3.1 $L^2$-integrable forms

We start with a fibrewise consideration. Let $(X,D=\sum _{i=1}^lD_i)$ be a smooth log pair, and set $X'=X\backslash D$. We consider a holomorphic line bundle L on X together with a metric h that is smooth on $X'$ and whose inverse has the asymptotic behaviour from equation (2.1)

$$ \begin{align*} h^{-1}|_{X'}= \exp(u)\cdot \frac{h_{L^{-1}}^{C^{\infty}}}{\prod_{i=1}^l{||\sigma_i||_i^2\log^2||\sigma_i||_i^2}}, \end{align*} $$

where the notation is as follows:

• • $h_{L^{-1}}^{C^{\infty }}$ is a smooth metric on $\mathscr {L}^{-1}$.

• • $||\sigma _i||_i$ is the norm of the canonical section cutting out $D_i$ with regard to a smooth metric s.t. $||\sigma _i||_i<1$.

• • u is a function in $\mathscr {C}^{k,\alpha }(X')$.

• • $\omega _{X'}:= -\mathrm {i} \partial \bar \partial \log (h)|_{X'}$ is a Poincaré type Kähler metric.

Then we can identify the holomorphic and locally $L^2$-integrable $(n,0)$-forms with values in L:

Proposition 1. We denote by $\mathscr {O}_{(2)}(\Omega ^n_{X'}(L|_{X'}),h|_{X'})$ the sheaf of holomorphic L-valued n-forms on $X'$, which are locally $L^2$ on X with respect to $h|_{X'}$. Then

$$ \begin{align*} \mathscr{O}_{(2)}(\Omega^n_{X'}(L|_{X'}),h|_{X'}) = \mathscr{O}(\Omega^n_X(\log D)(L)). \end{align*} $$

The proof follows immediately from a Laurent series argument together with the estimates of Poincaré type metrics: sections that are locally square integrable with respect to a metric with a Poincaré type metric extend holomorphically as forms with logarithmic poles to the given snc divisor D (and vice versa).$\Box $

Let $\mathscr {A}^{n,q}_{(2)}(L|_{X'})$ denote the sheaf on X of L-valued $(n,q)$-forms that are locally $L^2$ integrable with respect to $\omega _{X'}$ and $h|_{X'}$ and whose $\overline \partial $-exterior derivatives, taken in the current sense, are also locally $L^2$. We refer to [Reference Zucker25] or [Reference Brylinski and Zucker5] for more details on the $L^2$-complex of sheaves.

3.2 Quasi-coordinates and Hölder spaces

We recall from [Reference Cheng and Yau6] that a quasi-coordinate map is a holomorphic map from an open set $V \subset \mathbb {C}$ into $X'$ if it is of maximal rank everywhere in V. In this case, V together with the Euclidean coordinates of $\mathbb {C}^n$ is called a local quasi-coordinate of $X'$. According to [Reference Cheng and Yau6, Reference Kobayashi14, Reference Tian, Yau, Yau and Diego21], we have the following:

A complete Kähler manifold $(X',g)$, which admits a family $\mathscr {V}$ of local quasi-coordinates satisfying the conditions of the proposition, is called being of bounded geometry (of order $\infty $).

Although the coordinate system from Proposition 3 is not a coordinate system in the ordinary sense because of the covering map involved, it makes sense to talk about the components of a tensor field on $X'$ (or a neighbourhood $U(p)=(\Delta ^*)^k \times \Delta ^{n-k}$) with respect to these ‘coordinates’ $v^i$ by first lifting it to a tensor field $\Delta ^n$. The behaviour of the tensor on $U(p)$ can thus be examined by looking at the lifted function in a neighbourhood of $(1,\ldots ,1,*,\ldots ,*)$ in $\Delta $.

We mention here the transition of tensors from a local coordinate function $z^i$, where a component $D_i$ is given by $\{z^i=0\}$ to a quasi-coordinate $v^i$, $|v^i|< R <1$, which is given by

(3.1)$$ \begin{align} \frac{dz^i}{z^i\log|z^i|^2}= \frac{|v^i-1|^2}{(v^i-1)^2}\frac{dv^i}{1-|v^i|^2} \sim dv^i, \end{align} $$

because v is bounded away from $1$. The respective equation for $\partial /\partial z^1$ reads

(3.2)$$ \begin{align} z^i \log |z^i|^2 \frac{ \partial}{ \partial z^i} \sim \frac{ \partial}{ \partial v^i}. \end{align} $$

This transformation rule is used to derive estimates from Hölder regular functions and tensors that we will now define. The Hölder spaces $\mathscr {C}^{k,\alpha }(X')$ are defined in terms of quasi-coordinates for sufficiently large values of k and $0<\alpha <1$. The Hölder norms are computed in terms of the infinite number of quasi-coordinate systems. Following [Reference Cheng and Yau6, Reference Kobayashi14, Reference Tian, Yau, Yau and Diego21], we define

Definition 1. Let $k \in \mathbb {N}_0$ and $\alpha \in (0,1)$, and denote by $C^k(X')$ the space of k-times differentiable functions $u: X' \to \mathbb {C}$. For $u \in C^k(X')$, let

$$ \begin{align*} \Vert u \rVert_{k,\alpha} = \sup_{(V,v^1,\ldots,v^n) \in \mathscr{V}} \left( \sup _{z \in V} \sum_{|I|+|J|\leq k}{| \partial_v^J\overline{ \partial}_v^J u(z)|} + \sup_{z,z' \in V} \sum_{|I|+|J|=k} \frac{| \partial_v^I\overline{ \partial}_v^J u(z)- \partial_v^I\overline{ \partial}_v^J u(z')|}{|z-z'|^{\alpha}} \right) \end{align*} $$

be the $\mathscr {C}^{k,\alpha }$-norm of u, where $ \partial _v^I\overline { \partial }_v^J=\frac { \partial ^{|I|+|J|}}{ \partial v^I \partial \overline {v}^J}$. Then let

$$ \begin{align*} \mathscr{C}^{k,\alpha}:=\mathscr{C}^{k,\alpha}(X') := \{u \in C^k(X')\;:\; \Vert u \Vert_{k,\alpha} <\infty\} \end{align*} $$

be the function space of $\mathscr {C}^{k,\alpha }$ functions on $X'$ with respect to $\mathscr {V}$.

In a similar way, we can define $\mathscr {C}^{k,\alpha }$-tensors by pulling back via the quasi-coordinate maps. Exterior derivatives and covariant derivatives of $\mathscr {C}^{k,\alpha }$-tensors are of the same type (with k being replaced by $k-1$). The arguments from [Reference Kobayashi14] using Hölder spaces $\mathscr {C}^{k,\alpha }$ (with respect to quasi-coordinates) are valid for all large fixed numbers $k\geq k_0$, where $k_0$ denotes some minimal degree. During the computations, we will have to take derivatives, products and contractions of such tensors, arriving at $\mathscr {C}^{k,\alpha }$-tensors for some lower value of k. In each of these steps, we will have to increase the lower bound $k_0$. Only finitely many such steps will be necessary. We will increase $k_0$ tacitly.

Let $\omega _{X'}$ be the complete Poincaré type Kähler form from the previous section.

The proof follows immediately from the given uniform bounds of such tensors in terms of the above quasi-coordinate systems and the metric $\omega _{X'}$ with respect to the transition equations of the type in equation (3.1): take the pointwise norm of such a tensor with respect to the given metric. We consider the resulting function a bounded $\mathscr {C}^{k,\alpha }$-tensor (for some value of k), which means it is uniformly bounded in terms of quasi-coordinate systems (or any other coordinate systems). Finally, we use the boundedness of the volume of $X'$. $\Box $

3.3 Hodge theory on the open fibres

In this section, we summarise the Hodge theory on the complete, noncompact Kähler manifold $(X',\omega _{X'})$, for which we refer to the book of Marinescu and Ma [Reference Ma and Marinescu16, Chapter 3]. We consider the holomorphic vector bundle $E=K_{X'}\otimes L|_{X'}$ equipped with the hermitian metric $h|_{X'}$.

