{1}{2}\phi^{i\bar{i}}\left((\log v(m_{dd^{c}\phi}))_{i}[\phi^{j\bar{j}}G_{j}G_{\bar{j}}]_{\bar{i}}+(\log v(m_{dd^{c}\phi}))_{\bar{i}}[\phi^{j\bar{j}}G_{j}G_{\bar{j}}]_{i}\right)$
	$\displaystyle=\frac{1}{2}\phi^{j\bar{j}}\Big{(}G_{\bar{j}}\phi^{i\bar{i}}(\log v(m_{dd^{c}\phi}))_{i}G_{j\bar{i}}+G_{j}\phi^{i\bar{i}}(\log v(m_{dd^{c}\phi}))_{i}G_{\bar{j}\bar{i}}+G_{\bar{j}}\phi^{i\bar{i}}(\log v(m_{dd^{c}\phi}))_{\bar{i}}G_{ji}+G_{j}\phi^{i\bar{i}}(\log v(m_{dd^{c}\phi}))_{\bar{i}}G_{\bar{j}i}\Big{)}$
	$\displaystyle=2\operatorname{tr}_{\phi}(dG\wedge d^{c}[\operatorname{tr}_{\phi}(d\log v(m_{dd^{c}\phi})\wedge d^{c}G)])-\Re\left(\phi^{i\bar{i}}\phi^{j\bar{j}}(\log v(m_{dd^{c}\phi}))_{ij}G_{\bar{i}}G_{\bar{j}}\right)-\phi^{i\bar{i}}\phi^{j\bar{j}}(\log v(m_{dd^{c}\phi}))_{i\bar{j}}G_{\bar{i}}G_{j}$

and the inequality (B.2) can be expressed as

\begin{split}e^{-\frac{G}{2}}\Delta_{\phi,v}(e^{\frac{G}{2}}|dG|_{\phi}^{2})&\geq 2\operatorname{tr}_{\phi}(d\Delta_{\phi,v}G\wedge d^{c}G)+|\nabla^{\phi}\overline{\nabla^{\phi}}G|_{\phi}^{2}+\frac{1}{2}\Delta_{\phi,v}G\cdot|\nabla^{\phi}G|_{\phi}^{2}\\ &\quad-\Re\left(\phi^{i\bar{i}}\phi^{j\bar{j}}(\log v(m_{dd^{c}\phi}))_{ij}G_{\bar{i}}G_{\bar{j}}\right).\end{split}

(B.3)

Recall that

dm^{\xi_{\alpha}}_{dd^{c}\phi}=-i_{\xi_{\alpha}}dd^{c}\phi=-i_{\xi_{\alpha}}(2\sqrt{-1}\phi_{p\bar{q}}dz^{p}\wedge d\bar{z}^{q})=-2\sqrt{-1}\phi_{p\bar{q}}(\xi_{\alpha}^{\prime})^{p}d\bar{z}^{q}+2\sqrt{-1}\phi_{p\bar{q}}(\xi_{\alpha}^{\prime\prime})^{\bar{q}}dz^{p}.

Hence,

		$\displaystyle\partial_{ij}^{2}\log v(m_{dd^{c}\phi})=\partial_{i}\left\{\sum_{\alpha}(\partial_{\alpha}\log v)(m_{dd^{c}\phi})\cdot 2\sqrt{-1}\phi_{j\bar{\ell}}(\xi^{\prime\prime}_{\alpha})^{\bar{\ell}}\right\}$		(B.4)
		$\displaystyle=4\sum_{\alpha,\beta}(\partial_{\alpha\beta}^{2}\log v)(m_{dd^{c}\phi})\cdot[-\phi_{i\bar{q}}\phi_{j\bar{\ell}}(\xi^{\prime\prime}_{\beta})^{\bar{q}}(\xi^{\prime\prime}_{\alpha})^{\bar{\ell}}]+2\sum_{\alpha}(\partial_{\alpha}\log v)(m_{dd^{c}\phi})\cdot\sqrt{-1}[\phi_{j\bar{\ell}i}(\xi^{\prime\prime}_{\alpha})^{\bar{\ell}}+\phi_{j\bar{\ell}}(\xi^{\prime\prime}_{\alpha})^{\bar{\ell}}_{i}].$

Note that $\xi^{\prime}_{\alpha}$ is a holomorphic vector field for any $\alpha$ . Therefore, $(\xi^{\prime}_{\alpha})^{j}_{\bar{i}}=0=(\xi^{\prime\prime}_{\alpha})^{\bar{j}}_{i}$ for any $i,j\in\{1,\cdots,n\}$ .

