Calabi type functionals for coupled Kähler–Einstein metrics

Abstract

We introduce the coupled Ricci–Calabi functional and the coupled H-functional which measure how far a Kähler metric is from a coupled Kähler–Einstein metric in the sense of Hultgren–Witt Nyström. We first give corresponding moment weight type inequalities which estimate each functional in terms of algebraic invariants. Secondly, we give corresponding Hessian formulas for these functionals at each critical point, which have an application to a Matsushima type obstruction theorem for the existence of a coupled Kähler–Einstein metric.

1 Introduction

Hultgren–Witt Nyström [28] introduced the notion of a coupled Kähler–Einstein metric on a compact complex manifold of general type or a Fano manifold. In this paper, we mainly focus on the Fano case. Let X be an n-dimensional Fano manifold. A decomposition of the first Chern class $2\pi c_1(X)$ is a sum

$$\begin{aligned} 2\pi c_1(X)=\alpha _1 + \cdots + \alpha _N \end{aligned}$$

where each $\alpha _i$ is a Kähler class for X. Let $\theta _i\in \alpha _i$ be a reference Kähler metric, and let $V_i$ be the volume $\int _X\theta _i^n$ of $\alpha _i$. We define the set of tuples of Kähler potentials by

$$\begin{aligned} {\mathcal {M}}:= \prod _{i=1}^{N}{\mathcal {M}}_{i}:=\prod _{i=1}^{N}\{ \phi _i\in C^{\infty }(X;{\mathbb {R}}) | \omega _{\phi _i}:=\theta _i+\sqrt{-1}\partial {\overline{\partial }}\phi _i>0 \}, \end{aligned}$$

and identify a Kähler metric $\omega _{\phi _i}$ with its potential $\phi _i$. A tangent space of ${\mathcal {M}}$ is identified with $(C^{\infty }(X;{\mathbb {R}}))^N$. For any tuple of Kähler metrics $\Phi =(\phi _i)_{i=1}^N\in {\mathcal {M}}$, since the Ricci form $\textrm{Ric}(\omega _{\phi _i}):=-\sqrt{-1}\partial {\overline{\partial }}\log \omega _{\phi _i}^n$ and the sum $\sum _{i=1}^N\omega _{\phi _i}$ represent $2\pi c_1(X)$, there exists a unique smooth real function $f_i(\Phi )$ satisfying

$$\begin{aligned} \textrm{Ric}(\omega _{\phi _i})-\sum _{j=1}^N\omega _{\phi _j} = \sqrt{-1}\partial {\overline{\partial }}f_i(\Phi ) \quad \text {and}\quad \int _X(1-e^{f_i(\Phi )})\omega _{\phi _i}^n=0. \end{aligned}$$

(1)

In this paper, we call the tuple $(f_i(\Phi ))_{i=1}^N$ the Ricci potential for $\Phi =(\phi _i)_{i=1}^N$. Then, the tuple $\Phi =(\phi _i)_{i=1}^N$ is a coupled Kähler–Einstein metric for the decomposition $(\alpha _i)_{i=1}^N$ if every $f_i(\Phi )$ vanishes, that is,

$$\begin{aligned} \textrm{Ric}(\omega _{\phi _1})=\cdots = \textrm{Ric}(\omega _{\phi _N})=\sum _{i=1}^{N}\omega _{\phi _i}. \end{aligned}$$

Coupled Kähler–Einstein metrics were studied extensively in recent years [13, 14, 19,20,21,22, 27, 28, 31, 32, 36, 37]. One of the motivation to study comes from algebro-geometric stabilities. Indeed, Hultgren–Witt Nyström [28] introduced the notion of called K-polystability for $(X, (\alpha _i)_{i=1}^N)$ and showed that the existence of a coupled Kähler–Einstein metric implies it. Datar-Pingali [13] introduced a framework of geometric invariant theory for a coupled constant scalar curvature Kähler metric which is a generalization of a coupled Kähler–Einstein metric.

The well-known Calabi functional [6, 7], which is the $L^2$-norm of a scalar curvature, plays an important role for studies of a Kähler–Einstein metric and a constant scalar curvature Kähler metric. In this paper, we introduce two Calabi type functionals which measure how $\Phi$ is far from a coupled Kähler–Einstein metric. We first discuss moment weight type inequalities which give lower bounds of these functionals in terms of algebro-geometric stability invariants. Secondly, we discuss Hessians for these functionals at each critical point to obtain various corollaries including a new proof of a Matsushima type obstruction theorem for the existence of a coupled Kähler–Einstein metric.

Let us introduce two Calabi type functionals as follows.

$$\begin{aligned} R_c(\Phi )=\sum _{i=1}^N \frac{1}{V_{i}}\int _X (1-e^{f_i(\Phi )})^2\omega _{\phi _i}^n \quad \text {and}\quad H_c(\Phi )=\sum _{i=1}^N \frac{1}{V_{i}}\int _X f_i(\Phi )e^{f_i(\Phi )}\omega _{\phi _i}^n. \end{aligned}$$

In this paper, we call $R_c$ the coupled Ricci–Calabi functional and $H_{c}$ the coupled H-functional. These are non-negative functionals in ${\mathcal {M}}$ whose zeros are coupled Kähler–Einstein metrics (see the inequality (8)). When $N=1$, these functional are written as R and H, respectively, and are called the Ricci–Calabi functional and the H-functional, respectively. Functionals R and H were studied in [2, 15, 17, 23, 25, 30, 38,39,40] and, in particular, play important roles in the context of optimal destabilizers for a Fano manifold admitting no Kähler–Einstein metric.

1.1 Moment weight type inequalities

The Calabi functional for a polarized manifold has a lower bound in terms of the Donaldson–Futaki invariant [16]. Such an inequality is called the moment weight inequality since it already appears in geometric invariant theory as an inequality between the squared norm of a moment map and a Hilbert–Mumford weight. The Ricci–Calabi functional R and the H-Functional H satisfy a corresponding moment weight inequality [2, 15, 23,24,25]. The first results in this paper are two moment weight type inequalities for $R_c$ and $H_c$ which generalize these inequalities.

Theorem 1.1

We have

$$\begin{aligned} \inf _{\Phi \in {\mathcal {M}}}R_c(\Phi )^{1/2}\ge & {} \sup _{({\mathcal {X}},({\mathcal {L}}_i)_{i=1}^N)}\frac{-D_c({\mathcal {X}},({\mathcal {L}}_i)_{i=1}^N)}{\Vert ({\mathcal {X}},({\mathcal {L}}_i)_{i=1}^N) \Vert _2} \quad \text {and} \\{} & {} \quad \inf _{\Phi \in {\mathcal {M}}}H_c(\Phi ) \ge \sup _{({\mathcal {X}},({\mathcal {L}}_i)_{i=1}^N)}-H_c({\mathcal {X}},({\mathcal {L}}_i)_{i=1}^N). \end{aligned}$$

Here $({\mathcal {X}},({\mathcal {L}}_i)_{i=1}^N)$ in the above supremums runs through arbitrary test configuration of the decomposition $(X, (\alpha _{i})_{i=1}^{N})$ introduced in [28], $\Vert ({\mathcal {X}},({\mathcal {L}}_i)_{i=1}^N) \Vert _2$ is the $L^2$-norm, $D_c({\mathcal {X}},({\mathcal {L}}_i)_{i=1}^N)$ is the coupled Ding invariant, and $H_c({\mathcal {X}},({\mathcal {L}}_i)_{i=1}^N)$ is the coupled H-invariant. These notions are introduced in Sect. 2.

As a direct consequence of Theorem 1.1, a Fano manifold admitting a coupled Kähler–Einstein metric satisfies algebraic (semi-)stability conditions.

Corollary 1.2

Suppose the existence of a coupled Kähler–Einstein metric for the decomposition $(\alpha _{i})_{i=1}^{N}$ of $2\pi c_{1}(X)$. Then, we have

$$\begin{aligned} D_{c}({\mathcal {X}},({\mathcal {L}}_{i})_{i=1}^{N})\ge 0 \quad \text {and}\quad H_{c}({\mathcal {X}},({\mathcal {L}}_{i})_{i=1}^{N})\ge 0 \end{aligned}$$

for any test configuration $({\mathcal {X}},({\mathcal {L}}_{i})_{i=1}^{N})$ for $(X,(\alpha _{i})_{i=1}^{N})$.

When $N=1$, the equalities in Theorem 1.1 in fact hold. Dervan-Székelyhidi [15] showed the moment weight equality for H by applying the Kähler–Ricci flow together with deep results in [8, 10]. Hisamoto [25] showed corresponding equalities for R and H by using the inverse Monge–Ampère flow [12] and the Kähler–Ricci flow, respectively, together with a technique for multiplier ideal sheaves. When $N>1$, in order to establish the equality in Theorem 1.1, it is natural to consider the generalization of these flow, that is, the coupled inverse Monge–Ampère flow

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\phi _i(t) = 1-e^{f_i(\Phi (t))} \quad (i=1,\dots ,N) \end{aligned}$$

(2)

and the coupled Kähler–Ricci flow

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\phi _i(t) = -f_i(\Phi (t)) \quad (i=1,\dots ,N). \end{aligned}$$

(3)

However, little is known for these flows at present. For instance, the short time existence for each flow is true since they are parabolic. However, the long time existence is not established. In Sect. 3, we see that each flow is a gradient flow for $R_c$ and $H_c$, respectively (Corollary 3.5). They will present not only some applications to establish the equalities in Theorem 1.1 but also some interesting problems in geometric analysis.

