2.1 Deformation theory of cscK manifolds

In this section, we describe the deformations of the complex structure of a Kähler manifold with constant scalar curvature, following the notation of Székelyhidi [Reference Székelyhidi44, §3], [Reference Székelyhidi45, §6]. A similar description was also treated by Brönnle [Reference Brönnle5, Part 1], and the author [Reference Ortu41].

Let $(M, \omega )$ be a symplectic manifold. Define the infinite-dimensional manifold

$$\begin{align*}\mathscr{J} = \{J : TM \to TM \text{ almost complex structure compatible with } \omega \}. \end{align*}$$

We denote by $\mathscr {J}^{\mathrm {int}}$ the complex subspace of integrable almost complex structures. The tangent space at a point $J \in \mathscr {J}$ can be identified with

$$\begin{align*}T_J^{0,1}\mathscr{J} = \{\alpha \in \Omega^{0,1}(T^{1,0}M)\vert \omega(\alpha(u), v) + \omega(u, \alpha(x)) = 0\}. \end{align*}$$

Recall that a vector field $\xi _h$ is Hamiltonian with respect to $\omega $ if there exists $h \in C^\infty (M, \mathbb {R})$ such that $\omega (\xi _h, \cdot )= -\mathrm {d} h $ . On a Kähler manifold we can characterise a Hamiltonian vector field with Hamiltonian function h as $\xi _h =J\nabla ^g(h)$ , where $\nabla ^gh$ is the Riemannian gradient of h.

Let $\mathscr {G}$ be the group of Hamiltonian symplectomorphisms of $(M,\, \omega )$ . Then $\mathscr {G}$ acts on $\mathscr {J}$ by pull-back. The Lie algebra of $\mathscr {G}$ can be identified with the space of smooth functions on M with mean value zero, denoted $C_0^\infty (M)$ . The following theorem is due to Fujiki [Reference Fujiki19] and Donaldson [Reference Donaldson13].

Theorem 2.1. The action $\mathscr {G} \curvearrowright \mathscr {J}$ is Hamiltonian with moment map

(2.1) $$ \begin{align} \begin{aligned} \mathcal{S}: \mathscr{J} &\longrightarrow \mathrm{Lie}(\mathscr{G})^* \\ J &\longmapsto \mathrm{Scal}(\omega,J) - \widehat{S}. \end{aligned} \end{align} $$

Therefore, cscK metrics on M correspond to $J \in \mathscr {J}^{\mathrm {int}}$ such that $\mu (J) = 0$ . The function $\mathrm {Scal}(\omega ,J) - \widehat {S}$ is the Hermitian scalar curvature if J is nonintegrable, which coincides with the Riemannian scalar curvature in the integrable case. We view it as an element in $C_0^\infty (M)^*$ by identifying $\mathrm {Lie}(\mathscr {G})$ with its dual.

Let J be an integrable compatible complex structure on $(M,\omega )$ , and denote by $\mathcal {K}_J(\omega )$ the set of Kähler metrics in the same Kähler class of $\omega $ with respect to J, that is,

$$\begin{align*}\mathcal{K}_J(\omega) = \{\omega' \in [\omega] \mid \omega' = \omega + i \partial_J \bar{\partial}_J \phi \ \text{for some} \ \phi \in C^\infty(M, \mathbb{R})\}. \end{align*}$$

For fixed $J \in \mathscr {J}$ , consider the infinitesimal action of $\mathscr {G}$ on $\mathscr {J}$ , defined by the operator

(2.2) $$ \begin{align} \begin{aligned} P : C_0^\infty(M, \mathbb{R}) &\to T_J\mathscr{J} \\ h &\mapsto \mathcal{L}_{\xi_h} J, \end{aligned} \end{align} $$

where $\xi _h$ is the Hamiltonian vector field with function h; let $P^{\mathbb {C}}$ be its extension to $C_0^\infty (M, \mathbb {C})$ . The following proposition (see also [Reference Székelyhidi45, §6.1]) justifies the fact that instead of moving the Kähler metric inside the Kähler class with respect to a fixed J, one can think of $\omega $ as fixed and move the complex structure.

Proposition 2.2 [Reference Donaldson14, p.17]

For every $\omega _\phi \in \mathcal {K}_J(\omega )$ there exists an $f \in \mathrm {Diff}_0(M)$ such that $f^*\omega _\phi = \omega $ and $(M, \omega _\phi , J)$ is isomorphic to $(M, \omega , f^*J)$ . In particular, there is a map F that sends a Kähler potential $\phi $ to $F_{\phi }(J) = f^*J$ . The differential at $0$ of F is given by

$$ \begin{align*} \mathrm{d}_0F(\phi) = J\mathcal{L}_{\xi_{\phi}(\omega)}J. \end{align*} $$

Although a complexification of $\mathscr {G}$ does not genuinely exist, we can think of the image of the map F as an infinitesimal complexified orbit of the action of $\mathscr {G}$ on $\mathscr {J}$ . The proposition says that a variation of the Kähler form in a given Kähler class for J fixed corresponds to a variation of the complex structure J in the same $\mathscr {G}^c$ -orbit, for $\omega $ fixed.

The following result characterises P as the adjoint operator of the differential of the scalar curvature when varying the complex structure.

Lemma 2.3 [Reference Donaldson13, §4]

Let Q be the differential of the scalar curvature map (2.1) at a point $J \in \mathscr {J}$ . Then if $\alpha \in T_{J}\mathscr {J}$ and $\varphi \in C^\infty (M)$ ,

$$\begin{align*}\langle Q(\alpha), \varphi \rangle_{L^2} = \frac{1}{2}\langle \alpha, P(\varphi) \rangle_J \end{align*}$$

so $P^* = 2Q$ , and

$$\begin{align*}Q(P(\phi)) = \mathrm{Re}(\mathcal{D}^*\mathcal{D}\phi). \end{align*}$$

We next let $(M, L)$ be a compact Kähler manifold with an ample line bundle L, and assume that $\omega \in c_1(L)$ has constant scalar curvature with respect to the complex structure J. The deformations of J compatible with $\omega $ can be considered as elements of the vector space

(2.3) $$ \begin{align} \widetilde{H}^1 = \ker \lbrace\square := PP^*+(\bar{\partial}^*\bar{\partial})^2\rbrace \end{align} $$

on $T_J\mathscr {J}$ . Let K be the stabilizer of J with respect to the action of $\mathscr {G}$ on $\mathscr {J}$ . Then K is the group of Hamiltonian isometries of $(\omega , J)$ . The complexification $K^{\mathbb {C}}$ is the connected component of the identity of the group of automorphisms of M which lift to L, denoted $\mathrm {Aut}_0(M, L)$ . The group K (as well as its complexification) act on $\widetilde {H}^1$ by pull-back, and the origin is a fixed point of the action.

The following theorem is a symplectic version of Kuranishi’s theorem [Reference Kuranishi30] and an infinite-dimensional version of Luna’s slice theorem [Reference Luna32]: it provides a parametrisation of compatible infinitesimal deformations of J. A proof that takes into account the compatibility with the symplectic structure is explained in [Reference Chen and Sun7, Lemma 6.1].

Theorem 2.4. There exists a ball around the origin $V \subset \widetilde {H}^1$ and a K-equivariant embedding

(2.4) $$ \begin{align} \Psi : V \to \mathscr{J} \end{align} $$

such that $\Psi (0) = J$ and

1. 1. the $\mathscr {G}^c$ -orbit of every integrable complex structure near $J_0$ intersects the image of $\Psi $ ;

2. 2. If $x, x' \in V$ are in the same orbit for the complexified action of K, and $\Psi (x)$ is integrable, then $\Psi (x), \Psi (x')$ are in the same $\mathscr {G}^c$ -orbit;

3. 3. $\mathrm {Scal}(\omega , \Psi (x))$ is an element of the Lie algebra of K.

In particular, a moment map for the action of K on V is given by

(2.5) $$ \begin{align} x \mapsto \mathrm{Scal}(\omega, \Psi(x)) -\widehat{S}. \end{align} $$

The Kuranishi space is the base of a deformation family $\mathcal {M} \to V$ centred at J.

Definition 2.5. We say that the deformation family is complete if, for another deformation family $\mathcal {M}_1 \to V_1$ , there exists a map

$$\begin{align*}\tau : V_1 \to V \end{align*}$$

such that $\mathcal {M}_1 = \tau ^*\mathcal {M}$ . Moreover, if the differential of $\tau $ at the origin is unique we say that the deformation family is versal, and universal if $\tau $ itself is unique.

We remark that Kuranishi’s theorem [Reference Kuranishi30] also gives the existence of a versal deformation family centred at J which is complete for nearby complex structures.

2.2 Fibrations

In this section, we collect some results on Kähler metrics on holomorphic submersions. Let $\pi _X:X \to B$ be a holomorphic submersion of compact complex manifolds. Let n be the dimension of B, m the dimension of the fibres, so $\mathrm {dim}(X)=n+m$ . We also assume that the spaces $H^0(X_b, T^{1,0}X_b)$ of holomorphic vector fields on $X_b$ are isomorphic as Lie algebras for all b.

Let $\omega $ be a relatively Kähler metric on X and $\omega _B$ a Kähler metric on B. We can define a Kähler metric on X by taking the relative Kähler metric $\omega $ and adding a large multiple of the pull-back of the base metric:

$$ \begin{align*} \omega_k := \omega + k\omega_B \qquad k \gg0. \end{align*} $$

The relative symplectic form $\omega $ determines a splitting of the tangent space

(2.6) $$ \begin{align} TX = \mathcal{V} \oplus \mathcal{H}^{\omega}, \end{align} $$

where $\mathcal {V}_x = T_x X_{\pi (x)}$ is the tangent space to the fibre $X_x$ , and

$$ \begin{align*} \mathcal{H}^{\omega}_x = \{u \in T_xX| \omega(u, v)=0 \ \forall v \in \mathcal{V}_x\}. \end{align*} $$

In the context of symplectic fibrations $\omega $ is called a symplectic connection [Reference McDuff and Salamon34, Chapter 6] and the splitting (2.6) induces a splitting on all tensor bundles.

In local computations, we make the following notational conventions:

• $\{ w^1, \dots , w^m \}$ denote vertical holomorphic coordinates; indices are denoted with the letters $a, b, c, \dots $ ;

• $\{z^1, \dots , z^n\}$ denote holomorphic coordinates on the base; indices are denoted with the letters $i, j, k, \dots $ ;

• $\{\zeta ^1, \dots , \zeta ^{n+m}\}$ denote holomorphic coordinates on the total space; indices are denoted with the letters $p,q,r, \dots $ .

