2.1.1 Z-stability

We work throughout over the complex numbers, in order to preserve links with the complex differential geometry. We also fix a normal polarised variety $(X,L)$ of dimension n, with L an ample $\mathbb {Q}$ -line bundle. Normality implies that the canonical class $K_X$ of X exists as a Weil divisor, we always assume that $K_X$ exists as a $\mathbb {Q}$ -line bundle.

In addition to our ample line bundle, we will fix a stability vector, a unipotent cohmology class and a polynomial Chern form; we define these in turn.

Definition 2.1. A stability vector is a sequence of complex numbers

$$ \begin{align*}\rho = (\rho_0, \ldots, \rho_n) \in \mathbb{C}^{n+1}\end{align*} $$

such that $\rho _n = i= \sqrt {-1}$ .

The condition $\rho _n = i $ is a harmless normalisation condition which, when it is not satisfied, can be achieved by multiplying the stability vector by a fixed complex number. In Bridgeland stability, one normally assumes $\rho \in (\mathbb {C}^{*})^{n+1}$ ; this will be unnecessary for us.

Note that $\Theta '$ must satisfy

$$ \begin{align*}\overbrace{\Theta' \cdot \ldots \cdot \Theta'}^{j \text{ times}}=0\end{align*} $$

for $j \geq n+1$ . A typical example of a choice of $\Theta $ is to fix a class $\beta \in H^{1,1}(X,\mathbb {R})$ and set $\Theta = e^{-\beta }$ , which is analogous to a ‘B-field’ in Bridgeland stability.

Definition 2.3. A polynomial Chern form is a sum of the form

$$ \begin{align*}f(K_X) = \sum_{j=0}^n a_j K_X^j,\end{align*} $$

where $a_j \in \mathbb {C}$ and $K_X^j$ denotes the $j^{th}$ -intersection product $K_X \cdot \ldots \cdot K_X$ , viewed as a cycle. We always assume the normalisation condition $a_0 = a_1=1$ , and interpret $K_X^0 = 1$ as a cycle.

As mentioned above, in the current section we restrict ourselves to central charges only involving $c_1(X) = c_1(-K_X)$ , with the case of higher Chern classes postponed to Section 4.

Definition 2.4. A polynomial central charge is a function $Z: \mathbb {N} \to \mathbb {C}$ taking the form

$$ \begin{align*}Z_{k}(X,L) =\sum_{l=0}^n \rho_l k^{l} \int_X L^l \cdot f(K_X) \cdot \Theta,\end{align*} $$

for some $\rho $ and $\Theta $ . A central charge is a polynomial central charge with k fixed, such that $Z(X,L)\neq 0$ . We often set $\varepsilon = k^{-1}$ and denote the induced quantity by $Z_{\varepsilon }(X,L)$ .

We will sometimes simply call a polynomial central charge a central charge when the dependence on k is clear from context. The definition is motivated by an analogous definition of Bayer in the bundle setting [Reference Bayer3, Theorem 3.2.2]. For a polynomial central charge it is automatic that $Z_{k}(X,L)$ lies in the upper half plane in $\mathbb {C}$ for $k \gg 0$ since $\operatorname {\mathrm {Im}}(\rho _n)>0$ . Thus, we can make the following definition.

Definition 2.5. We define the phase of X to be

$$ \begin{align*}\varphi_k(X,L) = \arg Z_{k}(X,L),\end{align*} $$

the argument of the nonzero complex number. We denote this by $\varphi (X,L)$ when k is fixed, and for fixed $(X,L)$ often simply denote this by $\varphi $ .

Here, we consider $\arg $ as a function $\arg : \mathbb {C} \to \mathbb {R}$ by setting $\arg (1)=0$ . We now turn to our definition of stability, which depends on a choice of central charge Z. As in the definition of K-stability of polarised varieties, we require the notion of a test configuration, which is essentially a $\mathbb {C}^{*}$ -degeneration of $(X,L)$ to another polarised scheme.

A test configuration admits a canonical compactification to a family over $\mathbb {P}^1$ by equivariantly compactifying trivially over infinity [Reference Wang66, Section 3]. This compactification produces a flat family endowed with a $\mathbb {C}^{*}$ -action, which we abusively denote $(\mathcal {X},\mathcal {L}) \to \mathbb {P}^1$ , such that each fibre over $t \neq \infty \in \mathbb {P}^1$ is isomorphic to $(X,L)$ . The reason to compactify is that it allows us to perform intersection theory on the resulting projective variety $\mathcal {X}$ .

It will also be convenient to be able to consider classes on X as inducing classes on $\mathcal {X}$ , so we pass to a variety with a surjective map to X as follows. There is a natural equivariant birational map

$$ \begin{align*}f:(X\times\mathbb{P}^1,p_1^{*}L)\dashrightarrow (\mathcal{X},\mathcal{L}),\end{align*} $$

with $p_1: X \times \mathbb {P}^1 \to X$ the projection, so we take an equivariant resolution of indeterminacy of the form:

where we may assume $\mathcal {Y}$ is smooth. In particular, the unipotent cohomology class $\Theta $ on X involved in the definition of a central charge induces a class $(q \circ p_1)^{*}\Theta $ on $\mathcal {Y}$ , which we still denote $\Theta $ . The classes $\mathcal {L}$ and $K_{\mathcal {X}}$ on $\mathcal {X}$ induce also classes $r^{*}\mathcal {L}$ and $r^{*}K_{\mathcal {X}/\mathbb {P}^1}$ on $\mathcal {Y}$ , we in addition set $K_{\mathcal {X}/\mathbb {P}^1} = K_{\mathcal {X}} - \pi ^{*}K_{\mathbb {P}^1}$ to be the relative canonical class. Thus, to a given intersection number $L^d \cdot K_X^j \cdot U$ we can associate the intersection number on $\mathcal {Y}$ which we (slightly abusively) denote

$$ \begin{align*}\int_{\mathcal{X}} \mathcal{L}^{l+1} \cdot K_{\mathcal{X}/\mathbb{P}^1}^j \cdot \Theta= \int_{\mathcal{Y}} (r^{*}\mathcal{L})^{l+1}\cdot r^{*}(K_{\mathcal{X}/\mathbb{P}^1}^j ) \cdot \Theta,\end{align*} $$

which is computed in $\mathcal {Y}$ . In computing this intersection number, note that $\dim \mathcal {X} = \dim \mathcal {Y} = n+1$ . The following elementary result justifies the notation omitting $\mathcal {Y}$ .

Definition 2.7. Let $(\mathcal {X},\mathcal {L})$ be a test configuration and Z be a polynomial central charge. We define the central charge of $(\mathcal {X},\mathcal {L})$ to be

$$ \begin{align*}Z_k(\mathcal{X},\mathcal{L}) = \sum_{l=0}^n \frac{\rho_l k^{l}}{l+1}\int_{\mathcal{X}} \mathcal{L}^{l+1} \cdot f(K_{\mathcal{X}/\mathbb{P}^1}) \cdot \Theta,\end{align*} $$

and set $\varphi _k(\mathcal {X},\mathcal {L}) = \arg Z_{k}(\mathcal {X},\mathcal {L})$ when $Z_{k}(\mathcal {X},\mathcal {L}) \neq 0$ . Note that $f(K_{\mathcal {X}/\mathbb {P}^1}) = \sum _{j=0}^{n}a_jK_{\mathcal {X}/\mathbb {P}^1}^j$ arises from the polynomial Chern form. With k fixed we denote these by $Z(\mathcal {X},\mathcal {L})$ and $\varphi (\mathcal {X},\mathcal {L})$ , respectively.

The stability condition, for fixed k, is then the following.

The natural asymptotic notion is the following.

Definition 2.9. We say that $(X,L)$ is asymptotically Z-stable if for all nontrivial test configurations $(\mathcal {X},\mathcal {L})$ and for all $k \gg 0$ we have

$$ \begin{align*}\operatorname{\mathrm{Im}}\left(\frac{Z_{k}(\mathcal{X},\mathcal{L})}{Z_{k}(X,L)}\right)>0.\end{align*} $$

Asymptotic Z-polystability, semistability and instability are defined similarly.

Note that, as $\operatorname {\mathrm {Im}}(\rho _n)>0$ by assumption, both $Z_k(X,L)$ and $Z_k(\mathcal {X},\mathcal {L})$ are nonvanishing and lie in the upper half plane for $k \gg 0$ . Here, strictly speaking to ensure that $Z_k(\mathcal {X},\mathcal {L})$ lies in the upper half plane we may need to modify $\mathcal {L}$ to $\mathcal {L}+\mathcal {O}(m)$ for some $\mathcal {O}(m)$ pulled back from $\mathbb {P}^1$ ; this leaves the various stability inequalities unchanged by Lemma 2.4 below. Thus, asymptotic Z-stability can be rephrased as asking for all test configurations $(\mathcal {X},\mathcal {L})$ to have for $k \gg 0$

$$ \begin{align*}\varphi_k(\mathcal{X},\mathcal{L})> \varphi_k(X,L).\end{align*} $$

The factor $l+1$ in the definition of $Z_{k}(\mathcal {X},\mathcal {L})$ ensures that the key inequality defining stability is invariant under certain changes of $\mathcal {L}$ . For this, note that one can modify the polarisation of a test configuration $(\mathcal {X},\mathcal {L})$ by adding the pullback $\mathcal {O}(m)$ of the (m ^th tensor power of the) hyperplane line bundle from $\mathbb {P}^1$ for any j.

Lemma 2.4. The phase inequality remains unchanged under the addition of $\mathcal {O}(m)$ . That is,

$$ \begin{align*}\operatorname{\mathrm{Im}}\left(\frac{Z(\mathcal{X},\mathcal{L}+\mathcal{O}(m)))}{Z(X,L)}\right) = \operatorname{\mathrm{Im}}\left(\frac{Z(\mathcal{X},\mathcal{L})}{Z(X,L)}\right).\end{align*} $$

Proof. A single intersection number changes as

$$ \begin{align*}\int_{\mathcal{X}}(\mathcal{L}+\mathcal{O}(m))^{l+1} \cdot K_{\mathcal{X}/\mathbb{P}^1}^j\cdot \Theta = \int_{\mathcal{X}}\mathcal{L}^{l+1} \cdot K_{\mathcal{X}/\mathbb{P}^1}^j\cdot \Theta + m(l+1) \int_X L^l \cdot K_{X}^j\cdot \Theta,\end{align*} $$

since by flatness intersecting with $\mathcal {O}(1)$ can be viewed as intersecting with a fibre $\mathcal {X}_t \cong X$ for $t \neq 0$ , and $\mathcal {L}, K_{\mathcal {X}/\mathbb {P}^1}$ and $\Theta $ restrict to $L, K_X$ and $\Theta $ respectively on X. It follows that

$$ \begin{align*}Z(\mathcal{X},\mathcal{L}+\mathcal{O}(m)) = Z(\mathcal{X},\mathcal{L}) + mZ(X,L),\end{align*} $$

which means since $m \in \mathbb {Q}$ is real

$$ \begin{align*} \operatorname{\mathrm{Im}}\left(\frac{Z(\mathcal{X},\mathcal{L}+\mathcal{O}(m)))}{Z(X,L)}\right) &= \operatorname{\mathrm{Im}}\left(\frac{Z(\mathcal{X},\mathcal{L}) + mZ(X,L)}{Z(X,L)}\right), \\ &= \operatorname{\mathrm{Im}}\left(\frac{Z(\mathcal{X},\mathcal{L})}{Z(X,L)}\right).\\[-34pt] \end{align*} $$

Example 2.5. A central charge of special interest is

$$ \begin{align*} Z_k(X,L) &= -\int_X e^{-ikL}\cdot e^{-K_X}, \\ &= - \sum_{j=0}^n \frac{(-i)^j }{j!(n-j)!}\int_X(kL)^j\cdot (-K_X)^{n-j}. \end{align*} $$

This can be viewed as an analogue of the central charge on the Grothendieck group $K(X)$ (in the sense of Bridgeland stability) associated to the deformed Hermitian Yang–Mills equation on a holomorphic line bundle [Reference Collins and Yau12, Section 9].

We will not consider a completely arbitrary central charge in the present work, as we require that the large volume limit of our conditions is ‘nondegenerate’ in a suitable sense. Let $\Theta _1$ denote the $(1,1)$ -part of the unipotent cohomology class $\Theta \in \oplus _j H^{j,j}(X,\mathbb {C})$ .

The motivation for these definition is through the link with K-stability and its variants.

2.1.2 K-stability

The definition of asymptotic Z-stability given is motivated not only by the vector bundle theory, but also by the notion of K-stability of polarised varieties due to Tian and Donaldson [Reference Tian63, Reference Donaldson24]. As before, we take $(X,L)$ to be a normal polarised variety such that $K_X$ is a $\mathbb {Q}$ -line bundle.