For the operator $\bar \partial $ defined on the smooth, compactly supported forms $A_0^{0,q}(X',E) \subset L^{0,q}_{(2)}(X',E)$, we consider its formal adjoint $\bar \partial ^*$. For $s_1 \in L_{(2)}^{0,q}(X',E)$, we can calculate $\bar \partial s_1$ in the sense of currents: $\bar \partial s_1$ is the current defined by

$$ \begin{align*} {\langle}\bar \partial s_1,s_2{\rangle} = {\langle}s_1,\bar \partial^*s_2{\rangle} \quad \mbox{ for } s_2 \in A_0^{(0,q+1)}(X',E). \end{align*} $$

Then we have

Lemma 3.2 [Reference Ma and Marinescu16, Lemma 3.1.1].

The operator $\bar \partial _{\max }$ defined by

$$ \begin{align*} &\operatorname{Dom} (\bar \partial_{\max}) = \{s \in L_{(2)}^{0,q}(X',E)\; : \; \bar \partial s \in L_{(2)}^{0,q+1}(X',E)\}\\ & \bar \partial_{\max} s = \bar \partial s \quad \mbox{ for } s \in \operatorname{Dom}(\bar \partial_{\max}) \end{align*} $$

is a densely defined, closed extension called the maximal extension of $\bar \partial .$

In the sequel, we work with the maximal extension and simply write $\bar \partial = \bar \partial _{\max }$. The Hilbert space adjoint of $\bar \partial _{\max }$ is denoted by $\bar \partial ^*_H$. We note that $\bar \partial ^*_H \subset (\bar \partial ^*)_H$ and $\bar \partial _{\max } = (\bar \partial ^*_H)^*_H$. From now on, we work with $\bar \partial ^*_H$ and just write $\bar \partial ^*$.

The Laplacian $\Box =\bar \partial \bar \partial ^*+\bar \partial ^*\bar \partial $ is a densely defined, positive operator, so one can consider its Friedrichs extension. But in the context of $L^2$ cohomology, it is useful to consider another extension:

Proposition 4 [Reference Ma and Marinescu16, Proposition 3.1.2].

The operator defined by

$$ \begin{align*} & \operatorname{Dom} (\Box) = \{s \in \operatorname{Dom}(\bar \partial) \cap \operatorname{Dom}(\bar \partial^*)\; : \; \bar \partial s \in \operatorname{Dom}(\bar \partial^*), \; \bar \partial^* s \in \operatorname{Dom}(\bar \partial)\},\\ & \Box s = \bar \partial^* \bar \partial s + \bar \partial \bar \partial^* s \quad \mbox{ for } s \in \operatorname{Dom}(\Box) \end{align*} $$

is a positive self-adjoint extension of the Laplacian, called the Gaffney extension.

We note that this result relies on Gaffney’s generalisation of Stokes’ theorem that we will use tacitly during our computation:

Proposition 5 [Reference Gaffney9].

Let $(M,g)$ be an orientable complete Riemannian manifold of real dimension $2n$ whose Riemann tensor is of class $C^2$. Let $\gamma $ be a $(2n-1)$-form on M of class $C^1$ such that both $\gamma $ and $d \gamma $ are in $L^1$. Then

$$ \begin{align*} \int_M{d\gamma} =0. \end{align*} $$

We define the space of harmonic forms $\mathscr {H}^{0,q}(X',E) \subset L_{(2)}^{0,q}(X',E)$ by

$$ \begin{align*} \mathscr{H}^{0,q}(X',E) := \operatorname{Ker} (\Box) = \{s \in \operatorname{Dom}(\Box)\;:\: \Box s =0\}. \end{align*} $$

We see that

$$ \begin{align*} \mathscr{H}^{0,q}(X',E) = \operatorname{Ker}(\bar \partial) \cap \operatorname{Ker}(\bar \partial^*). \end{align*} $$

The qth $L^2$ Dolbeault cohomology is defined by

which is of course the same as the group $H^q_{(2)}(X',K_{X'}^p(L|_{X'}))$ from Proposition 2, because the $L^2$-cohomology can be also computed by smooth forms. From this, we also get

Proposition 6. The $L^2$-Dolbeault cohomology $H^{0,q}_{(2)}(X',E) $ is finite-dimensional, and we have

$$ \begin{align*} H^{0,q}_{(2)}(X',E) \cong \mathscr{H}^{0,q}(X',E). \end{align*} $$

Moreover, the images of $\bar \partial $ and $\bar \partial ^*$ are closed in $L_{(2)}(X',E)$, and thus there exists a constant $C>0$ such that

(3.3)$$ \begin{align} ||s||^2_{L^2} \leq C \left( ||\bar \partial s||^2_{L^2} + ||\bar \partial^* s||^2_{L^2}\right) \end{align} $$

for all $s \in \operatorname {Dom}(\bar \partial ) \cap \operatorname {Dom}{\bar \partial ^*}\cap L^{0,q}_{(2)}(X',E), \,s \perp \mathscr {H}^{0,q}(X',E)$.

Corollary 3. We have the strong Hodge decomposition, which is orthogonal:

$$ \begin{align*} L^{0,q}_{(2)}(X',E) &= \mathscr{H}^{0,q}(X',E) \oplus \operatorname{Im}(\Box) = \mathscr{H}^{0,q}(X',E) \oplus \operatorname{Im}(\bar \partial\bar \partial^*) \oplus \operatorname{Im}(\bar \partial^*\bar \partial),\\ \operatorname{Ker}(\bar \partial) \cap L^{0,q}_{(2)}(X',E) &= \mathscr{H}^{0,q}(X',E) \oplus \left( \operatorname{Im}(\bar \partial) \cap L^{0,q}_{(2)}(X',E) \right). \end{align*} $$

Moreover, there exists a bounded operator G on $L^{0,q}_{(2)}(X',E)$, called the Green operator, such that

$$ \begin{align*} \Box G = G \Box = \operatorname{Id} -H, \quad HG=GH=0, \end{align*} $$

where H is the orthogonal projection form $L^{0,q}_{(2)}(X',E)$ onto $\mathscr {H}^{0,q}(X',E)$.

Remark 3.3. By the usual elliptic regularity theory, we get that harmonic sections are in fact smooth. G maps smooth forms to smooth forms so that we can cut down the Hodge decomposition to the space of smooth $L^2$ sections:

$$ \begin{align*} A^{0,q}_{(2)}(X',E) = \mathscr{H}^{0,q}(X',E) \oplus \operatorname{Im}(\Box). \end{align*} $$

3.4 Families of logarithmic pairs

Let $(\mathscr {X},\mathscr {D})$ be a smooth log pair: that is, $\mathscr {X}$ a complex manifold and $\mathscr {D} \subset \mathscr {X}$ a reduced snc divisor. The boundary divisor $\mathscr {D}$ is written as a sum of its irreducible components $\mathscr {D} = \mathcal {D}_1+\ldots + \mathscr {D}_k$. If $I \subset \{1,\ldots ,k\}$ is any non-empty subset, we consider the intersection $\mathscr {D}_I:=\cap _{i \in I}\mathscr {D}_i$.