Remark 1.3

Very recently, Hashimoto [22] introduced a different framework of test configurations for a decomposition $(\alpha _{i})_{i=1}^{N}$ where $\alpha _{i}=2\pi c_{1}(L_{i})$ for a line bundle $L_{i}\rightarrow X$. The author expects that corresponding moment weight type inequalities hold in his framework.

1.2 Hessian formulas for functionals and its application to a Matsushima type obstruction theorem

In this paper, we call a critical point of $R_c$ a coupled Mabuchi soliton (cf. [26, 29, 40]) and call a critical point of $H_{c}$ a coupled Kähler–Ricci soliton (cf. [23]). In Sect. 3, we show that a tuple $\Phi =(\phi _i)_{i=1}^N\in {\mathcal {M}}$ is a coupled Mabuchi soliton if and only if the vector fields $\textrm{grad}_{\phi _i}e^{f_i(\Phi )}$ are holomorphic and $\textrm{grad}_{\phi _1}e^{f_1(\Phi )}=\cdots =\textrm{grad}_{\phi _N}e^{f_N(\Phi )}$. Similarly, we show that $\Phi =(\phi _i)_{i=1}^N\in {\mathcal {M}}$ is a coupled Kähler–Ricci soliton if and only if the vector fields $\textrm{grad}_{\phi _i}f_i(\Phi )$ are holomorphic and $\textrm{grad}_{\phi _1}f_1(\Phi )=\cdots =\textrm{grad}_{\phi _N}f_N(\Phi )$.

Examples of coupled Mabuchi solitons and coupled Kähler–Ricci solitons on Fano manifolds with large symmetry have already appeared in [14] (see also [27]). However, in that paper, the conditions $\textrm{grad}_{\phi _1}e^{f_1(\Phi )}=\cdots =\textrm{grad}_{\phi _N}e^{f_N(\Phi )}$ and $\textrm{grad}_{\phi _1}f_1(\Phi )=\cdots =\textrm{grad}_{\phi _N}f_N(\Phi )$ are not required for each definition. Their motivation is the construction of such metrics by proving $C^{0}$-estimate to a class of coupled Monge–Ampère equation, which is independent of the Calabi type functionals.

The second result in this paper shows that each critical metric is in fact a local minimum of the corresponding functional by giving the Hessian formulas at each critical point. Let ${\langle \!\hspace{0.02cm} \!\langle }\cdot , \cdot {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi }$ and ${\langle \!\hspace{0.02cm} \!\langle }\cdot , \cdot {\rangle \!\hspace{0.02cm}\!\rangle }^f_{\Phi }$ be Hermitian inner products on $(C^{\infty }(X;{\mathbb {C}}))^N$ defined by

$$\begin{aligned} {\langle \!\hspace{0.02cm} \!\langle }\varvec{u}, \varvec{v} {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi } = \sum _{i=1}^N \frac{1}{V_{i}} \int _X u_i \overline{v_i}\omega _{\phi _i}^n \quad \text {and}\quad {\langle \!\hspace{0.02cm} \!\langle }\varvec{u}, \varvec{v} {\rangle \!\hspace{0.02cm}\!\rangle }^f_{\Phi } = \sum _{i=1}^N \frac{1}{V_{i}} \int _X u_i \overline{v_i}e^{f_{i}(\Phi )}\omega _{\phi _{i}}^{n}. \end{aligned}$$

Let $P_{\Phi }$ and $P_{\Phi }^f$ be the operators acting on $T_{\Phi }{\mathcal {M}}$ defined by (9) and (10) in Sect. 3.1.

Theorem 1.4

At a coupled Mabuchi soliton $\Phi _R \in {\mathcal {M}}$, the Hessian of the coupled Ricci–Calabi functional $R_{c}$ is written as

$$\begin{aligned} \textrm{Hess}(R_c)(\delta \Phi _1, \delta \Phi _2) = 2{\langle \!\hspace{0.02cm} \!\langle }P_{\Phi _R}^f \overline{P_{\Phi _R}^f}\delta \Phi _1, \delta \Phi _2 {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi _R} = 2{\langle \!\hspace{0.02cm} \!\langle }\overline{P_{\Phi _R}^f} P_{\Phi _R}^f\delta \Phi _1, \delta \Phi _2 {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi _R} \end{aligned}$$

(4)

for any variations $\delta \Phi _1, \delta \Phi _2\in T_{\Phi _R}{\mathcal {M}}$. At a coupled Kähler–Ricci soliton $\Phi _H \in {\mathcal {M}}$, the Hessian of the coupled H-functional is written as

$$\begin{aligned} \textrm{Hess}(H_c)(\delta \Phi _1, \delta \Phi _2) = {\langle \!\hspace{0.02cm} \!\langle }P_{\Phi _R} \overline{P_{\Phi _R}}\delta \Phi _1, \delta \Phi _2 {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi _R}^f = {\langle \!\hspace{0.02cm} \!\langle }\overline{P_{\Phi _R}} P_{\Phi _R}\delta \Phi _1, \delta \Phi _2 {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi _R}^f \end{aligned}$$

(5)

for any variations $\delta \Phi _1, \delta \Phi _2\in T_{\Phi _H}{\mathcal {M}}$.

Here the operator $\overline{P_{\Phi _R}}$ (resp. $\overline{P_{\Phi _R}^f}$) in the above theorem is the complex conjugate of $P_{\Phi _R}$ (resp. $P_{\Phi _R}^f$). Since the operator $P_{\Phi }^f$ (resp. $P_{\Phi }$) is self-adjoint and non-negative with respective to ${\langle \!\hspace{0.02cm} \!\langle }\cdot , \cdot {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi }$ (resp. ${\langle \!\hspace{0.02cm} \!\langle }\cdot , \cdot {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi }^f$) (Proposition 3.1), it turns out the following.

Corollary 1.5

Operators $P_{\Phi _{R}}^f$ and $\overline{P_{\Phi _{R}}^f}$ (resp. $P_{\Phi _{H}}$ and $\overline{P_{\Phi _{H}}}$) are commutative. As a result, the composition $P_{\Phi _{R}}^f\overline{P_{\Phi _{R}}^f}$ (resp. $P_{\Phi _{H}}\overline{P_{\Phi _{H}}}$) is a self-adjoint non-negative operator with respect to ${\langle \!\hspace{0.02cm} \!\langle }\cdot , \cdot {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi _R}$ (resp. ${\langle \!\hspace{0.02cm} \!\langle }\cdot , \cdot {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi _H}^f$). In particular, each Hessian of $R_c$ and $H_c$ is non-negative at each critical point.

When $N=1$, the Hessian formula of the Ricci–Calabi functional at a Mabuchi soliton is obtained by the author [30]. On the other hand, Fong [17] gives the Hessian formula of the H-functional at any point by a tensor calculus. It seems to be technically difficult to apply the Fong’s tensor calculus for our case where $N>1$. A unifying technique which generalizes the author’s one in [30] gives the Hessian formulas for $R_c$ and $H_c$ in Theorem 1.4.

By Corollary 1.5, operators $P_{\Phi }^f$ and $\overline{P_{\Phi }^f}$ (resp. $P_{\Phi }$ and $\overline{P_{\Phi }}$) are commutative at a coupled Mabuchi soliton (resp. at a coupled Kähler–Ricci soliton). Applying this commutativity, we show a Matsushima type obstruction theorem for the existence of a coupled Mabuchi soliton and a coupled Kähler–Ricci soliton.

Theorem 1.6

Let X be a Fano manifold and ${\mathfrak {h}}(X)$ be the space of holomorphic vector fields. If X admits a coupled Mabuchi soliton $\Phi =(\phi _i)_{i=1}^N$, then ${\mathfrak {h}}(X)$ is, as a vector space, the direct sum

$$\begin{aligned} {\mathfrak {h}}(X)=\sum _{\lambda \ge 0} {\mathfrak {h}}_{\lambda }(X) \end{aligned}$$

where ${\mathfrak {h}}_{\lambda }(X)$ is the $\lambda$-eigenspace of the adjoint action of the holomorphic vector field $\textrm{grad}_{\phi _1}(1-e^{f_1(\Phi )}) = \cdots = \textrm{grad}_{\phi _N}(1-e^{f_N(\Phi )})$. If X admits a coupled Kähler–Ricci soliton $\Phi =(\phi _i)_{i=1}^N$, then ${\mathfrak {h}}(X)$ has the same decomposition as above where ${\mathfrak {h}}_{\lambda }(X)$ is the $\lambda$-eigenspace of the adjoint action of the holomorphic vector field $-\textrm{grad}_{\phi _1}f_1(\Phi ) = \cdots = -\textrm{grad}_{\phi _N}f_N(\Phi )$. Furthermore, in both cases, ${\mathfrak {h}}_0(X)$ coincides with the complexification of the Lie algebra of Killing vector fields for every $\omega _{\phi _i}$. In particular, ${\mathfrak {h}}_0(X)$ is reductive.

The above theorem gives a new proof of a Matsushima type obstruction theorem for the existence of a coupled Kähler–Einstein metric which is already proved by Hultgren–Witt Nyström [28] and Futaki-Zhang [20].

Corollary 1.7

Let X be a Fano manifold admitting a coupled Kähler–Einstein metric for a decomposition of $2\pi c_1(X)$. The holomorphic automorphism group $\textrm{Aut}(X)$ is reductive.

1.3 Organization

This paper is organized as follows. In Sect. 2, we introduce the notion of test configurations for a decomposition and some algebraic invariants to prove the moment weight type inequalities for $R_{c}$ and $H_{c}$. Some energy functionals and its slope formulas at infinity play an important role for the proof. In Sect. 3, we give the Hessian formulas of $R_{c}$ and $H_{c}$ to see that each critical point is a local minimum. As an application of the Hessian formulas, Matsushima type obstruction theorems for the existence of coupled Mabuchi solitons and coupled Kähler–Ricci solitons are proved.