Definition 2.11. We define the slope of $(X,L)$ to be the topological invariant, computed as an integral over X

$$ \begin{align*}\mu(X,L) = \frac{-K_X.L^{n-1}}{L^n}.\end{align*} $$

We further define the Donaldson–Futaki invariant of a test configuration $(\mathcal {X},\mathcal {L})$ to be

$$ \begin{align*}\operatorname{\mathrm{DF}}(\mathcal{X},\mathcal{L}) =\int_{\mathcal{X}}\left( \frac{n\mu(X,L)}{n+1}\mathcal{L}^{n+1} + \mathcal{L}^n.K_{\mathcal{X}/\mathbb{P}^1}\right).\end{align*} $$

We remark that this is not Donaldson’s original definition but rather is proven by Odaka and Wang to be an equivalent one [Reference Odaka50, Theorem 3.2] [Reference Wang66, Section 3] (see also [Reference Donaldson24, Proposition 4.2.1]).

The following is immediate from the definitions.

Lemma 2.6. K-semistability is equivalent to asymptotic Z-semistability where

$$ \begin{align*}Z_k(X,L) = \int_X (ik^nL^n -k^{n-1}K_X.L^{n-1}).\end{align*} $$

That is, with $\rho = (0,0, \ldots , -1,i)$ , $\Theta =0$ .

Of course, the same is true for K-stability and K-polystability, modulo our slightly nonstandard requirement that $K_{\mathcal {X}}$ is a $\mathbb {Q}$ -line bundle, which is irrelevant for K-semistability as in that situation one can assume $\mathcal {X}$ is smooth.

Example 2.7. K-semistability of maps can recovered as a special cases of Z-stability. Indeed, supposing $p: (X,L) \to (Y,H)$ is a map of polarised varieties, then setting

$$ \begin{align*}Z_{k}(X,L) = \int_X (ik^nL^n -k^{n-1}(K_X+p^{*}H).L^{n-1})\end{align*} $$

recovers the notion of K-semistability of the map p [Reference Dervan and Ross18, Definition 2.9]. That is, we take $\Theta $ to be (the class of) $p^{*}H$ .

Slightly more generally, twisted K-stability fits into this picture [Reference Dervan13, Definition 2.7], though this notion is less geometric than K-stability of maps and we hence do not discuss it. Similarly, the ‘fully degenerate’ case $a_j = 0$ for $j\leq n-1$ produces variants of J-stability [Reference Lejmi and Székelyhidi42, Section 2] and has links with Z-stability of holomorphic line bundles [Reference Dervan, McCarthy and Sektnan15, Conjecture 1.6]. In general, asymptotic Z-stability is related to K-stability as follows:

Proof. We only give the proof for K-semistability, as the proof is the same for the map type situation. By nondegeneracy, there is an expansion

$$ \begin{align*}Z_k(X,L) = k^n i\int_{X}L^n + k^{n-1} \rho_{n-1} \int_{X}K_X.L^{n-1} + O(k^{n-2}),\end{align*} $$

where we have used that $\Theta _1=0$ and that our normalisation for the polynomial Chern form assumes $a_0=a_1=1$ . Thus,

$$ \begin{align*}Z_k(\mathcal{X},\mathcal{L}) = \frac{i}{n+1}k^n\int_{X}\mathcal{L}^{n+1} + \frac{\rho_{n-1}}{n}k^{n-1} \int_{X}K_{\mathcal{X}/\mathbb{P}^1}.L^{n-1} + O(k^{n-2}),\end{align*} $$

meaning that

$$ \begin{align*}\operatorname{\mathrm{Im}}\left(\frac{Z_{k}(\mathcal{X},\mathcal{L})}{Z_{k}(X,L)}\right) =\frac{-\operatorname{\mathrm{Re}}(\rho_{n-1})}{nL^n}\operatorname{\mathrm{DF}}(\mathcal{X},\mathcal{L})k^{-1} + O(k^{-2}).\end{align*} $$

Thus, since $\operatorname {\mathrm {Re}}(\rho _{n-1})<0$ by nondegeneracy, the asymptotic Z-stability hypothesis demands that this be negative for $k \gg 0$ , forcing $\operatorname {\mathrm {DF}}(\mathcal {X},\mathcal {L}) \geq 0$ .

3.3.1 Moment maps in finite dimensions

Many geometric equations have an interpretation through moment maps; this has been especially influential for the cscK equation. We will give two ways of viewing the Z-critical equation as a moment map. The first is a finite-dimensional geometric interpretation, on the base of a holomorphic submersion, while the second is closer in spirit to the infinite-dimensional viewpoint of Fujiki–Donaldson for the cscK equation [Reference Fujiki30, Reference Donaldson23].

The setup is modelled on the situation of a test configuration $\pi : (\mathcal {X},\mathcal {L})\to \mathbb {C}$ for $(X,L)$ with smooth central fibre. The properties of interest are firstly that there is an $S^1$ -action on both $\mathbb {C}$ and $(\mathcal {X},\mathcal {L})$ , making $\pi $ an $S^1$ -equivariant map, secondly that all fibres over the open dense orbit under the associated $\mathbb {C}^{*}$ -action are isomorphic and thirdly that we may choose an $S^1$ -invariant relatively Kähler metric $\omega _{\mathcal {X}} \in c_1(\mathcal {L})$ . If we had considered test configurations over the unit disc $\Delta $ , the same properties would be true with the $\mathbb {C}^{*}$ -action meant only locally, in the sense that one only obtains an action induced by sufficiently small elements of the Lie algebra of $\mathbb {C}^{*}$ .

More generally, we consider a holomorphic submersion $\pi : (\mathcal {X},\mathcal {L}) \to B$ over a complex manifold B, with $\mathcal {L}$ a relatively ample $\mathbb {Q}$ -line bundle. We assume that B admits an effective action of a compact Lie group, which induces an effective local action of the complexification G of K. In addition, we assume that there is a K-action on $(\mathcal {X},\mathcal {L})$ making $\pi $ an equivariant map, and fix a K-invariant relatively Kähler metric $\omega _{\mathcal {X}} \in c_1(\mathcal {L})$ . We lastly assume that there is an open dense orbit associated to G such that all fibres are isomorphic to $(X,L)$ ; we denote this orbit as $\mathcal {X}^o \to B^o$ . In practice, we will apply these results to the special case of an isotrivial family.

Let v be a holomorphic vector field on $\mathcal {X}$ induced by an element of the Lie algebra $\mathfrak {k}$ of K. Denote by $h_v$ the function on $\mathcal {X}$ defined by the equation

(3.3) $$ \begin{align}\mathcal{L}_{Jv} \omega_{\mathcal{X}} = i\partial\overline{\partial} h,\end{align} $$

where J denotes the almost-complex structure of $\mathcal {X}$ and the differentials on the right-hand side are also computed on $\mathcal {X}$ . As in Section 3.1, we will refer to h as a Hamiltonian, even though $\omega _{\mathcal {X}}$ is only relatively Kähler. Note that $h_v$ does restrict to a genuine Hamiltonian for v on the fibres over B on which v induces a holomorphic vector field; these are the fibres over points for which the corresponding vector field on B vanishes.

We now fix the input of the Z-critical equation. Setting $\varepsilon = k^{-1}$ , our central charge can be written

$$ \begin{align*}Z_{\varepsilon}(X,L) =\sum_{l=0}^n \rho_l \varepsilon^{-l} \int_X L^l \cdot f(K_X) \cdot \Theta.\end{align*} $$

We have fixed a representative $\theta \in \Theta $ , and we assume that the form $G.\theta $ defined on the dense orbit $\mathcal {X}^o$ extends to a smooth form on $\mathcal {X}$ itself and denote this form abusively by $\theta $ , which is automatically a G-invariant closed form on $\mathcal {X}$ . The form $\omega _{\mathcal {X}}$ induces a metric on the relative holomorphic tangent bundle $T_{\mathcal {X}/B}$ , and hence on its top exterior power $-K_{\mathcal {X}/B}$ , and we denote the curvature of the latter metric as $\rho \in c_1(-K_{\mathcal {X}/B})$ .

We associate to $Z_{\varepsilon }(X,L)$ an $(n+1,n+1)$ -form on $\mathcal {X}$ as follows. We will define the $(n+1,n+1)$ -form on $\mathcal {X}$ linearly in the terms of this expression, and hence it is sufficient to define an $(n+1,n+1)$ -form associated to a term of the form $\int _X L^l \cdot K_X^j \cdot \Theta $ , to which we associate $\frac {1}{l+1}\omega _{\mathcal {X}}^{l+1}\wedge \rho ^j \wedge \theta .$ This induces a form $\tilde Z_{\varepsilon }(\mathcal {X},\mathcal {L})$ , and we set

(3.4) $$ \begin{align}\Omega_{\varepsilon} = \operatorname{\mathrm{Im}}\left(e^{-i\varphi_{\varepsilon}} \int_{\mathcal{X}/B} \tilde Z(\mathcal{X},\mathcal{L})\right)\end{align} $$

to be the associated fibre integral. By general properties of fibre integrals, this produces a closed $(1,1)$ -form on B. K-invariance of the forms on $\mathcal {X}$ and of the map $\pi : \mathcal {X} \to B$ imply that $\Omega _{\varepsilon }$ is K-invariant. In general, the form $\Omega _{\varepsilon }$ may not be Kähler, which in addition requires positivity. In our applications, $\Omega _{\varepsilon }$ will, however, be Kähler for $0<\varepsilon \ll 1$ .

We let $\omega _b$ denote the restriction of $\omega _{\mathcal {X}}$ to the fibre $\mathcal {X}_b$ over b and denote $\operatorname {\mathrm {Im}}(e^{-i\varphi _{\varepsilon }}\tilde Z_{\varepsilon }(\omega _b))$ the Z-critical operator computed on $\mathcal {X}_b$ with respect to $\omega _b$ . We similarly set $h_{v,b}$ be the restriction of a Hamiltonian $h_v$ to the fibre $\mathcal {X}_b$ . Define a map $\mu _{\varepsilon }: B \to \mathfrak {k}^{*}$ by

$$ \begin{align*}\langle \mu_{\varepsilon}, v \rangle(b) =-\frac{1}{2} \int_{\mathcal{X}_b} h_{v,b} \operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}}\tilde Z_{\varepsilon}(\omega_b)) \omega_b^n,\end{align*} $$

where $v \in \mathfrak {k}$ is viewed as inducing a holomorphic vector field on $\mathcal {X}$ to induce the Hamiltonian $h_v$ .

Here, we mean that the defining conditions of a moment map are satisfied, namely that

$$ \begin{align*}d\langle \mu_{\varepsilon}, v \rangle = - \iota_v \Omega_{\varepsilon},\end{align*} $$

and $\mu $ is K-equivariant when $\mathfrak {k}^{*}$ is given the coadjoint action; in general we emphasise that $\Omega _{\varepsilon }$ is not actually a symplectic form (although for $\varepsilon $ sufficiently small it will be in our applications, producing genuine moment maps). In the contraction $\iota _v \Omega _{\varepsilon }$ , we view $v\in \mathfrak {k}$ as inducing a holomorphic vector field on B.

Proof. We first show that the equation $d\langle \mu _{\varepsilon }, v \rangle = - \iota _v \Omega _{\varepsilon }$ holds. Note that it is enough to show that this holds on the dense locus $B^o$ since both sides of the equation extend continuously to B.

