Anticanonically balanced metrics and the Hilbert–Mumford criterion for the $\delta_{m}$-invariant of Fujita–Odaka

Canonical metrics on Fano manifolds have been a focus of intensive studies in recent years, particularly in relation to the stability notions in Geometric Invariant Theory (GIT) [MFK]. There are many important results, but we mention here only the ones that are directly related to the results in this paper. The existence of Kähler–Einstein metrics on a Fano manifold with no nontrivial holomorphic vector fields is equivalent to the stability condition called the uniform Ding stability [BBJ, Theorem A]. This stability condition is given in terms of objects called the test configurations, which is a certain class of degenerations of Fano manifolds, and plays the role of the $\mathbb{C}^{*}$-action in the Hilbert–Mumford criterion of stability [MFK]. The uniform Ding stability can be characterised by an invariant called the $\delta$-invariant introduced by Fujita–Odaka [fo18], in the sense that it is equivalent to $\delta>1$ by [bj20, Fujita2019, fo18]. In a recent paper, K. Zhang [Zhang21] directly proved that $\delta>1$ implies the existence of Kähler–Einstein metrics, without using the uniform Ding stability itself and based instead on a finite dimensional approximation (often called quantisation) which is briefly reviewed below; indeed, all the objects mentioned above have finite dimensional approximations in an appropriate sense.

Donaldson [donproj1] proved a foundational theorem that a Kähler metric of constant scalar curvature (such as Kähler–Einstein metrics) can be approximated by a sequence of Fubini–Study metrics called the balanced metrics, as long as the automorphism group is discrete. Combined with the theorem due to H. Luo [Luo] and S. Zhang [zhang96] (see also Phong–Sturm [PS03]), which proves that a balanced metric exists if and only if the manifold is Chow stable, Donaldson’s theorem implies that a Kähler manifold admitting a constant scalar curvature Kähler metric is asymptotically Chow stable [donproj1, Corollary 4]. The asymptotic Chow stability can be regarded as a finite dimensional approximation of the $K$-stability, which has a well-understood relationship to the Ding stability, in the sense that the Donaldson–Futaki invariant can be written as the limit of a sequence of Chow weights.

For Fano manifolds, we have a variant of balanced metrics that is called the anticanonically balanced metrics, which was introduced by Donaldson [donnum09, §2.2.2]. The analogue of Donaldson’s theorem [donproj1] above was established by Berman–Witt Nyström [BWN14], who proved that the existence of Kähler–Einstein metrics (resp. Kähler–Ricci $g$-solitons) implies that we can find a sequence of anticanonically balanced metrics that converges to the Kähler–Einstein metric (resp. the Kähler–Ricci $g$-soliton), in the sense of currents; see also [tak15] for the case of solitons. The convergence can in fact be improved to the smooth convergence by [tak19, Theorem 1.3 with $N=1$] and [Ioos20, Ioos21]. From the point of view of stability, Saito–Takahashi [st19] defined a notion of stability that can be regarded as an anticanonical version of the Chow stability, and proved that it is implied by the existence of anticanonically balanced metrics.

The $\delta$-invariant of Fujita–Odaka also has a finite dimensional approximation called the $\delta_{m}$-invariant, which originally appeared as an intermediate object for the definition of the $\delta$-invariant. Recently, Rubinstein–Tian–Zhang [rtz] proved that the $\delta_{m}$-invariant characterises the existence of anticanonically balanced metrics. The theorem of K. Zhang [Zhang21] mentioned above relies on this result.

While many foundational works so far established a deep understanding of the uniform Ding stability, the $\delta$-invariant, and the Kähler–Einstein metrics, the relationship between the finite dimensional analogues of these concepts does not seem to be complete in the sense that no GIT stability condition, given in terms of a Hilbert–Mumford type criterion involving test configurations, has been proved to fully characterise $\delta_{m}>1$ or the anticanonically balanced metrics. This is the question that is addressed in this paper.

The main result of this paper is the following, which can be regraded as the anticanonical version of the theorem by H. Luo [Luo] and S. Zhang [zhang96], and establishes the required correspondence between the anticanonically balanced metrics and the GIT stability.

One direction of the above result, i.e. the existence of anticanonically balanced metrics implying the stated stability condition, was proved by Saito–Takahashi [st19, Theorem 1.2]. Thus the main point of the above theorem is the stability implying the anticanonically balanced metrics, although the proof that we give in this paper easily establishes both directions.

The main application of the above theorem, combined with Rubinstein–Tian–Zhang [rtz, Theorem 2.3], is the following result.