2 Moment weight type inequalities

2.1 Test configurations and invariants

Following [28], we define the notion of test configurations for a decomposition $(\alpha _{i})_{i=1}^{N}$ of $2\pi c_{1}(X)$.

Definition 2.1

Let L be an ample line bundle over a projective manifold Y. A test configuration $({\mathcal {Y}},{\mathcal {L}})$ for (Y, L) is a normal scheme ${\mathcal {Y}}$, a flat surjective morphism ${\mathcal {Y}}\rightarrow {\mathbb {C}}$ and a relatively ample line bundle ${\mathcal {L}}$ together with a ${\mathbb {C}}^{*}$-action on ${\mathcal {L}}$ compatible with the standard ${\mathbb {C}}^{*}$-action on ${\mathbb {C}}$, such that the fiber over $1\in {\mathbb {C}}$ is equal to (Y, L).

An ${\mathbb {R}}$-line bundle is understood as a formal linear combination over ${\mathbb {R}}$ of line bundles. The following definition is based on the fact that any Kähler class $\alpha$ on a Fano manifold can be written as the first Chern class of an ${\mathbb {R}}$-line bundle, that is, $\alpha =c_{1}(\sum _{j}r_{j}L_{j}):=\sum _{j}r_{j}c_{1}(L_{j})$ for an ample line bundle $L_{j}$ and a positive real number $r_{j}$.

Definition 2.2

Let $\alpha$ be a Kähler class on a Fano manifold X. A test configuration $({\mathcal {X}},{\mathcal {L}})$ for $(X,\alpha )$ is defined as a test configuration in the sense of Definition 2.1 satisfying the following.

• (1) The scheme ${\mathcal {X}}$ is ${\mathbb {Q}}$-Gorenstein.

• (2) There exists an ${\mathbb {R}}$-line bundle $L=\sum _{j}r_{j}L_{j}$ over X satisfying $\alpha =c_{1}(L)$, where each $L_{j}$ is ample and $r_{j}>0$.

• (3) The line bundle ${\mathcal {L}}$ is written as $\sum _{j}r_{j}{\mathcal {L}}_{j}$ where ${\mathcal {L}}_{j}$ is a line bundle over ${\mathcal {X}}$ such that $({\mathcal {X}},{\mathcal {L}}_{j})$ is a test configuration for $(X,L_{j})$ in the sense of Definition 2.1.

Definition 2.3

Let $(\alpha _{i})_{i=1}^{N}$ be a decomposition of $2\pi c_{1}(X)$. A test configuration $({\mathcal {X}}, ({\mathcal {L}}_{i})_{i=1}^{N})$ for $(X, (\alpha _{i})_{i=1}^{N})$ is defined as follows.

• (1) The scheme ${\mathcal {X}}$ is ${\mathbb {Q}}$-Gorenstein.

• (2) For each i, the ${\mathbb {R}}$-line bundle ${\mathcal {L}}_{j}$ over ${\mathcal {X}}$ defines a test configuration $({\mathcal {X}},{\mathcal {L}}_{j})$ for $(X,\alpha _{i})$ in the sense of Definition 2.2.

• (3) The sum $\sum _{i=1}^{N}{\mathcal {L}}_{i}$ defines a test configuration $({\mathcal {X}},\sum _{i=1}^{N}{\mathcal {L}}_{i})$ for the Fano manifold $(X,-K_{X})$ in the sense of Definition 2.1.

Now we define some invariants appearing in the right hand side of moment weight type inequalities. Gluing each $({\mathcal {X}},{\mathcal {L}}_{i})$ with the trivial family, we have the unique ${\mathbb {C}}^{*}$-equivalent family $(\bar{{\mathcal {X}}},\bar{{\mathcal {L}}_{i}})$ over ${\mathbb {P}}^{1}$. This gives a compactification of a test configuration $({\mathcal {X}},({\mathcal {L}})_{i=1}^{N})$ for $(X,(\alpha _{i})_{i=1}^{N})$. Set

$$\begin{aligned} E({\mathcal {X}},{\mathcal {L}}_{i})=\frac{\bar{{\mathcal {L}}_{i}}^{n+1}}{(n+1)V_{i}} \end{aligned}$$

to define the coupled Ding invariant (cf. [2]) using the log canonical threshold

$$\begin{aligned} D_{c}\Bigl ({\mathcal {X}},({\mathcal {L}}_{i})_{i=1}^{N}\Bigr )= & {} L\Bigl ({\mathcal {X}},\sum _{i=1}^{N}{\mathcal {L}}_{i}\Bigr ) -\sum _{i=1}^{N}E({\mathcal {X}},{\mathcal {L}}_{i}) \\:= & {} \textrm{lct}_{(\bar{{\mathcal {X}}},{\mathcal {B}})}({\mathcal {X}}|_{t=0})-1 -\sum _{i=1}^{N}\frac{\bar{{\mathcal {L}}_{i}}^{n+1}}{(n+1)V_{i}} \end{aligned}$$

where ${\mathcal {X}}|_{t=0}$ is the fiber over $0\in {\mathbb {P}}^{1}$ and ${\mathcal {B}}$ is the boundary divisor uniquely determined by the properties ${\mathcal {B}}\sim _{{\mathbb {Q}}} -K_{\bar{{\mathcal {X}}}/{\mathbb {P}}^{1}}-\sum _{i=1}^{N}\bar{{\mathcal {L}}_{i}}$ and $\textrm{supp}{\mathcal {B}}\subset {\mathcal {X}}|_{t=0}$. We can consider the ${\mathbb {C}}^{*}$-action on the fiber over $t=0$ to describe $E({\mathcal {X}},{\mathcal {L}}_{i})$ in terms of the weight $\lambda _{i,1},\dots ,\lambda _{i,N_{i,k}}$ for the action on $H^{0}({\mathcal {X}}|_{t=0}, k{\mathcal {L}}_{i}|_{t=0})$ for fixed positive integer k, where $N_{i,k}:=\dim H^{0}({\mathcal {X}}|_{t=0}, k{\mathcal {L}}_{i}|_{t=0})$. It is well known that we have

$$\begin{aligned} E({\mathcal {X}},{\mathcal {L}}_{i})=\lim _{k\rightarrow \infty } \frac{\sum _{j=1}^{N_{i,k}}\lambda _{i,j}}{kN_{i,k}}. \end{aligned}$$

In view of this formula, the equality $E({\mathcal {X}},{\mathcal {L}}_{i}+c{\mathcal {X}}|_{t=0}) =E({\mathcal {X}},{\mathcal {L}}_{i})+c$ holds under the constant replacing ${\mathcal {L}}_{i}\mapsto {\mathcal {L}}_{i}+c{\mathcal {X}}|_{t=0}$. Setting $\hat{\lambda _{i}}=N_{i,k}^{-1}(\lambda _{i,1}+\cdots +\lambda _{i,N_{i,k}})$, we can define the $L^{p}$-norm

$$\begin{aligned} \Vert ({\mathcal {X}},{\mathcal {L}}_{i})\Vert _{p}= \lim _{k\rightarrow \infty }\Bigg [ \frac{\sum _{j=1}^{N_{i,k}}| \lambda _{i,j}-\hat{\lambda _{i}} |^{p}}{k^{p}N_{i,k}} \Bigg ]^{1/p} \end{aligned}$$

which is invariant under the replacing ${\mathcal {L}}_{i}\mapsto {\mathcal {L}}_{i}+c{\mathcal {X}}|_{t=0}$. We define the $L^{p}$-norm of $({\mathcal {X}},({\mathcal {L}})_{i=1}^{N})$ by

$$\begin{aligned} \Vert ({\mathcal {X}},({\mathcal {L}}_{i})_{i=1}^{N})\Vert _{p} =\Bigg [\sum _{i=1}^{N}\Vert ({\mathcal {X}},{\mathcal {L}}_{i})\Vert _{p}^{p}\Bigg ]^{1/p}. \end{aligned}$$

Finally, we define the coupled H-invariant (cf. [15])

$$\begin{aligned} H_{c}\Big ({\mathcal {X}},({\mathcal {L}}_{i})_{i=1}^{N}\Big )= & {} L\Big ({\mathcal {X}},\sum _{i=1}^{N}{\mathcal {L}}_{i}\Big ) -\sum _{i=1}^{N}F({\mathcal {X}},{\mathcal {L}}_{i}) \\:= & {} L\Big ({\mathcal {X}},\sum _{i=1}^{N}{\mathcal {L}}_{i}\Big ) -\sum _{i=1}^{N}\lim _{k\rightarrow \infty } \Big [ -\log \frac{1}{N_{i,k}}\sum _{j=1}^{N_{i,k}}e^{-\frac{\lambda _{i,j}}{k}} \Big ]. \end{aligned}$$

Note that Jensen’s inequality shows $H_{c}\Big ({\mathcal {X}},({\mathcal {L}}_{i})_{i=1}^{N}\Big ) \ge D_{c}\Big ({\mathcal {X}},({\mathcal {L}}_{i})_{i=1}^{N}\Big )$.