Lemma 2.7. Let $\alpha $ be a covariant $2$ -tensor of type $(1,1)$ . Then

$$ \begin{align*} \Lambda_{\omega_k} \alpha = \Lambda_{\mathcal{V}}\alpha + \frac{1}{k}\Lambda_{\omega_B}\alpha + O\left(k^{-2}\right), \end{align*} $$

where $\Lambda _{\omega _k}$ denotes the contraction with the Kähler metric $\omega _k$ , $\Lambda _{\omega _B}$ the contraction with the (pull-back of the) base metric $\omega _B$ and $\Lambda _{\mathcal {V}}$ the contraction with the vertical part of the relatively Kähler metric $\omega $ .

Proof. At each point $x \in X_b \subset X$ , the matrix $[\omega _k]$ is block-diagonal:

$$ \begin{align*} \begin{pmatrix} [\omega_b] &0\\ 0 & [\omega_{k, \mathcal{H}}] \end{pmatrix}. \end{align*} $$

Therefore in local coordinates around the point x

$$ \begin{align*} (\omega_k)^{p\bar{q}}\alpha_{p\bar{q}} = (\omega_b)^{a\bar{b}}\alpha_{a\bar{b}} + (\omega_{k,\mathcal{H}})^{i\bar{j}}\alpha_{i\bar{j}}. \end{align*} $$

The horizontal part of $\omega _k$ , denoted $\omega _{k, \mathcal {H}}$ splits as $\omega _{\mathcal {H}}+ k\omega _B$ , where $\omega _{\mathcal {H}}$ is the horizontal part of $\omega $ . Let $[\omega _{k, \mathcal {H}}]$ , $[\omega _{\mathcal {H}}]$ and $[\omega _B]$ be the matrices of coefficients of the two-forms $\omega _{k, \mathcal {H}}$ , $\omega _{\mathcal {H}}$ and $\omega _B$ respectively. We can write

$$ \begin{align*} [\omega_{k,\mathcal{H}}] = k \left( \frac{[\omega_{\mathcal{H}]} [\omega_B]^{-1}}{k} + \mathrm{Id}\right) [\omega_B], \end{align*} $$

where $\mathrm {Id}$ is the identity matrix and the base form $\omega _B$ induces a Riemannian metric on the horizontal tangent bundle, so its inverse is well defined. The inverse of the matrix $[\omega _{k,\mathcal {H}}]$ can be expanded in inverse powers of k as

$$ \begin{align*} \begin{aligned} [\omega_{k,\mathcal{H}}] ^ {-1} &= k^{-1}[\omega_B]^{-1}\left( \frac{[\omega_{\mathcal{H}}] [\omega_B]^{-1}}{k} + \mathrm{Id}\right)^{-1} \\ &=k^{-1}[\omega_B]^{-1} \sum_{i=0}^\infty \left( -\frac{[\omega_{\mathcal{H}}] [\omega_B]^{-1}}{k}\right)^i \\ &= \frac{1}{k} [\omega_B]^{-1} + O\left( k^{-2} \right). \end{aligned} \end{align*} $$

This implies the claim.

Let us now restrict to the case when the fibres of X each admit a cscK metric. The following lemma explains that one can define a relatively Kähler metric on the total space which is relatively cscK.

Under the relatively cscK assumption, we now explain how the function space $C^\infty (X, \mathbb {R})$ splits in a way that takes into account the fibration structure. We begin by considering the vertical Lichnerowicz operator,

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

defined fibrewise as $\left ( \mathcal {D}_{\mathcal {V}}^*\mathcal {D}_{\mathcal {V}} \varphi \right )|_{X_b} = \mathcal {D}_b^*\mathcal {D}_b \varphi |_{X_b}$ . It is a real operator since the fibrewise metric is cscK. By integrating a function $\varphi \in C^\infty (X, \mathbb {R})$ over the fibres of $\pi $ , we define a projection

$$ \begin{align*} \begin{aligned} C^\infty(X, \mathbb{R}) &\longrightarrow C^\infty(B, \mathbb{R}) \\ \varphi &\longmapsto \int_{X/B} \varphi \omega^m. \end{aligned} \end{align*} $$

Its kernel is given by the space $C^\infty _0(X, \mathbb {R})$ of functions that have fibrewise mean value zero. A key step in the study of optimal symplectic connections is that we can further split this space as follows.

Consider the real vector bundle $E \to B$ [Reference Dervan and Sektnan9, §3.1], whose fibre over $b \in B$ is the real finite-dimensional vector space $\mathrm {ker}_0(\mathcal {D}_b^*\mathcal {D}_b)$ of holomorphy potentials on the fibre $X_b$ with mean-value zero with respect to $\omega _b$ . It is well defined as a vector bundle since we assume that the complex dimension of the space $H^0(X_b, T^{1,0}X_b)$ of holomorphic vector fields on $X_b$ is independent of b [Reference Hallam24, Proposition 2.16]. The space of smooth global sections of E, denoted by $C^\infty (E)$ , is given by the kernel over the fibrewise mean-value-zero functions of the vertical Lichnerowicz operator $\mathcal {D}_{\mathcal {V}}^*\mathcal {D}_{\mathcal {V}}$ .

We can split $C^\infty _0(X)$ as

$$ \begin{align*} C^\infty_0(X, \mathbb{R}) = C^\infty(E) \oplus C^\infty(R), \end{align*} $$

where $C^\infty (R)$ is the fibrewise $L^2$ -orthogonal complement with respect to the fibre metric $\omega _b$ , that is, for all $\varphi \in \mathrm {ker}_0 \mathcal {D}_b^*\mathcal {D}_b$ , $\psi \in C^\infty (R)$

$$ \begin{align*} \langle \varphi , \psi \rangle_b := \int_{X_b} \varphi \psi \omega_b^m = 0. \end{align*} $$

So we obtain

(2.7) $$ \begin{align} C^\infty(X, \mathbb{R}) = C^\infty(B) \oplus C^\infty(E)\oplus C^\infty(R). \end{align} $$

We denote by $p_E : C^\infty (X) \to C^\infty (E)$ the projection.

In particular, if we change the relatively cscK metric $\omega $ to $\omega + i\partial \bar {\partial }\varphi $ with $\varphi \in \mathcal {K}_E$ , the vector bundles $E(\omega )$ and $E(\omega +i\partial \bar {\partial }\varphi )$ are isomorphic. The following result [Reference Dervan and Sektnan11, Lemma 4.20] relates the space $\mathcal {K}_E$ to the vector bundle $E\to B$ .

Lemma 2.11. Let $\varphi _t : [0, 1] \to \mathcal {K}_E$ be a smooth path. Then for all t

$$ \begin{align*} \dot{\varphi}_t \in C^\infty(B) \oplus C^\infty(E(\omega + i\partial\bar{\partial}\varphi_t)) \end{align*} $$

that is to say that, up to a function on the base, $\dot {\varphi _t}$ is a fibrewise holomorphy potential with respect to $\omega +i\partial \bar {\partial }\varphi _t$ .

2.3 Optimal symplectic connections

Let $(Y, H_Y)\to (B,L)$ be a holomorphic submersion with a relatively ample line bundle $H_Y \to Y$ . In this section we recall the theory of optimal symplectic connections [Reference Ortu40]. The following assumptions restrict the class of admissible fibrations to those whose fibres satisfy a stability property defined in terms of K-stability. More precisely we assume that:

1. (i) the fibres $Y_b$ are analytically K-semistable, which means by definition that they each admit a degeneration to a cscK manifold $X_b$ . We also assume that the deformation is compatible with the fibration structure in the following sense: there exists a holomorphic map $\varpi :(\mathcal {X}, \mathcal {H}) \to (B, L) \times S$ , parametrised by a disk S, such that for $s\ne 0$ , the family $(\mathcal {X}_s, \mathcal {H}_s) \to B$ is isomorphic to the original fibration $\pi _Y:(Y, H_Y) \to B$ and the central fibration at $s=0$ is a family $\pi _X:(X, H_X) \to B$ whose fibres are cscK;

2. (ii) the automorphism groups $\mathrm {Aut}_0(X_b, H_b)$ of the fibres are all isomorphic.

The first hypothesis is a stability assumption. We will refer to the submersion $X \to B$ as the relatively cscK degeneration of $Y\to B$ . The second hypothesis holds if and only if the spaces $H^0(X_b, T^{1,0}X_b)$ are isomorphic as Lie algebras, which we assumed in §2.2 to define the vector bundle $E\to B$ . It is needed to define optimal symplectic connection and it is a key hypothesis for the construction of the moduli space.

A relative version of Ehresmann’s theorem [Reference Ortu40, Proposition 4.5] implies that X and Y are diffeomorphic. Let M denote the underlying smooth manifold. Since the Chern classes are integral classes we have that $c_1(H_X)$ coincides with $c_1(H_Y)$ as cohomology classes on M. Since Y is a small deformation of X, the cohomology class $c_1(H_X)$ is of type $(1,1)$ on Y, so $\omega $ is a $(1,1)$ -form with respect to the complex structure of Y [Reference Huybrechts28, §6.1]. By Moser’s theorem [Reference Cannas da Silva6, Theorem 7.2], we can modify the complex structure of Y by a small diffeomorphism so that $\omega $ restricted to the fibres of $\pi _Y$ is compatible with the restriction of the complex structure. Thus we can assume that $\omega $ is relatively Kähler on Y.

Therefore we can view $Y\to B$ and $X\to B$ as the same relatively symplectic fibration $(M, \omega ) \to B$ with two different integrable almost complex structures J and I where $(\omega , I)$ is relatively cscK and $(\omega , J)$ is just relatively Kähler. The family $\mathcal {X} \to B\times S$ corresponds to a family of complex structures $\{J_s\}$ on $(M, \omega ) \to B$ , such that for $s \ne 0$ , $J_s$ is isomorphic to J and $J_0$ is isomorphic to I.