While the above is a result in algebraic geometry, it seems that no purely algebro-geometric proof is known at the moment of writing this paper; the proof that we give in §5.1 relies on Theorem 1.1 and [rtz, Theorem 2.3] which concern the anticanonically balanced metrics in differential geometry. Note also that, in the above corollary, the triviality of test configurations is defined in a way that is not commonly used recently; see Remark 3.9.

For the definitions of the terminologies that arise in the above results, the reader is referred to §2.2, §3.2, and §A.1.

We also extend these results to the Kähler–Ricci $g$-solitons (Theorem 5.6 and Corollary 5.8) and the coupled Kähler–Einstein metrics (Theorem 5.11 and Corollary 5.12). While the main argument of the proof carries over almost word by word for both cases, it turns out that we need to strengthen the stability condition for the coupled Kähler–Einstein case proposed by Hultgren–Witt Nyström [HWN19]. For this purpose, we first define the notion of a test configuration generated by the $\mathbb{C}^{*}$-actions of multiple test configurations (Definition 3.15). We then define a strengthened version of the coupled Ding invariant (Definition 3.16), and prove that it naturally arises as the asymptotic slope of the coupled Ding functional (Theorem 5.9). It seems natural to expect that this result can be applied to give further results for the coupled Kähler–Einstein metrics, as pointed out at the end of §3.4, but we shall only consider its implications to the anticanonically balanced metrics in this paper. The case of Kähler–Ricci $g$-solitons is also treated similarly, but the analogue of Corollary 1.2 is weaker than the full characterisation of $\delta^{g}_{m}>1$ (see Corollary 5.8 and §A.2) because of nontrivial holomorphic vector fields, which is analogous to the well-known phenomenon that varieties with nontrivial holomorphic vector fields are never (uniformly) Ding stable.

It should also be possible to consider the twisted version of above results as in [rtz, §4.3], but we decide not to treat such cases since Proposition 5.3, which is the key estimate in the proof of Theorem 1.1, seems more complicated with the twist term.

We start by reviewing the differential-geometric preliminaries in §2. While most of the materials in §2 are well-known, Lemma 2.18 for the coupled Ding functional seems to be a new result that plays an important role later. Algebro-geometric preliminaries are recalled in §3; again this is mostly a review of well-known results, but in §3.4 we define the notion of a test configuration generated by the $\mathbb{C}^{*}$-actions of multiple test configurations, and define the strengthened version of the coupled Ding invariant (Definition 3.16) which seems to be new. The relationship between these analytic and algebraic concepts are reviewed in §4, which is a summary of many well-established foundational results.

The proof of the main results is given in §5; we note that, almost as a by-product of the proof for the coupled version, we obtain a formula (Theorem 5.9) for the asymptotic slope of the coupled Ding functional along a coupled psh ray that does not seem to have been considered before and seems appropriate for the study of the coupled Kähler–Einstein metrics. While the invariants $\delta$ and $\delta_{m}$ are important objects that appear in Corollary 1.2, we only review them in Appendix A since the main body of this paper does not quite depend on these invariants.

Let $(X,-K_{X})$ be a Fano manifold of complex dimension $n$, polarised by the anticanonical bundle. Throughout this paper, we use the additive notation for the tensor product of line bundles and write $\mathrm{Vol}(X)$ for $\int_{X}c_{1}(-K_{X})^{n}$. We work with the very ample line bundle $-mK_{X}$, by choosing $m\in\mathbb{N}$ to be sufficiently large; we shall further assume that $m$ is sufficiently divisible for the coupled Kähler–Einstein case in §2.4. We fix a reference hermitian metric $h_{0}$ on $-mK_{X}$, rather than $-K_{X}$, and write $\omega_{0}\in c_{1}(-K_{X})$ for the associated Kähler metric re-scaled by $1/m$. This reference metric is assumed to satisfy various extra hypotheses according to the situation under consideration; we further assume $h_{0}$ to be a Fubini–Study metric defined in §2.2 for convenience, and moreover to be $T$-invariant when we consider the Kähler–Ricci $g$-solitons in §2.3. The coupled Kähler–Einstein case §2.4 is slightly more complicated as there will be an auxiliary reference metric (16), but the one in (18) is the relevant one which we denote by $h_{0}$. In any case, without loss of generality we may assume that $h_{0}$ satisfies all these required properties and fix the notation once and for all to streamline the notational convention.