2.2 Energy functionals and geodesic rays

We define some energy functionals on ${\mathcal {M}}$. For $\Phi =(\phi _{i})_{i=1}^{N}\in \prod _{i=1}^{N}(\textrm{PSH}(X,\theta _{i})\cap L^{\infty })$, the Monge–Ampère energy is defined by

$$\begin{aligned} E_{\theta _{i}}(\phi _{i})=\frac{1}{(n+1)V_{i}}\sum _{k=0}^{n} \int _{X}\phi _{i} \omega _{\phi _{i}}^{k} \wedge \theta _{i}^{n-k}. \end{aligned}$$

The functional $E_{\theta _{i}}$ satisfies $E_{\theta _{i}}(\phi _{i}+C)=E_{\theta _{i}}(\phi _{i})+C$ for any $C\in {\mathbb {R}}$. For $\Phi =(\phi _{i})_{i=1}^{N}\in \prod _{i=1}^{N}(\textrm{PSH}(X,\theta _{i})\cap L^{\infty })$, set

$$\begin{aligned} L(\Phi )=-\log \int _{X}e^{-\sum _{i=1}^{N}\phi _{i}}\theta _{0}^{n} \end{aligned}$$

where $\theta _{0}$ is a Kähler metric satisfying $\textrm{Ric}(\theta _{0})=\sum _{i=1}^{N}\theta _{i}$ and $\int _{X}\theta _{0}^{n}=1$. Hultgren–Witt Nyström [28] introduced the coupled Ding functional

$$\begin{aligned} D_{c}(\Phi ) =L(\Phi )-\sum _{i=1}^{N}E_{\theta _{i}}(\phi _{i}). \end{aligned}$$

For any smooth $\Phi \in {\mathcal {M}}$, by using the equality of probability measures

$$\begin{aligned} \frac{e^{-\sum _{i}\phi _{i}}\theta _{0}^{n}}{\int _{X}e^{-\sum _{i}\phi _{i}}\theta _{0}^{n}} =\frac{e^{f_{1}(\Phi )}\omega _{\phi _{1}}^{n}}{V_{1}} =\cdots =\frac{e^{f_{N}(\Phi )}\omega _{\phi _{N}}^{n}}{V_{N}}, \end{aligned}$$

(6)

we have the first variation formula

$$\begin{aligned} \delta D_{c}(\delta \Phi )= -\sum _{i=1}^{N}\frac{1}{V_{i}}\int _{X}\delta \phi _{i} \Bigl ( 1-e^{f_{i}(\Phi )} \Bigr )\omega _{\phi _{i}}^{n} \end{aligned}$$

(7)

which shows that a coupled Kähler–Einstein metric is a critical point of $D_{c}$.

In order to relate the invariants of test configurations and the energy functional, we introduce the notion of geodesic rays on the space of Kähler metrics.

Definition 2.4

Let $\theta$ is a Kähler form on X and $\Delta ^{*}$ be the punctured unit disc in ${\mathbb {C}}$. We identify $\theta$ with its lift to $X\times \Delta ^{*}$. Let $\phi (x,\tau )$ be an upper-semicontinuous, locally integrable, $S^{1}$-invariant function on $X\times \Delta ^{*}$ and $\Omega (x,\tau )$ be the (1, 1)-form on $X\times \Delta ^{*}$ defined by $\theta +\sqrt{-1}\partial {\overline{\partial }}\phi$. Then, $\phi ^{t}(x):=\phi (x, \tau ) \in \textrm{PSH}(X,\theta )$ with $t=-\log |\tau |^{2}$ is a subgeodesic ray if the restriction of $\Omega$ to $X\times \{\tau \}$ is semipositive for all $\tau$. Moreover, $\phi ^{t}(x)$ is a weak geodesic ray if it is a subgeodesic ray satisfying $\Omega ^{n+1}=0$ on $X\times \Delta ^{*}$.

The optimal $C^{1,1}$-regularity of the weak geodesic ray is proved by [11] (see also [34]).

For each locally bounded weak geodesic ray $\phi _{i}^{t}\in \textrm{PSH}(X,\theta _{i})$, the function $t\mapsto E_{\theta _{i}}(\phi _{i}^{t})$ is affine [3]. Since $\sum _{i=1}^{N}\phi _{i}^{t}$ is a locally bounded subgeodesic ray in $\textrm{PSH}(X,\sum _{i=1}^{N}\theta _{i})$, the function $t\mapsto L(\sum _{i=1}^{N}\phi ^{t}_{i})$ is convex [4]. Thus, the coupled Ding functional $D_{c}$ is convex along $(\phi _{i}^{t})_{i=1}^{N}$.

In view of [9, 33, 35], a test configuration $({\mathcal {X}},{\mathcal {L}})$ for a Fano manifold X with an ample line bundle L in the sense of Definition 2.1 defines a weak geodesic ray starting from a given Kähler potential. Note that it is equivalent to the rays constructed in [9, 33, 35], since it is known the uniqueness theorem for the completely degenerate complex Monge–Ampère equation [34]. It follows from an argument in [28, page 6786] that a test configuration $({\mathcal {X}},({\mathcal {L}}_{i})_{i=1}^{N})$ for $(X, (\alpha )_{i=1}^{N})$ in the sense of Definition 2.3 and a collection of given Kähler potentials $(\phi ^{0}_{i})_{i=1}^{N}\in {\mathcal {M}}$ induce a collection of weak geodesic rays $(\phi ^{t}_{i})_{i=1}^{N}\in \prod _{i=1}^{N}\textrm{PSH}(X,\theta _{i})$ for $t\ge 0$.

The slopes at infinity of energy functionals along $(\phi ^{t}_{i})_{i=1}^{N}$ play an important role in the moment weight type inequalities we are interested in. Results of Berman [2] showed

$$\begin{aligned} E({\mathcal {X}},{\mathcal {L}}_{i}) =\lim _{t\rightarrow \infty }\frac{E_{\theta _{i}}(\phi _{i}^{t})}{t} \quad \text {and}\quad D_{c}({\mathcal {X}},({\mathcal {L}}_{i})_{i=1}^{N}) =\lim _{t\rightarrow \infty }\frac{D_{c}((\phi _{i}^{t})_{i=1}^{N})}{t}. \end{aligned}$$

By the $C^{1,1}$-regularity of the weak geodesic ray $\phi _{i}^{t}$, the existence of the time derivative $\dot{\phi ^{t}_{i}}$ is guaranteed. Berndtsson [5] showed that the push forward probability measure $\textrm{DH}({\mathcal {X}},{\mathcal {L}}_{i}):=(\dot{\phi _{i}^{t}})_{*}(V_{i}^{-1}\omega _{\phi _{i}^{t}}^{n})$ on ${\mathbb {R}}$ is independent of t. Hisamoto [24] showed that the weak convergence of the spectral measure

$$\begin{aligned} \frac{1}{N_{i,k}}\sum _{j=1}^{N_{i,k}}\delta _{\frac{\lambda _{i,j}}{k}} \rightarrow \textrm{DH}({\mathcal {X}},{\mathcal {L}}_{i}) \end{aligned}$$

as $k\rightarrow \infty$ to obtain the equality

$$\begin{aligned} \Vert ({\mathcal {X}},{\mathcal {L}}_{i}) \Vert _{p} =\Bigg [\frac{1}{V_{i}}\int _{X}\Big | \dot{\phi _{i}^{t}}-E({\mathcal {X}},{\mathcal {L}}_{i}) \Big |^{p} \omega _{\phi _{i}^{t}}^{n}\Bigg ]^{1/p}. \end{aligned}$$

Consider the “virtual slope”

$$\begin{aligned} F(\dot{\phi _{i}^{t}})=-\log \frac{1}{V_{i}}\int _{X}e^{-\dot{\phi _{i}^{t}}}\omega _{\phi _{i}}^{n} =-\log \int _{{\mathbb {R}}}e^{-x}\textrm{DH}({\mathcal {X}},{\mathcal {L}}_{i}) \end{aligned}$$

to obtain the slope formula

$$\begin{aligned} H_{c}\Big ( {\mathcal {X}},({\mathcal {L}}_{i})_{i=1}^{N} \Big ) =\lim _{t\rightarrow \infty }\Bigg [ \frac{L(\sum _{i=1}^{N}\phi _{i}^{t})}{t}-\sum _{i=1}^{N}F(\dot{\phi _{i}^{t}}) \Bigg ]. \end{aligned}$$

2.3 Proof of Theorem 1.1

Proof of the moment weight type inequality for $R_{c}$. Fix any $\Phi =(\phi _{i})_{i=1}^{N}\in {\mathcal {M}}$ and any test configuration $({\mathcal {X}},({\mathcal {L}}_{i})_{i=1}^{N})$ for a decomposition $(\alpha _{i})_{i=1}^{N}$ of $2\pi c_{1}(X)$. Take weak geodesic rays $(\phi _{i}^{t})_{i=1}^{N}$ for $t\ge 0$ starting from $(\phi _{i})_{i=1}^{N}$ associated with $({\mathcal {X}},({\mathcal {L}}_{i})_{i=1}^{N})$.

By the convexity of the coupled Ding functional,

$$\begin{aligned} -D({\mathcal {X}},({\mathcal {L}}_{i})_{i=1}^{N}) =\lim _{t\rightarrow \infty }\frac{-D_{c}((\phi _{i}^{t})_{i=1}^{N})}{t} \le -\frac{d}{dt}\bigg |_{t=0}D_{c}((\phi _{i}^{t})_{i=1}^{N}). \end{aligned}$$

By the equality (7), the normalization of the Ricci potentials and the Schwartz inequality,

$$\begin{aligned} -\frac{\textrm{d}}{\textrm{d}t}\bigg |_{t=0}D_{c}((\phi _{i}^{t})_{i=1}^{N})= & {} \sum _{i=1}^{N}\frac{1}{V_{i}}\int _{X}\dot{\phi _{i}^{0}}\Big ( 1-e^{f_{i}(\Phi )} \Big ) \omega _{\phi _{i}}^{n}\\= & {} \sum _{i=1}^{N}\frac{1}{V_{i}}\int _{X} \Big (\dot{\phi _{i}^{0}}-E({\mathcal {X}},{\mathcal {L}}_{i})\Big )\Big ( 1-e^{f_{i}(\Phi )} \Big ) \omega _{\phi _{i}}^{n} \\\le & {} \Bigg [ \sum _{i=1}^{N}\frac{1}{V_{i}}\int _{X} \Big |\dot{\phi _{i}^{0}}-E({\mathcal {X}},{\mathcal {L}}_{i})\Big |^{2} \omega _{\phi _{i}^{0}}^{n} \Bigg ]^{1/2} \Bigg [ \sum _{i=1}^{N}\frac{1}{V_{i}}\int _{X}\Big ( 1-e^{f_{i}(\Phi )} \Big )^{2} \omega _{\phi _{i}}^{n} \Bigg ]^{1/2}\\= & {} \Vert ({\mathcal {X}},({\mathcal {L}}_{i})_{i=1}^{N})\Vert _{2} R_{c}(\Phi )^{1/2}. \end{aligned}$$

This completes the proof.