We fix a point $b \in B$ at which we wish to demonstrate the moment map equation and consider the orbit U of $b \in B$ under the G-action. We fix an isomorphism $(\mathcal {X}_b,\mathcal {L}_b) \cong (X,L)$ and simply write $\omega _b = \omega $ . The G-action induces an isomorphism

$$ \begin{align*}(\mathcal{X},\mathcal{L}) \cong (X \times U, L),\end{align*} $$

where $B^o \cong U \subset G$ is a submanifold. Since we only obtain a local action of G on B, U may not consist of all of G in general. The isomorphism $\mathcal {X} \cong X \times U$ is in addition compatible with the projections to $B^o$ . The relatively Kähler metric $\omega _{\mathcal {X}} \in c_1(\mathcal {L})$ thus induces a form $\omega _{X\times U}$ on $X \times U$ and we can define

$$ \begin{align*}E_Z: U \to \mathbb{R}\end{align*} $$

defined as the Z-energy with respect to the reference metric $\omega $ (or rather its pullback to $X \times U$ ) and the varying metric $\omega _{X \times U}$ on the fibres over U. Proposition 3.3 then implies that on $U \cong B^0$ we have

(3.5) $$ \begin{align}i\partial\overline{\partial} E_{Z_{\varepsilon}} = \Omega_{\varepsilon}. \end{align} $$

By Proposition 3.5, the derivative of the Z-energy along any path $\omega _t = \omega + i\partial \overline {\partial } \psi _t$ satisfies

$$ \begin{align*}\frac{d}{dt}E_{Z_{\varepsilon}}(\psi_t) = \int_X \dot \psi_t \operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}}\tilde Z_{\varepsilon}(\omega_t))\omega_t^n.\end{align*} $$

Considering the path $\omega _t$ defined above, the defining property of the Hamiltonian h means that $\dot \psi _0 = h_{v,b}$ on $\mathcal {X}_b \cong X$ . Thus,

$$ \begin{align*}\frac{d}{dt}\Bigr|_{t=0}\int_X\dot\psi_t\operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}}\tilde Z(\omega_t))\omega_t^n = \int_X h_{v,b}\operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}}\tilde Z(\omega))\omega^n.\end{align*} $$

But this is then all we need: By a standard calculation [Reference Székelyhidi59, Lemma 12]

$$ \begin{align*}\iota_v(i\partial\overline{\partial} E_{Z_{\varepsilon}}) = \frac{1}{2}d(Jv(E_{Z_{\varepsilon}})),\end{align*} $$

so it follows that

(3.6) $$ \begin{align}\iota_v(\Omega_{\varepsilon}) = \frac{1}{2}d(Jv(E_{Z_{\varepsilon}})) = \frac{1}{2}d\left(\frac{d}{dt}E_{Z_{\varepsilon}}(\exp(Jvt).p)\right) = \frac{1}{2}d\left(\frac{d}{dt}E_{Z_{\varepsilon}}(\psi_t)\right) = -d\langle \mu_{\varepsilon},h\rangle,\end{align} $$

proving the first defining property of a moment map with respect to v at the point b. But by continuity this implies that the same property holds on all of B.

What remains to prove is K-equivariance of $\mu _{\varepsilon }$ , which requires us to show for all $g \in K$

$$ \begin{align*}\langle \mu_{\varepsilon}(g.b),v\rangle = \langle \mu_{\varepsilon}(g.b),g^{-1}.v\rangle,\end{align*} $$

where K acts on $\mathfrak {k}$ by the adjoint action. However, the Hamiltonian on $\mathcal {X}$ with respect to $g^{-1}.v$ is simply the pullback $g^{*}h_v$ , meaning $g^{*}h_{v,g(b)} = h_{g^{-1}.v,b}$ . Thus, using K-invariance of $\omega _{\mathcal {X}}$ , the equality

$$ \begin{align*}\int_{\mathcal{X}_{g(b)}} h_{v,{g(b)}} \operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}}\tilde Z_{\varepsilon}(\omega_{g(b)})) \omega_{g(b)}^n = \int_{\mathcal{X}_b} g^{*}h_{v,g(b)} \operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}}\tilde Z_{\varepsilon}(\omega_b)) \omega_b^n\end{align*} $$

is enough to imply equivariance.

3.3.2 Moment maps in infinite dimensions

We next demonstrate how the Z-critical equation appears as a moment map in infinite dimensions. While this is important conceptually, it will also be used in order to understand the linearisation of the Z-critical equation. Analogously to the Fujiki–Donaldson moment map interpretation of the cscK equation [Reference Fujiki30, Reference Donaldson23], we fix a compact symplectic manifold $(M,\omega )$ and consider the space $\mathcal {J}(M,\omega )$ of almost complex structures compatible with $\omega $ . We refer to Scarpa [Reference Scarpa54, Section 1.3] and Gauduchon [Reference Gauduchon33, Section 8] for a good exposition of this space its properties. $\mathcal {J}(M,\omega )$ naturally has the structure of an infinite-dimensional complex manifold; its tangent space at $J \in \mathcal {J}(M,\omega )$ is given by

$$ \begin{align*}T_J\mathcal{J}(M,\omega)= \{A: TM \to TM \ | \ AJ+JA =0,\ \omega(\cdot , A \cdot) = \omega(A \cdot, \cdot)\},\end{align*} $$

with complex structure defined by $A \to JA$ on $T_J\mathcal {J}(M,\omega )$ . At an almost complex structure J, the tangent space can be identified with $\Omega ^{0,1}(TX^{1,0})$ , the space of $(0,1)$ -forms with values in holomorphic vector fields [Reference Scarpa54, p. 14].

We let $\mathcal {G}$ denote the group of exact symplectomorphisms of $(M,\omega )$ , which acts naturally on $\mathcal {J}(M,\omega )$ . The Lie algebra of $\mathcal {G}$ can be identified with $C^{\infty }_0(M)$ , the functions which integrate to zero, through the Hamiltonian construction. For $h \in C^{\infty }_0(M)$ , we denote by $v_h$ the associated Hamiltonian vector field. The infinitesimal action of $\mathcal {G}$ is then given by

(3.7) $$ \begin{align} \begin{aligned} P: C^{\infty}_0(X,\mathbb{R})& \to T_J\mathcal{J}(M,\omega), \\ P h &= \mathcal{L}_{v_h} J. \end{aligned} \end{align} $$

Under the identification of $T_J\mathcal {J}(M,\omega )$ with $\Omega ^{0,1}(TX^{1,0})$ , the operator P corresponds to the operator [Reference Scarpa54, Lemma 1.4.3]

(3.8) $$ \begin{align} \begin{aligned} \mathcal{D}: C^{\infty}_0(X,\mathbb{R})& \to \Omega^{0,1}(TX^{1,0}), \\ \mathcal{D} h &= \bar\partial\nabla^{1,0} h. \end{aligned} \end{align} $$

The operator $\mathcal {D}$ plays a central role in the theory of cscK metrics. Note, for example, that its kernel consists of functions generating global holomorphic vector fields; these are called holomorphy potentials.

Now, let $(X,L)$ be a smooth polarised variety with complex structure $J_X \in \mathcal {J}(M,\omega )$ . We assume for the moment that $\operatorname {\mathrm {Aut}}(X,L)$ is trivial and will later consider the other case of interest for our main results, namely that $\operatorname {\mathrm {Aut}}(X,L)$ is finite. We denote by $\mathcal {J}_X(M,\omega ) \subset \mathcal {J}(M,\omega )$ the set of $J' \in \mathcal {J}(M,\omega )$ such that there is a diffeomorphism $\gamma $ , which lies in the connected component of the identity inside the space of diffeomorphisms of M, with $\gamma \cdot J' = J_X$ . Thus, $\mathcal {J}_X(M,\omega )$ corresponds to complex structures producing manifolds biholomorphic to X. This space is discussed by Gauduchon [Reference Gauduchon33, Section 8.1]; for us an important point will be that $\mathcal {J}_X(M,\omega )$ is actually a complex submanifold of $\mathcal {J}(M,\omega )$ [Reference Gauduchon33, Proposition 8.2.3]. As in the work of Fujiki [Reference Fujiki30, Section 8], the space $\mathcal {J}(M,\omega )$ admits a universal family $(\mathcal {U},\mathcal {L}_{\mathcal {U}}) \to \mathcal {J}(M,\omega )$ which hence restricts to a family $(\mathcal {U},\mathcal {L}_{\mathcal {U}})$ over $\mathcal {J}_{X}(M,\omega )$ . The fibre over a complex structure $J_b \in \mathcal {J}(M,\omega )$ is simply the complex manifold $(M,J_b)$ .

We next induce a form $\theta _{\mathcal {U}}$ on $\mathcal {U} \to \mathcal {J}_{X}(M,\omega )$ associated to the form $\theta $ on $\mathcal {U}$ , using the fact that each fibre of $\mathcal {U}\to \mathcal {J}_X(M,\omega )$ is isomorphic to X. For any $B \subset \mathcal {J}_{X}(M,\omega )$ a finite-dimensional complex submanifold, the Fischer–Grauert theorem produces an isomorphism $\mathcal {U}|_B \cong X \times B$ commuting with the maps to B. One can extend this isomorphism to an isomorphism of line bundles

(3.9) $$ \begin{align} \Psi_B: (\mathcal{U}|_B, \mathcal{L}|_B) \cong (X,L) \times B,\end{align} $$

perhaps after shrinking B [Reference Newstead49, Lemma 5.10] (while the proof given by Newstead assumes algebraicity of B, it also holds in the holomorphic category [Reference Hallam35, Lemma 6.3]). Then as we have assumed $\operatorname {\mathrm {Aut}}(X,L)$ is actually trivial, the isomorphism $\Psi _B$ is actually unique. Pulling back $\theta $ on X via $\Psi _B$ induces a closed form $\theta _B$ on $\mathcal {U}|_B$ , and uniqueness then means that the forms $\theta _B$ glue to a closed form $\theta _{\mathcal {U}}$ on all of $\mathcal {U}$ . Similarly, using these isomorphisms, the Z-energy induces a function

(3.10) $$ \begin{align}E_Z: \mathcal{J}_X(M,\omega) \to \mathbb{R}\end{align} $$

after fixing the reference Kähler metric $\omega $ on X.

Denote by $\Omega _{\varepsilon }$ the family of closed $(1,1)$ -forms on $\mathcal {J}_X(M,\omega )$ given by

(3.11) $$ \begin{align} \Omega_{\varepsilon} =\operatorname{\mathrm{Im}}\left(e^{-i\varphi_{\varepsilon}} \int_{\mathcal{U}/\mathcal{J}_X(M,\omega)} \tilde Z_{\varepsilon}(\mathcal{U},\mathcal{L}_{\mathcal{U}})\right), \end{align} $$

where $\tilde Z_{\varepsilon }(\mathcal {U},\mathcal {L}_{\mathcal {U}})$ is defined just as in Equation (3.4) using the relatively Kähler metric $\omega _{\mathcal {X}} \in c_1(\mathcal {L})$ , the form $\rho \in c_1(-K_{\mathcal {U}/ \mathcal {J}_{X}(M,\omega )})$ induced by the relatively Kähler metric $\omega _{\mathcal {X}}$ and $\theta _{\mathcal {U}}$ . The forms $\Omega _{\varepsilon }$ are then closed $\mathcal {G}$ -invariant $(1,1)$ -forms which are not, however, positive in general.

The Z-critical operator can be viewed as a function

$$ \begin{align*}\operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}}\tilde Z): \mathcal{J}_X(M,\omega) \to C^{\infty}_0(X),\end{align*} $$

which we wish to demonstrate is a moment map with respect to the $\Omega _{\varepsilon }$ . Thus, we need to understand the behaviour of $\operatorname {\mathrm {Im}}(e^{-i\varphi _{\varepsilon }}\tilde Z)$ under a change in complex structure. We will use a similar idea to Section 3.3.1, namely to realise the Z-energy as a Kähler potential, which requires us to relate the change in complex structure to the change in metric structure. Consider a path $J_t \in \mathcal {J}_X(M,\omega )$ , and let $F_t \cdot J_t = J_X$ for $F_t$ diffeomorphisms of X. Then we obtain a corresponding path of Kähler metrics $F_t^{*}\omega = \omega _t = \omega +i\partial \overline {\partial } \psi _t$ compatible with $J_X$ . Then the key fact we need is that the path $J_t$ satisfies [Reference Székelyhidi61, p. 1083]

(3.12) $$ \begin{align} \frac{d}{dt}\Big|_{t=0} J_t = JP\dot \psi_0. \end{align} $$

Theorem 3.9. The map

$$ \begin{align*}\mu_{\varepsilon} = \operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}}\tilde Z): \mathcal{J}_X(M,\omega) \to C^{\infty}_0(X)\end{align*} $$

is a moment map for the $\mathcal {G}$ -action on $\mathcal {J}_X(M,\omega )$ with respect to the forms $\Omega _{\varepsilon }$ .

Here, the statement means that the moment map condition is satisfied, note again that $\Omega _{\varepsilon }$ may not actually be positive (hence Kähler) in general.

Proof. Fix a point $b \in \mathcal {J}_X(M,\omega )$ at which we wish to demonstrate the moment map property, and let $Ph$ be the tangent vector at b induced the element $h \in \operatorname {\mathrm {Lie}} \mathcal {G} \cong C^{\infty }_0(M)$ . We show that for any finite-dimensional complex submanifold $B \subset \mathcal {J}_X(M,\omega )$ containing $Ph$ , the moment map equality

$$ \begin{align*}-\iota_{Ph} \Omega_{\varepsilon} = d\langle \mu_{\varepsilon}, Ph \rangle\end{align*} $$

holds. The proof of this is essentially the same as that of Theorem 3.6.