If $s \in S$ is any point, set $X_s=f^{-1}(s)$ and $D_s:= \mathscr {D} \cap X_s$. Then $X_s$ is smooth and $(X_s,D_s)$ is a snc pair. Moreover, the number of irreducible components of $D_s$ is also k, which is the number of irreducible components of $\mathscr {D}$. This excludes phenomena like the deformation of the smooth hyperbola $\{wz=s \neq 0\}$ into the snc divisor $\{zw=0\}$ that has two components. The reason for this will become clear later. Moreover, the definition implies

The lemma says that we can trivialise the pair $(X_s,D_s)$ in a differentiable way. By restriction to $\mathscr {X}':=\mathscr {X}\setminus \mathscr {D}$, we find a smooth trivialisation of $\mathscr {X}'\cong X'\times S$.

Given a smooth log pair $(X,D)$ and a point x, there exists an open neighbourhood $U=U(x)$ and holomorphic coordinates $z_1, \dots ,z_n$ such that $D\cap U=\{z_1\cdots z_r\}$ for some $0\leq r \leq n$. The following is the relative analogue of this fact:

In the following, we will make use of these adapted coordinates tacitly. In the applications, we also consider the following families:

In particular, our families are logarithmic deformations in the sense of [Reference Kawamata12] and generalise the deformations considered in [Reference Schumacher18] to the case of singular snc divisors.

3.5 L ²-integrable Kodaira-Spencer forms

We consider a holomorphic family of smooth log pairs $\mathscr {D} \overset {i}{\hookrightarrow } \mathscr {X} \overset {f}{\rightarrow } S$ and a hermitian holomorphic line bundle $(\mathscr {L},h)$ on $\mathscr {X}$ whose inverse metric has Poincaré type singularities along $\mathscr {D}$, as already described in the introduction. Moreover, we recall that we have defined on $\mathscr {X}'$ the global $(1,1)$-form

$$ \begin{align*} \omega_{\mathscr{X}'} = -\mathrm{i}\partial\overline\partial (\log h)= \mathrm{i} \partial \bar \partial \log(h^{C^{\infty}}_{\mathscr{L}^{-1}})-\sum_{i=1}^l{\mathrm{i} \partial \bar \partial\log(||\sigma_i||^2 \log^2||\sigma_i||^2})+\mathrm{i} \partial \bar \partial u, \end{align*} $$

whose restrictions to the open fibres $X^{\prime }_s$ give a smooth family of Poincaré type Kähler forms $\omega _s = \omega _{\mathscr {X}'}|_{X^{\prime }_s}$.

By using the local description

$$ \begin{align*} \omega_{\mathscr{X}'}=\sqrt{-1} \left( g_{\alpha\overline{\beta}} dz^{\alpha} \wedge dz^{\overline{\beta}} + g_{i\overline{\beta}} ds^i\wedge dz^{\overline{\beta}}+ g_{\alpha \overline{\jmath}} dz^{\alpha} \wedge ds^{\overline\jmath} + g_{i\overline \jmath} ds^i\wedge ds^{\overline\jmath} \right) \end{align*} $$

with respect to the local coordinates $(z,s)$, we can say that the restrictions of $ \omega _{\mathscr {X}'}$ as well as restrictions of contractions depend in a $C^{\infty }$ way upon the parameter.

Lemma 3.7. The restrictions

$$ \begin{align*} \omega_{\mathscr{X}'}\llcorner (\partial/\partial s^i)|X^{\prime}_s &= g_{i\overline\beta}dz^{\overline\beta}|X^{\prime}_s\\ \omega_{\mathscr{X}'}\llcorner (\partial/\partial s^{\overline \jmath})|X^{\prime}_s &= g_{\alpha\overline \jmath}dz^{\alpha}|X^{\prime}_s\\ \omega_{\mathscr{X}'}\llcorner (\partial/\partial s^i\wedge ds^{\overline\jmath})|X^{\prime}_s&= g_{i\overline \jmath}|X^{\prime}_s \end{align*} $$

are $\mathscr {C}^{k,\alpha }$-tensors that depend in a $C^{\infty }$ way upon the parameter $s\in S$. In particular, they are smooth and $L^2$ integrable tensors $(\mathscr {A}_{(2)}$-tensors for short). The analogous statement holds if $\omega _{\mathscr {X}'}$ is replaced by $\widetilde \omega _{\mathscr {X}'}$, where the first two tensors do not change.

Horizontal lifts of tangent vectors from S to the total space $\mathscr {X}'$ are defined as perpendicular to the fibres of $\mathscr {X}' \to S$ with respect to $\omega _{\mathscr {X}'}$. Here we only need the property that the restrictions of $\omega _{\mathscr {X}'}$ to the fibres are positive definite.

We denote by $(g^{\overline \beta \alpha })$ the inverses of the metric tensors $g_{\alpha \overline \beta }$ of $\omega _s$ on the fibres $X^{\prime }_s$. Covariant derivatives are always taken with respect to these metrics. We will use both the semicolon notation and the $\nabla $-notation. We continue using Greek indices for tensors in fibre direction and $\partial _{\alpha } = \partial /\partial z^{\alpha }$.

The horizontal lift $v_i$ of a tangent vector $ \partial _i={\partial }/{\partial } s^i$ is a differentiable lift of $ \partial _i$ to $\mathscr {X}'$, which is orthogonal to the fibres with respect to the sesquilinear form $\omega _{\mathscr {X}'}$:

$$ \begin{align*} {\langle}v_i, \partial_{\alpha}{\rangle}_{\omega_{\mathscr{X}'}} =0 \quad \mbox{for all } \alpha=1\ldots n. \end{align*} $$

This is well defined since the form $\omega _{\mathscr {X}'}$ is positive when restricted to the fibres. In terms of the coefficients of $\omega _{\mathscr {X}}$, it is given by

$$ \begin{align*} v_i= \partial_i + a_i^{\alpha} \partial_{\alpha}, \end{align*} $$

where

$$ \begin{align*} a_i^{\alpha} = - g^{\overline{\beta}\alpha}g_{i\overline{\beta}}. \end{align*} $$

Lemma 3.8. Let $A_i = \overline \partial v_i|X^{\prime }_s$: that is,

$$ \begin{align*} A_i = A^{\alpha}_{i\overline\beta}\partial_{\alpha} dz^{\overline\beta}. \end{align*} $$

Then $A_i$ is of class $\mathscr {C}^{k,\alpha }$ and satisfies $\overline \partial A_i=0$. In particular, it lies in $A^{0,1}_{(2)}(X^{\prime }_s,T_{X^{\prime }_s})$.

The proof is the same as in [Reference Schumacher18, Lemma 3].$\Box $

The Kodaira-Spencer map for a family $\mathscr {D} \overset {i}{\hookrightarrow } \mathscr {X} \overset {f}{\rightarrow } S$

$$ \begin{align*} \rho_s:T_sS \to H^1_{(2)}(X^{\prime}_s, T_{X^{\prime}_s}) \end{align*} $$

was already defined in [Reference Schumacher18]. Analogous results like those of Section 3.1 hold for the sheaf of holomorphic vector fields:

The sequence $(\mathscr {A}^{0,\bullet }_{(2)}(\mathcal {T}_{X^{\prime }_s}),\overline {{\partial }})$ is a fine resolution of the sheaf of $L^2$ holomorphic vector fields $\mathscr {O}_{(2)}(T_{X^{\prime }_s})$, which is isomorphic to $T_{X_s}(-\log D_s):=(\Omega ^1_{X_s}(\log D_s))^{\vee }$.

In particular,

$$ \begin{align*} H^1_{(2)}(X^{\prime}_s, T_{X^{\prime}_s}) \cong H^1(X_s,T_{X_s}(-\log D_s)), \end{align*} $$

and the above explicit construction yields a $L^2$-Dolbeault representative $A^{\alpha }_{i\overline {\beta }}{\partial }_{\alpha } dz^{\overline {\beta }}$ of $\rho _s({\partial }/{\partial } s^i)$, where the Kodaira-Spencer map is taken as map

$$ \begin{align*} \rho_s:T_sS \to H^1_{(2)}(X^{\prime}_s,T_{X^{\prime}_s}). \end{align*} $$

We will also need the following fact that follows from the definition.