Before a proof of the moment weight type inequality for the coupled H-functional $H_{c}$, we give some remarks. For two probability measures $\mu$ and $\nu$ on X, the relative entropy is defined by

$$\begin{aligned} H(\mu | \nu )=\int _{X}\log \Big ( \frac{\mu }{\nu } \Big )\mu . \end{aligned}$$

In this terminology, $H_{c}(\Phi )$ is written as $\sum _{i=1}^{N}H(\mu _{\Phi } | \nu _{\phi _{i}})$ where $\mu _{\Phi }$ is one of the probability measures in the equality 6), that is,

$$\begin{aligned} H_{c}(\Phi ) =H\Big (\frac{e^{f_{1}(\Phi )}\omega _{\phi _{1}}^{n}}{V_{1}} \Big | \nu _{\phi _{1}}\Big ) +\cdots +H\Big (\frac{e^{f_{N}(\Phi )}\omega _{\phi _{N}}^{n}}{V_{N}} \Big | \nu _{\phi _{N}}\Big ) \end{aligned}$$

and where $\nu _{\phi _{i}}=\omega _{\phi _{i}}^{n}/V_{i}$. Note that the Csiszár–Kullback–Pinsker inequality yields the inequality

$$\begin{aligned} \sqrt{2H\Big (\frac{e^{f_{i}(\Phi )}\omega _{\phi _{i}}^{n}}{V_{i}}\Big | \nu _{\phi _{i}}\Big )} \ge \frac{1}{V_{i}}\int _{X}\big |1-e^{f_{i}(\Phi )}\big |\omega _{\phi _{i}}^{n} \end{aligned}$$

(8)

which shows that a zero point of $H_{c}$ is a coupled Kähler–Einstein metric. Note also that the relative entropy has an expression in terms of the Legendre duality as follows [1].

$$\begin{aligned} H(\mu | \nu )=\sup _{f\in C^{0}(X;{\mathbb {R}})} \Big ( \int _{X}f\mu -\log \int _{X}e^{f}\nu \Big ). \end{aligned}$$

Proof of the moment weight type inequality for $H_{c}$. We use the same notation as in the previous proof. By using the Legendre duality expression and the convexity of the function $L(\sum _{i=1}^{N}\phi _{i}^{t})$ for $t\ge 0$, we have

$$\begin{aligned} H_{c}(\Phi )\ge & {} \sum _{i=1}^{N}\Big (\int _{X}-\dot{\phi _{i}^{0}}\mu _{\Phi } -\log \int _{X}e^{-\dot{\phi _{i}^{0}}}\nu _{\phi _{i}}\Big )\\= & {} -\frac{d}{dt}\Big |_{t=0}L\Big (\sum _{i=1}^{N}\phi _{i}^{t}\Big )+\sum _{i=1}^{N}F(\dot{\phi _{i}^{0}})\\\ge & {} -\frac{L(\sum _{i=1}^{N}\phi _{i}^{t})}{t}+\sum _{i=1}^{N}F(\dot{\phi _{i}^{t}}). \end{aligned}$$

Taking $t\rightarrow \infty$, we get $H_{c}(\Phi )\ge -H_{c}({\mathcal {X}},({\mathcal {L}}_{i})_{i=1}^{N})$. This completes the proof.

3 Hessian formulas and its application

3.1 Hessian formulas

We fix notations to obtain Hessian formulas for the coupled Ricci–Calabi functional $R_c$ and the coupled H-functional $H_c$. For any $\Phi =(\phi _i)_{i=1}^N\in {\mathcal {M}}$, we write one of the probability measures in the equality (6) as $\mu _{\Phi }$. Let $\Delta _{\phi _i}$ be the negative Laplacian of the metric $\phi _i$, and let $P_{\Phi }$ and $P_{\Phi }^f$ be operators acting on $(C^{\infty }(X;{\mathbb {C}}))^N$ defined by

$$\begin{aligned} P_{\Phi }(\varvec{u})= \left( -\Delta _{\phi _i}u_i- \langle {\overline{\partial }}u_i, {\overline{\partial }}f_i(\Phi ) \rangle _{\phi _i} -\sum _{j=1}^N u_j + \int _X\sum _{j=1}^N u_j \mu _{\Phi } \right) _{i=1}^{N} \end{aligned}$$

(9)

and

$$\begin{aligned} P_{\Phi }^f(\varvec{u})= \left( \Bigl (-\Delta _{\phi _i}u_i- \langle {\overline{\partial }}u_i, {\overline{\partial }}f_i(\Phi ) \rangle _{\phi _i} -\sum _{j=1}^N u_j + \int _X\sum _{j=1}^N u_j \mu _{\Phi } \Bigr ) e^{f_i(\Phi )} \right) _{i=1}^{N} \end{aligned}$$

(10)

for $\varvec{u}=(u_i)_{i=1}^N \in (C^{\infty }(X;{\mathbb {C}}))^N$. Their complex conjugates are defined by

$$\begin{aligned} \overline{P_{\Phi }}(\varvec{u})=\overline{P_{\Phi }(\overline{\varvec{u}})} \quad \text {and}\quad \overline{P_{\Phi }^f}(\varvec{u})=\overline{P_{\Phi }^f(\overline{\varvec{u}})}. \end{aligned}$$

Recall the Hermitian inner products ${\langle \!\hspace{0.02cm} \!\langle }\cdot , \cdot {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi }$ and ${\langle \!\hspace{0.02cm} \!\langle }\cdot , \cdot {\rangle \!\hspace{0.02cm}\!\rangle }^f_{\Phi }$ on $(C^{\infty }(X;{\mathbb {C}}))^N$ are defined by

$$\begin{aligned} {\langle \!\hspace{0.02cm} \!\langle }\varvec{u}, \varvec{v} {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi } = \sum _{i=1}^N \int _X u_i \overline{v_i}\frac{\omega _{\phi _i}^n}{V_i} \quad \text {and}\quad {\langle \!\hspace{0.02cm} \!\langle }\varvec{u}, \varvec{v} {\rangle \!\hspace{0.02cm}\!\rangle }^f_{\Phi } = \sum _{i=1}^N \int _X u_i \overline{v_i}\mu _{\Phi }. \end{aligned}$$

The followings are basic properties for the operator $P_{\Phi }$ and the inner product ${\langle \!\hspace{0.02cm} \!\langle }\cdot , \cdot {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi }^f$.

Proposition 3.1

[37, Proposition 2.4]

• (1) The operator $P_{\Phi }$ is self-adjoint with respect to the inner product ${\langle \!\hspace{0.02cm} \!\langle }\cdot , \cdot {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi }^f$.

• (2) The operator $P_{\Phi }$ is non-negative, and the Kernel ${\text {Ker}}P_{\Phi }$ is equal to

  $$\begin{aligned} \{ (u_i)_{i=1}^N\!\in \!(C^{\infty }(X;{\mathbb {C}}))^N | \textrm{grad}_{\phi _1}u_1\!=\!\cdots =\textrm{grad}_{\phi _N}u_N\!=:\!V \text { and }\text { { V} is holomorphic} \}, \end{aligned}$$

  where $\textrm{grad}_{\phi _i}u_i$ is a type (1, 0) gradient vector field on X defined by

  $$\begin{aligned} i_{(\textrm{grad}_{\phi _i}u_i)}\omega _{\phi _i}=\sqrt{-1}{\bar{\partial }}u_i. \end{aligned}$$

Note that the same properties as in the above proposition holds for $P_{\Phi }^f$ and ${\langle \!\hspace{0.02cm} \!\langle }\cdot , \cdot {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi }$.

We give the first variation formula of the Ricci potential to obtain that of $R_c$ and $H_c$. Note that the variation $\delta \Phi$ of $\Phi \in {\mathcal {M}}$ is in $T_{\Phi }{\mathcal {M}}$ and it is identified with an element in $(C^{\infty }(X;{\mathbb {R}}))^N$.

Lemma 3.2

The first variation in the Ricci potential at $\Phi =(\phi _{i})_{i=1}^{N}\in {\mathcal {M}}$ is given by

$$\begin{aligned} \delta f_i(\delta \Phi )= -\Delta _{\phi _i}\delta \phi _i -\sum _{j=1}^N\delta \phi _j + \int _X\sum _{j=1}^N\delta \phi _j \mu _{\Phi } \end{aligned}$$

(11)

for any variation $\delta \Phi = (\delta \phi _i)_{i=1}^N \in T_{\Phi }{\mathcal {M}}$.