The definition of optimal symplectic connections involves various curvature quantities and a description of the complex structure of Y as a deformation of the one on X. To describe the deformations of the complex structure I, we begin by considering the space $\mathscr {J}_\pi $ of almost complex structures compatible with $\omega $ and such that $\mathrm {d}\pi \circ J = J_B \circ \mathrm {d}\pi $ . The tangent space at I to $\mathscr {J}_\pi $ can be identified with

$$\begin{align*}T_{I}^{0,1}\mathscr{J}_\pi = \{\alpha \in \Omega^{0,1}(\mathcal{V}^{1,0}) | \omega_F (\alpha\cdot, \cdot) + \omega_F (\cdot, \alpha\cdot) = 0\}, \end{align*}$$

where $\omega _F$ is the vertical part of $\omega $ . In local holomorphic coordinates an element $\alpha \in T_{J}^{0,1}\mathscr {J}_\pi $ is written as

$$\begin{align*}\alpha = \alpha^a_{\ \bar{b}} \partial_{w^a}\otimes \mathrm{d} \bar{w}^b + \alpha^a_{\ \bar{j}}\partial_{w^a}\otimes \mathrm{d} \bar{z}^j, \end{align*}$$

where [Reference Ortu40, Lemma 4.1]

(2.8) $$ \begin{align} \alpha^{a}_{\ \bar{j}} = \alpha^a_{\ \bar{c}}(\omega_{F})^{d\bar{c}} (\omega)_{d\bar{j}}. \end{align} $$

Consider the map

$$ \begin{align*} \begin{aligned} P_{\mathcal{V}}:C^\infty_0(X, \mathbb{R}) &\longrightarrow T_{I}^{0,1}\mathscr{J}_\pi \\ \varphi &\longmapsto \bar{\partial} (\mathrm{grad}^{\omega_F}\varphi)^{1,0}, \end{aligned} \end{align*} $$

which is the relative version of the map (2.2). Let $\widetilde {H}^1_{\mathcal {V}}$ be the kernel of the elliptic operator

$$\begin{align*}\square_{\mathcal{V}} = P_{\mathcal{V}}P_{\mathcal{V}}^* + (\bar{\partial}^*\bar{\partial})^2, \end{align*}$$

where the adjoint is computed with respect to any Kähler metric on X which restricts to $\omega _F$ vertically. The operator $\square _{\mathcal {V}}$ is fibrewise elliptic because $P_{\mathcal {V}}P_{\mathcal {V}}^*$ is elliptic in the fibre direction, and globally elliptic because $(\bar {\partial }^*\bar {\partial })^2$ is. The space $\widetilde {H}^1_{\mathcal {V}}$ is the space of integrable first-order deformations of I. Let $K_\pi $ be the group of biholomorphisms of I which restrict to an isometry on each fibre, with respect to the fibrewise metric defined by $(\omega , I)$ . The group $K_\pi $ acts on $\widetilde {H}^1_{\mathcal {V}}$ by pull-back. The following theorem is a fibrewise version of Theorem 2.4.

Theorem 2.12 [Reference Ortu40, Theorem 4.7]

There exists a neighbourhood of the origin $V_\pi \subset \widetilde {H}^1_{\mathcal {V}}$ and a $K_\pi $ -equivariant map

(2.9) $$ \begin{align} \Phi : V_\pi \rightarrow \mathscr{J}_\pi \end{align} $$

such that:

1. 1. $\Phi (0) = I$ ;

2. 2. If $v_1, v_2 \in V_\pi $ and $v_1 \in K_b^{\mathbb {C}}\cdot v_2$ , and if $\Phi (v_1)$ is integrable, then $\Phi (v_1)$ and $\Phi (v_2)$ are isomorphic and for all $b \in B$ the complex structure $\Phi (v_1) |_{X_b}$ is in the same $\mathscr {G}_b^c$ -orbit as $\Phi (v_2) |_{X_b}$ ;

3. 3. For any $J \in \mathscr {J}_\pi $ integrable close to I, there exists $J'$ in the image of $\Phi $ such that, for all b, $J_b'$ is in the same $\mathscr {G}_b^c$ -orbit as $J_b$ ;

4. 4. $\mathrm {Scal}_{\mathcal {V}}\left (\omega _X, \Phi (x)\right ) \in C^\infty (E, I)$ .

We now describe the curvature quantities involved in the definition of optimal symplectic connections. The symplectic connection $\omega $ determines the following curvature quantities:

1. (i) the symplectic curvature is a two-form on B with values in the fibrewise Hamiltonian vector fields defined for $v_1, v_2 \in \mathfrak {X}(B)$ as

   $$ \begin{align*} F_{\mathcal{H}}(u_1, u_2) = [u_1^\sharp, u_2^\sharp]^{\mathrm{vert}}, \end{align*} $$
   where $u_j^\sharp $ denotes the horizontal lift. Let $\gamma $ be the map which associates to a fibrewise Hamiltonian vector field its fibrewise Hamiltonian function with fibrewise mean value zero. Thus we consider $\gamma ^*(F_{\mathcal {H}})$ , which is a two-form on B with values in $C^\infty _0(Y, \mathbb {R})$ , and we pull it back to Y;

2. (ii) the curvature $\rho $ of the Hermitian connection induced by $(\omega , I)$ on the top wedge power $\bigwedge \nolimits ^{m}\mathcal {V}$ . We will primarily consider its purely horizontal part $\rho _{\mathcal {H}}$ ;

3. (iii) the curvature of the deformation family $\nu $ , which is defined as follows. On each fibre $X_b$ of the relatively cscK degeneration $X \to B$ , let $\widetilde {H}^1_b$ be the deformation space (2.3) and $V_b$ the Kuranishi space described in Theorem 2.4. Let $v_b \in \widetilde {H}^1_b$ and denote by $\mathbf {v}_b$ a vector field on $V_b$ such that $\mathbf {v}_b\vert _{0} = v_b$ . For any $f \in \mathfrak {k}_b$ let

   $$\begin{align*}A_f v_b := -(\mathcal{L}_{\mathrm{grad}^\omega f} \mathbf{v}_b)\vert_0. \end{align*}$$
   Define $\nu _b$ as
   $$ \begin{align*} \langle \nu_b(v_b), f \rangle = \frac{1}{2}\Omega_0(A_f v_b, v_b). \end{align*} $$
   In particular, $\nu _b$ is the moment map for the action of the group $K_b$ on the tangent space at the origin to $V_b$ . Therefore we can define a global section $\nu $ of the vector bundle $E \to B$ of relatively holomorphy potentials of $X \to B$ , as
   (2.10) $$ \begin{align} \nu(b) = \nu_b(v_b). \end{align} $$
   Given $x \in V_\pi $ , $v \in \widetilde {H}^1_{\mathcal {V}}$ , we will write $\nu _x(v)$ when we want to underline the dependence of the map $\nu $ on the complex structure $\Phi (x)$ and on the deformation v.

Definition 2.13 [Reference Ortu40, Definition 3.3]

A relatively Kähler metric $\omega $ on $Y \to B$ is called an optimal symplectic connection if

(2.11) $$ \begin{align} p_E \left(\Delta_{\mathcal{V}}(\Lambda_{\omega_B} \gamma^*(F_{\mathcal{H}}))+ \Lambda_{\omega_B}\rho_{\mathcal{H}}\right) + \frac{\lambda}{2}\nu=0, \end{align} $$

for a positive number $\lambda $ . In the following, we will use the notation $\Theta (\omega , J) = \Delta _{\mathcal {V}}(\Lambda _{\omega _B} \gamma ^*(F_{\mathcal {H}}))+ \Lambda _{\omega _B}\rho _{\mathcal {H}}$ .

Equation (2.11) is a second-order elliptic equation on the vector bundle $E \to B$ . It arises by expanding the scalar curvature of the Kähler metric $(\omega _k, J_s)$ in inverse powers of k and powers of s and then relating the two parameters to obtain a single expansion. The left-hand side of (2.11) is then the projection onto $C^\infty (E(\omega , I))$ of the sub-leading order term of such an expansion. The equation with $\nu =0$ is the condition for an optimal symplectic connection in the sense of [Reference Dervan and Sektnan9], where all the fibres are required to be cscK.

The linearisation of the equation at a solution is given by the operator [Reference Ortu40, §5.3]

$$\begin{align*}\widehat{\mathcal{L}} = \mathcal{R}^*\mathcal{R} + \mathcal{A}^*\mathcal{A} \end{align*}$$

on $C^\infty (E)$ , where

(2.12) $$ \begin{align} \mathcal{R}(\varphi_E) = \bar{\partial}_B \nabla_{\mathcal{V}}^{1,0} \varphi_E \end{align} $$

and

(2.13) $$ \begin{align} \mathcal{A}(\varphi_E) = \mathrm{d}_0\Phi\left(A_{\varphi_E} v\right). \end{align} $$

The adjoint is computed with respect to $\omega _F + \omega _B$ . Here $\nabla _{\mathcal {V}}^{1,0} \varphi _E$ is a section of the holomorphic tangent bundle; the vertical part of $\bar {\partial } \nabla _{\mathcal {V}}^{1,0} \varphi _E$ vanishes since $\varphi _E \in C^\infty (E)$ and the horizontal part is denoted by the expression (2.12). The operator (2.12) can be described as follows [Reference Dervan and Sektnan9, §4.3]: let $\mathcal {D}_k^*\mathcal {D}_k$ be the Lichnerowicz operator with respect to the Kähler metric $\omega _k$ . It admits a power series expansion in negative powers of k:

$$ \begin{align*} \mathcal{D}_k^*\mathcal{D}_k = \mathcal{L}_0 + k^{-1}\mathcal{L}_1 + O\left(k^{-2}\right), \end{align*} $$

where $\mathcal {L}_0$ is the vertical Lichnerowicz operator $\mathcal {D}_{\mathcal {V}}^*\mathcal {D}_{\mathcal {V}}$ . Then for $\varphi , \psi $ fibrewise holomorphy potentials

$$ \begin{align*} \int_X \varphi \mathcal{L}_1(\psi) \omega^m \wedge \omega_B^n = \int_X \langle \mathcal{R}\varphi, \mathcal{R}\psi \rangle_{\omega_F+\omega_B} \omega^m\wedge \omega_B^n. \end{align*} $$

This means that the operator $\mathcal {R}^*\mathcal {R}$ can actually be seen as $p_E \circ \mathcal {L}_1$ restricted to $\mathcal {C}^\infty _E(X)$ . The kernel of $\mathcal {R}$ , thus of $\mathcal {R}^*\mathcal {R}$ , consists of fibrewise holomorphy potentials which are global holomorphy potentials on X with respect to $\omega _k$ , and it is independent of k. The operator (2.13) is described as

(2.14) $$ \begin{align} \langle \mathrm{d}_v \nu (\mathcal{L}_{\nabla_{\mathcal{V}}{\varphi}}v), \psi \rangle = \int_X \langle \mathrm{d}_0 \Phi \left(A_{\varphi}v\right), \mathrm{d}_0 \Phi\left(A_\psi v\right) \rangle_{\omega_F} \omega_F^m \wedge \omega_B^n, \end{align} $$

where $\varphi , \psi \in C^\infty (E)$ . A function $\psi \in C^\infty (E)$ is in the kernel of A if and only if $\psi $ is a fibrewise holomorphy potential with respect to all $J_s$ , that is, $\psi \in C^\infty (E, J_s)$ . Thus the kernel of $\widehat {\mathcal {L}}$ consists of those functions $\psi \in C^\infty (E, J_0)$ such that $\bar {\partial }_s (\nabla _{s, \mathcal {V}}^{1,0}\psi )= 0$ for all s.

3.1 Openness of the setting

Given a fibration $\pi _Y: (Y, H_Y) \to (B, L)$ with analytically K-semistable fibres, we have introduced in (i) the assumption that there exists a degeneration of $\pi _Y$ to a fibration $\pi _X : (X, H_X) \to (B, L)$ such that the fibres of $\pi _X$ are cscK. We have explained in §2.2 that we can consider $Y\to B$ and $X\to B$ as the same symplectic fibration $\pi : (M, \omega ) \to B$ , and the degeneration as a deformation of a complex structure J to I, where $(\omega , I)$ has fibrewise constant scalar curvature. The goal of this section is to understand for which deformations $J'$ of J we can still find a relatively cscK degeneration $X'$ , and to construct such a degeneration.