Perhaps after shrinking B, the family $(\mathcal {U}_B,\mathcal {L}_B) \to B$ satisfies

(3.13) $$ \begin{align} (\mathcal{U}_B, \mathcal{L}_B) \cong (X,L) \times B \end{align} $$

by the argument of Equation (3.9). We thus obtain a function

$$ \begin{align*}E_Z: B \to \mathbb{R}\end{align*} $$

by Equation 3.10, which by the argument of Theorem 3.6 satisfies

$$ \begin{align*}i\partial\overline{\partial} E_Z = \Omega_{\varepsilon},\end{align*} $$

an equality of $(1,1)$ -forms on B. Since this holds for each B, it also holds on $\mathcal {J}_X(M,\omega ).$

Consider a path $J_{b_t} \in B$ of almost complex structures such that the induced tangent vector at $t=0$ is given by $JPh$ . Then we obtain a corresponding path of Kähler metrics $\omega _t = \omega + i\partial \overline {\partial } \psi _t$ through the isomorphism of Equation (3.13), and Equation (3.12) implies that $\dot \psi _0 = h$ . It follows that

$$ \begin{align*}\frac{d}{dt}\Big|_{t=0} E_Z(J_t) = \int_X h \operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}}\tilde Z(J_b))\omega^n,\end{align*} $$

which means that as functions on $\mathcal {J}_X(M,\omega )$ we have

$$ \begin{align*}\langle dE_Z, JPh\rangle = \int_X h \operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}}\tilde Z(J_b))\omega^n = -\langle \mu_{\varepsilon}(b), Ph\rangle.\end{align*} $$

Then the same argument as Equation (3.6) implies that

$$ \begin{align*}\iota_{Ph} \Omega_{\varepsilon} = \iota_{Ph} i\partial\overline{\partial} E_Z = -d\langle \mu_{\varepsilon}(b), Ph\rangle,\end{align*} $$

which proves the defining equation of the moment map.

The $\mathcal {G}$ -action on $\operatorname {\mathrm {Lie}}(\mathcal {G})$ is the adjoint action, which corresponds to pullback of Hamiltonians [Reference Scarpa54, Equation (1.5)]. Then equivariance follows by the same argument as Theorem 3.6.

While positivity is not guaranteed for all $\varepsilon $ , it will be important to have positivity for finite-dimensional submanifolds and for $\varepsilon $ sufficiently small.

Proof. Fujiki proves that the form

$$ \begin{align*}\Omega_0 = - \int_{\mathcal{U}/\mathcal{J}(M,\omega)} \rho \wedge\omega^n + \frac{n}{n+1}\mu(X,L) \int_{\mathcal{U}/\mathcal{J}(M,\omega)}\omega^{n+1}\end{align*} $$

is actually a Kähler metric on $\mathcal {J}(M,\omega )$ and agrees with the usual Kähler metric on $\mathcal {J}(M,\omega )$ used in the moment map interpretation of the scalar curvature on $\mathcal {J}(M,\omega )$ [Reference Fujiki30, Theorem 8.3]. Thus, since

$$ \begin{align*}Z_{\varepsilon}(X,L) = \varepsilon^{-n}\int_X (iL^n +\operatorname{\mathrm{Re}}(\rho_{n-1})\varepsilon K_X.L^{n-1}) + O(\varepsilon^{-n+2}),\end{align*} $$

we have

$$ \begin{align*}\Omega_{\varepsilon} = \operatorname{\mathrm{Im}}\left(e^{-i\varphi_{\varepsilon}} \int_{\mathcal{X}/B} \tilde Z(\mathcal{X},\mathcal{L})\right)= -\varepsilon\operatorname{\mathrm{Re}}(\rho_{n-1})n\Phi^{*}\Omega + O(\varepsilon^2),\end{align*} $$

which implies the result.

One can also prove Fujiki’s result, namely the equality of the fibre integral $\Omega _0$ and the usual Kähler metric on $\mathcal {J}_X(M,\omega )$ , directly, giving another proof. By [Reference Gauduchon33, Equation 8.1.10], the tangent space $T_J\mathcal {J}_X(M,\omega )$ is spanned by elements of the form $Ph, JPh$ , for $h \in C^{\infty }_0(M,\omega )$ . But it follows from the argument of Theorem 3.9 that the moment map for the $\mathcal {G}$ -action on $\mathcal {J}_X(M,\omega )$ is given by the scalar curvature. But then since

$$ \begin{align*}\iota_{Ph}\Omega_{\mathcal{J}} = \iota_{Ph}\Omega_0\end{align*} $$

for all $h \in C^{\infty }_0(M)$ , it follows that the forms actually agree on $\mathcal {J}_X(M,\omega )$ .

In particular, if $B \subset \mathcal {J}_X(M,\omega )$ is a complex submanifold invariant under $K\subset \mathcal {G}$ , we obtain a genuine sequence of moment maps $\mu _{\varepsilon }$ for $\varepsilon \ll 0$ with respect to genuine Kähler metrics $\Omega _{\varepsilon }$ .

Remark 3.11. In the case $\operatorname {\mathrm {Aut}}(X,L)$ is nontrivial, but still finite, we denote by $G =\operatorname {\mathrm {Aut}}(X,L)$ , assume $\theta $ is G-invariant and work G-equivariantly. Let $\mathcal {J}_X(M,\omega )^G$ denote the fixed locus of the G-action on $\mathcal {J}_X(M,\omega ).$ Then while the isomorphisms

$$ \begin{align*}(\mathcal{U}_B, \mathcal{L}_B) \cong (X,L) \times B\end{align*} $$

of Equation 3.13, which were used to construct the form $\theta _{\mathcal {U}}$ on $\mathcal {U}$ and function $E_Z$ on $\mathcal {J}_X(M,\omega )$ are no longer unique, they are unique up to the action of G. But since $\theta $ is G-invariant by assumption, working on $\mathcal {J}_X(M,\omega )^G$ instead allows us to construct functions $E_Z$ on $\mathcal {J}_X(M,\omega )^G$ and a form $\theta _{\mathcal {U}^G}$ on $\mathcal {U}^G \to \mathcal {J}_X(M,\omega )^G$ . The proof of the moment map property is then identical to the case G is trivial.

3.4.1 Understanding the model operator

Let

$$ \begin{align*}\mathcal{F}_{\varepsilon}: C^{\infty}(X,\mathbb{R}) \to \mathbb{R}\end{align*} $$

denote the linearisation of the Z-critical operator $G_{\varepsilon }$ . In order to understand the mapping properties of $\mathcal {F}_{\varepsilon }$ , we will compare it to a simpler model operator. Much as with the linearisation of the scalar curvature, a key operator will be the operator

$$ \begin{align*}\mathcal{D}\psi = \bar\partial \nabla^{1,0}\psi,\end{align*} $$

as mentioned in Equation (3.8), whose kernel $\ker \mathcal {D} $ consists of functions inducing holomorphic vector fields on X. We denote the vector space of such functions, namely the holomorphy potentials, by $\mathfrak {k}$ ; we include the constant functions in our definition. The space of holomorphy potentials of integral zero is isomorphic to the Lie algebra of the automorphism group $\operatorname {\mathrm {Aut}}(X,L)$ . Letting $\mathcal {D}^{*}$ be the $L^2$ -adjoint of $\mathcal {D}$ with respect to the inner product induced by $\omega $ , the Lichnerowicz operator is given by $\mathcal {D}^{*}\mathcal {D}$ ; this is a fourth-order elliptic linear partial differential operator, whose kernel consists of holomorphy potentials [Reference Székelyhidi60, Definition 4.3]. It is then well-known that the linearisation of the scalar curvature at a cscK metric is given by $-\mathcal {D}^{*}\mathcal {D}$ [Reference Székelyhidi60, Lemma 4.4].

Another important term involved in the model operator is a sixth-order elliptic operator, defined as follows. As the vector bundle $TX^{1,0}$ is a holomorphic vector bundle, it admits a $\bar \partial $ -operator; we let $\bar \partial ^{*}$ denote its $L^2$ -adjoint. We will then also consider the operator $\mathcal {D}^{*}\bar \partial ^{*}\bar \partial \mathcal {D}$ , which can also be written

$$ \begin{align*}\nabla^{1,0*}(\bar\partial^{*}\bar\partial)^2 \nabla^{1,0} =\nabla^{1,0*}\Delta_{\bar \partial}^2 \nabla^{1,0},\end{align*} $$

where $\Delta _{\bar \partial }$ denotes the $\bar \partial $ -Laplacian. In particular, its symbol agrees with that of $\Delta ^3$ .

We will also need to consider two further operators $H_1, H_2$ , which are arbitrary self-adjoint operators satisfying for $j=1,2$

$$ \begin{align*}\int_X \gamma H_j \psi \omega^n = \int_X (\mathcal{D}\gamma, \mathcal{D}\psi)_{g_j} d\mu_j,\end{align*} $$

where each $d\mu _j$ is a smooth $(n,n)$ -form and each

$$ \begin{align*}g_j: \Gamma(T^{1,0}X\otimes \Omega^{0,1}(X)) \otimes \Gamma(T^{1,0}X\otimes \Omega^{0,1}(X)) \to \mathbb{R}\end{align*} $$

is a smooth bilinear pairing but not necessarily a metric. Our model operator will then take the form

(3.14) $$ \begin{align}\mathcal{G}_{\varepsilon} = c_0\mathcal{D}^{*}\mathcal{D} +\varepsilon(c_1\mathcal{D}^{*}\bar \partial^{*}\bar\partial\mathcal{D} + H_1) + \varepsilon^2(c_2\mathcal{D}^{*}\bar \partial^{*}\bar\partial\mathcal{D} + H_2), \end{align} $$

where $c_0$ and $c_1$ are strictly positive. Note that this is a self-adjoint elliptic operator for $\varepsilon $ sufficiently small, as its symbol agrees with that of $\varepsilon c_1 \Delta ^3 + \varepsilon ^2 c_2 \Delta ^3$ , which is elliptic for $\varepsilon $ sufficiently small since $c_1>0$ . As we explain in Remark 3.16, the $\varepsilon ^2$ -term is included as the estimates we prove will only allow us to perturb the operator by an $O(\varepsilon ^3)$ -term while retaining the relevant mapping properties.

We now work with Sobolev spaces $L^2_k$ for some large k. We let $ \mathfrak {k}_{k, \perp }^2$ denote the $L^2$ -orthogonal complement of the holomorphy potentials inside $L^2_k$ . Note that the holomorphy potentials themselves are actually smooth, being the kernel of the elliptic operator $\mathcal {D}^{*}\mathcal {D}$ , but we will sometimes also denote the space of holomorphy potentials as $\mathfrak {k}_{k}^2$ when considered as a subspace of $L^2_k$ .

Lemma 3.15. There is a constant $c>0$ such that for all sufficiently small $\varepsilon $ and for all $\psi \in \mathfrak {k}_{k, \perp }^2 $ we have

$$ \begin{align*} \langle \psi, \mathcal{G}_{\varepsilon} \psi\rangle_{L^2} \geq c\|\psi\|^2_{L^2}. \end{align*} $$

Furthermore, the kernel of $\mathcal {G}_{\varepsilon }$ consists of holomorphy potentials.

Proof. We first consider the operator

$$ \begin{align*}c_0\mathcal{D}^{*}\mathcal{D} + \varepsilon H_1 + \varepsilon^2 H_2.\end{align*} $$

The desired bound for the operator $\mathcal {D}^{*}\mathcal {D}$ is well-known: There is a constant $c'$ such that for all $\psi \in \mathfrak {k}_{k, \perp }^2 $ we have

$$ \begin{align*}\langle \psi, \mathcal{D}^{*}\mathcal{D} \psi\rangle_{L^2} \geq c'\|\psi\|^2_{L^2};\end{align*} $$

see, for example, Brönnle [Reference Brönnle7, Lemma 37]. We can obtain uniform bounds for $j=1,2$

$$ \begin{align*}-C_1(\mathcal{D}\gamma, \mathcal{D}\psi)_{\omega} \leq (\mathcal{D}\gamma, \mathcal{D}\psi)_{g_j} \leq C_j (\mathcal{D}\gamma, \mathcal{D}\psi)_{\omega}\end{align*} $$

for some $C_j>0$ , independent of $\psi , \gamma $ and hence can obtain uniform bounds for some possibly different $C_j$

$$ \begin{align*}-C_j\int_X(\mathcal{D}\gamma, \mathcal{D}\psi)_{\omega} \omega^n \leq \int_X(\mathcal{D}\gamma, \mathcal{D}\psi)_{g_j}d\mu_j \leq C_j \int_X(\mathcal{D}\gamma, \mathcal{D}\psi)_{\omega}\omega^n.\end{align*} $$

Here, we view $\omega $ as inducing a metric on $TX^{1,0}\otimes \Omega ^{0,1}.$ It follows that for $\varepsilon $ sufficiently small we have a bound

$$ \begin{align*}\langle \psi, c_0\mathcal{D}^{*}\mathcal{D} \psi + \varepsilon H_1 + \varepsilon H_2\psi\rangle_{L^2} \geq c\|\psi\|^2_{L^2}\end{align*} $$

for some $c>0$ .