We define

$$\mathcal{H}:=\{\phi\in C^{\infty}(X,\mathbb{R})\mid\omega_{0}+\sqrt{-1}\partial\bar{\partial}\phi/2\pi>0\}$$

for the space of smooth Kähler metrics in $c_{1}(-K_{X})$. Note that a hermitian metric $e^{-m\phi}h_{0}$ on $-mK_{X}$, with $\phi\in\mathcal{H}$, corresponds to the Kähler metric $\omega_{\phi}:=\omega_{0}+\sqrt{-1}\partial\bar{\partial}\phi/2\pi$.

An elementary yet important observation is that a hermitian metric on $-K_{X}$ naturally defines a volume form on $X$; more precisely, a hermitian metric on $-K_{X}$ is a smooth positive section of $\mathcal{H}om_{C^{\infty}_{X}}((-K_{X})\otimes\overline{(-K_{X})},\mathbb{C})$, which corresponds to the one of $K_{X}\otimes\overline{K_{X}}$, i.e. a volume form. To clarify the notation, we shall write $d\mu_{\phi}$ for the volume form on $X$ defined by the hermitian metric $e^{-m\phi}h_{0}$ on $-mK_{X}$ with $\phi\in\mathcal{H}$; note that we have

$$d\mu_{\phi}=e^{-\phi}d\mu_{0}.$$

(1)

It is convenient to re-scale $h_{0}$ by a nonzero constant if necessary so that $\int_{X}d\mu_{0}=1$.

Another notational convention that we use is

$$\fint_{X}fd\mu_{\phi}:=\left(\int_{X}d\mu_{\phi}\right)^{-1}\int_{X}fd\mu_{\phi}$$

for an integrable function $f$ on $X$, noting that this is just a re-scaling so as to have $\fint_{X}d\mu_{\phi}=1$.

We recall the following definition.

A straightforward computation reveals that a critical point of $\mathscr{D}$ is precisely the metric that satisfies $\omega^{n}_{\phi}=d\mu_{\phi}$, which is equivalent to the Kähler–Einstein equation

$$\mathrm{Ric}(\omega_{\phi})=\omega_{\phi}.$$

An important result due to Berndtsson [Ber09a, ber15] is that $\mathscr{L}$ is convex along psh (plurisubharmonic) rays in $\mathcal{H}$, which we recall later (Theorem 4.2) in the form that is necessary for the proof of our main results. Note that Berndtsson’s convexity implies that the critical point, i.e. the Kähler–Einstein metric, is unique [ber15, Theorems 1.2 and 5.1], giving an alternative proof of the result originally obtained by Bando–Mabuchi [BanMab] that can be generalised to more singular situations.

Note finally that the convention $\int_{X}d\mu_{0}=1$ implies $\mathscr{L}(0)=\mathscr{E}(0)=\mathscr{D}(0)=0$.

An important subset of $\mathcal{H}$ is the Fubini–Study metrics given in terms of the Kodaira embedding

$$\iota:X\hookrightarrow\mathbb{P}(H^{0}(X,-mK_{X})^{\vee})$$

defined as follows. Suppose that we have a positive definite hermitian form $H$ on $H^{0}(X,-mK_{X})$. This naturally induces a hermitian metric on the hyperplane bundle $\mathcal{O}(1)$ over $\mathbb{P}(H^{0}(X,-mK_{X})^{\vee})$, and hence on $-mK_{X}=\iota^{*}\mathcal{O}(1)$ by pullback. By taking the $m$-th root, we get a hermitian metric on $-K_{X}$ and write $\mathrm{FS}(H)$ for the corresponding Kähler potential in $\mathcal{H}$. Noting that we may write $H=e^{A^{*}}e^{A}$ for some $A\in\mathfrak{gl}(H^{0}(X,-mK_{X}))$, the set $\mathcal{B}_{m}$ of positive definite hermitian forms on $H^{0}(X,-mK_{X})$ can be identified with the right coset space

$$\mathcal{B}_{m}=U(N_{m})\backslash GL(N_{m},\mathbb{C}),$$

(2)

where $N_{m}:=\dim_{\mathbb{C}}H^{0}(X,-mK_{X})$. Thus, the construction as above defines the Fubini–Study map [donproj2, §2]

$$\mathrm{FS}:\mathcal{B}_{m}\to\mathcal{H},$$

which is injective [Lempert2021, Theorem 1.1]. We write $\mathcal{H}_{m}:=\mathrm{FS}(\mathcal{B}_{m})$ for its image, and the elements of $\mathcal{H}_{m}$ are called the Fubini–Study metrics. We also note that we have the Hilbert map [donnum09, §2.2],

$$\mathrm{Hilb}_{\mu}:\mathcal{H}\to\mathcal{B}_{m},$$

defined by associating $\phi\in\mathcal{H}$ to the (re-scaled) $L^{2}$-inner product $\frac{N_{m}}{\int_{X}d\mu_{\phi}}\int_{X}e^{-m\phi}h_{0}(\cdot,\cdot)d\mu_{\phi}$ on $H^{0}(X,-mK_{X})$.