Lemma 3.9. Let $A_i^{\alpha \sigma }=g^{\overline {\beta } \sigma }A_{i\overline {\beta }}^{\alpha }$. Then

$$ \begin{align*} A_i^{\alpha \sigma}=A_i^{\sigma \alpha}. \end{align*} $$

Finally, the pointwise inner product with respect to $\omega _{\mathscr {X}'}$ of two horizontal lifts

$$ \begin{align*} \varphi_{i\overline{\jmath}} := {\langle}v_i,v_j{\rangle}_{\omega_{\mathscr{X}'}}, \end{align*} $$

called the geodesic curvature, has an expression

$$ \begin{align*} \varphi_{i\overline{\jmath}} = g_{i\overline{\jmath}} - g_{\alpha\overline{\jmath}}g_{i\overline{\beta}}g^{\alpha\overline{\beta}} \end{align*} $$

so that we conclude from Lemma 3.7 that this function lies in $\mathscr {C}^{k,\alpha }$.

3.6 Bundles of L ²-integrable forms

Now we consider the $L^2$ condition on the total space $\mathscr {X}$. Because the form $\omega _{\mathscr {X}'}$ is fibrewise positive, for any $s_0\in S$ after replacing S by a neighbourhood (contractible and Stein), there exists a positive hermitian form $\omega _S$ such that $\widetilde {\omega }^{\prime }_{\mathscr {X}} = \omega _{\mathscr {X}'}+f^*\omega _S$ is a positive form on $\mathscr {X}'$.

The previous arguments imply that the statements of Proposition 1 and Proposition 2 hold for the total spaces $(\mathscr {X}',\widetilde \omega _{\mathscr {X}'})$. The sheaves of locally $L^2$ sections $\mathscr {O}_{(2)}(\Omega ^n_{\mathscr {X}'}(\mathscr {L}|_{\mathscr {X}'}), \widetilde \omega _{\mathscr {X}'},h|_{\mathscr {X}'})$ on the whole total space $\mathscr {X}$ are equal to $\Omega ^n_{\mathscr {X}}(\log \mathscr {D})(\mathscr {L})$, and an analogous fine resolution in terms of square-integrable forms exists. However, the sheaves $\Omega ^n_{\mathscr {X}}(\log \mathcal {D})$ of log n-forms on the total space $\mathscr {X}$ are not suitable for our methods.

Instead, we will need the coherent sheaf $\Omega ^n(\log \mathcal {D})_{\mathscr {X}/S}(\mathscr {L})$ and the sheaves

$$ \begin{align*} \mathscr{O}_{(2)}(\Omega^n_{\mathscr{X}'/S}(\mathscr{L}|_{\mathscr{X}'}), \widetilde{\omega}_{\mathscr{X}'},h|_{\mathscr{X}'}) \end{align*} $$

of relative $\mathscr {L}$-valued holomorphic n-forms that are square-integrable with respect to the Kähler form $\widetilde {\omega }_{\mathscr {X}'}= \omega _{\mathscr {X}'}+ f^*\omega _S$ and the hermitian metric $h|_{\mathscr {X}'}$, where integrability does not depend upon the choice of a hermitian form $\omega _S$.

Let

$$ \begin{align*} \mathscr{A}^{0,q}_{(2)}(\Omega^n_{\mathscr{X}'/S}(\mathscr{L}),\widetilde{\omega}_{\mathscr{X}'},h|_{\mathscr{X}'}) \end{align*} $$

denote the sheaf of $(0,q)$-currents on the total space $\mathscr {X}'$with values in the coherent sheaf $\Omega ^n_{\mathscr {X}'/S}(\mathscr {L})$ that are square-integrable along with their exterior $\overline {{\partial }}$-derivatives.

Proof. To simplify notation, we assume that $\dim S=1$. We note that

$$ \begin{align*} \widetilde\omega_{\mathscr{X}'}^{n+1} = \sqrt{-1}\omega_{\mathscr{X}'/S}^n\;\varphi\; ds\wedge d\overline{s} + \omega_{\mathscr{X}'/S}^n \wedge f^*\omega_S \end{align*} $$

and the function $\varphi $ is (locally with regard to S uniformly) bounded. For an open set $U \subset \mathscr {X}$, we thus have for the $L^2$-norm of a section $u \in \Omega ^n_{\mathscr {X}'/S}(\mathscr {L})$ that

$$ \begin{align*} \int_U{|u|^2_{\omega_{\mathscr{X}'/S}, h_{\mathscr{L}}}\;\widetilde\omega_{\mathscr{X}'}^{n+1}} \sim \int_{f(U)}\left(\int_{X_s\cap U}{(u\wedge \overline{u})_h}\right)\omega_S. \end{align*} $$

The first statement follows from Fubini’s theorem, the fact that nearby fibres and bundles are quasi-isometric, and Proposition 1. For the second statement, we decompose the vector bundle $\Omega ^n_{\mathscr {X}'/S}(\mathscr {L}|_{\mathscr {X}'})$ locally as a sum of line bundles and apply [Reference Fujiki8, Prop. 2.1]; also compare to [Reference Fujiki8, p.870]. This shows that we have a resolution. By Proposition 2, fibrewise we find cut-off functions with bounded differentials. Because, in the differentiable sense, we have a product situation, this also holds on the total space. Hence, the sheaves of locally $L^2$ integrable smooth sections are again fine.

3.7 Fibre integrals and Lie derivatives

Given our family $f: \mathscr {X}' \to S$ of open complex manifolds $X^{\prime }_s$ of dimension n and a relative $(n,n)$-form $\eta $ on $\mathscr {X}'$ that is smooth there and integrable along the fibres, the fibre integral

$$ \begin{align*} \int_{\mathscr{X}'/S}{\eta} \end{align*} $$

gives a function on the base S (see [Reference Schumacher19, Sect. 2.1] and [Reference Greub, Halperin and Vanstone10, Ch. VII] for the general definition of fibre integrals). In our case, the components $H^{\overline {l} k}$ of the metric tensor on the direct image are defined by such fibre integrals, where $\eta $ is given by the inner product/wedge product of the sections $\psi ^k$:

$$ \begin{align*} \eta \,(s) = \psi^k|_{X^{\prime}_s} \cdot \psi^{\overline{l}}|_{X^{\prime}_s} \;dV = \mathrm{i}^{n^2} \; (\psi^k|_{X^{\prime}_s} \wedge \psi^{\overline{l}}|_{X^{\prime}_s})_h. \end{align*} $$

We want to show that these fibre integrals give smooth functions on the base so that we indeed get a smooth hermitian metric on the direct image we consider. Thus, if $s^1,\ldots ,s^m$ are local holomorphic coordinates on the base, we need to compute the derivatives

$$ \begin{align*} \frac{ \partial}{ \partial s^k}\int_{X_s}\eta \quad \mbox{for }1\leq i \leq r \quad\mbox{ and } \quad \frac{ \partial}{ \partial s^{\overline{l}}}\int_{X_s}\eta, \quad \mbox{for } 1\leq l \leq r. \end{align*} $$

This can be done by using Lie derivatives:

Lemma 3.10. For $1\leq k \leq m$, let $v_k$ be the horizontal lift of $ \partial / \partial s^k$. We write $ \partial / \partial s^{\overline {l}}$ for $ \partial / \partial \overline {s^l}$ and $v_{\overline {l}}$ for $\overline {v_l}$. Then

$$ \begin{align*}\frac{ \partial}{ \partial s^k}\int_{X^{\prime}_s}\eta = \int_{X^{\prime}_s}L_{v_k}(\eta) \quad \mbox{and} \quad \frac{ \partial}{ \partial s^{\overline{l}}}\int_{X^{\prime}_s}\eta = \int_{X^{\prime}_s}L_{v_{\overline{l}}}(\eta), \end{align*} $$

where $L_{v_k}$ and $L_{v_{\overline {l}}}$ denotes the Lie derivative in the direction of $v_k$ and $v_{\overline {l}}$, respectively.