The remaining terms are nonnegative for $\varepsilon $ sufficiently small. Indeed, for $\varepsilon $ sufficiently small the coefficient $\varepsilon c_1 + \varepsilon ^2c_2$ is positive and

$$ \begin{align*}\langle \psi,(\varepsilon c_1 + \varepsilon^2c_2)\mathcal{D}^{*}\bar \partial^{*}\bar\partial\mathcal{D}\psi \rangle_{L^2}^2= (\varepsilon c_1 + \varepsilon^2c_2)\| \bar \partial^{*}\mathcal{D}\psi \|_{L^2} \geq 0.\end{align*} $$

It follows that

$$ \begin{align*}\langle \psi, \mathcal{G}_{\varepsilon} \psi\rangle_{L^2} \geq c\|\psi\|^2_{L^2},\end{align*} $$

as required.

What remains is to characterise the kernel of $\mathcal {G}_{\varepsilon }$ . Note that certainly $\mathfrak {k} \subset \ker \mathcal {G}_{\varepsilon }$ since $\mathfrak {k} = \ker \mathcal {D}.$ Otherwise, we may write $\psi \in L^2_k$ as $\psi = \psi _{\mathfrak {k}_{k}^2} + \psi _{\mathfrak {k}_{k, \perp }^2}$ where $ \psi _{\mathfrak {k}_{k}^2} \in \mathfrak {k}_{k}^2$ and $\psi _{\mathfrak {k}_{k, \perp }^2} \in \mathfrak {k}_{k, \perp }^2$ are $L^2$ -orthogonal and we may assume $\psi _{\mathfrak {k}_{k, \perp }^2} \neq 0$ , and we see that

$$ \begin{align*} \langle \psi, \mathcal{G}_{\varepsilon} \psi\rangle_{L^2} = \langle \psi_{\mathfrak{k}_{k, \perp}^2}, \mathcal{G}_{\varepsilon} \psi_{\mathfrak{k}_{k, \perp}^2}\rangle_{L^2} \geq c\|\psi_{\mathfrak{k}_{k, \perp}^2}\|^2_{L^2}>0, \end{align*} $$

where we have used that

$$ \begin{align*} \langle \psi_{\mathfrak{k}_{k,}^2}, \mathcal{G}_{\varepsilon} \psi_{\mathfrak{k}_{k, \perp}^2}\rangle = 0 \end{align*} $$

since $\mathcal {G}_{\varepsilon }$ is self-adjoint and $\mathcal {G}_{\varepsilon } \psi _{\mathfrak {k}_{k}^2} = 0$ .

Corollary 3.9. For sufficiently small $\varepsilon $ , the operator

$$ \begin{align*}\mathcal{G}_{\varepsilon}: \mathfrak{k}_{k, \perp}^2 \to \mathfrak{k}_{k-6, \perp}^2\end{align*} $$

is an isomorphism. Furthermore, the induced map

$$ \begin{align*} \hat{\mathcal{G}}_{\varepsilon}: L^2_k \times \mathfrak{k} &\to L^2_{k-6}, \\ (\psi, h) &\to \mathcal{G}_{\varepsilon}\psi + h \end{align*} $$

is surjective and admits a right inverse.

Proof. We first show that $\mathcal {G}_{\varepsilon }$ does actually send $\mathfrak {k}_{k, \perp }^2$ to $\mathfrak {k}_{k-6, \perp }^2$ . In fact, for any $\psi \in L^2_k$ and any $h \in \mathfrak {k}$ we have

$$ \begin{align*}\langle h, \mathcal{G}_{\varepsilon} \psi\rangle_{L^2} = 0\end{align*} $$

again by self-adjointness of $\mathcal {G}_{\varepsilon }$ . Since

$$ \begin{align*}\mathcal{G}_{\varepsilon}: \mathfrak{k}_{k, \perp}^2 \to \mathfrak{k}_{k-6, \perp}^2\end{align*} $$

has trivial kernel by Lemma 3.15, it is a a self-adjoint elliptic partial differential operator with trivial kernel, hence is an isomorphism by the Fredholm alternative.

Surjectivity of the induced map $\hat {\mathcal {G}}_{\varepsilon }: L^2_k \times \mathfrak {k} \to L^2_{k-6},$ is an immediate consequence, while a right inverse can be constructed explicitly. Indeed, since the operator $\mathcal {G}_{\varepsilon }: \mathfrak {k}_{k, \perp }^2 \to \mathfrak {k}_{k-6, \perp }^2$ is an isomorphism, it admits some inverse $\mathcal {G}_{\varepsilon }^{-1}: \mathfrak {k}_{k-6, \perp }^2 \to \mathfrak {k}_{k, \perp }^2$ . Write $\psi \in \mathfrak {k}_{k-6, \perp }^2 $ as $\psi $ as $\psi _{\mathfrak {k}_{k-6}^2} + \psi _{\mathfrak {k}_{k-6, \perp }^2}$ where $ \psi _{\mathfrak {k}_{k-6}^2} \in \mathfrak {k}_{k-6}^2$ and $\psi _{\mathfrak {k}_{k-6, \perp }^2} \in \mathfrak {k}_{k-6, \perp }^2$ are $L^2$ -orthogonal. Note that $\psi _{\mathfrak {k}_{k-6}^2}$ is actually smooth as it is a holomorphy potential. Then a right inverse is given by

(3.15) $$ \begin{align} \mathcal{M}_{\varepsilon}(\psi) = (\mathcal{G}_{\varepsilon}^{-1}\psi_{\mathfrak{k}_{k-6}^2}, \psi_{\mathfrak{k}_{k-6}^2}). \end{align} $$

We next obtain an operator norm of the inverse operator $\mathcal {G}_{\varepsilon }^{-1}: \mathfrak {k}_{k, \perp }^2 \to \mathfrak {k}_{k-6, \perp }^2$ . We will use elliptic regularity estimates for this, so it is more convenient to consider the rescaled operator $\varepsilon ^{-1}\mathcal {G}_{\varepsilon }$ so that the ellipticity constants are actually uniformly bounded in $\varepsilon $ ; here, we recall that ellipticity follows from the fact that the sixth-order coefficient of $\mathcal {G}_{\varepsilon }$ is $(\varepsilon c_1+\varepsilon ^2 c_2) \Delta ^3$ , where we have assumed $c_1>0$ , so scaling by $\varepsilon ^{-1}$ gives a family of operators whose ellipticity constants are actually bounded independently of $\varepsilon $ .

Proposition 3.10 [Reference Taylor62, Chapter 5, Theorem 11.1].

There is a constant $c>0$ such that for any $\psi \in \mathfrak {k}_{k-6, \perp }^2$ and for all sufficiently small $\varepsilon $ there is a bound of the form

$$ \begin{align*}\|(\varepsilon^{-1}\mathcal{G}_{\varepsilon})^{-1} \psi\|_{L_{k}^2} \leq c\varepsilon^{-1}\left( \|(\varepsilon^{-1}\mathcal{G}_{\varepsilon})^{-1} \psi\|_{L^2} + \|\psi\|_{L_{k-6}^2}\right).\end{align*} $$

The point here is that our model operator $(\varepsilon ^{-1}\mathcal {G}_{\varepsilon })^{-1}$ has uniformly bounded ellipticity constants, but the norm of the coefficients of the equation are actually only bounded uniformly by $C\varepsilon ^{-1}$ for some constant C and hence are blowing up as $\varepsilon \to 0$ . Explicitly, the term which is blowing up is the leading term $\varepsilon ^{-1}\mathcal {D}^{*}\mathcal {D}$ . In this situation, one obtains an elliptic regularity estimate where the coefficient in the bound is $c\varepsilon ^{-1}$ . We learned that such a elliptic regularity estimate holds from an observation of Hashimoto for general elliptic operators [Reference Hashimoto36, p. 800]; the dependence of the coefficient in the bound on the norm of the coefficients is standard for second-order elliptic operators [Reference Gilbarg and Trudinger34, p. 92].

Corollary 3.11. There is a bound of the form

$$ \begin{align*}\|\mathcal{G}_{\varepsilon}^{-1}\|_{op} \leq C\varepsilon^{-2}\end{align*} $$

for the operator $\mathcal {G}_{\varepsilon }^{-1}: \mathfrak {k}_{k-6, \perp }^2 \to \mathfrak {k}_{k, \perp }^2$ , for some $C>0$ .

Proof. Let $\psi \in \mathfrak {k}_{k-6, \perp }^2$ , and set $\gamma = \mathcal {G}_{\varepsilon }^{-1} \psi , $ so that $\mathcal {G}_{\varepsilon } \gamma = \psi .$ The elliptic regularity estimate gives

$$ \begin{align*}\frac{\|(\varepsilon^{-1} \mathcal{G}_{\varepsilon})^{-1}\psi\|_{L^2_k}}{\|\psi\|_{L^2_{k-6}}} \leq c\varepsilon^{-1} + c\varepsilon^{-1}\frac{ \|(\varepsilon^{-1}\mathcal{G}_{\varepsilon})^{-1} \psi\|_{L^2}}{\|\psi\|_{L^2_{k-6}}} = c\varepsilon^{-1} +c \frac{ \|\mathcal{G}_{\varepsilon}^{-1} \psi\|_{L^2}}{\|\psi\|_{L^2_{k-6}}} .\end{align*} $$

By Cauchy–Schwarz, we have

$$ \begin{align*}\|\gamma\|_{L^2}\|\mathcal{G}_{\varepsilon}\gamma\|_{L^2} \geq \langle \gamma, \mathcal{G}_{\varepsilon}\gamma\rangle_{L^2},\end{align*} $$

so the bound

$$ \begin{align*}\langle \gamma, \mathcal{G}_{\varepsilon} \gamma\rangle_{L^2} \geq \tilde c\|\gamma\|^2_{L^2}\end{align*} $$

for some $\tilde c>0$ given by Lemma 3.15 implies

$$ \begin{align*}\|\mathcal{G}_{\varepsilon} \gamma\|_{L^2} \geq \tilde c \|\gamma\|_{L^2}.\end{align*} $$

Thus,

$$ \begin{align*}\frac{ \|\mathcal{G}_{\varepsilon}^{-1} \psi\|_{L^2}}{\|\psi\|_{L^2_{k-6}}} \leq \frac{ \|\mathcal{G}_{\varepsilon}^{-1} \psi\|_{L^2}}{\|\psi\|_{L^2}} = \frac{ \|\gamma \|_{L^2}}{\|\mathcal{G}_{\varepsilon}\gamma\|_{L^2}} \leq \tilde c^{-1}.\end{align*} $$

It follows that

$$ \begin{align*}\frac{\|(\varepsilon^{-1} \mathcal{G}_{\varepsilon})^{-1}\psi\|_{L^2_k}}{\|\psi\|_{L^2_{k-6}}} \leq c \varepsilon^{-1} + c(\tilde c^{-1}) \leq C\varepsilon^{-1}\end{align*} $$

for $\varepsilon $ sufficiently small and some $C>0$ , as required.

Recall that a right inverse to the induced map

$$ \begin{align*}\hat{\mathcal{G}}_{\varepsilon}: L^2_k \times \mathfrak{k} &\to L^2_{k-6}, \\ (\psi, h) &\to \mathcal{G}_{\varepsilon}\psi + h \end{align*} $$

is given through Equation (3.15) by

$$ \begin{align*}\mathcal{M}_{\varepsilon}(\psi) = (\mathcal{G}_{\varepsilon}^{-1}\psi_{\mathfrak{k}_{k-6}^2}, \psi_{\mathfrak{k}_{k-6}^2}),\end{align*} $$

where $ \psi _{\mathfrak {k}_{k-6}^2} \in \mathfrak {k}$ is the $L^2$ -projection of $\psi $ onto $\mathfrak {k}$ .

Corollary 3.12. There is a bound on the operator norm of $\mathcal {M}_{\varepsilon }$ of the form

$$ \begin{align*}\|\mathcal{M}_{\varepsilon}^{-1}\|_{op} \leq C\varepsilon^{-2}\end{align*} $$

for some $C>0$ .

We will eventually be interested in perturbations of $\hat {\mathcal {G}}_{\varepsilon }$ . The following is then a consequence of standard linear algebra (see for example [Reference Brönnle7, Lemma 4.3] for the result in linear algebra).