We begin by working locally in B. Let $\mathcal {U} \subseteq B$ be a coordinate open subset of B. The fibre over the origin of $\mathcal {U}$ , denoted $M_0$ , has a constant scalar curvature metric $(\omega _0, I_0)$ . Let K be the group of Hamiltonian isometries of $(\omega _0, I_0)$ and let $\widetilde {H}^1_0$ be the vector space (2.3) parametrising first-order deformations of $(M_0, \omega _0, I_0)$ . Since $(\omega _0, I_0)$ has constant scalar curvature, the complexification of K is the group

$$\begin{align*}G=\mathrm{Aut}_0(M_0, H_0). \end{align*}$$

We will employ the following definition of GIT-stability for the action of a reductive group on an affine space [Reference Newstead38, §3] [Reference Hoskins27, §4.5, §4.6].

Definition 3.1. Let $\mathbb {A}^d$ be an affine space of dimension d and G a reductive affine group acting on it. Let $(z_1, \dots , z_d)$ be a system of coordinates on $\mathbb {A}^d$ . The space $\mathbb {A}^d$ can be embedded in the projective space $\mathbb {P}^d$ as a coordinate chart with the map

$$\begin{align*}(z_1, \dots, z_d) \mapsto [1:z_1: \dotsc :z_d]. \end{align*}$$

We extend the action of G to an action on $\mathbb {P}^d$ by acting trivially on the first coordinate. Let $x \in \mathbb {A}^d$ and let $\widehat {x}$ be its image in $\mathbb {P}^d$ . We say that x is

1. (i) semistable if there exists a nonconstant homogeneous polynomial in $[z_0: \dotsc : z_d]$ on $\mathbb {P}^d$ which is G-invariant and does not vanish at $\widehat {x}$ ;

2. (ii) polystable if it is semistable and its orbit is closed;

3. (iii) stable if it is semistable, its orbit is closed and its stabiliser is finite.

In particular, the polynomial $P(\mathbf {z}) = z_0$ is a G-invariant homogeneous polynomial that does not vanish at any point of $\mathbb {A}^d$ . So every point of $\mathbb {A}^d$ is semistable.

We apply the definition of stability to the vector space $\widetilde {H}^1_0$ . The complex structure $I_0$ corresponds to the origin in the Kuranishi space $V_0$ , which is fixed by the action of G. Therefore its orbit is closed and its stabiliser is the group G itself, so it is a polystable point. A key result for our construction is the following.

Let $V_0$ be the subspace of the Kuranishi space which parametrises integrable almost complex structures; it is a locally closed analytic subspace of $\widetilde {H}^1_0$ because it is defined by the vanishing of the Nijenhuis tensor. Thus the family $X \to B$ can be described locally over $\mathcal {U}$ as a family $\{ x_b \}$ in $V_0$ . The family $Y \to B$ can be described locally over $\mathcal {U}$ as a family $\{ y_b\mid b \in \mathcal {U} \}$ of points such that for each b the closure of the G-orbit of $y_b$ contains the polystable point $x_b$ [Reference Chen and Sun7, Theorem 1.3].

The key step in the construction is the following: by our hypothesis ii, the automorphism group of $(M_b, H_b)$ is isomorphic to G for all $b \in B$ . Therefore the points $\{ x_b \mid b \in \mathcal {U} \}$ are all fixed by the action of G and are hence polystable. By Lemma 3.3, $x_b$ is the only polystable orbit in the closure of the orbit of $y_b$ . Let $V_0^+$ be the set of all semistable points that have a fixed point in the closure of their orbit, it follows that the map

(3.1) $$ \begin{align} \begin{aligned} F: V_0^+ &\to V_0 \end{aligned} \end{align} $$

that maps a semistable point to the corresponding fixed point is well-defined. Note that the action of G on $V_0^+$ is effective, since by definition if $x \in V_0^+$ , then the closure of the orbit of x is in $V_0^+$ .

Proof. The space $V_0$ is an open subset of the vector space $\widetilde {H}^1$ . Let d be the dimension of $\widetilde {H}^1$ , so each point $z \in V_0$ has coordinates

$$\begin{align*}\left(z_1, \dots, z_d\right). \end{align*}$$

Let us begin with the case when G is isomorphic to a complex torus ${(\mathbb {C}^*)}^r$ . The fixed points of the action can be described by the vanishing of the coordinates

$$\begin{align*}z_{i_1} = \dots = z_{i_h} = 0 \quad\text{for}\quad i_1, \dots, i_h \in \{1, \dots, d\}, \end{align*}$$

thus they form an analytic subspace of $V_0$ .

By the Hilbert-Mumford criterion [Reference Mumford, Fogarty and Kirwan36, Theorem 2.1], a point $y_j$ is semistable if it is semistable for the action of every 1-parameter subgroup of the group G. Let $\rho (t): \mathbb {C^*} \hookrightarrow G$ be a 1-parameter subgroup. The action of $\rho (t)$ can be written as $(t^{a_1}, \dots , t^{a_d})$ , where the numbers $a_j$ are the weights of the action. Then $V_0$ splits into a sum of weight spaces $V_0^{\mathrm {pos}} \oplus V_0^{\mathrm {fix}} \oplus V_0^{\mathrm {neg}}$ , where $\rho (t)$ acts on $V_0^{\mathrm {pos}}$ with positive weights, on $V_0^{\mathrm {neg}}$ with negative weights and fixes the subspace $V_0^{\mathrm {fix}}$ . A semistable point that has a fixed point in the closure of its orbit is described by the following condition: if a coordinate in $V_0^{\mathrm {pos}}$ is nonzero, then all coordinates in $V_0^{\mathrm {neg}}$ vanish. Applying this condition to all 1-parameter subgroups yields a set of polynomial equations that define the semistable points in $V_0^+$ . Thus, the semistable points correspond to an analytic subset of $V_0$ . The map F is the projection onto the set described by $\{ z_{i_1} = \dots \ = z_{i_h} = 0\}$ , thus it is holomorphic.

Let now G be any reductive group, and let y be a semistable point and x be a polystable and fixed point in the closure of its orbit. The fixed points of G form a vector subspace $V_0^{\mathrm {fix}}$ also in this case. Moreover there exists a 1-parameter subgroup $\lambda _x : \mathbb {C}^* \hookrightarrow G$ such that

$$\begin{align*}\lim_{t\to 0} \lambda_x(t)\cdot y = x. \end{align*}$$

It follows that the map $y \mapsto x$ is the projection onto the vector subspace of fixed points for $\lambda _x$ , hence it is holomorphic. Consider the composite map

(3.2) $$ \begin{align} V_0 \to V_0 \overset{pr}{\to} V_0^{\mathrm{fix}} \end{align} $$

where the first map is the projection onto the vector subspace of fixed points for $\lambda _x$ and the second map is the projection onto the subspace of fixed points for the whole group G. The map is holomorphic because it is a composition of holomorphic projections. We prove that it coincides with the map (3.1). Let $\widetilde {x}' = \lim _{t\to 0} \lambda _x(t)\cdot y'$ . Then

$$\begin{align*}\overline{G\cdot \widetilde{x}'} \subseteq \overline{G\cdot y'}. \end{align*}$$

The unique polystable orbit contained in $\overline {G\cdot \widetilde {x}'}$ is also a polystable orbit in $\overline {G\cdot y'}$ , so it must coincide with the fixed-point orbit $\{x'\}$ . Thus flowing along the orbit of $\widetilde {x}'$ amounts to projecting onto the subspace of fixed points, and so the map (3.2) maps any point $y'$ to the fixed point $x'$ in the closure of its orbit.

Let $V_\pi $ be the Kuranishi space of the fibration $X \to B$ defined in Theorem 2.12. Consider the subspace

$$\begin{align*}V^+_\pi := \{y \in V_\pi \mid y\vert_{X_b} \in V_b^+\}. \end{align*}$$

We remark that $V^+_\pi $ depends on the complex structure of the reference fibration $X\to B$ and its deformation $Y\to B$ . We denote by $\mathscr {J}^+_\pi $ the image of $V_\pi ^+$ via the relative Kuranishi map (2.9).

Proof. Let $\mathcal {U}_1$ and $\mathcal {U}_2$ be two open coordinate subsets of B with nonempty intersection and let $0_1$ be the origin of $\mathcal {U}_1$ and $0_2$ be the origin of $\mathcal {U}_2$ . Denote by $I_1$ and $I_2$ the complex structures of the fibres $X_{0_1}$ and $X_{0_2}$ , and $G_1$ , $G_2$ their groups of automorphisms. By assumption, $G_1$ and $G_2$ are isomorphic, and we will denote them by G. Recall that the Kuranishi space is versal and more specifically that it is a complete deformation space for the nearby fibres. We use the versality of the Kuranishi map to glue the spaces $V_1^+$ and $V_2^+$ constructed in Lemma 3.4 to a variety $V^+$ and prove that $V_\pi ^+$ is obtained as the intersection of said variety with the relative Kuranishi space $V_\pi $ . More precisely, versality of the Kuranishi space means that there is a map

$$\begin{align*}\tau_{21} : V_2 \to V_1, \end{align*}$$

not necessarily unique.

The map $\tau _{21}$ can be taken to be G-equivariant. In fact, the G-equivariance can be traced back to the proof of Kuranishi’s Theorem 2.4. The map $\tau _{21}$ is defined using the implicit function theorem, which can be applied to a K-equivariant map to provide an implicit inverse function which is K-equivariant. Since G is the complexification of K, we obtain that $\tau _{21}$ is G-equivariant. The equivariance also implies that the image of $V_2^+$ is $V_1^+$ , so we can restrict $\tau _{21}$ to

$$\begin{align*}\widetilde{\tau}_{21}:V_2^+ \to V_1^+. \end{align*}$$

Moreover, the map $\widetilde {\tau }_{21}$ has an inverse that is constructed by reversing the roles of $V_1$ and $V_2$ , so it is an isomorphism. In fact, although the map $\tau _{21}$ is not canonical, the restriction to $\widetilde {\tau }_{21}$ is fixed by the reference K-semistable fibration $Y\to B$ . Each Kuranishi space $V_b$ is a complex subspace of the vector space $\widetilde {H}_b^1$ , described as the kernel of the elliptic operator $P_bP_b^*+(\bar {\partial }_b^*\bar {\partial }_b)^2$ (2.3). So we can use the isomorphism $\widetilde {\tau }_{21}$ to glue the spaces $V_b^+$ to a subvariety $V^+$ of the kernel of the fibrewise elliptic operator

$$\begin{align*}P_{\mathcal{V}}P_{\mathcal{V}}^*+ (\bar{\partial}^*_{\mathcal{V}}\bar{\partial}_{\mathcal{V}})^2. \end{align*}$$

Therefore the intersection

$$\begin{align*}V_\pi^+ = V^+ \cap V_\pi \end{align*}$$

is a locally closed subvariety of $V_\pi $ .