Combining (B.3), (B.4), and the fact that $(\xi^{\prime\prime}_{\alpha})^{\bar{\ell}}_{i}=0$ , under normal coordinates with respect to $\phi$ at $x_{0}$ , we get

	$\displaystyle e^{-\frac{G}{2}}\Delta_{\phi,v}(e^{\frac{G}{2}}\|dG\|_{\phi}^{2})$	$\displaystyle\geq 2\operatorname{tr}_{\phi}(d\Delta_{\phi,v}G\wedge d^{c}G)+\|\nabla^{\phi}\overline{\nabla^{\phi}}G\|_{\phi}^{2}+\frac{1}{2}\Delta_{\phi,v}G\cdot\|\nabla^{\phi}G\|_{\phi}^{2}$
		$\displaystyle\quad+4\Re\sum_{\alpha,\beta}(\partial_{\alpha\beta}^{2}\log v)(m_{dd^{c}\phi})(\xi^{\prime\prime}_{\beta})^{\bar{i}}(\xi^{\prime\prime}_{\alpha})^{\bar{j}}G_{\bar{i}}G_{\bar{j}}$
		$\displaystyle\geq 2\operatorname{tr}_{\phi}(d\Delta_{\phi,v}G\wedge d^{c}G)+\|\nabla^{\phi}\overline{\nabla^{\phi}}G\|_{\phi}^{2}+\frac{1}{2}\Delta_{\phi,v}G\cdot\|\nabla^{\phi}G\|_{\phi}^{2}-4C_{v}^{4}C_{\xi}\|dG\|_{\omega}^{2}$

where $C_{\xi}>0$ is a constant such that $|\xi|^{2}_{\omega}\leq C_{\xi}$ . Using the second equation $\Delta_{\phi,v}G=-w(m_{dd^{c}\phi})$ , we then derive

	$\displaystyle\operatorname{tr}_{\phi}(d\Delta_{\phi,v}G\wedge d^{c}G)$	$\displaystyle=-\operatorname{tr}_{\phi}(dw(m_{dd^{c}\phi})\wedge d^{c}G)=-\sum_{\alpha}w_{\alpha}(m_{dd^{c}\phi})\operatorname{tr}_{\phi}(dm_{dd^{c}\phi}^{\xi_{\alpha}}\wedge d^{c}G)$
		$\displaystyle=\sum_{\alpha}w_{\alpha}(m_{dd^{c}\phi})\Re(\xi^{\prime}_{\alpha})^{i}G_{i}\geq-C_{\xi}C_{w}\|dG\|_{\omega}\geq-C_{1}-(\Delta\phi)\cdot\|dG\|_{\phi}^{2}$

where $C_{1}>0$ depending only on $C_{\xi},C_{v}$ . Then

e^{-\frac{G}{2}}\Delta_{\phi,v}(e^{\frac{G}{2}}|dG|_{\phi}^{2})\geq-2C_{1}-C_{2}(\Delta\phi)|dG|_{\phi}^{2}+|\nabla^{\phi}\overline{\nabla^{\phi}}G|_{\phi}^{2}

(B.5)

where $C_{2}>0$ depend only on $C_{\xi},C_{v},C_{w}$ .

Next, we compute, under normal coordinates with respect to $\phi_{i\bar{j}}$ ,

	$\displaystyle\Delta_{\phi,v}(\Delta\phi)$	$\displaystyle=\Delta_{\phi}(\Delta\phi)+\operatorname{tr}_{\phi}(d\log v(m_{dd^{c}\phi})\wedge d^{c}\Delta\phi)$
		$\displaystyle=\operatorname{tr}_{\omega}(\sqrt{-1}\partial\bar{\partial}\log\det(\phi_{i\bar{j}}))+\operatorname{tr}_{\phi}(d\log v(m_{dd^{c}\phi})\wedge d^{c}\Delta\phi)$
		$\displaystyle=\Delta G-\Delta\log v(m_{dd^{c}\phi})+\operatorname{tr}_{\phi}(d\log v(m_{dd^{c}\phi})\wedge d^{c}\Delta\phi)$
		$\displaystyle\geq\Delta G-\sum_{\alpha,\beta}(\partial_{\alpha\beta}^{2}\log v)(m_{dd^{c}\phi})\langle dm_{dd^{c}\phi}^{\xi_{\alpha}},dm_{dd^{c}\phi}^{\xi_{\beta}}\rangle-C_{3}\Delta\phi$
		$\displaystyle\geq\Delta G-2C_{v}^{2}(\Delta\phi)^{2}-C_{3}\Delta\phi,$