We can write down the Fubini–Study metric more explicitly as follows. We choose a reference hermitian form $H_{0}\in\mathcal{B}_{m}$ once and for all, and we define the reference metric $h_{0}$ on $-mK_{X}$ to be the Fubini–Study metric defined by $H_{0}$. More precisely, writing $\tilde{h}_{0}$ for the hermitian metric on the hyperplane bundle over $\mathbb{P}(H^{0}(X,-mK_{X})^{\vee})$ defined by $H_{0}$, we write

$$h_{0}:=\iota^{*}\tilde{h}_{0}.$$

(3)

With this reference metric chosen, the Fubini–Study metric $\mathrm{FS}(H)\in\mathcal{H}_{m}$ defined by $H\in\mathcal{B}_{m}$ can be written as

$$\mathrm{FS}(H)=\frac{1}{m}\log\sum_{i=1}^{N_{m}}\left\|s^{H}_{i}\right\|^{2}_{h_{0}}\in\mathcal{H},$$

(4)

where $\{s^{H}_{i}\}_{i=1}^{N_{m}}$ is any $H$-orthonormal basis for $H^{0}(X,-mK_{X})$. Note that $\mathrm{FS}(H)$ can be characterised as the unique element in $\mathcal{H}$ such that $h_{H}:=\exp(-m\mathrm{FS}(H))h_{0}$ satisfies $\sum_{i=1}^{N_{m}}\left\|s^{H}_{i}\right\|^{2}_{h_{H}}=1$ over $X$.

We observe that $\mathcal{B}_{m}=U(N_{m})\backslash GL(N_{m},\mathbb{C})$ is a Riemannian symmetric space with respect to the natural bi-invariant metric. An important object is a one-parameter family in $\mathcal{H}_{m}$ defined by the geodesic in $\mathcal{B}_{m}$ with respect to the bi-invariant metric, defined more explicitly as follows.

By abuse of terminology, the resulting family $\{\mathrm{FS}(H_{t})\}_{t\geq 0}\subset\mathcal{H}_{m}$ emanating from $h_{0}$ is also called the Bergman geodesic ray. The formula (4) immediately yields

$$\mathrm{FS}(H_{t})=\frac{1}{m}\log\sum_{i=1}^{N_{m}}\left\|e^{tA}s_{i}\right\|^{2}_{h_{0}},$$

(6)

where $\{s_{i}\}_{i=1}^{N_{m}}$ is any $H_{0}$-orthonormal basis for $H^{0}(X,-mK_{X})$, which we may of course take to be the reference basis, by noting that $\{e^{tA}s_{i}\}_{i=1}^{N_{m}}$ is an $H_{t}$-orthonormal basis.

We recall the following functional defined by Berman–Witt Nyström [BWN14, §4.2.2], which “quantises” the Ding functional.

The computation of the asymptotic slope of $\mathscr{D}_{m}$ along Bergman geodesic rays will be of crucial importance in what follows, and hence we record an elementary result for the derivative of $\mathscr{D}_{m}$.

Note that the above lemma immediately implies

$$\frac{d^{2}}{dt^{2}}\mathscr{E}_{m}(H_{t})=0.$$

(7)

Recall that the Bergman function $\rho_{m}(\mathrm{FS}(H))\in C^{\infty}(X,\mathbb{R})$ (also called the distortion function in [BBGZ, §7.3]) is defined as follows: writing $\{\sigma_{i}\}_{i=1}^{N_{m}}$ for an orthonormal basis for $H^{0}(X,-mK_{X})$ with respect to $\frac{\mathrm{Vol}(X)}{N_{m}}\mathrm{Hilb}_{\mu}\circ\mathrm{FS}(H)$, we define

$$\rho_{m}(\mathrm{FS}(H)):=\sum_{i=1}^{N_{m}}\|\sigma_{i}\|^{2}_{h_{H}},$$

(8)

where $h_{H}:=\exp(-m\mathrm{FS}(H))h_{0}$. The following well-known result is contained in [BBGZ, §7.3], and also stated slightly implicitly in [st19, tak19].