Proof

The derivation of the first equation in (1) shows $\delta f_i(\delta \Phi )= -\Delta _{\phi _i}\delta \phi _i -\sum _{j=1}^N\delta \phi _j+C$ for some constant C. The constant C is equal to $\int _X\sum _{j=1}^N\delta \phi _j \mu _{\Phi }$ since

$$\begin{aligned} 0 =\int _{X}\delta (e^{f_{i}(\Phi )}\omega _{\phi _{i}})/V_{i} =\int _X\delta f_i(\delta \Phi )\mu _{\Phi }+ \int _X\Delta _{\phi _i}\delta \phi _i\mu _{\Phi }. \end{aligned}$$

$\square$

Lemma 3.3

The first variations in $R_c$ and $H_c$ at $\Phi \in {\mathcal {M}}$ are given by

$$\begin{aligned} \delta R_c(\delta \Phi ) = 2{\langle \!\hspace{0.02cm} \!\langle }P_{\Phi }^f(e^{f_1(\Phi )}, \dots , e^{f_N(\Phi )}), \delta \Phi {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi } = 2{\langle \!\hspace{0.02cm} \!\langle }\overline{P_{\Phi }^f}(e^{f_1(\Phi )}, \dots , e^{f_N(\Phi )}), \delta \Phi {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi } \end{aligned}$$

and

$$\begin{aligned} \delta H_c(\delta \Phi ) = {\langle \!\hspace{0.02cm} \!\langle }P_{\Phi }(f_1(\Phi ), \dots , f_N(\Phi )), \delta \Phi {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi }^f = {\langle \!\hspace{0.02cm} \!\langle }\overline{P_{\Phi }}(f_1(\Phi ), \dots , f_N(\Phi )), \delta \Phi {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi }^f \end{aligned}$$

for any variation $\delta \Phi \in T_{\Phi }{\mathcal {M}}$.

Remark 3.4

In Lemma 3.3, the first variation in the coupled Ricci–Calabi functional is also expressed as $2{\langle \!\hspace{0.02cm} \!\langle }P_{\Phi }(e^{f_1(\Phi )}, \dots , e^{f_N(\Phi )}), \delta \Phi {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi }^f$. However, the expression in Lemma 3.3 is technically crucial for the proof of Theorem 1.4.

Proof

For any variation $\delta \Phi =(\delta \phi _{i})_{i=1}^{N}$, direct computations together with Lemma 3.2 show

$$\begin{aligned} \delta \int _X (1-e^{f_i(\Phi )})^2 \frac{\omega _{\phi _i}^n}{V_i}= & {} \int _X 2 e^{f_i(\Phi )}\delta f_i(\Phi ) \mu _{\Phi } + \int _X e^{f_i(\Phi )}(\Delta _{\phi _i} \delta \phi _i) \mu _{\Phi } \\= & {} -\int _X e^{f_i(\Phi )}\Delta _{\phi _i}\delta \phi _i \mu _{\Phi } -2\int _X e^{f_i(\Phi )}\sum _{j=1}^N\delta \phi _j \mu _{\Phi } \\{} & {} +2\int _X e^{f_i(\Phi )} \mu _{\Phi }\int _X\sum _{j=1}^N\delta \phi _je^{f_i(\Phi )} \mu _{\Phi } \end{aligned}$$

and

$$\begin{aligned} \delta \int _X f_i(\Phi ) e^{f_i(\Phi )} \frac{\omega _{\phi _i}^n}{V_i}= & {} \int _X \delta f_i(\delta \Phi ) \mu _{\Phi } + \int _X f_i(\Phi ) \delta f_i(\delta \Phi ) \mu _{\Phi } +\int _X f_i(\Phi ) (\Delta _{\phi _i} \delta \phi _i) \mu _{\Phi } \\= & {} -\int _X \Delta _{\phi _i}\delta \phi _i\mu _{\Phi } - \int _X f_i(\Phi ) \sum _{j=1}^N \delta \phi _j \mu _{\Phi } + \int _X f_i(\Phi ) \mu _{\Phi } \int _X \sum _{j=1}^N \delta \phi _j\mu _{\Phi }. \end{aligned}$$

By the integration by parts, we have

$$\begin{aligned} \int _X e^{f_i(\Phi )}\Delta _{\phi _i}\delta \phi _i \mu _{\Phi } =2\int _X\delta \phi _i(\Delta _{\phi _i}e^{f_i(\Phi )}+\langle {\overline{\partial }}e^{f_i(\Phi )},{\overline{\partial }}f_i(\Phi ) \rangle _{\phi _i})\mu _{\Phi } \end{aligned}$$

to show that

$$\begin{aligned} \delta R_c(\delta \Phi )= & {} 2\sum _{i=1}^N \int _X\delta \phi _i(-\Delta _{\phi _i}e^{f_i(\Phi )}-\langle {\overline{\partial }}e^{f_i(\Phi )},{\overline{\partial }}f_i(\Phi ) \rangle _{\phi _i})\mu _{\Phi } \\{} & {} - 2\int _X\sum _{j=1}^N e^{f_j(\Phi )} \sum _{i=1}^N \delta \phi _i \mu _{\Phi } + 2\int _X\sum _{j=1}^N e^{f_j(\Phi )} \mu _{\Phi } \int _X \sum _{i=1}^N \delta \phi _i\mu _{\Phi } \\= & {} 2{\langle \!\hspace{0.02cm} \!\langle }P_{\Phi }^f(e^{f_1(\Phi )}, \dots , e^{f_N(\Phi )}), \delta \Phi {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi }. \end{aligned}$$

On the other hand, by the integration by parts, we have

$$\begin{aligned} \int _X\Delta _{\phi _i}\delta \phi _i\mu _{\Phi } =\int _X\delta \phi _i(\Delta _{\phi _i}f_i(\Phi )+\langle {\overline{\partial }}f_i(\Phi ),{\overline{\partial }}f_i(\Phi ) \rangle _{\phi _i})\mu _{\Phi }, \end{aligned}$$

to show that

$$\begin{aligned} \delta H_c(\delta \Phi )= & {} \sum _{i=1}^N \int _X\delta \phi _i(-\Delta _{\phi _i}f_i(\Phi )-\langle {\overline{\partial }}f_i(\Phi ),{\overline{\partial }}f_i(\Phi ) \rangle )\mu _{\Phi } \\{} & {} - \int _X\sum _{j=1}^N f_j(\Phi ) \sum _{i=1}^N \delta \phi _i \mu _{\Phi } + \int _X\sum _{j=1}^N f_j(\Phi ) \mu _{\Phi } \int _X \sum _{i=1}^N \delta \phi _i\mu _{\Phi } \\= & {} {\langle \!\hspace{0.02cm} \!\langle }P_{\Phi }(f_1(\Phi ), \dots , f_N(\Phi )), \delta \Phi {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi }^f. \end{aligned}$$

Similarly, we have

$$\begin{aligned} \int _Xe^{f_i(\Phi )}\Delta _{\phi _i}\delta \phi _i\mu _{\Phi } =2\int _X\delta \phi _i(\Delta _{\phi _i}e^{f_i(\Phi )} +\overline{\langle {\overline{\partial }} e^{f_i(\Phi )},{\overline{\partial }} f_i(\Phi ) \rangle }_{\phi _i})\mu _{\Phi } \end{aligned}$$

and

$$\begin{aligned} \int _X\Delta _{\phi _i}\delta \phi _i\mu _{\Phi } =\int _X\delta \phi _i(\Delta _{\phi _i}f_i(\Phi ) +\overline{\langle {\overline{\partial }} f_i(\Phi ),{\overline{\partial }} f_i(\Phi ) \rangle }_{\phi _i})\mu _{\Phi } \end{aligned}$$

which yield the equalities

$$\begin{aligned} \delta R_c(\delta \Phi ) = 2{\langle \!\hspace{0.02cm} \!\langle }\overline{P_{\Phi }^f}(e^{f_1(\Phi )}, \dots , e^{f_N(\Phi )}), \delta \Phi {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi } \end{aligned}$$

and

$$\begin{aligned} \delta H_c(\delta \Phi )={\langle \!\hspace{0.02cm} \!\langle }\overline{P_{\Phi }}(f_1(\Phi ), \dots , f_N(\Phi )), \delta \Phi {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi }^f. \end{aligned}$$

This completes the proof. $\square$

Therefore, Proposition 3.1 and Lemma 3.3 show that a pair $\Phi =(\phi _i)_{i=1}^N\in {\mathcal {M}}$ is a coupled Mabuchi soliton if and only if the vector fields $\textrm{grad}_{\phi _i}e^{f_i(\Phi )}$ are holomorphic and $\textrm{grad}_{\phi _1}e^{f_1(\Phi )}=\cdots =\textrm{grad}_{\phi _N}e^{f_N(\Phi )}$. Similarly, $\Phi =(\phi _i)_{i=1}^N\in {\mathcal {M}}$ is a coupled Kähler–Ricci soliton if and only if the vector fields $\textrm{grad}_{\phi _i}f_i(\Phi )$ are holomorphic and $\textrm{grad}_{\phi _1}f_1(\Phi )=\cdots =\textrm{grad}_{\phi _N}f_N(\Phi )$.

Now we prove the Hessian formulas. The following argument generalizes the authors’s one in [30].