Proof. The statement is well-known when the fibres are compact [Reference Naumann17, Lemma 1]. We only have to show that $L_{v_k}(\eta )$ and $L_{v_{\overline {l}}}(\eta )$ are square integrable. Then the statement follows from the dominated convergence theorem. We only present it for the first. We are using local holomorphic coordinates $z^1,\ldots ,z^n$ near a point $p \in D_s$ such that $D_s=\{z^1\cdots z^k=0\}$. Because $\psi ^k,\psi ^l \in H^0(\mathscr {X},\Omega ^n(\log \mathcal {D})_{\mathscr {X}/S}(\mathscr {L}))$ and due to the assumption on the metric h, we have that

$$ \begin{align*} \eta_{1\ldots n \overline{1} \ldots \overline{n}} = O\left(\prod_{i=1}^k{\log^2(|z^i|^2)}\right), \end{align*} $$

where

$$ \begin{align*} \eta = \eta_{1\ldots n \overline{1} \ldots \overline{n}} \; dz^1 \wedge \ldots \wedge dz^n \wedge dz^{\overline{1}} \wedge \dots \wedge dz^{\overline{n}}. \end{align*} $$

Now $v_k = \partial / \partial _k + a_k^{\alpha }\, \partial / \partial z^{\alpha }$ so that

$$ \begin{align*} L_{v_k}\eta = \left(\eta_{1\ldots n, \overline{1}, \dots \overline{n};k} + \sum_{\alpha=1}^na_k^{\alpha}\eta_{1\ldots n \overline{1} \ldots \overline{n};\alpha} + \sum_{\alpha=1}^n{a_{k;\alpha}^{\alpha} \eta_{1\ldots n \overline{1}\dots \overline{n}}} \right) \;dz^1\wedge \ldots \wedge dz^n \wedge dz^{\overline{1}}\wedge \ldots \wedge dz^{\overline{n}}. \end{align*} $$

Here, $;$ denotes the covariant derivative. Now, because of

$$ \begin{align*} a_k^{\alpha} = O(|z^{\alpha}| \log |z^{\alpha}|) \quad \mbox{ for } \quad 1\leq \alpha \leq k, \end{align*} $$

we see that $L_{v_k}(\eta )$ is indeed integrable.

We see that we can iterate this process so that the fibre integral gives a smooth function on S. But this means the $L^2$-metric is indeed a smooth metric on $f_*{(\Omega ^n(\log \mathcal {D})_{\mathscr {X}/S}(\mathscr {L})})$. We note here that the square integrability of $L_v(\eta )$ will also follow from Lemma 6.2.

Before we go to the computation of the curvature, which is the most technical part of the article, let us consider the two applications.

6.1 Setup

By polarisation, it is sufficient to treat the case where $\dim S=1$ for the computation of the curvature, which simplifies the notation. Therefore, we set $s=s^1, v_s=v_1$, and so on. We write $s,\overline {s}$ for the indices $1,\overline {1}$ so that

$$ \begin{align*} v_s = \partial_s + a_{s}^{\alpha} \partial_{\alpha} \end{align*} $$

and

$$ \begin{align*} A_s = A_{s\overline{\beta}}^{\alpha} \partial_{\alpha}dz^{\overline{\beta}}. \end{align*} $$

We assume local freeness of the sheaf $f_*(\Omega ^n(\log \mathscr {D}))_{\mathscr {X}/S}(\mathscr {L})$. According to Corollary 4, we can represent local sections of this sheaf by holomorphic sections of $(\Omega ^n(\log \mathcal {D}))_{\mathscr {X}'/S}(\mathscr {L}|_{\mathscr {X}'})$, which restrict to holomorphic and square integrable $(n,0)$-forms on the open fibres $X^{\prime }_s$. We denote such a section by $\psi $. In local coordinates, we have

$$ \begin{align*} \psi|_{X^{\prime}_s} &= \psi_{\alpha_1 \ldots \alpha_n}dz^{\alpha_1}\wedge \ldots dz^{\alpha_n}\\ &= \psi_{A_n}dz^{A_n}, \end{align*} $$

where $A_n=(\alpha _1,\ldots ,\alpha _n)$. The $\bar \partial $-closedness of $\psi $ means

(6.1)$$ \begin{align} \psi_{A_n;s} = 0 \quad \mbox{and} \quad \psi_{A_n;\overline{\beta}} = 0 \quad \mbox{for all} \quad A_n, 1\leq \beta \leq n. \end{align} $$

6.2 Cup product

Definition 4. Let $s \in S$ and $A=A_{s\overline {\beta }}^{\alpha }(z,s) \partial _{\alpha }dz^{\overline {\beta }}$ be the Kodaira-Spencer form on the fibre $X^{\prime }_s$. The wedge product, together with the contraction, defines a map

$$ \begin{align*} A_{i\overline{\beta}}^{\alpha} \partial_{\alpha}dz^{\overline{\beta}}\cup \; : H^0(X_s,\Omega^n_{X_s}(\log D_s)(L|_{X_s})) \to A^{0,1}_{(2)}(X^{\prime}_s,\Omega^{n-1}_{X^{\prime}_s}(L|_{X^{\prime}_s})), \end{align*} $$

which can be described locally by

$$ \begin{align*} &\quad\left( A_{i\overline{\delta}}^{\gamma} \partial_{\gamma}dz^{\overline{\delta}}\right) \cup \left(\psi_{\alpha_1 \ldots \alpha_n}\;dz^{\alpha_1}\wedge \ldots \wedge dz^{\alpha_n} \right)\\ &= A_{i\overline{\beta}}^{\gamma} \psi_{\gamma\alpha_1\ldots \alpha_{n-1}}\; dz^{\overline{\beta}}\wedge dz^{\alpha_1} \wedge \ldots \wedge dz^{\alpha_{n-1}}. \end{align*} $$

The fact that $A_i \cup \psi $ is indeed square integrable will be proved in Lemma 6.2.

6.3 Lie derivatives

Now we choose a local frame $\{\psi ^1,\ldots ,\psi ^r\}$ according to Corollary 4. The components of the metric tensor $H^{\overline {l} k}$ for $f_*(\Omega ^n(\log \mathscr {D}))_{\mathscr {X}/S}(\mathscr {L})$ on the base space S are given by

$$ \begin{align*} H^{\overline{l} k}(s):={\langle}\psi^k,\psi^l{\rangle} :={\langle}\psi^k|_{X^{\prime}_s},\psi^l|_{X^{\prime}_s}{\rangle} = \int_{X^{\prime}_s}{\psi^k_{A_n}\psi^{\overline{l}}_{\overline{B}_n}g^{\overline{B}_nA_n} h|_{X^{\prime}_s}\,dV}. \end{align*} $$

We also write

$$ \begin{align*} \psi^k \cdot \psi^{\overline{l}}= \psi^k_{A_n}\psi^{\overline{l}}_{\overline{B}_n}g^{\overline{B}_nA_n} h|_{X^{\prime}_s} \end{align*} $$

for the pointwise inner product of $L|_{X^{\prime }_s}$-valued $(n,0)$-forms. Here and in the following, we write g for the hermitian metric associated to the complete Kähler from $\omega _s$. When we compute derivatives with respect to the base of these fibre integrals, we apply Lie derivatives with respect to the horizontal lifts of the tangent vectors according to Lemma 3.10. This considerably simplifies the computation. To break up the Lie derivative of the pointwise inner product (which is a relative $(n,n)$-form), we need to introduce Lie derivatives of relative differential forms with values in a line bundle. This can be done by using the hermitian connection $\nabla $ on $\Lambda ^{n,0}T^*_{\mathscr {X}'/S}\otimes \mathscr {L}|_{\mathscr {X}'}$ induced by the Chern connections on $(T_{X^{\prime }_s},\omega _{X_s})$ and $(L_s,h_s)$. We define the Lie derivative of $\psi $ with respect to the horizontal lift v by using Cartan’s formula

(6.2)$$ \begin{align} L_v\psi := L_v (\psi_{\mathscr{X}'/S}) := \left(\delta_v\circ \nabla + \nabla \circ \delta_v\right)\psi \end{align} $$

and similar for the Lie derivative with respect to $\overline {v}$.