The item (i) of Proposition 2.6 can be regarded as the “zero of the moment map” condition, and $\mathscr{D}_{m}$ can be regarded as the Kempf–Ness type functional. It is thus natural to expect that the existence of the anticanonically balanced metric can be characterised by a Hilbert–Mumford type criterion in Geometric Invariant Theory. The appropriate criterion, as it turns out, is the $F$-stability defined by Saito–Takahashi [st19] that we review in Definition 3.8. For more details on the anticanonically balanced metrics, the reader is referred to [BBGZ, §7].

We finally note that Berndtsson’s convexity (recalled later in Theorem 4.2) and (7) immediately imply that a critical point of $\mathscr{D}_{m}$ must be the global minimum over $\mathcal{B}_{m}$.

We start with the preliminary materials on the automorphism group of Fano manifolds. We write $\mathrm{Aut}_{0}(X)$ for the identity component of the automorphism group of $X$. First observe that $\mathrm{Aut}_{0}(X)=\mathrm{Aut}_{0}(X,-K_{X})$ is a linear algebraic group, since the action $\mathrm{Aut}_{0}(X)\curvearrowright X$ naturally lifts to the anticanonical bundle, which is ample since $X$ is Fano. This in turn implies that we have the action $\mathrm{Aut}_{0}(X)\curvearrowright\mathbb{P}(H^{0}(X,-mK_{X})^{\vee})$ which preserves the image $\iota(X)\subset\mathbb{P}(H^{0}(X,-mK_{X})^{\vee})$ by the equivariant embedding theorem [Kambayashi] (see also [CG, §5.1]), for any $m\in\mathbb{N}$ such that $-mK_{X}$ is very ample. More precisely, for $m$ large enough there exists a faithful rational representation

$$\theta:\mathrm{Aut}_{0}(X)\hookrightarrow GL(H^{0}(X,-mK_{X})),$$

(9)

which is unique up to the choice of the linearisation (i.e. an overall constant multiple by $\mathbb{C}^{*}$), such that it is equivariant with respect to $\iota$, i.e. for any $x\in X$ and $a\in\mathrm{Aut}_{0}(X)$ we have

$$\iota(a\cdot x)=\theta(a)\cdot\iota(x).$$

We observe that an element of $GL(H^{0}(X,-mK_{X}))$ which preserves $\iota(X)\subset\mathbb{P}(H^{0}(X,-mK_{X})^{\vee})$ must be contained in the image of $\theta$ above.

In what follows, we shall mostly regard $\mathrm{Aut}_{0}(X)$ as a closed subgroup of $GL(H^{0}(X,-mK_{X}))$ by (9), and suppress $\theta$ in the notation. Likewise, its Lie algebra $\mathfrak{aut}(X)$ will be regarded as a Lie subalgebra of $\mathfrak{gl}(H^{0}(X,-mK_{X}))$. We define a maximal compact subgroup $K_{0}:=\mathrm{Aut}_{0}(X)\cap U(N_{m})$ of $\mathrm{Aut}_{0}(X)$, where the unitarity is defined with respect to the reference hermitian form $H_{0}\in\mathcal{B}_{m}$. Note that $K_{0}$ is the isometry group of the reference Kähler metric $\omega_{0}$ by Lemma 2.9 and [Lempert2021, Theorem 1.1]. We also define a reductive subgroup $\mathrm{Aut}_{0}(X)_{r}:=K_{0}^{\mathbb{C}}$ of $\mathrm{Aut}_{0}(X)$ by its complexification, and note that $\mathrm{Aut}_{0}(X)$ can be written as a semidirect product of its unipotent radical and $\mathrm{Aut}_{0}(X)_{r}$ by the Jordan–Chevalley decomposition. We write $\mathfrak{aut}(X)_{r}$ for the Lie algebra of $\mathrm{Aut}_{0}(X)_{r}$, and note that if $A\in\mathfrak{aut}(X)$ is $H_{0}$-hermitian then it must be contained in $\mathfrak{aut}(X)_{r}$.