Proof of Theorem 1.4

We first compute the variation $(\delta _{\Phi } P_{\Phi }^f)(e^{f_1(\Phi )}, \dots , e^{f_N(\Phi )})$ at a coupled Mabuchi soliton $\Phi =(\phi _i)_{i=1}^N\in {\mathcal {M}}$ to obtain the Hessian of the coupled Ricci–Calabi functional

$$\begin{aligned} \frac{1}{2} \textrm{Hess}(R_c)(\delta \Phi ,\delta \Psi )= & {} {\langle \!\hspace{0.02cm} \!\langle }\delta _{\Phi }(P_{\Phi }^f(e^{f_1(\Phi )}, \dots , e^{f_N(\Phi )})), \delta \Psi {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi } \\= & {} {\langle \!\hspace{0.02cm} \!\langle }(\delta _{\Phi }P_{\Phi }^f)(e^{f_1(\Phi )}, \dots , e^{f_N(\Phi )}), \delta \Psi ) {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi } \\{} & {} + {\langle \!\hspace{0.02cm} \!\langle }P_{\Phi }^f(\delta _{\Phi }(e^{f_1(\Phi )}), e^{f_2(\Phi )}, \dots , e^{f_N(\Phi )}), \delta \Psi ) {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi } \\{} & {} + \cdots \\{} & {} + {\langle \!\hspace{0.02cm} \!\langle }P_{\Phi }^f(e^{f_1(\Phi )}, \dots , e^{f_{N-1}(\Phi )}, \delta _{\Phi }( e^{f_N(\Phi )})), \delta \Psi ) {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi } \end{aligned}$$

where $\delta _{\Phi }$ stands for the variation along $\delta \Phi =(\delta \phi _i)_{i=1}^N$ at $\Phi$ and $\delta \Psi$ is another variation at $\Phi$. Now we have the holomorphic vector field

$$\begin{aligned} Z_R:=\textrm{grad}_{\phi _1}e^{f_1(\Phi )}=\cdots =\textrm{grad}_{\phi _N}e^{f_N(\Phi )}, \end{aligned}$$

since $\Phi =(\phi _i)_{i=1}^N$ is a coupled Mabuchi soliton. Note that $Z_R$ is also expressed as

$$\begin{aligned} Z_R=\textrm{grad}_{\phi _1+t \delta \phi _1}(e^{f_1(\Phi )}+t Z_R(\delta \phi _1))=\cdots =\textrm{grad}_{\phi _N+t \delta \phi _N}(e^{f_1(\Phi )}+t Z_R(\delta \phi _N)). \end{aligned}$$

Indeed, the equality $i_{Z_R}(\omega _{\phi _i}+t\sqrt{-1}\partial {\overline{\partial }}\delta \phi _i)=\sqrt{-1}{\overline{\partial }}(e^{f_i(\Phi )}+t Z_R (\delta \phi _i))$ holds for each i and for any small $t\in (-\varepsilon ,\varepsilon )$. Set $\Phi _t=(\phi _i+t\delta \phi _i)_{i=1}^N\in {\mathcal {M}}$ as a perturbation of $\Phi$. Since $Z_{R}\in {\text {Ker}}P^{f}_{\Phi _{t}}$ by Proposition 3.1, we then have

$$\begin{aligned} P_{\Phi _t}^f( e^{f_1(\Phi )}+t Z_R(\delta \phi _1), \dots , e^{f_N(\Phi )}+t Z_R(\delta \phi _N) )=0. \end{aligned}$$

Thus, the derivative of the above equation at $t=0$ yields the equality

$$\begin{aligned} (\delta _{\Phi } P_{\Phi }^f)(e^{f_1(\Phi )}, \dots , e^{f_N(\Phi )})= & {} - P_{\Phi }^f (Z_R(\delta \phi _1), e^{f_2(\Phi )}, \dots , e^{f_N(\Phi )}) \nonumber \\{} & {} - \cdots \nonumber \\{} & {} - P_{\Phi }^f (e^{f_1(\Phi )}, \dots ,e^{f_{N-1}(\Phi )}, Z_R(\delta \phi _N)). \end{aligned}$$

(12)

Therefore, by using Eq. (12), the formula $Z_R(\delta \phi _i)= \overline{\langle {\overline{\partial }} \delta \phi _i, {\overline{\partial }}(e^{f_i(\Phi )}) \rangle }_{\phi _i}$ and the formula of the derivative of $e^{f_i(\Phi )}$ in Lemma 3.2, we obtain

$$\begin{aligned}{} & {} \delta _{\Phi }(P_{\Phi }^f(e^{f_1(\Phi )}, \dots , e^{f_N(\Phi )})) \\{} & {} \quad = P_{\Phi }^f (-\overline{\langle {\overline{\partial }} \delta \phi _1, {\overline{\partial }}(e^{f_1(\Phi )}) \rangle }_{\phi _1} + \delta _{\Phi }(e^{f_1(\Phi )}), \dots , -\overline{\langle {\overline{\partial }} \delta \phi _N, {\overline{\partial }}(e^{f_N(\Phi )}) \rangle }_{\phi _N} + \delta _{\Phi }(e^{f_N(\Phi )})) \\{} & {} \quad = P_{\Phi }^f\overline{P_{\Phi }^f}(\delta \Phi ). \end{aligned}$$

Similarly, $\delta _{\Phi }(\overline{P_{\Phi }^f}(e^{f_1(\Phi )}, \dots , e^{f_N(\Phi )})) = \overline{P_{\Phi }^f}P_{\Phi }^f(\delta \Phi ).$ This completes the proof of the Hessian formula for the coupled Ricci–Calabi functional.

In order to prove the Hessian formula for the coupled H-functional $H_c$, we follow the same argument as above to obtain

$$\begin{aligned} (\delta _{\Phi } P_{\Phi })(f_1(\Phi ), \dots , f_N(\Phi ))= & {} - P_{\Phi } (Z_H(\delta \phi _1), f_2(\Phi ), \dots , f_N(\Phi )) \nonumber \\{} & {} - \cdots \nonumber \\{} & {} - P_{\Phi } (f_1(\Phi ), \dots , f_{N-1}(\Phi ), Z_H(\delta \phi _N)) \end{aligned}$$

(13)

where $\Phi =(\phi _i)_{i=1}^N\in {\mathcal {M}}$ is a coupled Kähler–Ricci soliton, $Z_H$ is the holomorphic vector field $\textrm{grad}_{\phi _1}f_1(\Phi )=\cdots =\textrm{grad}_{\phi _N}f_N(\Phi )$ and $\delta \Phi :=(\delta \phi _i)_{i=1}^N$ is a variation at $\Phi$. By equation (13), the formula $Z_H(\delta \phi _i) =\overline{\langle {\overline{\partial }} \delta \phi _i, {\overline{\partial }}(f_i(\Phi )) \rangle }_{\phi _i}$ and Lemma 3.2, we have

$$\begin{aligned} \textrm{Hess}(R_c)(\delta \Phi ,\delta \Psi )= & {} {\langle \!\hspace{0.02cm} \!\langle }(\delta _{\Phi }P_{\Phi })(f_1(\Phi ), \dots , f_N(\Phi )), \delta \Psi ) {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi }^f \\{} & {} + {\langle \!\hspace{0.02cm} \!\langle }P_{\Phi }(\delta _{\Phi }f_1(\Phi ), f_2(\Phi ), \dots , f_N(\Phi )), \delta \Psi ) {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi }^f \\{} & {} + \cdots \\{} & {} + {\langle \!\hspace{0.02cm} \!\langle }P_{\Phi }(f_1(\Phi ), \dots , f_{N-1}(\Phi ), \delta _{\Phi }f_N(\Phi )), \delta \Psi ) {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi }^f \\= & {} {\langle \!\hspace{0.02cm} \!\langle }P_{\Phi }\overline{P_{\Phi }}(\delta \Phi ), \delta \Psi {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi }^f, \end{aligned}$$

where $\delta \Psi$ is another variation at $\Phi$. Similarly $\textrm{Hess}(H_c)(\delta \Phi , \delta \Psi ) ={\langle \!\hspace{0.02cm} \!\langle }\overline{P_{\Phi }}P_{\Phi }(\delta \Phi ), \delta \Psi {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi }^f$. This completes the proof of Theorem 1.4.

Proof of Corollary 1.5

This is a consequence of Theorem  1.4 and the non-negativity of $P_{\Phi }$ (resp. $P_{\Phi }^f$) with respect to ${\langle \!\hspace{0.02cm} \!\langle }\cdot , \cdot {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi }^f$ (resp. ${\langle \!\hspace{0.02cm} \!\langle }\cdot , \cdot {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi }$) in Proposition 3.1.

To end this subsection, we discuss the coupled flows introduced in Sect. 1. Lemma 3.3 shows the following.

Corollary 3.5

The coupled Ricci–Calabi functional $R_{c}$ is monotonically decreasing along a coupled inverse Monge–Ampére flow in the sense of (3). The coupled H-functional $H_{c}$ is monotonically decreasing along a coupled Kähler–Ricci flow in the sense of (2).

Proof

Set $F_{R}=F_{R}(t)=(e^{f_{i}(\Phi _{R}(t))})_{i=1}^{N}$ where $\Phi _{R}=\Phi _{R}(t)\in {\mathcal {M}}$ is a coupled inverse Monge–Ampére flow. Set $F_{H}=F_{H}(t)=(f_{i}(\Phi _{H}(t))_{i=1}^{N}$ where $\Phi _{H}=\Phi _{H}(t)\in {\mathcal {M}}$ is a coupled Kähler–Ricci flow. Lemma 3.3 shows

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}R_{c}(\Phi _{R})=-2{\langle \!\hspace{0.02cm} \!\langle }P^{f}_{\Phi _{R}}(F_{R}), F_{R} {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi _{R}} \quad \text {and}\quad \frac{\textrm{d}}{\textrm{d}t}H_{c}(\Phi _{H})=-{\langle \!\hspace{0.02cm} \!\langle }P_{\Phi _{H}}(F_{H}), F_{H} {\rangle \!\hspace{0.02cm}\!\rangle }_{\Phi _{H}}^{f} \end{aligned}$$

which are both non-positive by Proposition 3.1. This completes the proof. $\square$

Since a coupled Mabuchi soliton is a self-similar solution of a coupled inverse Monge–Ampére flow $\Phi _{R}(t)$, Corollary 1.5 suggests that for a Fano manifold admitting a coupled Mabuchi soliton, this flow $\Phi _{R}(t)$ starting from any metric converges to a coupled Mabuchi soliton in some sense. The same statement for a coupled Kähler–Ricci flow and a coupled Kähler–Ricci soliton is expected to hold.