where $C_{3}$ depend only on $n$ and a constant $C>0$ such that $-C\omega\leq\mathcal{L}_{\xi_{\alpha}}\omega\leq C\omega$ . The fourth line follows from the same estimate (A.6).

Then we obtain

\displaystyle\Delta_{\phi,v}(e^{\frac{G}{2}}|dG|_{\phi}^{2}+K\Delta\phi)

\displaystyle\geq-C_{1}e^{\frac{G}{2}}-C_{2}(\Delta\phi)e^{\frac{G}{2}}|dG|_{\phi}^{2}+e^{\frac{G}{2}}|\nabla^{\phi}\overline{\nabla^{\phi}}G|_{\phi}^{2}+K\Delta G-2C_{v}^{2}(\Delta\phi)^{2}-C_{3}\Delta\phi.

Recall that $|\nabla^{\phi}\overline{\nabla^{\phi}}G|_{\phi}^{2}=\sum_{i,j}\phi^{i\bar{i}}\phi^{j\bar{j}}|G_{i\bar{j}}|^{2}$ and $\|G\|_{L^{\infty}}\leq C_{4}$ . By Cauchy–Schwarz inequality, we have the following estimate

\displaystyle e^{\frac{G}{2}}|\nabla^{\phi}\overline{\nabla^{\phi}}G|_{\phi}^{2}+e^{C_{4}}\frac{K^{2}}{4}(\Delta\phi)^{2}\geq e^{-C_{4}}\sum_{i,j}\phi^{i\bar{i}}\phi^{j\bar{j}}|G_{i\bar{j}}|^{2}+e^{C_{4}}\frac{K^{2}}{4}\sum_{i,j}\phi_{i\bar{i}}\phi_{j\bar{j}}\geq\sum_{i,j}K|G_{i\bar{j}}|\geq K\sum_{i}K|G_{i\bar{i}}|\geq K|\Delta G|.

Moreover, since $\Delta\phi\geq c$ ; hence, we derive the following estimate

\Delta_{\phi,v}u=\Delta_{\phi,v}(e^{\frac{G}{2}}|dG|_{\phi}^{2}+K\Delta\phi)\geq-C_{5}\cdot K\cdot(\Delta\phi)\cdot u,

for a constant $C_{5}>0$ depending only on $n$ , $C_{v}$ , $C_{w}$ and some sufficient large $K>0$ depending on $C_{v},C_{1},C_{3},C_{4},c$ and $\|G\|_{L^{\infty}(B_{1}(0))}$ . ∎

Using Lemma B.1, one can obtain Theorem 2.3 following an argument in Chen–Cheng’s article [CC21a]:

Proof of Theorem 2.3.

Let $\eta$ be a positive smooth function with compact support in $B_{1}(0)$ , $\eta\equiv 1$ on $B_{3/4}(0)$ and $0\leq\eta\leq 1$ . We first claim that there is a positive constant

D_{1}=D_{1}(n,\|G\|_{L^{\infty}(B_{1}(0))},v,w,\|\Delta\phi\|_{L^{n+1}(B_{1}(0))},C_{\xi},C_{\eta})

such that $\||dG|_{\phi}^{2}\|_{L^{1}(B_{3/4}(0))}\leq D_{1}$ , where $C_{\eta}>0$ is a constant such that $|d\eta|_{\omega},|dd^{c}\eta|_{\omega}\leq C_{\eta}$ and $C_{\xi}>0$ is a constant such that $|\xi_{\alpha}|_{\omega}\leq C_{\xi}$ .