For any connected subgroup $K$ of $K_{0}$, we define

$$\mathcal{B}_{m}^{K}:=\{H\in\mathcal{B}_{m}\mid\text{$H$ commutes with all elements in $K$.}\},$$

where we identified $H\in\mathcal{B}_{m}$ with a hermitian endomorphism with respect to the reference basis. Note that we have $u\cdot H=H$ for all $u\in K$ and $H\in\mathcal{B}_{m}^{K}$, with the action $u\cdot H$ given by (10), since $u\cdot H=u^{\dagger}Hu=u^{-1}Hu$ for all $u\in K$ by $K\subset U(N_{m})$, noting that $u^{\dagger}$ agrees with the $H_{0}$-hermitian conjugate $u^{*}$ since $u$ commutes with $H$. Defining $\mathcal{H}^{K}$ for the set of Kähler potentials that are invariant under the $K$-action and setting $\mathcal{H}_{m}^{K}:=\mathcal{H}^{K}\cap\mathcal{H}_{m}$, we thus get $\mathrm{FS}(\mathcal{B}_{m}^{K})\subset\mathcal{H}^{K}_{m}$ by Lemma 2.9. Just as in (2), we can write $\mathcal{B}^{K}_{m}$ as a right coset space

$$\mathcal{B}_{m}^{K}=C(K)\backslash C(K)^{\mathbb{C}}$$

where $C(K)$ is the centraliser of $K$ in $U(N_{m})$ and $C(K)^{\mathbb{C}}$ is its complexification. In particular, $\mathcal{B}^{K}_{m}$ is a Riemannian symmetric space with respect to the bi-invariant metric.

We now define the Kähler–Ricci $g$-solitons, following [BWN14, §2.2] (see also [rtz, §6.2]). Let $T$ be a (compact) real torus in $\mathrm{Aut}_{0}(X)$, and suppose that $(X,-K_{X})$ admits a Hamiltonian (with respect to $\omega_{0}$) $T$-action which is also holomorphic; note that this necessarily implies that $T$ is a subgroup of $K_{0}$. We write $T^{\mathbb{C}}$ for the complexification of $T$, $\mathrm{Aut}_{0}(X,T^{\mathbb{C}})$ for the elements in $\mathrm{Aut}_{0}(X)$ that commute with $T^{\mathbb{C}}$, and $\mathfrak{aut}(X,T^{\mathbb{C}})$ for its Lie algebra. Just as in Lemma 2.9, we have a natural action of $\mathrm{Aut}_{0}(X,T^{\mathbb{C}})$ on $\mathcal{B}_{m}^{T}$ which is an isometry with respect to the bi-invariant metric and consistent with the natural action on $\mathcal{H}^{T}$. As before, we define its reductive subgroup $\mathrm{Aut}_{0}(X,T^{\mathbb{C}})_{r}$ by the complexification of $\mathrm{Aut}_{0}(X,T^{\mathbb{C}})\cap U(N_{m})$, whose Lie algebra is denoted by $\mathfrak{aut}(X,T^{\mathbb{C}})_{r}$.

The action $T\curvearrowright(X,-K_{X})$, with the Kähler form $\omega_{\phi}$ ($\phi\in\mathcal{H}^{T}$), defines a moment map

$$m_{\phi}:X\to\mathfrak{t}^{\vee}\cong\mathbb{R}^{\dim T},$$

and its image $P:=m_{\phi}(X)$ is a compact convex polytope in $\mathbb{R}^{\dim T}$ known as the Delzant polytope. The Duistermaat–Heckman measure $\nu$ is the measure on $\mathbb{R}^{\dim T}$, supported on $P$, defined by

$$\nu:=(m_{\phi})_{*}\left(\frac{1}{\mathrm{Vol}(X)}\frac{\omega^{n}_{\phi}}{n!}\right)$$

which is known to be absolutely continuous and independent of $\phi$ [DuiHec82].

We can define $\mathrm{MA}_{g}(\phi)$ for a more general singular potential $\phi$ as in [BWN14, §2.2] but we only consider the smooth case. Note that the definition of $\nu$ means that we have the volume normalisation

$$\int_{X}\mathrm{MA}_{g}(\phi)=\int_{P}g\nu.$$

An equivalent way of writing down the above equation is

$$\frac{\mathrm{MA}_{g}(\phi)}{\int_{P}g\nu}=e^{-\phi+f_{\phi}}\frac{d\mu_{\phi}}{\int_{X}d\mu_{\phi}},$$

where $f_{\phi}$ is the Ricci potential, which is the unique ($T$-invariant) smooth function on $X$ which satisfies

$$\mathrm{Ric}(\omega_{\phi})=\omega_{\phi}+\sqrt{-1}\partial\bar{\partial}f_{\phi}\quad\text{and}\quad\fint_{X}e^{f_{\phi}}d\mu_{\phi}=1.$$

The above description in terms of the $g$-Monge–Ampère measure can be easily extended to allow for more singular solutions, as explained in [BWN14, §2.2].