The following lemma shows that we can glue the local fibration constructed in Lemma 3.4 to a global fibration over B.

Proof. Let $\mathcal {U} \subseteq B$ be an open coordinate subset. By the relative Kuranishi Theorem 2.4, for each $b\in \mathcal {U}$ there exists a point $y^{\prime }_b \in V_0$ such that $\Phi _0(y^{\prime }_b)$ is in the same $\mathcal {G}^c$ -orbit of $J^{\prime }_b$ . The fact that the map (3.1) is holomorphic implies that we can find polystable points $\{x^{\prime }_b\}$ such that $\{\Phi _0(x^{\prime }_b)\}$ are a holomorphic family of cscK complex structures over $\mathcal {U}$ that are deformations of $\{J^{\prime }_b\}$ . Then we can construct a local relatively cscK fibration from the pullback diagram

(3.3)

where $p_{\mathcal {U}} : \mathcal {M}_{\mathcal {U}} \to V_0$ is Kuranishi’s versal family and $i(b) = x_b'$ .

Now we glue the local fibrations to a fibration $X'\to B$ . Let $\mathcal {U}_1$ , $\mathcal {U}_2$ and G be as in Lemma 3.6. The associated Kuranishi maps are denoted respectively by $\Phi _1:V_1\to \mathscr {J}$ and $\Phi _2:V_2\to \mathscr {J}$ .

For $b \in \mathcal {U}_1 \cap \mathcal {U}_2$ , consider the complex structure $J_b'$ . Since $J_b'$ can be regarded as a deformation of both $I_1$ and $I_2$ , there exist points $y^{\prime }_{b,1} \in V_1$ and $y^{\prime }_{b,2} \in V_2$ such that

$$\begin{align*}\Phi_1(y^{\prime}_{b,1}) = J_b' = \Phi_2( y^{\prime}_{b,2} ). \end{align*}$$

The diagram (3.3) produces two fibrations $X_1' \to \mathcal {U}_1$ and $X_2' \to \mathcal {U}_2$ . In order to glue the local fibrations, we need to prove that there is an isomorphism

$$\begin{align*}\phi_{12}:X^{\prime}_1\vert_{\mathcal{U}_1 \cap \mathcal{U}_2} \overset{\sim}{\rightarrow} X^{\prime}_2\vert_{\mathcal{U}_1 \cap \mathcal{U}_2} \end{align*}$$

and that it satisfies the cocycle condition

$$\begin{align*}\phi_{23} \circ\phi_{12}=\phi_{13} \end{align*}$$

on a triple intersection $\mathcal {U}_1\cap \mathcal {U}_2\cap \mathcal {U}_3$ of open subsets of B. In particular, the first condition produces a global compact complex manifold $X'$ , while the cocycle condition implies that $X'$ admits a submersion onto B.

To prove the existence of the isomorphism $\phi _{12}$ we use again the fact that the Kuranishi space induces complete deformations on the nearby fibres. The map

$$\begin{align*}\tau_{21} : V_2 \to V_1 \end{align*}$$

is such that $\mathcal {M}_2 = \tau _{21}^*\mathcal {M}_1$ . In particular, we have the following diagram

If we show that the diagram is commutative, that is, $\tau _{21}\circ i_2 = i_1$ , then

$$\begin{align*}X_2'\vert_{\mathcal{U}_1\cap\mathcal{U}_2} = i_2^*\mathcal{M}_2 \simeq (\tau_{21}\circ i_2 )^* \mathcal{M}_1 = i_1^*\mathcal{M}_1 = X_1'\vert_{\mathcal{U}_1\cap\mathcal{U}_2}. \end{align*}$$

The commutativity follows from the fact that the map $\tau _{21}$ is G-equivariant. Indeed the points $i_1(b)$ and $i_2(b)$ are defined as the fixed-point limits of semistable orbits. Moreover, an equivariant map between two spaces on which there is an action of the same group G sends fixed points to fixed points and the closure of the orbit of $i_1(b)$ to the closure of the orbit of $i_2(b)$ .

We now show the cocycle condition. Let $\mathcal {U}$ be a triple intersection $\mathcal {U}_1\cap \mathcal {U}_2\cap \mathcal {U}_3$ and consider the following diagram

On $V_3$ we have two families pulled-back from $V_1$ , namely $\tau _{31}^* \mathcal {M}_1$ and $ (\tau _{21}\circ \tau _{32})^*\mathcal {M}_1$ . They induce two distinct families on $\mathcal {U}$ , pulled-back using $i_3$ . Although in general it is not true that $\tau _{31}$ is equal to the composition $\tau _{21}\circ \tau _{32}$ , the commutativity of the arrows proved above implies that

$$\begin{align*}\tau_{21}\circ\tau_{32}\circ i_3 = \tau_{31}\circ i_3. \end{align*}$$

Therefore the two families $\tau _{31}^* \mathcal {M}_1$ and $ (\tau _{21}\circ \tau _{32})^*\mathcal {M}_1$ coincide.

3.2 Openness of the space of optimal symplectic connections

Let $Y \to B$ be a fibration with K-semistable fibres, and assume that it degenerates to a fibration $X \to B$ with cscK fibres, in the sense of §2.2. Let $Y'\to B$ be a deformation of $Y \to B$ in $\mathscr {J}^+_\pi $ . Then $Y'\to B$ admits a degeneration to $X' \to B$ , whose fibres are cscK, as explained in §3.1. The goal of this section is to show that if Y admits an optimal symplectic connection, then $Y'$ also does.

We denote by I the complex structure of X, by J the complex structure of Y, and we assume that $Y \to B$ is represented by $v_0 \in V_\pi $ . We also assume that $(\omega , J)$ is an optimal symplectic connection. As before, let $V_\pi ^+$ be the subvariety of $V_\pi $ which describes the family of complex structures $\mathscr {J}_\pi ^+$ .

Proof. Let us consider a potential $\varphi \in \mathcal {K}_E(I)$ and a path $\{ \varphi _t\}$ in $\mathcal {K}_E(I)$ from 0 to $\varphi $ . [Reference Hallam24, Theorem 3.3] guarantees that this path exists and that it is smooth. We define the relatively cscK metrics

$$ \begin{align*} \omega_{t} = \omega + 2i\partial\bar{\partial}\varphi_t \end{align*} $$

and the Kähler metrics

$$ \begin{align*} \omega_{k,t} = \omega_{t}+ k\omega_B, \end{align*} $$

where the $\partial , \bar {\partial }$ operators are taken with respect to the relatively cscK complex structure I. From Proposition 2.11, we have that $\dot {\varphi }_t \in C^\infty (E(\omega _{t}, I))\oplus C^\infty (B)$ , for all t. Thus the fibrewise Hamiltonian vector fields

$$ \begin{align*} \eta_{t} :=\mathrm{grad}^{\omega_{t}}{\dot{\varphi}_t} \end{align*} $$

are well-defined. Consider the vertical vector fields

$$ \begin{align*} \xi_t := (\nabla^{g_t}\dot{\varphi}_t)_{\mathcal{V}} = (I\mathrm{grad}^{\omega_{t}}{\dot{\varphi}_t})_{\mathcal{V}}. \end{align*} $$

Then fibrewise

(3.4) $$ \begin{align} \frac{\mathrm{d}}{\mathrm{d} t} \omega_{t} = -\mathcal{L}_{I\eta_{t}}\omega_{t} = \mathcal{L}_{\xi_t}\omega_{t}. \end{align} $$

Let $\{f_t, t\in [0,1]\}$ be the isotopy of the time-dependent vector field $\xi _t$ , that is, the collection of diffeomorphisms of M such that

$$ \begin{align*} \frac{\mathrm{d}}{\mathrm{d} t} f_t = \xi_t(f_t), \quad f_0= \mathrm{id}. \end{align*} $$

Since $\xi _t$ is vertical, $f_t \in \mathrm {Diffeo}(M, \pi )$ . Recall the following property [Reference Cannas da Silva6, §6.4], for a smooth family $\sigma _t$ of tensors:

$$ \begin{align*} \frac{\mathrm{d}}{\mathrm{d} t} f_t^*\sigma_t = f_t^* \left( \mathcal{L}_{\xi_t} \sigma_t + \frac{\mathrm{d}\sigma_t}{\mathrm{d} t}\right), \end{align*} $$

where $\xi _t$ and $f_t$ are a time-dependent vector field and its isotopy, respectively. Applying it to $\omega _{t}$ gives the fibrewise relation

$$ \begin{align*} \frac{\mathrm{d}}{\mathrm{d} t} f_t^*\omega_{t} = 0, \end{align*} $$

which implies

$$ \begin{align*} f_t^* \omega_{t} = f_0^*\omega = \omega. \end{align*} $$

Define $J_t = f_t^*I$ . Then, for $t=1$ , the two metrics $g(\omega _X, f_1^*I)$ and $g(\omega _X+2i\partial \bar {\partial }\varphi , I)$ are fibrewise isometric, that is,

$$ \begin{align*} (M, \omega, f_1^*I) \simeq (M, \omega+ 2i\partial\bar{\partial}\varphi, I) \end{align*} $$

as relative Kähler manifolds with fibrewise constant scalar curvature.

Thus, given an integrable relatively cscK complex structure I in $\mathscr {J}_\pi $ , we have a map

(3.5) $$ \begin{align} \begin{aligned} F_\pi : \mathcal{K}_E(I) &\longrightarrow \mathscr{J}_\pi \\ \varphi &\longmapsto f_1^*I =: F_\pi(\varphi, I) \end{aligned} \end{align} $$

which locally parametrises all integrable complex structures in the same diffeomorphism class of I that are fibrewise cscK with respect to the fixed $\omega $ . Its differential at the origin is

$$ \begin{align*} \begin{aligned} \mathrm{d}_0F_\pi(\dot{\varphi}) &= \frac{\mathrm{d}}{\mathrm{d} t}\bigg|_{t=0} J_t = \frac{\mathrm{d}}{\mathrm{d} t}\bigg|_{t=0} f_t^*I = \frac{\mathrm{d}}{\mathrm{d} t}\bigg|_{t=0}\mathcal{L}_{\xi_t} I = \mathcal{L}_{(I\mathrm{grad}^{\omega}{\dot{\varphi}})_{\mathcal{V}}} I =\\ &= -\frac{1}{2} \bar{\partial} (\mathrm{grad}^{\omega}{\dot{\varphi}})_{\mathcal{V}}, \end{aligned} \end{align*} $$

where, again by Proposition 2.11, $\dot {\varphi } \in C^\infty (E(\omega , I)) \oplus C^\infty (B)$ .