To see the claim, set $C_{G}>0$ a constant such that $|G|\leq C_{G}$ and $e^{-G}\leq C_{G}$ on $B_{1}(0)$ . Then we have

	$\displaystyle\int_{B_{3/4}(0)}\|dG\|_{\phi}^{2}\omega^{n}\leq\int_{B_{1}(0)}\eta\|dG\|_{\phi}^{2}e^{-G}v(m_{dd^{c}\phi})(dd^{c}\phi)^{n}\leq C_{G}C_{v}\int_{B_{1}(0)}\eta dG\wedge d^{c}G\wedge(dd^{c}\phi)^{n-1}$
	$\displaystyle=C_{G}C_{v}\int_{B_{1}(0)}-\eta Gdd^{c}G\wedge(dd^{c}\phi)^{n-1}-Gd\eta\wedge d^{c}G\wedge(dd^{c}\phi)^{n-1}$
	$\displaystyle=C_{G}C_{v}\int_{B_{1}(0)}\eta Gn[w(m_{dd^{c}\phi})(dd^{c}\phi)^{n}+d\log v(m_{dd^{c}\phi})\wedge d^{c}G\wedge(dd^{c}\phi)^{n-1}]$
	$\displaystyle\qquad+\frac{1}{2}C_{G}C_{v}\int_{B_{1}(0)}G^{2}dd^{c}\eta\wedge(dd^{c}\phi)^{n-1}$
	$\displaystyle\leq ne^{C_{G}}C_{G}^{2}C_{v}^{2}C_{w}\int_{B_{1}(0)}\omega^{n}+\frac{1}{2}C_{G}^{3}C_{v}C_{\eta}\int_{B_{1}(0)}(\Delta\phi)^{n-1}\omega^{n}$
	$\displaystyle\qquad-\frac{1}{2}nC_{G}C_{v}\left[\int_{B_{1}(0)}G^{2}\eta dd^{c}\log v(m_{dd^{c}\phi})\wedge(dd^{c}\phi)^{n-1}+G^{2}d\log v(m_{dd^{c}\phi})\wedge d^{c}\eta\wedge(dd^{c}\phi)^{n-1}\right]$
	$\displaystyle\leq ne^{C_{G}}C_{G}^{2}C_{v}^{2}C_{w}\int_{B_{1}(0)}\omega^{n}+\frac{1}{2}C_{G}^{3}C_{v}C_{\eta}\int_{B_{1}(0)}(\Delta\phi)^{n-1}\omega^{n}+\frac{1}{2}nC_{G}C_{v}\left[\int_{B_{1}(0)}e^{C_{G}}C_{G}^{2}C_{v}^{3}C_{\xi}^{2}(\Delta\phi)\omega^{n}+e^{C_{G}}C_{G}^{2}C_{v}^{3}C_{\xi}C_{\eta}\omega^{n}\right]$
	$\displaystyle\leq D_{1}.$

Hence, the claim follows. We provide more details about the inequality next to the last one. Since

\partial_{\bar{j}}\log v(m_{dd^{c}\phi})=\frac{1}{v(m_{dd^{c}\phi})}\sum_{\alpha}v_{\alpha}(m_{dd^{c}\phi})(-2\sqrt{-1}\phi_{k\bar{j}}(\xi_{\alpha}^{\prime})^{k}),

we have $\operatorname{tr}_{\phi}(d\log v(m_{dd^{c}\phi})\wedge d^{c}\eta)\leq C_{v}^{2}C_{\xi}C_{\eta}$ ; thus,

d\log v(m_{dd^{c}\phi})\wedge d^{c}\eta\wedge(dd^{c}\phi)^{n}\leq C_{v}^{2}C_{\xi}C_{\eta}(dd^{c}\phi)^{n}=C_{v}^{2}C_{\xi}C_{\eta}\frac{e^{G}}{v(m_{dd^{c}\phi})}\omega^{n}\leq e^{C_{G}}C_{v}^{3}C_{\xi}C_{\eta}\omega^{n}.