Taking Lie derivatives is not type-preserving. We have the type decomposition for $\psi =\psi ^k$ or $\psi =\psi ^l$ and $v=v_s$

$$ \begin{align*} L_v\psi = L_v\psi' + L_v\psi", \end{align*} $$

where $L_v\psi '$ is of type $(n,0)$ and $L_v\psi "$ is of type $(n-1,1)$. In local coordinates, we have

(6.3)

(6.4)

One justification for using Lie derivatives is given by the following lemma, which allows us to express some components of the Lie derivatives as cup products with the Kodaira-Spencer form:

Lemma 6.1. We have

(6.5)$$ \begin{align} L_v\psi" = A_s \cup \psi, \end{align} $$

and it is primitive on the fibres.

Proof. First, we note

To prove that $A_s \cup \psi $ is primitive, we have to show that $\Lambda _s (A_s\cup \psi )=0$, where $\Lambda _s$ is the dual Lefschetz operator with respect to the Kähler form

$$ \begin{align*} \omega_s = \sqrt{-1}g_{\alpha\overline{\beta}}\;dz^{\alpha}\wedge dz^{\overline{\beta}}. \end{align*} $$

We have

$$ \begin{align*} (\Lambda_s (A_s \cup \psi))_{\alpha_2 \ldots \alpha_{n-1}} = g^{\overline{\beta}_n \alpha_1} A^{\alpha}_{s\overline{\beta}_n}\psi_{\alpha\,\alpha_1\ldots\alpha_{n-1}} = A^{\alpha\alpha_1}_s\psi_{\alpha\,\alpha_1\ldots\alpha_{n-1}}. \end{align*} $$

But now, because $A^{\alpha \alpha _1}_s=A^{\alpha _1\alpha }_s$ by Lemma 3.9 and $\psi _{\alpha \,\alpha _1\ldots \alpha _{n-1}}$ is skew-symmetric, we get that

$$ \begin{align*} (\Lambda_s (A_s \cup \psi))_{\alpha_2 \ldots \alpha_{n-1}} dz^{\alpha_2} \wedge \ldots \wedge dz^{\alpha_{n-1}} =0.\\[-36pt] \end{align*} $$

Similarly, we have a type decomposition for the Lie derivative along $\overline {v} = v_{\overline {s}}$

$$ \begin{align*} L_{\overline{v}}\psi = L_{\overline{v}}\psi' + L_{\overline{v}}\psi", \end{align*} $$

where $L_{\overline {v}}\psi '$ is of type $(n,0)$ and $L_{\overline {v}}\psi "$ is of type $(n+1,n-1)$ and hence vanishes by degree reasons. In local coordinates, this is

(6.6)

From this, we infer that $L_{\overline {v}}\psi =L_{\overline {v}}\psi '=0$ because $\psi $ is holomorphic. The type decomposition can be verified using the definition given by equation (6.2). We refer the reader to [Reference Naumann17] for verification.

Proof. We use local coordinates $z^1,\ldots ,z^n$ in a neighbourhood U of a point $p \in D_s \subset X_s$, where $D_s \cap U = \{z^1\cdots z^k=0\}$. We now have

$$ \begin{align*}\psi_{A_n} = O\left(\frac{1}{|z^1\cdots z^k|}\right) \end{align*} $$

and

$$ \begin{align*} a_s^{\alpha} = O(|z^{\alpha}| \log |z^{\alpha}|) \quad \mbox{for} \quad 1 \leq \alpha \leq k \quad \mbox{else} \quad a_s^{\alpha} = O(1). \end{align*} $$

From the local expression (6.3), we thus see that

$$ \begin{align*} (L_v\psi)^{\prime}_{A_n} = O\left(\frac{1}{|z^1\cdots z^k|}\right), \end{align*} $$

so it is again square integrable.

To prove that $L_v\psi "$ is square integrable is more complicated. We first look at the order of $A^{\alpha }_{s\overline {\beta }}$:

$$ \begin{align*} A^{\alpha}_{\overline{\beta}} &= O(1) \quad \mbox{for} \quad 1\leq \alpha=\beta \leq k \quad \mbox{or } \quad k+1 \leq \alpha,\beta \leq n.\\ A^{\alpha}_{\overline{\beta}} &= O\left(\frac{1}{|z^{\beta}|\log |z^{\beta}|}\right) \quad \mbox{for} \quad \alpha>k \mbox{ and } \beta\leq k. \\ A^{\alpha}_{\overline{\beta}} &= O\left(|z^{\alpha}|\log |z^{\alpha}|\right) \quad \mbox{for} \quad \alpha \leq k \mbox{ and } \beta > k. \\ A^{\alpha}_{\overline{\beta}} &= O\left(\frac{|z^{\alpha}|\log |z^{\alpha}|}{|z^{\beta}|\log |z^{\beta}|}\right) \quad \mbox{for} \quad 1 \leq \alpha\neq \beta \leq k. \end{align*} $$

To prove that $L_v\psi "=A_s \cup \psi $ is $L^2$-integrable means to verify that

$$ \begin{align*} \int_{X^{\prime}_s}{(A_s \cup \psi) \wedge \overline{(A_s \cup\psi)}} \end{align*} $$

is finite because the form is primitive by Lemma 6.1. For this we first note that the sum in the expression of $(A_s \cup \psi )_{\overline {\beta }_n\alpha _1\ldots \alpha _{n-1}}$ reduces to

$$ \begin{align*} A^{\alpha_n}_{s\;\overline{\beta}_n}\psi_{\alpha_n\alpha_1\ldots\alpha_{n-1}}. \end{align*} $$

The only really critical term in $(A_s \cup \psi )_{\overline {\beta }_n\alpha _1\ldots \alpha _{n-1}}$ occurs if $\alpha _n \in \{k+1,\dots ,n\}$, $\beta _n \in \{1,\ldots ,k\}$ and $\beta _n$ is among the $\alpha _1,\ldots , \alpha _{n-1}$, because in this case the order in $z^{\beta _n}$ is

$$ \begin{align*} \frac{1}{|z^{\beta_n}|^2 \log |z^{\beta_n}|}. \end{align*} $$

But then, in the above integral, this term can only be paired with another term that contains neither $dz^{\beta _n}$ nor $dz^{\overline {\beta }_n}$. So we see that the product $(A_s \cup \psi ) \wedge \overline {(A_s \cup \psi )}$ remains integrable.