Now let $v_0 \in V_\pi $ be the deformation of the complex structure which generates the family $Y \to B$ and let $\Phi $ be the relative Kuranishi map (2.9). Then we can extend the map (3.5) to a map

$$\begin{align*}\begin{aligned} F^{\prime}_\pi : \mathcal{K}_E(I) &\to \widetilde{H}^1_{\mathcal{V}} \\ \varphi &\mapsto f_1^*v_0 =: F_\pi' (\varphi, I, v_0). \end{aligned} \end{align*}$$

Its differential at the origin computed at $\dot {\varphi } \in C^\infty (E(\omega , I)) \oplus C^\infty (B)$ is

(3.6) $$ \begin{align} \mathrm{d}_0F^{\prime}_\pi (\dot{\varphi}) = \frac{\mathrm{d}}{\mathrm{d} t}\bigg\vert_{t=0} f_t^*v_0 = \mathcal{L}_{(I\mathrm{grad}^{\omega}{\dot{\varphi}})_{\mathcal{V}}} v_0. \end{align} $$

Definition 3.9. We denote by $\mathcal {P}$ the set of triples $(\varphi , x, v) \in C^\infty (M)\times TV_\pi $ such that $\varphi \in \mathcal {K}_E(\Phi (x))$ and x is $K_\pi ^{\mathbb {C}}$ -polystable. The optimal symplectic connection operator is the map

$$\begin{align*}\begin{aligned} \mathcal{G} : \mathcal{P} &\to C^\infty(X) \\ (\varphi, x, v) &\mapsto p_{E(\varphi, x)} \left( \Theta(\omega, F_\pi(\varphi,\Phi(x)))\right) + \frac{\lambda}{2}\nu_{\varphi, x}\left(F^{\prime}_\pi(\varphi, \Phi(x), v)\right). \end{aligned} \end{align*}$$

In this expression $E(\varphi , x)$ is the vector bundle of fibre holomorphy potentials with respect to the Kähler structure $(\omega , F_\pi (\varphi , \Phi (x)))$ and $\nu _{\varphi ,x}$ is the map (2.10) computed with respect to the complex structure $F_\pi (\varphi ,\Phi (x))$ .

We now compute the differential of $\mathcal {G}$ along the $\varphi $ -variable computed at $(0,0,v_0)$ . We write

$$\begin{align*}\mathcal{G} (\varphi, x, v) = \mathcal{G}_1(\varphi, x) + \mathcal{G}_2(\varphi, x, v), \end{align*}$$

and we split the computation into two separate lemmas.

Lemma 3.10. Let $\varphi \in C^\infty (E(\omega , I)) \oplus C^\infty (B)$ . The differential along the first variable of $\mathcal {G}_1$ is

$$\begin{align*}D_1 \mathcal{G}_1 \vert_{(0,0, v_0)}(\varphi) = -\frac{1}{2}\mathcal{R}^*\mathcal{R}(\varphi). \end{align*}$$

Proof. Let $\{J_t\}$ be a family of relatively cscK complex structures compatible with $\omega $ , such that $J_0 = I$ and consider the scalar curvature of $(\omega _k, J_t)$ ,

(3.7) $$ \begin{align} \mathrm{Scal}(\omega_k, J_t) = \mathrm{Scal} (\omega_k, I) + t\frac{\mathrm{d}}{\mathrm{d} t}\bigg|_{t=0} \mathrm{Scal}(\omega_k, J_t) + O(t^2). \end{align} $$

Let $\alpha = \partial _{t=0} J_t$ , and define

$$ \begin{align*} Q_k(\alpha) := \frac{\mathrm{d}}{\mathrm{d} s}|_{s=0} \mathrm{Scal}(\omega_k, J_t). \end{align*} $$

From Lemma 2.3 we obtain

$$ \begin{align*} \begin{aligned} Q_k(\alpha) &= \mathrm{Im} \left((g_k)^{p\bar{q}} \ \nabla_p \nabla_a \alpha^a_{\ \bar{q}}\right) = \mathrm{Re} \left((\omega_k)^{p\bar{q}} \ \nabla_p \nabla_a \alpha^a_{\ \bar{q}}\right). \end{aligned} \end{align*} $$

We need to compute the sub-leading order term of $Q_k$ along the differential of the map $F_\pi $ (3.5)

Let us consider $\alpha = \bar {\partial } \left (\mathrm {grad}^\omega {\varphi _E}\right )^{1,0}_{\mathcal {V}}$ , where $\varphi _E \in C^\infty _E(X)$ is a fibrewise holomorphy potential. Then

(3.8) $$ \begin{align} Q_k(\alpha) = k^{-1} \mathrm{Re}\left( \mathcal{R}^*\mathcal{R}(\varphi_E)\right) + O\left(k^{-2}\right), \end{align} $$

where $\mathcal {R}(\varphi _E) = \bar {\partial }_B \mathrm {grad}_{\mathcal {V}} \varphi _E$ and the adjoint is computed with respect to $\omega _F + \omega _B$ . As explained in §2.3, the operator $\mathcal {R}^*\mathcal {R}$ can actually be seen as $p_E \circ \mathcal {L}_1$ restricted to $\mathcal {C}^\infty (E(\omega ,I))$ . Its kernel consists of fibrewise holomorphy potentials which are global holomorphy potentials on X with respect to $\omega _k$ .

A local coordinate expression for $\alpha $ is

$$ \begin{align*} \begin{aligned} \alpha = \bar{\partial} \left(Y_{\varphi_E}\right)^{1,0}_{\mathcal{V}} &= \partial_{\bar{z}^j} \left(\omega_F^{a\bar{b}} \partial_{\bar{w}^b}\varphi_E \right)\partial_{w^a} \otimes \mathrm{d} \bar{z}^j = \nabla_{\bar{j}} \left(\omega_F^{a\bar{b}} \nabla_{\bar{b}}\varphi_E \right)\partial_{w^a} \otimes \mathrm{d} \bar{z}^j. \end{aligned} \end{align*} $$

In particular, since the potential $\varphi _E$ is a fibrewise holomorphy potential, the component of $\alpha $ with the covariant index of vertical type vanishes. Therefore, using Lemma 2.7,

$$ \begin{align*} \begin{aligned} Q_k(\alpha) &= k^{-1}\mathrm{Re} \left( \omega_B^{i\bar{j}} \ \nabla_i \nabla_a \nabla_{\bar{j}} \left(\omega_F^{a\bar{b}} \nabla_{\bar{b}}\varphi_E \right) \right) + O\left(k^{-2}\right) =\\ &= k^{-1}\mathrm{Re} \left( \omega_B^{i\bar{j}} \ \omega_F^{a\bar{b}} \ \nabla_i \nabla_a \nabla_{\bar{j}} \nabla_{\bar{b}}\varphi_E \right) + O\left(k^{-2}\right) =\\ &=k^{-1} \mathrm{Re}\left( \mathcal{R}^*\mathcal{R}(\varphi_E)\right) + O\left(k^{-2}\right). \end{aligned} \end{align*} $$

A similar computation also holds if we consider, instead of a potential $\varphi $ in $C^\infty (E(\omega , I))$ , a map $\varphi \in C^\infty (E(\omega , I))\oplus C^\infty (B)$ , so that $\alpha =\bar {\partial }(Y_\varphi )^{1,0}_{\mathcal {V}}$ . This choice amounts to considering an element in the image of the differential of the map $F_\pi $ (3.5). In this case

$$ \begin{align*} Q_k(\alpha) = \mathrm{Re}(\mathcal{L}_0\varphi) + \frac{1}{k}\mathrm{Re}(\mathcal{L}_1\varphi) + O\left(k^{-2}\right), \end{align*} $$

where $\mathcal {L}_0(\varphi )=\mathcal {L}_0(\varphi _B) =0$ , since $\varphi _B$ is constant when restricted to a fibre, so we obtain the same equation as (3.8). To obtain the correct coefficient in the claimed expression note that

$$\begin{align*}\mathrm{d}_0F_\pi(\varphi) = -\frac{1}{2}\alpha.\\[-34pt] \end{align*}$$

Lemma 3.11. Let $\varphi \in C^\infty (E(\omega , I)) \oplus C^\infty (B)$ . The differential along the first variable of $\mathcal {G}_2$ is

$$\begin{align*}D_1 \mathcal{G}_2 \vert_{(0,0, v_0)}(\varphi) = -\mathcal{A}^*\mathcal{A}(\varphi). \end{align*}$$

Proof. We compute

$$\begin{align*}\frac{\mathrm{d}}{\mathrm{d} t}\bigg\vert_{t=0} \nu_t \left(f_t^*v_0\right), \end{align*}$$

where $\nu _t$ is the map $\nu $ computed with respect to the complex structure $f_t^*\Phi (0)$ . Now, $f_t$ is the isotopy of the vector field $\xi _t = (I\mathrm {grad}^{\omega _t}\dot {\varphi }_t)_{\mathcal {V}}$ , where $\dot {\varphi }_t$ is in $C^\infty (E_t)\oplus C^\infty (B)$ and $\dot {\varphi }_0 = \varphi $ . In the expression of $\xi _t$ we are fixing the complex structure I and varying the Kähler form $\omega _t$ . In particular, $\xi _t$ is a fibrewise holomorphic vector field with respect to I. This implies that $f_t^*I = I$ , so $\nu _t = \nu $ . Therefore

$$\begin{align*}\frac{\mathrm{d}}{\mathrm{d} t}\bigg\vert_{t=0} \nu_t \left(f_t^*v_0\right) = \mathrm{d}_{v_0}\nu \left(\frac{\mathrm{d}}{\mathrm{d} t}\bigg\vert_{t=0}f_t^*v_0 \right). \end{align*}$$

Using the expression (3.6) we obtain

$$\begin{align*}\mathrm{d}_{v_0}\nu \left(\frac{\mathrm{d}}{\mathrm{d} t}\bigg\vert_{t=0}f_t^*v_0 \right) = \mathrm{d}_{v_0}\nu \left( \mathcal{L}_{(I\mathrm{grad}^{\omega}{\varphi})_{\mathcal{V}}} v_0 \right). \end{align*}$$

The right-hand side can be written as $-\mathcal {A}^*\mathcal {A}(\varphi )$ following the description (2.14). The minus sign follows from the relation

$$\begin{align*}(I\mathrm{grad}^{\omega}{\varphi})_{\mathcal{V}} = -\nabla_{\mathcal{V}}\varphi, \end{align*}$$

where $\nabla _{\mathcal {V}}$ is the vertical Riemannian gradient.

We define the operator

(3.9) $$ \begin{align} \begin{aligned} \mathcal{G} : \mathcal{P}^{2,l}&\longrightarrow W^{2, l-2}(X) \\ (\varphi, x,v) &\longmapsto p_{E(x,\varphi)} \left(\Theta(\omega, F_\pi (\varphi,\Phi(x)))\right) + \frac{\lambda}{2}\nu_{\varphi, x} \left(F_\pi'(\varphi, \Phi(x),v)\right), \end{aligned} \end{align} $$

where $\mathcal {P}^{2,l}$ is the space defined in Definition 3.9, but the functions are considered to be in the Sobolev space $W^{2,l}(X)$ instead of smooth.