Following [BWN14, §2.6], we define a functional $\mathscr{D}^{g}:\mathcal{H}^{T}\to\mathbb{R}$ by

$$\mathscr{D}^{g}(\phi):=\mathscr{L}(\phi)-\mathscr{E}^{g}(\phi),$$

where $\mathscr{E}^{g}$ is defined by its first variation as

$$\delta\mathscr{E}^{g}|_{\phi}(\psi):=\int_{X}\psi\mathrm{MA}_{g}(\phi)=\int_{X}\psi\frac{g(m_{\phi})}{\mathrm{Vol}(X)}\frac{\omega^{n}_{\phi}}{n!}.$$

(11)

That $\mathscr{E}^{g}$ is well-defined is proved by Berman–Witt Nyström [BWN14, Lemma 2.14], where the case of the Kähler–Ricci solitons was originally proved by X. Zhu [Zhu2000, Lemma 3.1]. It then follows that $\phi\in\mathcal{H}^{T}$ is a critical point of $\mathscr{D}^{g}$ if and only if it is a Kähler–Ricci $g$-soliton. See [BWN14, §2.4] for more details.

The above functional can be “quantised”, as proposed by [BWN14, §4]. First we write

$$P_{m}:=\{\lambda\in\mathrm{Hom}(T^{\mathbb{C}},\mathbb{C}^{*})\mid\lambda\text{ appears as a weight of }T^{\mathbb{C}}\curvearrowright H^{0}(X,-mK_{X})\}.$$

It is well-known that $\frac{1}{m}P_{m}\subset P=m_{\phi}(X)$, as in [Lah19, Lemma 13]. We have the weight decomposition

$$H^{0}(X,-mK_{X})=\bigoplus_{\lambda\in P_{m}}R_{m,\lambda}$$

(12)

where $T^{\mathbb{C}}$ acts with weight $\lambda\in P_{m}$ on $R_{m,\lambda}$. We also write

$$N_{m,\lambda}:=\dim_{\mathbb{C}}R_{m,\lambda}$$

and

$$\overline{g_{m}}:=\frac{1}{N_{m}}\sum_{\lambda\in P_{m}}g(\lambda/m)N_{m,\lambda}.$$

Note that, for the Bergman geodesic ray $\{H_{t}\}_{t\geq 0}\subset\mathcal{B}^{T}_{m}$ as defined in (5), we have

$$\frac{d}{dt}\mathscr{E}^{g}_{m}(H_{t})=\frac{1}{mN_{m}\overline{g_{m}}}\sum_{\lambda\in P_{m}}g(\lambda/m)\mathrm{tr}\left((A+A^{*})|_{R_{m,\lambda}}\right).$$

(13)

Just as we saw in Proposition 2.6, we can characterise the critical point of $\mathscr{D}^{g}_{m}$ by the “zero of the moment map” condition. Suppose that we write $\left\{s_{\alpha}^{(\lambda)}\right\}_{\alpha=1}^{N_{m,\lambda}}$ for an $H_{0}$-orthonormal basis for each $R_{m,\lambda}$, noting that the weight subspaces are orthogonal to each other with respect to $H_{0}$ which is $T$-invariant. Then $H=e^{-A^{*}}e^{-A}\in\mathcal{B}^{T}_{m}$ is a critical point of $\mathscr{D}^{g}_{m}$ if and only if

$$\fint_{X}\frac{h_{0}\left(e^{A}s^{(\lambda)}_{\alpha},e^{A}s^{(\lambda)}_{\beta}\right)}{\sum_{\lambda^{\prime}\in P_{m}}\sum_{\gamma=1}^{N_{m,\lambda^{\prime}}}\left\|e^{A}s^{(\lambda^{\prime})}_{\gamma}\right\|^{2}_{h_{0}}}d\mu_{\mathrm{FS}(H)}-\frac{g(\lambda/m)}{mN_{m}\overline{g_{m}}}\delta_{\alpha\beta}=0,$$

for all $\alpha,\beta=1,\dots,N_{m,\lambda}$ and all $\lambda\in P_{m}$; note that $A$ preserves the weight decomposition (12) as $H\in\mathcal{B}^{T}_{m}$. We can also characterise the above as the metric for which the $g$-Bergman function [BWN14, §4.2.1] is constant, by arguing exactly as in Proposition 2.6.