3.2 An application to Matsushima type obstruction theorem

As an application of the commutativity of the operators in Corollary 1.5, we prove Theorem 1.6.

Proof of Theorem 1.6

We first fix a coupled Mabuchi soliton $\Phi =(\phi _i)_{i=1}^N$. Since operators $P_{\Phi }^f$ and $\overline{P_{\Phi }^f}$ are commutative by Corollary 1.5, $\overline{P_{\Phi }^f}\in \textrm{End}({\text {Ker}}P_{\Phi }^f)$. Let $E_{\lambda }$ be the $\lambda$-eigenspace of $\overline{P_{\Phi }^f}$ in ${\text {Ker}}P_{\Phi }^f$. Then, we have

$$\begin{aligned} {\text {Ker}}P_{\Phi }^f = \sum _{\lambda \ge 0}E_{\lambda }. \end{aligned}$$

(14)

Note that, by Proposition 3.1, every $V\in {\mathfrak {h}}(X)$ is written as $V=\textrm{grad}_{\phi _1}u_1=\cdots = \textrm{grad}_{\phi _N}u_N$ for some $(u_i)_{i=1}^N \in {\text {Ker}}P_{\Phi }^f$ which are unique up to additive constants. Thus, setting

$$\begin{aligned} {\mathfrak {h}}_{\lambda }(X) =\{ V\in {\mathfrak {h}}(X) | V=\textrm{grad}_{\phi _1}u_1=\cdots = \textrm{grad}_{\phi _N}u_N \quad \text {and}\quad (u_i)_{i=1}^N\in E_{\lambda } \} \end{aligned}$$

and using the relation (14), we have the decomposition

$$\begin{aligned} {\mathfrak {h}}(X)=\sum _{\lambda \ge 0}{\mathfrak {h}}_{\lambda }(X). \end{aligned}$$

(15)

Here we claim that ${\mathfrak {h}}_{\lambda }(X)$ is the $\lambda$-eigenspace of the adjoint action of $Z_R:=\textrm{grad}_{\phi _1}(1-e^{f_1(\Phi )}) = \cdots = \textrm{grad}_{\phi _N}(1-e^{f_N(\Phi )})$. To see this, fix an element $V=\textrm{grad}_{\phi _1}u_1=\cdots = \textrm{grad}_{\phi _N}u_N$ in ${\mathfrak {h}}_{\lambda }(X)$ where $(u_i)_{i=1}^N\in E_{\lambda }$ and observe

$$\begin{aligned}{} & {} \lambda (u_1, \dots , u_N) \\{} & {} \quad = \overline{P_{\Phi }^f} (u_1, \dots , u_N) \\{} & {} \quad = (\overline{P_{\Phi }^f} - P_{\Phi }^f) (u_1, \dots , u_N) \\{} & {} \quad = (-\overline{\langle {\overline{\partial }} {\overline{u}}_{1}, {\overline{\partial }} e^{f_1(\Phi )} \rangle }_{\phi _1} + \langle {\overline{\partial }}u_1, {\overline{\partial }}e^{f_1(\Phi )} \rangle _{\phi _1}, \dots , -\overline{\langle {\overline{\partial }} {\overline{u}}_N, {\overline{\partial }} e^{f_N(\Phi )} \rangle }_{\phi _N} + \langle {\overline{\partial }}u_N, {\overline{\partial }}e^{f_N(\Phi )} \rangle _{\phi _N}) \\{} & {} \quad = (\{e^{f_1(\Phi )}, u_1 \}_{\phi _1}, \dots , \{e^{f_N(\Phi )}, u_N \}_{\phi _N}) \end{aligned}$$

where $\{\cdot ,\cdot \}_{\phi _i}$ is the Poisson bracket defined by $\{u,v\}_{\phi _i}=(\textrm{grad}_{\phi _i}v)u-(\textrm{grad}_{\phi _i}u)v$. Since the map $u\rightarrow -\textrm{grad}_{\phi _i}u$ is a complex Lie algebra homomorphism from $(C^{\infty }(X;{\mathbb {C}}), \{\cdot ,\cdot \}_{\phi _i})$ to $(\Gamma (TX), [\cdot ,\cdot ])$ where $[\cdot ,\cdot ]$ is the Lie bracket defined by $[Z,W]=ZW-WZ$, it follows that

$$\begin{aligned} \lambda V= & {} \textrm{grad}_{\phi _i}\lambda u_i \\= & {} \textrm{grad}_{\phi _i}\{e^{f_i(\Phi )}, u\}_{\phi _i} \\= & {} [-\textrm{grad}_{\phi _i}e^{f_i(\Phi )}, \textrm{grad}_{\phi _i}u ] \\= & {} [ Z_R, V]. \end{aligned}$$

This shows ${\mathfrak {h}}_{\lambda }(X)$ is the $\lambda$-eigenspace of the adjoint action of $Z_R$.

We next focus on ${\mathfrak {h}}_0(X)$. Since $E_0={\text {Ker}}P_{\Phi }^f \cap {\text {Ker}}\overline{P_{\Phi }^f}$, then both the real part $(\textrm{Re}u_i)_{i=1}^N$ and the imaginary part $(\textrm{Im}u_i)_{i=1}^N$ of $(u_i)_{i=1}^N\in E_0$ are in $E_0$ again. It follows that

$$\begin{aligned} E_0=\{ (u_i)_{i=1}^N \in (\sqrt{-1}C^{\infty }(X;{\mathbb {R}}))^N | \textrm{grad}_{\phi _1}u_1=\cdots =\textrm{grad}_{\phi _N}u_N \in {\mathfrak {h}}(X) \}\otimes {\mathbb {C}} \end{aligned}$$

and thus

$$\begin{aligned} {\mathfrak {h}}_0(X)= \{ V+{\overline{V}} | (u_i)_{i=1}^N \in (\sqrt{-1}C^{\infty }(X;{\mathbb {R}}))^N \quad \text {and}\quad V= \textrm{grad}_{\phi _1}u_1=\cdots =\textrm{grad}_{\phi _N}u_N \} \otimes {\mathbb {C}}. \end{aligned}$$

According to [18, Lemma 2.3.8], the vector filed $V+{\overline{V}}=\textrm{grad}_{\phi _i}u_i+\overline{\textrm{grad}_{\phi _i}u_i}$ as above is killing with respect to the Kähler metric $\omega _{\phi _i}$. Therefore, ${\mathfrak {h}}_0(X)$ is the complexification of the Lie algebra of Killing vector fields for $\omega _{\phi _i}$. This completes the proof for the case when $\Phi =(\phi _i)_{i=1}^N$ is a coupled Mabuchi soliton.

When $\Phi =(\phi _i)_{i=1}^N$ is a coupled Kähler–Ricci soliton, we follow the same argument as above to obtain the decomposition

$$\begin{aligned} {\mathfrak {h}}(X)=\sum _{\lambda \ge 0}{\mathfrak {h}}_{\lambda }(X), \end{aligned}$$

where ${\mathfrak {h}}_{\lambda }(X) =\{ V\in {\mathfrak {h}}(X) | V=\textrm{grad}_{\phi _1}u_1=\cdots = \textrm{grad}_{\phi _N}u_N \quad \text {and}\quad (u_i)_{i=1}^N\in E_{\lambda } \}$ and $E_{\lambda }$ is the $\lambda$-eigenspace of $\overline{P_{\Phi }}$ in ${\text {Ker}}P_{\Phi }$. In order to finish the proof, we only check that ${\mathfrak {h}}_{\lambda }(X)$ is the $\lambda$-eigenspace of the adjoint action of $Z_H:=-\textrm{grad}_{\phi _1}f_1(\Phi )=\cdots =-\textrm{grad}_{\phi _N}f_N(\Phi )$. For any $V=\textrm{grad}_{\phi _1}u_1=\cdots = \textrm{grad}_{\phi _N}u_N$ in ${\mathfrak {h}}_{\lambda }(X)$ where $(u_i)_{i=1}^N\in E_{\lambda }$, we have

$$\begin{aligned}{} & {} \lambda (u_1, \dots , u_N) \\{} & {} \quad = \overline{P_{\Phi }} (u_1, \dots , u_N) \\{} & {} \quad = (\overline{P_{\Phi }} - P_{\Phi }) (u_1, \dots , u_N) \\{} & {} \quad = (-\overline{\langle {\overline{\partial }} {\overline{u}}_1, {\overline{\partial }} f_1(\Phi ) \rangle }_{\phi _1} + \langle {\overline{\partial }}u_1, {\overline{\partial }}f_1(\Phi ) \rangle _{\phi _1}, \dots , -\overline{\langle {\overline{\partial }} {\overline{u}}_N, {\overline{\partial }} f_N(\Phi ) \rangle }_{\phi _N} + \langle {\overline{\partial }}u_N, {\overline{\partial }}f_N(\Phi ) \rangle _{\phi _N}) \\{} & {} \quad = (\{f_1(\Phi ), u_1 \}_{\phi _1}, \dots , \{f_N(\Phi ), u_N \}_{\phi _N}). \end{aligned}$$

This yields the equality $\lambda V = [Z_H, V].$ The remaining proof is very similar to the case of coupled Mabuchi solitons.