Corollary 3.13. Suppose $L_{\varepsilon }: L^2_k \to L^2_{k-6}$ is a sequence of bounded operators with $\|L_{\varepsilon }\|_{op} \leq K$ for some K independent of $\varepsilon $ . Then for all sufficiently small $\varepsilon $ the operator

$$ \begin{align*}(\psi, h) \to \hat{\mathcal{G}}_{\varepsilon}\psi + \varepsilon^3 L_{\varepsilon}\psi + h\end{align*} $$

is surjective and admits a right inverse $\tilde {\mathcal {M}}_{\varepsilon }$ . Moreover, there is a constant $C>0$ such that

$$ \begin{align*}\|\tilde{\mathcal{M}}_{\varepsilon}^{-1}\|_{op} \leq C \varepsilon^{-2}.\end{align*} $$

3.4.2 The approximate solution

We now assume that $\omega \in c_1(L)$ is cscK. Lemma 2.13 then implies that we have

$$ \begin{align*}\operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}} \tilde Z_{\varepsilon}(\omega)) = O(\varepsilon^2).\end{align*} $$

In order for our model linear operator to be a good approximation of the genuine linearised operator, we will need to consider a better approximation to a Z-critical Kähler metric. Since we are considering the general case when the Lichnerowicz operator $\mathcal {D}^{*}\mathcal {D}$ may have nontrivial kernel, or equivalently the case when $\operatorname {\mathrm {Aut}}(X,L)$ may not be discrete, rather than finding approximate Z-critical Kähler metrics, we will instead try to find a $\omega _{\varepsilon }$ approximately solving the condition that

(3.16) $$ \begin{align}\operatorname{\mathrm{Im}}(e^{-\varphi_{\varepsilon}}\tilde Z_{\varepsilon}(\omega_{\varepsilon})) \in \ker \mathcal{D}_{\varepsilon},\end{align} $$

where $\mathcal {D}_{\varepsilon } = \overline \partial \nabla ^{1,0}_{\varepsilon }$ is defined using $\omega _{\varepsilon }$ . That is to say, the function $\operatorname {\mathrm {Im}}(e^{-\varphi _{\varepsilon }}\tilde Z_{\varepsilon }(\omega _{\varepsilon }))$ is a holomorphy potential with respect to $\omega _{\varepsilon }$ . To this end, we recall that if $\nu $ is a Kähler potential and if h is the holomorphy potential with respect to $\omega $ for some holomorphic vector field, then the function

(3.17) $$ \begin{align}h + \frac{1}{2}\langle \nabla \nu, \nabla h\rangle\end{align} $$

is the holomorphy potential with respect to the Kähler metric $\omega _{\nu } = \omega +i\partial \overline {\partial } \nu $ (see, for example, [Reference Székelyhidi59, Lemma 12]).

Analogously to Corollary 3.9, the operator

$$ \begin{align*} L^2_k \times \mathfrak{k} &\to L^2_{k-4}, \\ (\psi, h) &\to \mathcal{D}^{*}\mathcal{D}\psi + h \end{align*} $$

is surjective. Although we have worked in Sobolev spaces, since the operator is elliptic, the same holds for smooth functions. Thus, given $e \in C^{\infty }(X)$ , there is a pair $(\psi ,h)$ with

(3.18) $$ \begin{align} \mathcal{D}^{*}\mathcal{D}\psi + h = e. \end{align} $$

Lemma 3.17. Suppose $\omega $ is a cscK metric. Then for any fixed m there is a sequence $\psi _j$ and holomorphy potentials $h_j$ such that

$$ \begin{align*}\operatorname{\mathrm{Im}}\left(e^{-i\varphi_{\varepsilon}}\tilde Z\left(\omega+ \sum_{j=1}^{m} \varepsilon^j i\partial\overline{\partial} \psi_j\right)\right) = \sum_{j=2}^{m+1} \varepsilon^2\left(h_j + \frac{1}{2}\left\langle h_j, \sum_{i=l}^m \varepsilon^l\psi_l\right\rangle \right)+ O(\varepsilon^{m+2}).\end{align*} $$

These are approximate solutions to Equation (3.16).

Proof. The linearisation of the scalar curvature at a cscK metric is the operator $-\mathcal {D}^{*}\mathcal {D}$ [Reference Székelyhidi60, Lemma 4.4]. As we have assumed $\omega $ is cscK, we have

$$ \begin{align*}\operatorname{\mathrm{Im}}\left(e^{-i\varphi_{\varepsilon}}\tilde Z_{\varepsilon}(\omega)\right) = e_2 \varepsilon^2 + O(\varepsilon^3).\end{align*} $$

By right invertibility of the Lichnerowicz operator there is a function $\psi _2$ and a holomorphy potential $h_2 \in \mathfrak {k}$ such that

$$ \begin{align*}(\operatorname{\mathrm{Re}}(\rho_{n-1})L^n)\mathcal{D}^{*}\mathcal{D}\psi_1 = e_2 - h_2.\end{align*} $$

Since $\mathcal {F}_{\varepsilon } = \varepsilon (\operatorname {\mathrm {Re}}(\rho _{n-1})L^n)\mathcal {D}^{*}\mathcal {D} + O(\varepsilon ^2)$ , it follows that

$$ \begin{align*}\operatorname{\mathrm{Im}}\left(e^{-i\varphi_{\varepsilon}}\tilde Z_{\varepsilon}\left(\omega+\varepsilon \partial\overline{\partial} \psi_1\right)\right) = h_2\varepsilon^2+ O(\varepsilon^3).\end{align*} $$

Next, consider the error term

$$ \begin{align*}\operatorname{\mathrm{Im}}\left(e^{-i\varphi_{\varepsilon}}\tilde Z_{\varepsilon}\left(\omega+\varepsilon \partial\overline{\partial} \psi_1\right)\right) - \varepsilon^2\left(h_2 +\frac{1}{2} \langle \nabla h_2, \nabla \varepsilon\psi_1 \rangle\right) = e_3\varepsilon^3.\end{align*} $$

We continue by applying Equation (3.18) to find a function $\psi _2$ and a holomorphy potential $h_3$ such that

$$ \begin{align*}(\operatorname{\mathrm{Re}}(\rho_{n-1})L^n)\mathcal{D}^{*}\mathcal{D}\psi_2 = e_2 - h_3.\end{align*} $$

Again, since the leading order linear operator is $(\operatorname {\mathrm {Re}}(\rho _{n-1})L^n)\mathcal {D}^{*}\mathcal {D}$ , it follows that

$$ \begin{align*}\operatorname{\mathrm{Im}}\left(e^{i-\varphi_{\varepsilon}}\tilde Z_{\varepsilon}\left(\omega+i\partial\overline{\partial}(\varepsilon \psi_1+\varepsilon^2 \psi_2)\right)\right) =\left(h_2 + \frac{1}{2} \langle \nabla h_2, \nabla \varepsilon\psi_1 \rangle\right) \varepsilon^2 +h_3\varepsilon^3 +O(\varepsilon^4).\end{align*} $$

In particular,

$$ \begin{align*}\operatorname{\mathrm{Im}}\left(e^{-\varphi_{\varepsilon}}\tilde Z_{\varepsilon}\left(\omega+i\partial\overline{\partial}(\varepsilon \psi_1+\varepsilon^2 \psi_2)\right)\right) = \sum_{j=2}^{3} \varepsilon^2\left(h_j + \frac{1}{2}\left\langle h_j, \sum_{l=1}^2 \varepsilon^l\psi_l\right\rangle \right) + O(\varepsilon^4).\end{align*} $$

Iterating this process gives the result.

We will only require the approximate solution

(3.19) $$ \begin{align}\omega_{\varepsilon} = \omega+\sum_{j=1}^{3}\varepsilon^j i\partial\overline{\partial} \psi_j,\end{align} $$

which satisfies

$$ \begin{align*}\operatorname{\mathrm{Im}}\left(e^{-i\varphi_{\varepsilon}}\tilde Z_{\varepsilon}\left(\omega+\sum_{j=1}^{3}\varepsilon^j i\partial\overline{\partial} \psi_j\right)\right) =\sum_{j=1}^3 \varepsilon^2\left(h_j + \frac{1}{2}\left\langle \nabla h_j, \nabla\left(\sum_{i=1}^3 \varepsilon^j\psi_j\right)\right\rangle \right)+ O(\varepsilon^{5}).\end{align*} $$

We then set

$$ \begin{align*}\gamma_{\varepsilon} = \sum_{j=1}^{3}\varepsilon^j i\partial\overline{\partial} \psi_j,\end{align*} $$

so that if h is a holomorphy potential with respect to $\omega $ , then $h+\frac {1}{2}\left \langle \nabla h,\nabla \gamma _{\varepsilon }\right \rangle $ is a holomorphy potential with respect to $\omega _{\varepsilon }$ by Equation (3.17).

We return to the model operator $\mathcal {G}_{\varepsilon }$ , however, now defined with respect to the approximate solution $\omega _{\varepsilon }$ . In order to understand its properties, for clarity we consider the Kähler metric $\omega _{\delta }$ the approximate solution to order $O(\delta ^5)$ given by Equation (3.19) (namely, we replace $\varepsilon $ with $\delta $ ). Denote by $\mathfrak {k}_{\delta }$ the space of holomorphy potentials with respect to $\omega _{\delta }$ . Then the results we have already established imply that for each fixed ${\delta }$ , the operator

$$ \begin{align*} \begin{aligned}\hat{\mathcal{G}}_{\varepsilon,{\delta}}: L^2_k \times \mathfrak{k}_{\delta} &\to L^2_{k-6},\\ (\psi, h) &\to \mathcal{G}_{\varepsilon,{\delta}}\psi + h\end{aligned} \end{align*} $$

is surjective for $\varepsilon $ sufficiently small.

We claim that one can take the $\varepsilon $ for which surjectivity of $\hat {\mathcal {G}}_{\varepsilon ,{\delta }}$ holds to be independent of ${\delta }$ for ${\delta }$ sufficiently small. More precisely, we claim that there is an $\varepsilon _0$ and a ${\delta }_0$ such that $\hat {\mathcal {G}}_{\varepsilon ,{\delta }}$ is surjective for all ${\delta } \leq {\delta }_0$ and $\varepsilon \leq \varepsilon _0$ . But this follows since in the ‘eigenvalue bound’ of Lemma 3.15

$$ \begin{align*}\langle \psi, \mathcal{G}_{\varepsilon,{\delta}}\psi\rangle_{L^2} \geq c_{\delta}\|\psi\|^2_{L^2},\end{align*} $$

for $\psi $ orthogonal to $\mathfrak {k}_{\delta }$ , the value $c_{\delta }$ is actually continuous in ${\delta }$ . Similar continuity statements in ${\delta }$ then further imply that the right inverse $\mathcal {M}_{\varepsilon , \delta }: L^2_{k-6} \to L^2_k \times \mathfrak {k}_{\delta }$ has operator norm which satisfies a uniform bound

$$ \begin{align*}\|\mathcal{M}_{\varepsilon,{\delta}}\|_{op} \leq C \varepsilon^{-2},\end{align*} $$

where C is independent of both $\delta $ and $\varepsilon $ . Here, the continuity used is in the elliptic regularity estimate of Proposition 3.10. It follows that we can take ${\delta }=\varepsilon $ and obtain a bound with respect to the approximate solution $\omega _{\varepsilon }$ . We will rephrase this in a form in which we will use these results.

Corollary 3.14. Denote by $\mathcal {G}_{\varepsilon }$ model operator with respect to the approximate solution $\omega _{\varepsilon }$ . Then the operator

$$ \begin{align*} \tilde{\mathcal{G}}_{\varepsilon}: L^2_k \times \mathfrak{k}_{\varepsilon} &\to L^2_{k-6}, \\ (\psi, h) &\to \mathcal{G}_{\varepsilon}\psi + h+\frac{1}{2}\langle \nabla h, \nabla \gamma_{\varepsilon}\rangle \end{align*} $$

is surjective and admits a right inverse $\tilde {\mathcal {M}}_{\varepsilon }$ . There is a bound on the operator norm of $\tilde {\mathcal {M}}_{\varepsilon }$ of the form $\|\tilde {\mathcal {M}}_{\varepsilon }\|_{op} \leq C \varepsilon ^{-2}$ .

Thus, if $L_{\varepsilon }: L^2_k \to L^2_{k-6}$ is a sequence of operators satisfying a uniform bound $\|L_{\varepsilon }\|_{op} \leq K$ independent of $\varepsilon $ , then the operator

$$ \begin{align*}(\psi, h) \to \mathcal{G}_{\varepsilon}\psi + h +\frac{1}{2}\langle \nabla h, \nabla \gamma_{\varepsilon}\rangle + \varepsilon^3 L_{\varepsilon}\end{align*} $$

is surjective and right invertible. The resulting right inverse also has operator norm satisfying a uniform bound by $C'\varepsilon ^{-2}$ for some $C'>0$ .

3.4.3 Understanding the expansion of the operator

We next consider some general aspects of the structure of the Z-critical equation. We will consider its expansion in powers of $\varepsilon $ , and to match with what we have considered it will be convenient to consider the ‘rescaled’ equation

$$ \begin{align*}-\varepsilon^{-1}\operatorname{\mathrm{Im}}\left(\frac{\tilde{Z}_{\varepsilon}(\omega)}{Z_{\varepsilon}(X,L)}\right) = \operatorname{\mathrm{Re}}(\rho_{n-1})L^nS(\omega) + O(\varepsilon)\end{align*} $$

so that if $\omega $ is a cscK metric its linearisation takes the form $- \operatorname {\mathrm {Re}}(\rho _{n-1})L^n\mathcal {D}^{*}\mathcal {D}+O(\varepsilon )$ . We will be interested in understanding the terms of order $\varepsilon $ and $\varepsilon ^2$ ; controlling these will allow us to see the full linearised operator as a perturbation of the sum involving only terms of order up to $\varepsilon ^2$ which will be sufficiently by Corollary 3.14. We will begin only by considering $\omega $ and will then later consider the approximate solution $\omega _{\varepsilon }$ .