On the other hand, for the term involving $dd^{c}\log v(m_{dd^{c}\phi})$ ,

	$\displaystyle\partial_{i\bar{j}}^{2}\log v(m_{dd^{c}\phi})=\partial_{i}\left\{\sum_{\alpha}(\partial_{\alpha}\log v)(m_{dd^{c}\phi})(-2\sqrt{-1}\phi_{k\bar{j}}(\xi_{\alpha}^{\prime})^{k})\right\}$
	$\displaystyle=4\sum_{\alpha,\beta}(\partial_{\alpha\beta}^{2}\log v)(m_{dd^{c}\phi})\phi_{i\bar{q}}\phi_{k\bar{j}}(\xi^{\prime}_{\beta})^{\bar{q}}(\xi_{\alpha}^{\prime})^{k}+2\sum_{\alpha}(\partial_{\alpha}\log v)(m_{dd^{c}\phi})(-\sqrt{-1})[\phi_{k\bar{j}i}(\xi^{\prime}_{\alpha})^{k}+\phi_{k\bar{j}}(\xi^{\prime}_{\alpha})^{k}_{i}].$

Note that $\mathcal{L}_{\xi_{\alpha}}dd^{c}\phi=0$ from the $G$ -invariant assumption. Therefore,

	$\displaystyle 0$	$\displaystyle=\frac{1}{2}\mathcal{L}_{\xi}dd^{c}\phi=\frac{1}{2}d(i_{\xi}dd^{c}\phi)=d\left(\sqrt{-1}\phi_{k\bar{\ell}}(\xi^{\prime})^{k}d\bar{z}^{\ell}-\sqrt{-1}\phi_{k\bar{\ell}}(\xi^{\prime\prime})^{\bar{\ell}}dz^{k}\right)$
		$\displaystyle=\left(\sqrt{-1}\phi_{k\bar{\ell}}(\xi^{\prime})^{k}_{i}dz^{i}\wedge d\bar{z}^{\ell}-\sqrt{-1}\phi_{k\bar{\ell}}(\xi^{\prime\prime})^{\bar{\ell}}_{\bar{j}}d\bar{z}^{j}\wedge dz^{k}\right)-\sqrt{-1}\phi_{k\bar{\ell}}(\xi^{\prime\prime})^{\bar{\ell}}_{i}dz^{i}\wedge dz^{k}+\sqrt{-1}\phi_{k\bar{\ell}}(\xi^{\prime})^{k}_{\bar{j}}d\bar{z}^{j}\wedge d\bar{z}^{\ell}$
		$\displaystyle=\sqrt{-1}\left(\phi_{\gamma\bar{\beta}}(\xi^{\prime})^{\gamma}_{\alpha}+\phi_{\alpha\bar{\delta}}(\xi^{\prime\prime})^{\bar{\delta}}_{\bar{\beta}}\right)dz^{\alpha}\wedge d\bar{z}^{\beta}-\sqrt{-1}\phi_{k\bar{\ell}}(\xi^{\prime\prime})^{\bar{\ell}}_{i}dz^{i}\wedge dz^{k}+\sqrt{-1}\phi_{k\bar{\ell}}(\xi^{\prime})^{k}_{\bar{j}}d\bar{z}^{j}\wedge d\bar{z}^{\ell}$

and this implies $\phi_{\gamma\bar{\beta}}(\xi^{\prime})^{\gamma}_{\alpha}+\phi_{\alpha\bar{\delta}}(\xi^{\prime\prime})^{\bar{\delta}}_{\bar{\beta}}=0$ . In the normal coordinates of $dd^{c}\phi$ , we get

\displaystyle\phi^{i\bar{j}}\phi_{k\bar{j}}(\xi^{\prime}_{\alpha})^{k}_{i}=\frac{1}{2}\left(\phi^{i\bar{j}}\phi_{k\bar{j}}(\xi^{\prime}_{\alpha})^{k}_{i}+\overline{\phi^{i\bar{j}}\phi_{\ell\bar{j}}(\xi^{\prime}_{\alpha})^{\ell}_{i}}\right)=\frac{1}{2}\phi^{i\bar{j}}\left(\phi_{k\bar{j}}(\xi^{\prime}_{\alpha})^{k}_{i}+\phi_{i\bar{\ell}}(\xi^{\prime\prime}_{\alpha})^{\bar{\ell}}_{\bar{j}}\right)=0.

This yields that $\operatorname{tr}_{\phi}dd^{c}\log v(m_{dd^{c}\phi})\leq C_{v}^{2}C_{\xi}^{2}(\Delta\phi)$ and

dd^{c}\log v(m_{dd^{c}\phi})\wedge(dd^{c}\phi)^{n-1}\leq C_{v}^{2}C_{\xi}^{2}(\Delta\phi)(dd^{c}\phi)^{n}\leq e^{C_{G}}C_{v}^{3}C_{\xi}^{2}(\Delta\phi)\omega^{n}.