Proposition 3.12. Let $\pi _X: X \to B$ be a holomorphic submersion with a fibrewise cscK structure $(\omega , I)$ and let $\pi _Y:Y\to B$ be a deformation of $X\to B$ with complex structure J. Let $V_\pi $ be the Kuranishi space based at I and let $v_0 \in V_\pi $ represent the complex structure J. Assume that

1. (i) $(\omega , J)$ is an optimal symplectic connection;

2. (ii) the relative automorphism group

   $$\begin{align*}\mathrm{Aut}(\pi_Y) := \{f \in \mathrm{Aut}(Y, H_Y) \mid f \circ \pi = \pi\} \end{align*}$$
   is discrete.

3. (iii) the group $\mathrm {Aut}(X_b, H_b)$ is independent of B, and will be denoted G.

Then for any small deformation v of $v_0$ in $V_\pi ^+$ there exists $x \in V_\pi $ such that $(\omega , \Phi (x))$ is relatively cscK, and a Kähler potential $\varphi $ such that

$$\begin{align*}\omega+ i \partial\bar{\partial}\varphi \end{align*}$$

is an optimal symplectic connection with respect to the complex structure $J'$ corresponding to v, where the $\partial $ , $\bar {\partial }$ operators are with respect to $\Phi (x)$ .

Proof. The proof consists of proving that the operator 3.9 is an elliptic operator with a trivial kernel so that we can apply the implicit function theorem. We note that this can be done even though $V_\pi ^+$ may be a singular complex space. Indeed, let $\mathcal {X} \to B\times V_\pi ^+$ be a family of holomorphic submersions such that the fibre over $0 \in V_\pi ^+$ is $X \to B$ . By the Kuranishi theorem [Reference Kuranishi30, p.149] we can locally consider a smooth trivialisation of the family over $V_\pi ^+$ such that the complex structures of the fibrations $\mathcal {X}_x \to B \times \{x\}$ form a smooth family $\{J_s\}$ .

As before, let I denote the complex structure of X. As we assume that $(\omega , I, v_0)$ is an optimal symplectic connection, $\mathcal {G}(0,0, v_0) = 0$ . The derivative with respect to the first component, given by Lemma 3.10 and 3.11, is

$$ \begin{align*} \mathrm{d}_1 \mathcal{G}_{(0,0, v_0)} (\varphi) = -\mathcal{R}^*\mathcal{R}(\varphi) - \lambda\mathcal{A}^*\mathcal{A}(\varphi) = -\widehat{\mathcal{L}}(\varphi). \end{align*} $$

The hypothesis on the automorphism group implies that the kernel of the linearisation is empty, so the implicit function theorem guarantees that there exists a path

$$\begin{align*}(\varphi(x,v), x, v) \in \mathcal{P}^{2,l} \end{align*}$$

such that locally around $(0,0,v_0)$

(3.10) $$ \begin{align} \mathcal{G}(\varphi(x,v), x, v) = 0. \end{align} $$

The function $\varphi (x,v)$ is smooth by the standard theory of regularity of solutions to elliptic partial differential equations, applied to the operator $\mathcal {G}$ [Reference Besse2, Theorem 41]. Therefore, solutions to (3.10) produce optimal symplectic connections.

3.3 The moduli space of optimal symplectic connections

Let $(Y, H_Y) \to B$ be a relatively K-semistable holomorphic submersion with a degeneration to a relatively cscK fibration $(X, H_X)\to B$ . Assume that we have a relatively Kähler metric $(\omega , J)$ on Y that is an optimal symplectic connection. Let W be the subset of the Kuranishi space $V_\pi $ that corresponds to fibrations satisfying the hypotheses of Proposition 3.12. Then W is an open subset of the locally closed analytic subvariety $V_\pi ^+$ described in §3.1 by Proposition 3.12.

In particular, under our assumption $\mathrm {Aut}(\pi _Y)$ is a finite discrete group. Let $V_{\pi _Y}$ be the Kuranishi space of the fibration $\pi _Y$ and let

$$\begin{align*}\tau : V_{\pi_Y} \to V_\pi \end{align*}$$

be the map given by completeness of the Kuranishi space. If we denote $\tau ^{-1}W =: W_Y$ , then $W_Y$ is a locally closed subvariety of $V_{\pi _Y}$ .

Let $\mathcal {Y} \to B \times W_Y$ be the Kuranishi family of fibrations which admit an optimal symplectic connection, with central fibration $Y \to B$ . The quotient

(3.11) $$ \begin{align} W_Y / \mathrm{Aut}(\pi_Y) \end{align} $$

is a local complex space and it is Hausdorff since it is the quotient of a variety by a finite group. We now explain that we can glue the quotients (3.11) to obtain a global Hausdorff moduli space $\mathcal {M}$ of fibrations that admit an optimal symplectic connection.

The following result builds on [Reference Fujiki and Schumacher22, Proposition 6.5] and [Reference Fujiki and Schumacher21, Lemma 3.8].

The global definition of $\mathcal {M}$ relies on the following lemma, whose proof follows from [Reference Arbarello, Cornalba and Griffiths1, XI.6].

Proof. We can prove the lemma for the entire relative Kuranishi space $V_{\pi _Y}$ . The statement follows from:

1. (i) For any $y\in V_{\pi _Y}$ , the automorphism group $\mathrm {Aut}(\pi _{Y_y})$ is contained in $\mathrm {Aut}(\pi _Y)$ ;

2. (ii) For any $y\in V_{\pi _Y}$ there exists an $\mathrm {Aut}(\pi _{Y_y})$ -invariant open neighbourhood $\mathcal {U}_y$ such that any isomorphism between fibres of $\mathcal {Y}\vert _{\mathcal {U}_y}\to \mathcal {U}_y$ is induced by an element of $\mathrm {Aut}(\pi _{Y_y})$ .

We begin by proving (ii). Assume by contradiction that there exist two sequences $\{y_n\}$ and $\{w_n\}$ both converging to y. Assume that there exist $g_n$ in $\mathrm {Aut}(\pi _{Y_y}) \setminus \mathrm {Aut}(\pi _Y)$ such that $g_n\cdot w_n =y_n$ . Then by Lemma 3.15 there exists $g \in \mathrm {Aut}(\pi _{Y_y})$ such that $g_n \to g$ . Up to replacing $g_n$ with $g_n g^{-1}$ and $w_n$ with $g^{-1}w_n$ we may assume that g is the identity of $\mathrm {Aut}(\pi _{Y_y})$ . Consider the map

$$\begin{align*}\begin{aligned} \mathrm{Aut}(\pi_Y)\times V_{\pi_Y} &\to \widetilde{\mathrm{H}}^1_{\mathcal{V}}\\ (\eta,y) &\mapsto \eta\cdot y. \end{aligned} \end{align*}$$

By Theorem 2.12, this is a local biholomorphism at $(\mathrm {id},y)$ . So we have $F(\mathrm {id},y_n) = F(g_n,w_n)$ . Hence $g_n = \mathrm {id}$ , a contradiction.

The claim (i) then follows exactly as in [Reference Arbarello, Cornalba and Griffiths1, p.204]. We report the proof for completeness. Set

$$\begin{align*}I = \{(y,g) \in W_Y \times \mathrm{Aut}(\pi_{Y_y})\}. \end{align*}$$

Then I is equal to $\mathrm {Isom}(\mathcal {Y},\mathcal {Y},B)$ . Up to shrinking $W_Y$ we can assume that any connected component $I'$ of I intersects $\{0\}\times \mathrm {Aut}(\pi _Y)$ . Let $\widehat {I} = \{(y,g) \in I' \mid g \in \mathrm {Aut}(\pi _Y)\}$ . Then $\widehat {I}$ is nonempty and Zariski-closed. It follows from (ii) that $\widehat {I}$ contains an open set. Then $\widehat {I}=I'$ , which concludes the proof.

To glue the charts (3.11), we use the completeness of the Kuranishi space. Let $Y_1 \to B$ be another relatively K-semistable fibration which admits an optimal symplectic connection and is close to $Y\to B$ . Then the Kuranishi theorem gives a map

$$\begin{align*}\tau: V_{\pi_{Y_1}} \to V_{\pi_Y} \end{align*}$$

such that a family $\mathcal {Y}_1 \to B \times V_{\pi _{Y_1}}$ is isomorphic to the pull-back via $\tau $ of $\mathcal {Y} \to B\times V_{\pi _Y}$ . Consider the composition of this map with the inclusion $i : W_{Y_1} \to V_{\pi _{Y_1}}$ :

$$\begin{align*}\tau \circ i : W_{Y_1} \to V_{\pi_Y}. \end{align*}$$

Let $y_1 \in W_{Y_1}$ . Then $y_1$ represents a fibration with an optimal symplectic connection, that is, the image of $y_1$ via the Kuranishi map is a complex structure $J_{y_1}$ such that $(\omega , J_{y_1})$ is an optimal symplectic connection. Therefore, there is a representative y of $J_{y_1}$ in $V_{\pi _Y}$ which belongs to $W_Y$ , so

$$ \begin{align*} \alpha:=\tau\circ i : W_{Y_1} \to W_{Y}. \end{align*} $$

Lemma 3.16 allows us to pass to the quotient and obtain an isomorphism

(3.12) $$ \begin{align} \widetilde{\alpha} : W_{Y_1}/\mathrm{Aut}({\pi_{Y_1}}) \to W_Y/\mathrm{Aut}(\pi_Y), \end{align} $$

which is uniquely determined (while $\alpha $ itself might not be). Indeed, the inverse is constructed by reverting the roles of Y and $Y_1$ . Therefore, we can use it to glue the local charts to give $\mathcal {M}$ the structure of a complex space.

Proof. The countability follows from [Reference Fujiki and Schumacher21, Theorem 7.3]. We prove the Hausdorff property. Let $\mathcal {Y} \to B \times W_Y$ and $\mathcal {Y}_1\to B\times W_{Y_1}$ be two families of fibrations which admit an optimal symplectic connection. Let $y_t \to \bar {y}$ be a sequence in $W_Y$ and $y_{1t} \to \bar {y}_1$ be a sequence in $W_{Y_1}$ and assume that $Y_{y_t}$ is isomorphic to $Y_{1,y_{1t}}$ as fibrations over B. Following the proof of [Reference Fujiki18, Proposition 10], we show that $Y_{\bar {y}}\to B$ is isomorphic to $Y_{1,\bar {y}_1}$ . Let

$$\begin{align*}\widehat{W} = W_Y \times W_{Y_1}, \end{align*}$$

and let $\widehat {\mathcal {Y}} \to B\times \widehat {W}$ and $\widehat {\mathcal {Y}}_1 \to B \times \widehat {W}$ be the pull-back of $\mathcal {Y}$ and $\mathcal {Y}_1$ using the projections of $\widehat {W}$ onto the first and second component respectively. Consider

$$\begin{align*}\Sigma = \{ (y, y_1) \in \widehat{W} \mid Y_y \to B \ \text{is isomorphic to} \ Y_{1, y_1} \to B\}. \end{align*}$$

It follows from the properness of $\mathrm {Isom}(\widehat {\mathcal {Y}}, \widehat {\mathcal {Y}}_1, B)$ that $\Sigma $ is a locally closed analytic subvariety of $\widehat {W}$ . Therefore $(\bar {y}, \bar {y}_1) \in \Sigma $ , which concludes the proof.