We use our assumptions that:

1. (i) $\theta _1=0 = \theta _2=\theta _3=0$ . The condition on $\theta _1$ is used so that the leading order term in the expansion is the scalar curvature, rather than the twisted scalar curvature, while the conditions on $\theta _2$ and $\theta _3$ are of a more technical nature and allow us to understand the $\varepsilon ^2$ -term of the linearised operator. We expect that the conditions on $\theta _2$ and $\theta _3$ can be removed.

2. (ii) $\operatorname {\mathrm {Re}}(\rho _{n-1})<0$ , $\operatorname {\mathrm {Re}}(\rho _{n-2})>0$ and $\operatorname {\mathrm {Re}}(\rho _{n-3})=0$ . The condition on $\operatorname {\mathrm {Re}}(\rho _{n-1})$ is essentially a sign convention, what is really needed is that these two real parts have opposite sign. This is essential to the analysis and is used in the $L^2$ -bound for the model operator proved in Lemma 3.15. The condition on $\operatorname {\mathrm {Re}}(\rho _{n-3})$ is a technical assumption which we expect can be removed.

As in Lemma 2.13 we write $Z_{\varepsilon }(X,L) = r_{\varepsilon }e^{i\varphi _{\varepsilon }},$ so that

$$ \begin{align*}\operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}(X,L)} \tilde{Z}_{\varepsilon}(\omega)) &= r_{\varepsilon}(X,L) \operatorname{\mathrm{Im}}\left(\frac{\tilde{Z}_{\varepsilon}(\omega)}{Z_{\varepsilon}(X,L)}\right), \\ &= r_{\varepsilon}(X,L) \frac{\operatorname{\mathrm{Im}} \tilde{Z}_{\varepsilon}(\omega) \operatorname{\mathrm{Re}} Z_{\varepsilon}(X,L) - \operatorname{\mathrm{Re}} \tilde{Z}_{\varepsilon}(\omega) \operatorname{\mathrm{Im}} Z_{\varepsilon}(X,L) }{\operatorname{\mathrm{Re}} Z_{\varepsilon}(X,L)^2 + \operatorname{\mathrm{Im}} Z_{\varepsilon}(X,L)^2}, \end{align*} $$

where we recall

$$ \begin{align*}Z_{\varepsilon}(X,L) &= iL^n{\varepsilon}^{-n} + \rho_{n-1}L^{n-1}.K_X{\varepsilon}^{-n+1} + \rho_{n-2}L^{n-2}.K_X^2\varepsilon^{-n+2} +\ldots,\\ \tilde{Z}_{\varepsilon}(\omega) &= i - \rho_{n-1}\frac{\operatorname{\mathrm{Ric}}\omega\wedge\omega^{n-1}}{\omega^n}{\varepsilon} + O({\varepsilon}^2).\end{align*} $$

Here, we have used our assumptions

Our equation takes the form

$$ \begin{align*}\operatorname{\mathrm{Im}}\left(\frac{\tilde{Z}_{\varepsilon}(\omega)}{Z_{\varepsilon}(X,L)}\right) = \frac{\operatorname{\mathrm{Im}} \tilde{Z}_{\varepsilon}(\omega) \operatorname{\mathrm{Re}} Z_{\varepsilon}(X,L) - \operatorname{\mathrm{Re}} \tilde{Z}_{\varepsilon}(\omega) \operatorname{\mathrm{Im}} Z_{\varepsilon}(X,L) }{\operatorname{\mathrm{Re}} Z_{\varepsilon}(X,L)^2 + \operatorname{\mathrm{Im}} Z_{\varepsilon}(X,L)^2},\end{align*} $$

where explicitly

$$ \begin{align*}Z_{\varepsilon}(X,L) &= iL^n{\varepsilon}^{-n} +\rho_{n-1}\alpha_1{\varepsilon}^{-n+1} +\rho_{n-2} \alpha_2\varepsilon^{-n+2} +\rho_{n-1} \alpha_3\varepsilon^{-n+3} + O(\varepsilon^{-n+4}),\\ \tilde{Z}_{\varepsilon}(\omega) &= i +\rho_{n-1} \tilde{\alpha}_1{\varepsilon} +\rho_{n-2} \tilde{\alpha}_2{\varepsilon}^2 + \rho_{n-3} \tilde{\alpha}_3\varepsilon^3+O(\varepsilon^4),\end{align*} $$

and where $\alpha _1 = L^{n-1}.K_X$ , $\alpha _2 = L^{n-2}.K_X^2$ , $\alpha _3 = L^{n-3}.K_X^3 $ , while

$$ \begin{align*}\tilde{\alpha}_1 &= -\frac{\operatorname{\mathrm{Ric}}\omega\wedge\omega^{n-1}}{\omega^n}, \qquad \tilde \alpha_2 = \frac{\operatorname{\mathrm{Ric}}\omega^2\wedge\omega^{n-2}}{\omega^n} - \frac{2}{n-1}\Delta \frac{\operatorname{\mathrm{Ric}}\omega\wedge\omega^{n-1}}{\omega^n}, \\ \tilde \alpha_3 &= -\frac{ \operatorname{\mathrm{Ric}}\omega^3\wedge\omega^{n-3}}{\omega^n} + \frac{3}{n-2}\Delta \frac{\operatorname{\mathrm{Ric}}\omega^2\wedge\omega^{n-2}}{\omega^n}.\end{align*} $$

The factor

$$ \begin{align*}\frac{r_{\varepsilon}(X,L)}{\operatorname{\mathrm{Re}} Z_{\varepsilon}(X,L)^2 + \operatorname{\mathrm{Im}} Z_{\varepsilon}(X,L)^2}\end{align*} $$

plays only a minor role in our expansion of $\operatorname {\mathrm {Im}}\left (\frac {\tilde Z_{\varepsilon }(\omega )}{Z_{\varepsilon }(X,L)}\right )$ . Indeed, we will have good control over the leading order two terms in $\varepsilon $ , while the third-order (for our rescaled equation) $\varepsilon ^2$ term will require the most care to manage. So we can ignore this factor in controlling the linearisation. In addition, all relevant terms below have a uniform factor of $L^n$ arising from the leading order term of the expansion $Z_{\varepsilon }(X,L) = iL^n{\varepsilon }^{-n}+\ldots $ , and we also omit this uniform factor. Thus, we need only understand the leading order three terms in the expansion of

$$ \begin{align*}\operatorname{\mathrm{Im}} \tilde Z_{\varepsilon}(\omega) \operatorname{\mathrm{Re}} Z_{\varepsilon}(X,L) - \operatorname{\mathrm{Re}} \tilde Z_{\varepsilon}(\omega) \operatorname{\mathrm{Im}} Z_{\varepsilon}(X,L).\end{align*} $$

Recall that we have assumed $\theta _1=\theta _2=\theta _3=0$ .

We see that the leading order term is

$$ \begin{align*}\varepsilon^{-n+1}\operatorname{\mathrm{Re}}(\rho_{n-1})\left(L^{n-1}.K_X + \frac{\operatorname{\mathrm{Ric}}\omega\wedge \omega^{n-1}}{\omega^n}\right).\end{align*} $$

For the $\varepsilon ^{-n+2}$ -term, we will for the moment only be interested in the degree six operator, which we see is given by

$$ \begin{align*}-\varepsilon^{-n+2}\frac{2\operatorname{\mathrm{Re}}(\rho_{n-2})}{n-1}\Delta \left( \frac{\operatorname{\mathrm{Ric}}\omega\wedge\omega^{n-1}}{\omega^n}\right).\end{align*} $$

For the $\varepsilon ^{-n+3}$ -term, we see that the sixth-order component is given by, for some topological constant c

(3.20) $$ \begin{align}-\frac{3\operatorname{\mathrm{Re}}(\rho_{n-3})}{n-2}\Delta \left(\frac{\operatorname{\mathrm{Ric}}\omega^2\wedge\omega^{n-2}}{\omega^n} \right)+ c\operatorname{\mathrm{Im}}(\rho_{n-2})\Delta \left( \frac{\operatorname{\mathrm{Ric}}\omega\wedge\omega^{n-1}}{\omega^n}\right).\end{align} $$

In particular, if $\operatorname {\mathrm {Re}}(\rho _{n-3})=0$ , the first of these two terms vanishes.

While we have considered $\omega $ rather than the approximate solution $\omega _{\varepsilon } = \omega +i\partial \overline {\partial } \gamma _{\varepsilon }$ , essentially the same statements hold using $\omega _{\varepsilon }$ . If we write $\alpha _{j,\varepsilon }$ for the coefficients of $\varepsilon ^j$ in $\tilde Z_{\varepsilon }(\omega _{\varepsilon })$ , then we still have

$$ \begin{align*}\tilde Z_{\varepsilon}(\omega_{\varepsilon}) = i +\rho_{n-1} \tilde\alpha_{1,\varepsilon}{\varepsilon} +\rho_{n-2} \tilde \alpha_{2,\varepsilon}{\varepsilon}^2\tilde \alpha_2 + \rho_{n-3} \tilde \alpha_{3,\varepsilon}\varepsilon^3+O(\varepsilon^4),\end{align*} $$

implying the linearisation has similar properties up to order $\varepsilon ^4$ but for example with the leading order term replaced with

$$ \begin{align*}\varepsilon^{-n+1}\operatorname{\mathrm{Re}}(\rho_{n-1})\left(L^{n-1}.K_X + \frac{\operatorname{\mathrm{Ric}}\omega_{\varepsilon}\wedge \omega_{\varepsilon}^{n-1}}{\omega_{\varepsilon}^n}\right).\end{align*} $$

3.4.4 Properties of the linearisation

We now turn to the linearisation of the Z-critical equation. The aim is to compare the linearisation at the approximate solution $\omega +i\partial \overline {\partial } \gamma _{\varepsilon }$ to the model operator $\mathcal {G}_{\varepsilon }$ and in particular to use Corollary 3.14 to infer properties of the genuine linearised operator.

We begin with a general result. We fix a K-equivariant Kähler metric $\omega \in c_1(L)$ , not assumed to be cscK and denote by $\mathcal {F}_{\varepsilon }$ the linearisation of the operator

$$ \begin{align*}\psi \to \operatorname{\mathrm{Im}}\left(e^{-i\varphi_{\varepsilon}}\tilde Z_{\varepsilon}\left(\omega+ i\partial\overline{\partial} \psi \right)\right).\end{align*} $$

Denote also $\mathfrak {k}$ the space of holomorphy potentials with respect to $\omega $ .

Proposition 3.15. For all $0<\varepsilon \ll 1$ , the map

$$ \begin{align*}\hat{\mathcal{F}}_{\varepsilon}: L^2_k \times \mathfrak{k} &\to L^2_{k-6}, \\ (\psi, h) &\to \mathcal{F}_{\varepsilon}\psi - \langle \nabla \operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}}\tilde{Z}_{\varepsilon}(\omega)), \nabla \psi \rangle + h\end{align*} $$

is surjective. In addition exists a right inverse $\hat {\mathcal {P}}_{\varepsilon }$ of $\hat {\mathcal {F}}_{\varepsilon }$ whose operator norm satisfies a bound of the form $\|\hat {\mathcal {P}}_{\varepsilon }\|_{op} \leq C \varepsilon ^{-2}$ .

We recall our assumption, which will be used in the proof, that $(X,L)$ is a degeneration of a polarised manifold with discrete automorphism group.

Remark 3.18. To compare Proposition 3.15 to a well-known result in Kähler geometry, recall that the scalar curvature operator $\psi \to S(\omega +i\partial \overline {\partial } \psi )$ has linearisation [Reference Székelyhidi60, Lemma 4.4]

$$ \begin{align*}\psi \to -\mathcal{D}^{*}\mathcal{D}\psi + \langle \nabla S(\omega), \nabla\psi\rangle,\end{align*} $$

so subtracting $\langle \nabla S(\omega ), \nabla \psi \rangle $ leads to an operator whose kernel is precisely given by $\mathfrak {k}$ . Thus, adding h leads to a surjective operator, mirroring Proposition 3.15.