We also remark that $\Delta_{\phi,v}f=\frac{1}{2v(m_{dd^{c}\phi})}\partial_{\alpha}(v(m_{dd^{c}\phi})g_{\phi}^{\alpha\beta}\partial_{\beta}f)$ where $\alpha,\beta$ are real coordinates and $[(g_{\phi})_{\alpha\beta}]_{1\leq\alpha,\beta\leq 2n}$ is the Riemannian metric associate to $dd^{c}\phi$ . Therefore, it follows from Lemma B.1, that

\partial_{\alpha}(v(m_{dd^{c}\phi})g_{\phi}^{\alpha\beta}\partial_{\beta}u)\geq-2Cv(m_{dd^{c}\phi})(\Delta\phi)u.

Using [CC21a, Lem. 6.3, arXiv version]), we derive that

\|u\|_{L^{\infty}(B_{1/2}(0))}\leq D_{2}(\|u\|_{L^{1}(B_{3/4}(0))}+1)\leq C,

(B.6)

where $D_{2}$ depends on $C_{v}$ , $C_{w}$ , $p$ , $\|S\|_{L^{\infty}}$ , $\|\Delta\phi\|_{L^{p}(B_{1}(0))}$ , $\|\operatorname{tr}_{\phi}\omega\|_{L^{p}(B_{1}(0))}$ . Recall that $u=e^{\frac{G}{2}}|dG|_{\phi}^{2}+K\Delta\phi$ ; hence we get

\|u\|_{L^{1}(B_{3/4}(0))}\leq e^{C_{G}/2}\||dG|^{2}_{\phi}\|_{L^{1}(B_{3/4}(0))}+K\|\Delta\phi\|_{L^{1}(B_{3/4}(0))}\leq e^{C_{G}/2}D_{1}+K\|\Delta\phi\|_{L^{1}(B_{3/4}(0))}.

Then combining this with (B.6) implies the uniform $L^{\infty}$ -estimates of $|dG|_{\phi}$ and $\Delta\phi$ on $B_{1/2}(0)$ which only depend on $C_{v}$ , $C_{w}$ , $p$ , $\|S\|_{L^{\infty}}$ , $\|\Delta\phi\|_{L^{p}(B_{1}(0))}$ , $\|\operatorname{tr}_{\phi}\omega\|_{L^{p}(B_{1}(0))}$ . Then the standard Evans–Krylov estimate and bootstrapping argument imply higher order estimates on $B_{1/2}(0)$ for $\phi,G$ which depend on $C_{v}$ , $C_{w}$ , $p$ , $\|S\|_{L^{\infty}}$ , $\|\Delta\phi\|_{L^{p}(B_{1}(0))}$ , $\|\operatorname{tr}_{\phi}\omega\|_{L^{p}(B_{1}(0))}$ . ∎

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	$\displaystyle e^{-\frac{G}{2}}\Delta_{\phi}(e^{\frac{G}{2}}\|dG\|_{\phi}^{2})$	$\displaystyle\geq\phi^{i\bar{i}}(\partial_{i}\Delta_{\phi}G\cdot\partial_{\bar{i}}G+\partial_{\bar{i}}\Delta_{\phi}G\cdot\partial_{i}G)-\phi^{i\bar{i}}\phi^{j\bar{j}}G_{i\bar{j}}G_{i}G_{\bar{j}}$
		$\displaystyle\quad+\phi^{i\bar{i}}\phi^{j\bar{j}}(\log v(m_{dd^{c}\phi}))_{i\bar{j}}G_{i}G_{\bar{j}}+\phi^{i\bar{i}}\phi^{j\bar{j}}G_{i\bar{j}}G_{i}G_{\bar{j}}+\|\nabla^{\phi}\overline{\nabla^{\phi}}G\|_{\phi}^{2}+\frac{1}{2}\Delta_{\phi}G\cdot\|dG\|_{\phi}^{2}$
		$\displaystyle=2\operatorname{tr}_{\phi}(d\Delta_{\phi}G\wedge d^{c}G)+\phi^{i\bar{i}}\phi^{j\bar{j}}(\log v(m_{dd^{c}\phi}))_{i\bar{j}}G_{i}G_{\bar{j}}+\|\nabla^{\phi}\overline{\nabla^{\phi}}G\|_{\phi}^{2}+\frac{1}{2}\Delta_{\phi}G\cdot\|dG\|_{\phi}^{2}.$