We have proved the following.

4.1 The determinant line bundle for the Weil-Petersson metric

We construct a line bundle on $W_Y$ , and hence on the moduli space, such that the Weil-Petersson metric represents its first Chern class. To do so, we appeal to the theory of Deligne pairings [Reference Elkik16], [Reference Elkik17], [Reference Boucksom, Hisamoto and Jonsson4, §1]. Let $M \to B$ be a flat, projective morphism between complex analytic spaces of relative dimension d and consider $d+1$ line bundles $L_0, \dots , L_d$ on M. The push-forward of the intersection product of $L_0, \dots , L_d$ is an isomorphism class of line bundles on B, represented by the cohomology class

(4.8) $$ \begin{align} \int_{M/B} c_1(L_0) \wedge \dots \wedge c_1(L_d). \end{align} $$

The Deligne pairing of $L_0, \dots , L_d$ , denoted by $ \langle L_0 , \dots , L_d \rangle _{M/B} $ , is a canonical choice of a line bundle on B such that (4.8) is its first Chern class. The construction is symmetric, multilinear and functorial. Moreover, if $h_0, \dots , h_d$ are Hermitian metrics on $L_0, \dots , L_d$ respectively, the theory provides a metric $\langle h_0, \dots , h_d\rangle _{M/B}$ on $\langle L_0 , \dots , L_d \rangle _{M/B} $ . Denoting by $\omega _0, \dots , \omega _d$ the curvature forms of $h_0, \dots , h_d$ respectively, the curvature of $\langle h_0, \dots , h_d\rangle _{M/B}$ is given by the fibre integral

(4.9) $$ \begin{align} \int_{M/B} \omega_0 \wedge \dots \wedge \omega_d. \end{align} $$

The fibre integral formula for the Weil-Petersson metric on $W_Y$ of Lemma 4.5 is a special case of the expression (4.9). To describe it as the curvature form of a line bundle on $W_Y$ , we first recall the following result of Fujiki and Schumacher for the k-dependent Weil-Petersson metric [Reference Fujiki and Schumacher22, Theorem 11.1], which we write using the Deligne pairing notation.

Proposition 4.9. The k-dependent Weil-Petersson type Kähler metric $\Omega _{k}$ represents the first Chern class of the line bundle

(4.10) $$ \begin{align} \langle-K_{\mathcal{Y}/W_Y}, \mathcal{L}_{k}^{n+m}\rangle_{\mathcal{Y}/W_Y} - \frac{1}{n+m+1}\frac{-K_Y\cdot (H_Y+kL)^{n+m-1}}{(H_Y+kL)^{n+m}}\langle \mathcal{L}_{k}^{n+m+1}\rangle_{\mathcal{Y}/W_Y}, \end{align} $$

where the constant

$$\begin{align*}\widehat{S}_{Y} := \frac{-K_Y\cdot (H_Y+kL)^{n+m-1}}{(H_Y+kL)^{n+m}} \end{align*}$$

is the average scalar curvature of Y, and hence of each $Y_t$ , with respect to the metric $\omega + k\omega _B$ .

We use Proposition 4.9 to prove the following.

Proposition 4.10. There exists a line bundle $\mathcal {D}(Y)$ on $W_Y$ whose first Chern class is represented by the Weil-Petersson metric (4.2), given as the sum of the line bundles

$$\begin{align*}\begin{aligned} &\mathcal{D}_0(Y) = -\langle -K_{\mathcal{Y}/B\times W_Y}, \widehat{\mathcal{H}}^{m}, L^{n} \rangle_{\mathcal{Y}/{W_Y}}, \\ & \mathcal{D}_2(Y)= \frac{1}{n+m+1}\frac{-K_{Y/B}\cdot L^{n}\cdot H_Y^{m-1}}{L^n\cdot H_Y^m}\left(\langle \widehat{\mathcal{H}}^{m+1}, L^{n} \rangle_{\mathcal{Y}/{W_Y}} + \langle \widehat{\mathcal{H}}^{m+2}, L^{n-1} \rangle_{\mathcal{Y}/{W_Y}}\right), \\ &\mathcal{D}_3(Y)=-\langle -K_B, \widehat{\mathcal{H}}^{m+1}, L^{n-1} \rangle_{\mathcal{Y}/{W_Y}}, \\ &\mathcal{D}_4(Y)= \frac{\left(-K_{Y/B} \cdot H_Y^{m-1}\cdot L^n\right)\left(H_Y^{m+1}\cdot L^{n-1}\right)}{(n+m+1)(H_Y^m\cdot L^n)^2}\langle \widehat{\mathcal{H}}^{m+1}, L^{n} \rangle_{\mathcal{Y}/{W_Y}}, \end{aligned} \end{align*}$$

defined using the Deligne pairing.

Proof. Let $ \widehat {\mathcal {H}} \to \mathcal {Y} $ be the relatively ample line bundle induced by each relative polarisation $H_t \to Y_t$ . More precisely, we consider the fibration $\mathcal {Y} \to W_Y$ as the composition of

$$\begin{align*}\mathcal{Y} \to B \times W_Y \to W_Y \end{align*}$$

so $\widehat {\mathcal {H}}$ is relatively ample with respect to the first projection. Moreover, since the base B is fixed, we can pull-back L to $\mathcal {Y}$ and consider it as a line bundle on $\mathcal {Y}$ . So we can define a relatively ample line bundle $\mathcal {L}_k$ over $\mathcal {Y}$ as

(4.11) $$ \begin{align} \mathcal{L}_k = \widehat{\mathcal{H}} \otimes kL, \end{align} $$

whose fibre over $Y_t$ is $\mathcal {L}_k\vert _{t} = H_t \otimes kL$ . Its first Chern class contains the relative metric $\omega _{\mathcal {Y},k}$ , where $\widehat {\omega }$ is in $c_1(\widehat {\mathcal {H}})$ . We define a line bundle $\mathcal {D}(Y)$ on $W_Y$ by using the fibre integral formula (4.7) and the expansion in powers of k of the line bundle (4.10). Expanding in k the intersection product $\mathcal {L}_k^{n+m}$ , $\mathcal {L}_k^{n+m+1}$ and the expression (4.10) we obtain that the Weil-Petersson metric $\Omega _{WP}$ represents the first Chern class of the line bundle $\mathcal {D}(Y)$ given as the tensor product of the line bundles $\mathcal {D}_0(Y),\mathcal {D}_2(Y),\mathcal {D}_3(Y), \mathcal {D}_4(Y)$ above. Indeed, by expanding the expression (4.10) in powers of k we see that the sum of the leading order term and the sub-leading order term is given by

$$\begin{align*}\mathcal{D}_0(Y)+\mathcal{D}_1(Y)+\mathcal{D}_2(Y)+\mathcal{D}_3(Y)+\mathcal{D}_4(Y), \end{align*}$$

where $\mathcal {D}_1(Y)$ is given by

$$\begin{align*}\mathcal{D}_1(Y) =- \langle-K_{\mathcal{Y}/B\times W_Y}, \widehat{\mathcal{H}}^{m+1}, L^{n-1} \rangle_{\mathcal{Y}/{W_Y}}. \end{align*}$$

However, its first Chern class is represented by the term (4.6), which does not appear in the fibre integral formula for the relative Weil-Petersson metric of Lemma 4.5. This concludes the proof.

We have constructed a line bundle $\mathcal {D}_Y$ on $W_Y$ whose first Chern class is the Weil-Petersson metric $\Omega _{WP}$ . Let now $p : W_Y \to \mathcal {M}$ be the composition of the maps

$$\begin{align*}W_Y \to W_Y/\mathrm{Aut}(\pi_Y) \hookrightarrow \mathcal{M}, \end{align*}$$

where the first map is the quotient by the group action and the second map is the inclusion of a local chart in the moduli space. It follows from Lemma 3.16 that the orbits of the $\mathrm {Aut}(\pi _Y)$ -action on $W_Y$ correspond to isomorphic manifolds in the family $\mathcal {Y} \to W_Y$ . Therefore the line bundle $\mathcal {D}(Y)$ is invariant for the action of $\mathrm {Aut}(\pi _Y)$ and thus descends to a line bundle $\widetilde {\mathcal {D}}(Y)$ on the quotient $W_Y/\mathrm {Aut}(\pi _Y)$ .

Proof. Let $W_{Y_1}/\mathrm {Aut}(\pi _{Y_1})$ and $W_{Y}/\mathrm {Aut}(\pi _{Y})$ be two local charts of $\mathcal {M}$ with nonempty intersection. Then using completeness of the Kuranishi space there exists an isomorphism $\widetilde {\alpha }: W_{Y_1}/\mathrm {Aut}(\pi _{Y_1}) \to W_Y/\mathrm {Aut}(\pi _Y)$ (3.12) which preserves the relative polarisation (4.11) and the submersions onto the base B. By functoriality of the Deligne pairings, the pull-back $\widetilde {\alpha }^*\widetilde {\mathcal {D}}(Y_1)$ is then isomorphic to $\widetilde {\mathcal {D}}(Y)$ . Therefore, on the intersection of $W_Y/\mathrm {Aut}(\pi _{Y})$ and $W_{Y_1}/\mathrm {Aut}(\pi _{Y_1})$ there is a morphism of line bundles $\chi : \widetilde {\mathcal {D}}(Y_1) \overset {\sim }{\to } \widetilde {\mathcal {D}}(Y)$ . Let $\varphi _Y : \widetilde {\mathcal {D}}(Y) \to W_Y/\mathrm {Aut}(\pi _{Y}) \times \mathbb {C}$ and $\varphi _{Y_1} : \widetilde {\mathcal {D}}(Y_1) \to W_{Y_1}/\mathrm {Aut}(\pi _{Y_1}) \times \mathbb {C}$ be local trivialisations and, on the intersection,

$$\begin{align*}\psi_{Y_1Y} := \varphi_Y\circ\chi\circ\varphi_{Y_1}^{-1}. \end{align*}$$

The map $\psi _{Y_1Y}$ , viewed as a function on $\mathbb {C}$ , is invertible, with inverse $\psi _{YY_1}$ . Indeed, the map $\chi $ is an isomorphism because $\widetilde {\alpha }$ is. The same argument proves that the cocycle condition holds.

The following corollary is a consequence of Theorem 4.11 and of [Reference Fujiki and Schumacher22, Proposition 1.7].