We need the following lemma:

Lemma 6.3. The Lie derivative of the volume element $dV=\omega _s^n/n!$ along the horizontal lift v vanishes: that is,

$$ \begin{align*} L_v( dV)=0. \end{align*} $$

Proof. It suffices to show that the $(1,1)$ component of $L_v(g_{\alpha \overline {\beta }})$ vanishes, which implies $L_v(\det (g_{\alpha \overline {\beta }}))=0.$ We have

$$ \begin{align*} L_v(g_{\alpha\overline{\beta}})_{\alpha\overline{\beta}} = g_{\alpha\overline{\beta},s} + a_{s}^{\gamma}g_{\alpha\overline{\beta};\gamma} + a_{s;\alpha}^{\gamma}g_{\gamma\overline{\beta}} = -a_{s\overline{\beta};\alpha} + a_{s;\alpha}^{\gamma}g_{\gamma\overline{\beta}}=0.\\[-36pt] \end{align*} $$

6.4 Main part of the computation

We start computing the curvature by computing the first-order variation. Using Lie derivatives, the pointwise inner products can be broken up:

Proposition 8.

$$ \begin{align*} \frac{ \partial}{ \partial s} {\langle}\psi^k,\psi^l{\rangle} = {\langle}L_v\psi^k,\psi^l{\rangle}, \end{align*} $$

where $ \partial / \partial s$ denotes a tangent vector on the base S and v its horizontal lift and analogous for $ \partial / \partial \overline {s}$.

Proof. We first apply Lemma 3.10 and get that

$$ \begin{align*} \frac{ \partial}{ \partial s} {\langle}\psi^k,\psi^l{\rangle} (s) = \int_{X^{\prime}_s}{L_v(\psi^k \cdot \psi^{\overline{l}}\; dV)} = \int_{X^{\prime}_s}{L_v(\psi^k \cdot \psi^{\overline{l}})\; dV} \end{align*} $$

by Lemma 6.3. Now it follows by a direct computation (see [Reference Naumann17, Prop.1]) that

$$ \begin{align*} L_v(\psi^k \cdot \psi^{\overline{l}}) = L_v\psi^k \cdot \psi^{\overline{l}} + \psi^k \cdot L_v\psi^{\overline{l}} \end{align*} $$

so that

$$ \begin{align*} \frac{ \partial}{ \partial s} {\langle}\psi^k,\psi^l{\rangle} = {\langle}L_v\psi^k,\psi^l{\rangle} + {\langle}\psi^k,L_{\overline{v}}\psi^l{\rangle} = {\langle}L_v\psi^k,\psi^l{\rangle} \end{align*} $$

because $L_{\overline {v}}\psi ^l=0$.

The above proposition is the primary reason for the use of Lie derivatives. For later computations, we need to compare Laplacians:

Lemma 6.4. We have the following relation on the space $A^{p,q}_{(2)}(X^{\prime }_s,L_s)$:

(6.7)$$ \begin{align} \Box_{\partial} - \Box_{\bar \partial} = (n-p-q)\cdot \operatorname{id}. \end{align} $$

In particular, the harmonic forms $\psi \in A^{n,0}_{(2)}(X^{\prime }_s,L_s)$ are also harmonic with respect to $ \partial $, which is the $(1,0)$- part of the hermitian connection on $A^{n,0}_{(2)}(X^{\prime }_s,L_s)$.

Proof. The Bochner-Kodaira-Nakano identity says (on the fibre $X^{\prime }_s$)

$$ \begin{align*} \Box_{\bar \partial} - \Box_{\partial} = \left[\sqrt{-1}\Theta(L_s),\Lambda\right]. \end{align*} $$

But by definition, we have $\omega _{X_s}=\sqrt {-1}\Theta (L_s)$. Furthermore, it holds (see [Reference Demailly7, Cor.VI.5.9])

$$ \begin{align*} \left[L_{\omega},\Lambda_{\omega}\right]u = (p+q-n)\,u \quad \mbox{for } u \in \mathscr{A}^{p,q}(X^{\prime}_s,L_s).\\[-36pt] \end{align*} $$

Next, we start to compute the second-order derivative of $H^{\overline {l} k}$ and begin with

$$ \begin{align*} \frac{ \partial}{ \partial s} H^{\overline{l} k} = {\langle}L_v \psi^k,\psi^l{\rangle}. \end{align*} $$

We obtain

$$ \begin{align*} \partial_{\overline{s}} \partial_s {\langle}\psi^k,\psi^l{\rangle} &= {\langle}L_{\overline{v}}L_v\psi^k,\psi^l{\rangle} + {\langle}L_v\psi^k,L_v\psi^l{\rangle}\\ &= {\langle}(L_{[\overline{v},v]} + \Theta(L|_{\mathscr{X}'})_{\overline{v} v})\psi^k, \psi^l{\rangle} + {\langle}L_vL_{\overline{v}}\psi^k,\psi^l{\rangle} + {\langle}L_v\psi^k,L_v\psi^l{\rangle}\\ &= {\langle}(L_{[\overline{v},v]} + \Theta(L|_{\mathscr{X}'})_{\overline{v} v})\psi^k, \psi^l{\rangle} + \partial_s{\langle}L_{\overline{v}}\psi^k,\psi^l{\rangle} - {\langle}L_{\overline{v}}\psi^k,L_{\overline{v}}\psi^l{\rangle} + {\langle}L_v\psi^k,L_v\psi^l{\rangle}. \end{align*} $$

Because of $L_{\overline {v}}\psi ^k \equiv 0$, as we just saw, we get

(6.8)$$ \begin{align} \partial_{\overline{s}} \partial_s {\langle}\psi^k,\psi^l{\rangle} = {\langle}(L_{[\overline{v},v]} + \Theta(L|_{\mathscr{X}'})_{\overline{v} v})\psi^k,\psi^l{\rangle} + {\langle}L_v\psi^k,L_v\psi^l{\rangle}. \end{align} $$

We will see below that the smooth $(n,0)$-form $(L_{[\overline {v},v]} + \Theta (L|_{\mathscr {X}'})_{\overline {v} v})\psi ^k$ is indeed square integrable, which justifies that $L_{\overline {v}}L_v \psi ^k$ is square integrable, too.

Now we treat each term on the right-hand side of equation (6.8) separately. For the first summand, we have

Lemma 6.5.

(6.9)$$ \begin{align} L_{[\overline{v},v]} + \Theta(L_{\mathscr{X}'})_{\overline{v} v}= [-\varphi^{;\alpha} \partial_{\alpha} + \varphi^{;\overline{\beta}} \partial_{\overline{\beta}},\rule{0.3cm}{0.4pt}] - \varphi \cdot \operatorname{id}, \end{align} $$

where the bracket $[w,\rule {0.3cm}{0.4pt}]$ stands for a Lie derivative along the vector field w.

Proof. We first compute the vector field $[\overline {v},v]$:

$$ \begin{align*} [\overline{v},v] &= [ \partial_{\overline{s}} + a_{\overline{s}}^{\overline{\beta}} \partial_{\overline{s}}, \partial_s + a_s^{\alpha} \partial_{\alpha}] \\ &= \left( \partial_{\overline{s}}(a_s^{\alpha}) + a_{\overline{s}}^{\overline{\beta}}a_{a|\overline{\beta}}^{\alpha} \right) \partial_{\alpha} - \left( \partial_s(a_{\overline{s}}^{\overline{\beta}}) + a_s^{\alpha}a_{\overline{s}|\alpha}^{\overline{\beta}} \right) \partial_{\overline{\beta}}. \end{align*} $$

Now we have

$$ \begin{align*} \partial_{\overline{s}}(a_s^{\alpha}) &= - \partial_{\overline{s}}(g^{\overline{\beta}\alpha}g_{s\overline{\beta}}) = g^{\overline{\beta}\sigma}g_{\sigma\overline{s}|\overline{\tau}}g^{\overline{\tau}\alpha}g_{s\overline{\beta}} - g^{\overline{\beta}\alpha}g_{s\overline{\beta}|s}\\ &= g^{\overline{\beta}\sigma}a_{\overline{s}\sigma;\overline{\tau}}g^{\overline{\tau}\alpha}a_{s\overline{\beta}} - g^{\overline{\beta}\alpha}g_{s\overline{s};\overline{\beta}}. \end{align*} $$