Just as we pointed out for $\mathscr{D}_{m}$ in §2.2, a critical point of $\mathscr{D}^{g}_{m}$ is necessarily the global minimum over $\mathcal{B}^{T}_{m}$; in fact it attains the global minimum over $\mathcal{B}_{m}$, as pointed out in [rtz, Proof of Theorem 2.11].

Suppose that we have a $k$-tuple of ample $\mathbb{Q}$-line bundles $L_{1},\dots,L_{k}$ over a Fano manifold $X$ such that

$$-K_{X}=L_{1}+\cdots+L_{k}.$$

We also write $(X,-K_{X};L_{1},\dots,L_{k})$ for the above data.

Coupled Kähler–Einstein metrics were introduced by Hultgren–Witt Nyström [HWN19] as a generalisation of the Kähler–Einstein metrics which is compatible with the above “decomposition” of $-K_{X}$, and have been actively studied recently. We recall here some basic results established in [HWN19], and also its relationship to the geometric quantisation as given by Takahashi [tak19]; in fact we will complement it by adding some new materials that were not discussed in [tak19], which turns out to be important later.

We choose a positive integer $m$ to be sufficiently large and divisible so that each of $-mK_{X}$, $mL_{1},\dots,mL_{k}$ is a very ample line bundle and that the natural multiplication map

$$H^{0}(X,mL_{1})\otimes\cdots\otimes H^{0}(X,mL_{k})\to H^{0}(X,mL_{1}+\cdots+mL_{k})=H^{0}(X,-mK_{X})$$

is surjective (see e.g. [LazarsfeldI, Example 1.2.22]). An elementary observation, which follows from taking the dual of the above, is that we have a sequence of embeddings

$$X\hookrightarrow\mathbb{P}(H^{0}(X,-mK_{X})^{\vee})\hookrightarrow\mathbb{P}(H^{0}(X,mL_{1})^{\vee}\otimes\cdots\otimes H^{0}(X,mL_{k})^{\vee})$$

(14)

which turns out to be important. We write $\iota_{\mathrm{coupled}}$ for the composition of the two embeddings above. Note that the hyperplane bundle over $\mathbb{P}(H^{0}(X,mL_{1})^{\vee}\otimes\cdots\otimes H^{0}(X,mL_{k})^{\vee})$ is pulled back by $\iota_{\mathrm{coupled}}$ to $-mK_{X}$ since the second embedding is linear.

On the other hand, since $mL_{1},\dots,mL_{k}$ are very ample, for each $i=1,\dots,k$ we have the Kodaira embedding $\iota_{i}:X\hookrightarrow\mathbb{P}(H^{0}(X,mL_{i})^{\vee})$. For each $i$ we fix a reference hermitian form $H_{i,0}$ for $H^{0}(X,mL_{i})$, and define $\tilde{h}_{i,0}$ to be the Fubini–Study metric on $\mathbb{P}(H^{0}(X,mL_{i})^{\vee})$ with respect to $H_{i,0}$, which is pulled back to a hermitian metric $h_{i,0}:=\iota_{i}^{*}\tilde{h}_{i,0}$ on $mL_{i}$ by $\iota_{i}$, entirely analogously to (3) in §2.2. We write $\theta_{i}\in c_{1}(L_{i})$ for the associated Kähler metric, scaled by $1/m$ to be in $c_{1}(L_{i})$. As in (4), for a general positive definite hermitian form $H_{i}$ on $H^{0}(X,mL_{i})$ we define

$$\mathrm{FS}_{i}(H_{i}):=\frac{1}{m}\log\sum_{j=1}^{N_{i,m}}\left\|s^{H_{i}}_{j}\right\|^{2}_{h_{i,0}},$$

(15)

where $\{s^{H_{i}}_{j}\}_{j=1}^{N_{i,m}}$ is any $H_{i}$-orthonormal basis for $H^{0}(X,mL_{i})$ and we wrote

$$N_{i,m}:=\dim_{\mathbb{C}}H^{0}(X,mL_{i}).$$

We pick an auxiliary reference hermitian metric on $-mK_{X}$ to be

$$h^{\prime}_{0}:=h_{1,0}\otimes\cdots\otimes h_{k,0},$$

(16)

with the associated volume form $d\mu^{\prime}_{0}$ which we may assume has unit volume over $X$ by re-scaling; a slightly subtle point is that, while this is a metric on $-mK_{X}$ that is naturally determined by the reference hermitian metrics on $mL_{1},\dots,mL_{k}$, we need another reference metric $h_{0}$ on $-K_{X}$, defined later in (18).