The proof will use the moment map techniques developed in Section 3.3.2. We continue to denote by $\mathcal {J}_X(M,\omega )$ the space of complex structures biholomorphic to the reference complex structure J and recall the closed $(1,1)$ -forms $\Omega _{\varepsilon }$ defined on $\mathcal {J}_X(M,\omega )$ through Equation (3.11). Any functions $u, v \in C^{\infty }(X,\mathbb {R})$ induce tangent vectors on $\mathcal {J}_X(M,\omega )$ through the assignment $u \to Pu$ of Equation (3.7); the same as true for functions in $L^2_k$ . As in Section 3.4.6, this process can be integrated, associating to $\psi $ a new complex structure $F_{\psi }(J)$ . We will use that the differential of the map $\psi \to F_{\psi }(J)$ at $\psi =0$ is [Reference Székelyhidi61, Equation 3]

$$ \begin{align*}\psi \to JP(\psi).\end{align*} $$

Proof of Proposition 3.15.

We use many of the ideas of Section 3.3 to understand the general properties of the linearised operator. Consider $\omega _t = \omega + t i\partial \overline {\partial } v$ so that the derivative of

$$ \begin{align*}\int_X u \operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}} \tilde Z_{\varepsilon}(\omega_t)) \omega_t^n\end{align*} $$

is given by

(3.21) $$ \begin{align} \frac{d}{dt}\int_X u \operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}} \tilde Z_{\varepsilon}(\omega_t)) \omega_t^n = \int_X u \mathcal{F}_{\varepsilon}v \omega^n + \int_X u\operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}} \tilde Z_{\varepsilon}(\omega)) \Delta v \omega^n.\end{align} $$

We are interested in the first of these terms, but the advantage of this perspective is that from the proof of Theorem 3.9 we know that for each t

$$ \begin{align*}\frac{d}{ds}\Big|_{s=0}E_Z(tv + su) = \int_X u \operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}}\tilde Z(\omega_t))\omega_t^n,\end{align*} $$

so that

$$ \begin{align*}\frac{d^2}{dtds}\Big|_{s,t=0}E_Z(tv + su) = \int_X u \mathcal{F}_{\varepsilon}v \omega^n + \int_X u\operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}} \tilde Z_{\varepsilon}(\omega)) \Delta v \omega^n.\end{align*} $$

It follows that the integral on the right-hand side, considered as a pairing on functions, is actually symmetric.

We need to identify the $\varepsilon ^2$ and $\varepsilon ^3$ terms in the expansion of $\mathcal {F}_{\varepsilon }$ in order to compare it to the model operator $\mathcal {G}_{\varepsilon }$ . For this, we will link with the space $\mathcal {J}_X(M,\omega )$ and the moment map interpretation of the Z-critical equation established in Section 3.3. We first consider the case $\operatorname {\mathrm {Aut}}(X,L)$ is discrete, which allows us to use the results of Section 3.3, which were proven under that assumption. Our functions $u,v$ can be viewed as inducing tangent vectors to $\mathcal {J}_X(M,\omega )$ at the point $J_X$ and we see from Equation (3.6) that

(3.22) $$ \begin{align} \Omega_{\varepsilon}(Pu, JPv) = \frac{d}{dt}\Big|_{t=0}\int_X u \operatorname{\mathrm{Im}}(e^{-\varphi_{\varepsilon}} \tilde Z_{\varepsilon}(J_t)) \omega^n, \end{align} $$

where we emphasise that we take the perspective that the complex structure is changing but the symplectic form $\omega $ is fixed.

We next compare this to the linearisation with fixed complex structure and varying symplectic structure. Let $f_t$ be the diffeomorphisms of X such that $f_t^{*}\omega _t = \omega $ and $f_t \cdot J = J$ . Then $f_t^{*}\omega _t^n = \omega ^n$ , while $f_t^{*}\operatorname {\mathrm {Im}}(e^{-\varphi _{\varepsilon }}\tilde Z_{\varepsilon } (\omega _t) )=\operatorname {\mathrm {Im}}(e^{-i\varphi _{\varepsilon }} \tilde Z_{\varepsilon }(J_t)).$ We also need to understand the infinitesimal change in u as we pull back along $f_t$ , for which we need to understand the construction of $f_t$ in more detail. As we only need to understand the infinitesimal construction of $f_t$ near $t=0$ , it suffices to note that $f_t$ is given by taking the gradient flow along a path of vector fields $\nu _t$ on X such that $\nu _0$ is the Hamiltonian vector field associated with the function v. Thus, the infinitesimal change in u is simply the Lie derivative

$$ \begin{align*}\mathcal{L}_{\nu_0} u = \langle \nabla u, \nabla v\rangle,\end{align*} $$

where we have used the relationship between the Poisson bracket of functions (that is, the pairing of the induced Hamiltonian vector fields with respect to $\omega $ ) and the inner products of the Riemannian gradients. That is,

$$ \begin{align*} \frac{d}{dt}\Big|_{t=0}\int_X u \operatorname{\mathrm{Im}}(e^{-\varphi_{\varepsilon}} \tilde Z_{\varepsilon}(J_t)) \omega^n =\frac{d}{dt}\Big|_{t=0}& \int_X u \operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}} \tilde Z_{\varepsilon}(\omega_t)) \omega_t^n \\ &- \int_X \langle \nabla u, \nabla v\rangle \operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}}\tilde Z_{\varepsilon} (\omega) )\omega^n. \end{align*} $$

We now use Equation (3.21), from which it follows that

$$ \begin{align*}\frac{d}{dt}\Big|_{t=0}&\int_X u \operatorname{\mathrm{Im}}(e^{-\varphi_{\varepsilon}} \tilde Z_{\varepsilon}(J_t)) \omega^n = \int_X u \mathcal{F}_{\varepsilon}v \omega^n \\ &+ \int_X u\operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}} \tilde Z_{\varepsilon}(\omega)) \Delta v \omega^n - \int_X \langle \nabla u, \nabla v\rangle \operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}}\tilde Z_{\varepsilon} (\omega) )\omega^n. \end{align*} $$

Since the final two terms on the right-hand side sum to $-\int _X u\langle \nabla \operatorname {\mathrm {Im}}(e^{-i\varphi _{\varepsilon }} \tilde Z_{\varepsilon }(\omega )),\nabla v\rangle \omega ^n ,$ we have

$$ \begin{align*} \Omega_{\varepsilon}(Pu,JPv) &= \frac{d}{dt}\Big|_{t=0}\int_X u \operatorname{\mathrm{Im}}(e^{-\varphi_{\varepsilon}} \tilde Z_{\varepsilon}(J_t)) \omega^n \\ &= \int_X u \mathcal{F}_{\varepsilon}v \omega^n- \int_X u\langle \nabla \operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}} \tilde Z_{\varepsilon}(\omega)), \nabla v\rangle\omega^n.\end{align*} $$

Thus, the operator

(3.23) $$ \begin{align} (u, v) \to \int_X u (\mathcal{F}_{\varepsilon}v -\langle \nabla \operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}} \tilde Z_{\varepsilon}(\omega)), \nabla v\rangle)\omega^n \end{align} $$

is a self-adjoint operator which only depends on $Pu, Pv$ . As this is true for all $\varepsilon $ , it is true for each term in the associated expansion in powers of $\varepsilon $ .

When $\operatorname {\mathrm {Aut}}(X,L)$ is not discrete, we use the key assumption that $(X,L)$ is a degeneration of a polarised manifold with discrete automorphism group. That is, $(X,L)$ is the central fibre of a test configuration for a polarised manifold with discrete automorphism group (to compare with our previous notation, we are considering $(X,L)$ to be what was previously denoted $(\mathcal {X}_0,\mathcal {L}_0)$ ). Thus, we obtain a family $J_t$ of complex structures on the fixed underlying smooth manifold M converging to $J_0$ , the complex structure inducing X. Since the linearisation satisfies Equation (3.23) for each t, the same equation holds at $t=0$ . In particular self-adjointness, and dependence only on $Pu, Pv$ hold also with respect to $J_0$ as well.

We use the results of Section 3.4.3 to identify the $\varepsilon , \varepsilon ^2$ and $\varepsilon ^3$ terms in the expansion of the operator

$$ \begin{align*}v \to \mathcal{F}_{\varepsilon}v -\langle \nabla \operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}} \tilde Z_{\varepsilon}(\omega)), \nabla v\rangle),\end{align*} $$

in order to compare them to the model operator. By what we have just proven, this operator must be self-adjoint, and the pairing

$$ \begin{align*}(u, v) \to \int_X u (\mathcal{F}_{\varepsilon}v -\langle \nabla \operatorname{\mathrm{Im}}(e^{-i\varphi_{\varepsilon}} \tilde Z_{\varepsilon}(\omega)), \nabla v\rangle)\omega^n\end{align*} $$

can only depend on $\mathcal {D} u$ and $\mathcal {D} v$ , due to the identification of Equation (3.8).

The leading order $\varepsilon $ -term is given by $-\operatorname {\mathrm {Re}}(\rho _{n-1})\mathcal {D}^{*}\mathcal {D}$ since the leading order $\varepsilon $ -term in the expansion of $\operatorname {\mathrm {Im}}(e^{-\varphi _{\varepsilon }} \tilde Z_{\varepsilon }(\omega ))$ is simply the scalar curvature. The sixth-order operator in the $\varepsilon ^2$ -term arises from linearising $\frac {2\operatorname {\mathrm {Re}}(\rho _{n-1})}{n(n-1)}\Delta S(\omega )$ , meaning that the linearisation inherits a term of the form $-\frac {2\operatorname {\mathrm {Re}}(\rho _{n-1})}{n(n-1)}\Delta \mathcal {D}^{*}\mathcal {D}$ . As we know the $\varepsilon ^2$ -term only depends on $\mathcal {D} u,\mathcal {D} v$ , the difference between the $\varepsilon ^2$ -term and $-\frac {2\operatorname {\mathrm {Re}}(\rho _{n-1})}{n(n-1)} (\bar \partial ^{*}\mathcal {D})^{*} \bar \partial ^{*} \mathcal {D}$ must be a fourth-order operator depending only on $\mathcal {D} u, \mathcal {D} v$ as both are of the form $-\frac {2\operatorname {\mathrm {Re}}(\rho _{n-1})}{n(n-1)} \Delta ^3$ plus some fourth-order operator. In particular, the $\varepsilon ^2$ -term must be of the form $c_1 \mathcal {D}^{*}\bar \partial ^{*}\bar \partial \mathcal {D}+ H_{1}$ , where

$$ \begin{align*}\int_X u H_{1} v \omega^n = \int_X (\mathcal{D} u, \mathcal{D} v)_{g_{1}} d\mu_{1},\end{align*} $$

and where $d\mu _{1}$ is a smooth $(n,n)$ -form and

$$ \begin{align*}g_{1}: \Gamma(T^{1,0}X\otimes \Omega^{0,1}(X)) \otimes \Gamma(T^{1,0}X\otimes \Omega^{0,1}(X)) \to \mathbb{R}\end{align*} $$

is a smooth bilinear pairing but not necessarily a metric. In particular, this is of the same form as the $\varepsilon ^2$ -term of our model operator of Equation (3.14) computed with respect to $\omega .$

We finally show that the $\varepsilon ^3$ -term of our linearisation takes the same form as the model operator, for which we use that $\operatorname {\mathrm {Re}}(\rho _{n-3}) = 0$ and $\theta _3=0$ . From Equation (3.20), it follows that the only sixth-order term arises from linearising a multiple of $\operatorname {\mathrm {Im}}(\rho _{n-2}) \Delta S(\omega )$ , which contributes one term which is involved in the $\varepsilon ^3$ -term of the model operator. The remaining order terms are fourth order and so again are given by some $H_2$ of the same form as $H_1$ .

What we have demonstrated is that the linearised operator agrees with the model operator to order $\varepsilon ^3$ . In particular Corollary 3.13 applies to give the statement of the Proposition.

In general, we wish to solve the equation

$$ \begin{align*}\operatorname{\mathrm{Im}}\left(e^{-i\varphi_{\varepsilon}}\tilde Z_{\varepsilon}\left(\omega+ i\partial\overline{\partial} \psi \right)\right) - f - \frac{1}{2}\langle \nabla \psi, \nabla f\rangle = 0,\end{align*} $$

for $f \in \mathfrak {k}$ and $\psi $ a Kähler potential. The linearisation of this operator is given by

$$ \begin{align*}d\mathcal{S}_{0,f}(\psi, h) = \mathcal{F}_{\varepsilon} \psi - \frac{1}{2}\langle \nabla \psi, \nabla f\rangle - h.\end{align*} $$

The following is an immediate consequence of Proposition 3.15.

Corollary 3.16. For all $0 < \varepsilon \ll 1$ , the operator

$$ \begin{align*}(\psi, h) \to d\mathcal{S}_{0,f}(\psi, h) + \frac{1}{2}\left\langle \nabla \psi, \nabla\left(f - 2\operatorname{\mathrm{Im}}\left(e^{-i\varphi_{\varepsilon}}\tilde Z_{\varepsilon}(\omega)\right) \right)\right\rangle\end{align*} $$

is surjective, admits a right inverse and the operator norm of the inverse is bounded by $C\varepsilon ^{-2}$ for some $C>0$ .