	$\displaystyle e^{-\frac{G}{2}}\Delta_{\phi,v}(e^{\frac{G}{2}}\|dG\|_{\phi}^{2})$	$\displaystyle\geq 2\operatorname{tr}_{\phi}(d\Delta_{\phi,v}G\wedge d^{c}G)+\|\nabla^{\phi}\overline{\nabla^{\phi}}G\|_{\phi}^{2}+\frac{1}{2}\Delta_{\phi,v}G\cdot\|\nabla^{\phi}G\|_{\phi}^{2}$
		$\displaystyle\quad+4\Re\sum_{\alpha,\beta}(\partial_{\alpha\beta}^{2}\log v)(m_{dd^{c}\phi})(\xi^{\prime\prime}_{\beta})^{\bar{i}}(\xi^{\prime\prime}_{\alpha})^{\bar{j}}G_{\bar{i}}G_{\bar{j}}$
		$\displaystyle\geq 2\operatorname{tr}_{\phi}(d\Delta_{\phi,v}G\wedge d^{c}G)+\|\nabla^{\phi}\overline{\nabla^{\phi}}G\|_{\phi}^{2}+\frac{1}{2}\Delta_{\phi,v}G\cdot\|\nabla^{\phi}G\|_{\phi}^{2}-4C_{v}^{4}C_{\xi}\|dG\|_{\omega}^{2}$

Weighted cscK metrics on Kähler varieties

Abstract.

Key words and phrases:

1991 Mathematics Subject Classification:

Introduction

Setting (GS).

Condition (A).

Theorem A.

Corollary B.

Acknowledgements.

1. Preliminaries

1.1. Pluripotential theory on normal Kähler varieties

Definition 1.1.

1.1.1. Finite energy class

1.2. Weighted cscK metrics

1.2.1. Automorphism group and holomorphic vector fields

1.2.2. Weighted setting

Definition 1.2.

1.2.3. Weighted cscK equations

1.2.4. Weighted Mabuchi functional

Definition 1.3.

Lemma 1.4.

Lemma 1.5.

Proof.

1.2.5. Weighted extremal metrics

Lemma 1.6 ([BJT24, Lem. 3.34]).

Definition 1.7.

1.2.6. Extremal Kähler metrics

1.2.7. Extension on ℰ1,T\mathcal{E}^{1,T} and coercivity

Definition 1.8.

1.3. Weighted variational formalism in the singular setting

1.3.1. Reduced automorphism group and moment maps

1.3.2. Singular weighted cscK metrics

Definition 1.9.

Definition 1.10.

2. A priori estimates

2.1. L∞L^{\infty}-estimates

Theorem 2.1.

Proof.

2.2. Local LpL^{p}-estimate for Laplacian

Proposition 2.2.

Proof.

2.3. Local higher-order estimates

Theorem 2.3.

Theorem 2.4.

3. Existence of singular weighted cscK metrics

3.1. Setup

Setting 3.1.

Theorem 3.2.

3.2. Proof of Theorem A

Theorem 3.3.

Proof.

Lemma 3.4.

Proof.

3.2.1. Singular cscK and extremal metrics

Proof of Corollary B.

Theorem 3.5.

3.3. Constructing examples of singular cscK metrics

Lemma 3.6.

Proof.

3.3.1. Construct singular cscK on blowups of singular KEs

Example 3.7 (Isolated cone singularities).

3.3.2. Mixing construction with smoothing

Appendix A Weighted Aubin–Yau inequality

Lemma A.1.

Remark A.2.

Proof.

Appendix B Weighted local Chen–Cheng’s 𝒞2\mathcal{C}^{2}-estimate

Lemma B.1.

Proof.

Proof of Theorem 2.3.

References

1.2.7. Extension on $\mathcal{E}^{1,T}$ and coercivity

2.1. $L^{\infty}$ -estimates

2.2. Local $L^{p}$ -estimate for Laplacian

Appendix B Weighted local Chen–Cheng’s $\mathcal{C}^{2}$ -estimate