Because of $\varphi = g_{s\overline {s}} - g_{\alpha \overline {s}}g_{s\overline {\beta }}g^{\overline {\beta }\alpha }$, the coefficient of $ \partial _{\alpha }$ is $g^{\overline {\beta }\alpha }\varphi _{;\overline {\beta }}=\varphi ^{;\alpha }$. In the same way, we get the coefficient of $ \partial _{\overline {\beta }}$. Next, we need to compute the contribution of the connection on $L|_{\mathscr {X}'}$. Because of $\sqrt {-1}[ \partial ,\bar \partial ]=\sqrt {-1}\Theta (L)|_{\mathscr {X}'}=\omega _{\mathscr {X}'}$, we have

$$ \begin{align*} \Theta(L|_{\mathscr{X}'})_{\overline{v} v} &= -\Theta(L|_{\mathscr{X}'})_{v \overline{v}}\\ &= -\left(g_{s\overline{s}} + a_{\overline{s}}^{\overline{\beta}}g_{s\overline{\beta}} + a_s^{\alpha} g_{\alpha \overline{s}} + a_{\overline{s}}^{\overline{\beta}}a_s^{\alpha}g_{\alpha\overline{\beta}}\right)\\ &= -\varphi.\\[-36pt] \end{align*} $$

Lemma 6.6.

(6.10)$$ \begin{align} {\langle}(L_{[\overline{v},v]}+ \Theta(L|_{\mathscr{X}'})_{\overline{v} v})\psi^k,\psi^l{\rangle} = -{\langle}\varphi \cdot\psi^k,\psi^l{\rangle} = -\int_{X^{\prime}_s}{\varphi\cdot(\psi^k\cdot\psi^{\overline{l}})\,dV}. \end{align} $$

Proof. The $ \partial $-closedness of $\psi ^k$ means

Thus

It is clear that $(\varphi ^{;\alpha } \partial _{\alpha }\cup \psi ^k)$ is square integrable, because $\varphi |_{X^{\prime }_s}$ lies in $\mathscr {C}^{k,\alpha }(X^{\prime }_s)$. Moreover, it guarantees that this form lies in the domain of $ \partial $. This leads to

$$ \begin{align*} {\langle}[\varphi^{;\alpha} \partial_{\alpha},\psi^k_{A_n}],\psi^l{\rangle} &= {\langle}[\varphi^{;\alpha} \partial_{\alpha},\psi^k_{A_n}]',\psi^l{\rangle}\\ &= {\langle} \partial\left(\varphi^{;\alpha} \partial_{\alpha}\cup \psi^k\right),\psi^l{\rangle} = {\langle}\varphi^{;\alpha} \partial_{\alpha}\cup \psi^k, \partial^*\psi^l{\rangle} =0. \end{align*} $$

Note that by Gaffney’s theorem, Proposition 5, the formal adjoint of $ \partial $ is equal to the adjoint operator.

In the same way, we get

$$ \begin{align*} {\langle}[\varphi^{;\overline{\beta}} \partial_{\overline{\beta}},\psi^k_{A_n}],\psi^l{\rangle} =0.\\[-36pt] \end{align*} $$

The following proposition contains important identities that allow us to obtain an intrinsic expression for the curvature:

Proposition 9.

(6.11)$$ \begin{align} \bar \partial(L_v\psi^k)'&= \partial(A_s\cup\psi^k), \end{align} $$

(6.12)$$ \begin{align} \bar \partial^*(L_v\psi^k)'&= 0, \end{align} $$

(6.13)$$ \begin{align} \partial^*(A_s \cup \psi^k)&= 0. \end{align} $$

We note that here the operators $ \partial ,\bar \partial , \partial ^*$ and $\bar \partial ^*$ mean the fibrewise operators, because we are always dealing with relative forms. For a proof, we refer to [Reference Naumann17, Appendix A]. We see from the proof of Lemma 6.2 that $\bar \partial (L_v \psi ^k)'$ is again square integrable.

Now we look at the second term in equation (6.8) and decompose it into its two types

$$ \begin{align*} {\langle}L_v\psi^k,L_v\psi^l{\rangle} &= {\langle}(L_v\psi^k)',(L_v\psi^l)'{\rangle} - {\langle}(L_v\psi^k)",(L_v\psi^l)"{\rangle}\\ &= {\langle}(L_v\psi^k)',(L_v\psi^l)'{\rangle} - {\langle}A_s \cup \psi^k, A_s \cup \psi^l{\rangle} \end{align*} $$

because of equation (6.5).

Now let $G_{ \partial }$ and $G_{\bar \partial }$ be the Green operators on the spaces $A^{p,q}_{(2)}(X^{\prime }_s,L|_{X^{\prime }_s})$ with respect to $\Box _{\partial }$ and $\Box _{\bar \partial }$, respectively. According to Lemma 6.4, they coincide for $p+q=n$. We use normal coordinates (of the second kind) at a given point $s_0 \in S$. The condition $( \partial / \partial s)H^{\overline {l} k}|_{s_0}=0$ for all $k,l$ means for $s=s_0$ the harmonic projection

$$ \begin{align*} H((L_v\psi^k)')=0 \end{align*} $$

vanishes for all k. Thus, using the identity $\operatorname {id} = H + G_{\bar \partial }\Box _{\bar \partial }$, we can write

$$ \begin{align*} (L_v\psi^k)'=G_{\bar \partial}\Box_{\bar \partial}(L_v\psi^k)' = G_{\bar \partial}\bar \partial^*\bar \partial(L_v\psi^k)' = \bar \partial^*G_{\bar \partial} \partial(A_s\cup \psi^k) \end{align*} $$

by equations (6.12) and (6.11). Because the form $\bar \partial (L_v\psi ^k)'= \partial (A_s \cup \psi ^k)$ is of type $(n,1)$, we have $G_{\bar \partial }=(\Box _{\partial } + 1)^{-1}$ on such forms by Lemma 6.4. We proceed by

$$ \begin{align*} {\langle}(L_v\psi^k)',(L_v\psi^l)'{\rangle} &= {\langle}\bar \partial^*G_{\bar \partial} \partial(A_s \cup \psi^k),(L_v\psi^l)'{\rangle}\\ &= {\langle}G_{\bar \partial} \partial(A_s \cup \psi^k), \partial(A_s \cup \psi^l){\rangle}\\ &= {\langle}(\Box_{\partial} + 1)^{-1} \partial(A_s \cup \psi^k), \partial(A_s \cup \psi^l){\rangle}\\ &= {\langle} \partial^*(\Box_{\partial} + 1)^{-1} \partial(A_s \cup \psi^k),A_s \cup \psi^l{\rangle}. \end{align*} $$

Again, we used Gaffney’s theorem. Now using equation (6.13) gives

$$ \begin{align*} {\langle}(L_v\psi^k)',(L_v\psi^l)'{\rangle} &= {\langle}(\Box_{\partial} + 1)^{-1}\Box_{\partial}(A_s \cup \psi^k),A_s \cup \psi^l{\rangle}\\ &= {\langle}(\Box_{\partial} + 1)^{-1}(\Box_{\partial} + 1 - 1) (A_s \cup \psi^k),A_s \cup \psi^l{\rangle}\\ &= {\langle}A_s \cup \psi^k,A_s \cup \psi^l{\rangle} - {\langle}(\Box_{\partial} + 1)^{-1}(A_s \cup \psi^k),A_s \cup \psi^l{\rangle}. \end{align*} $$

Altogether, we have

Lemma 6.7.

(6.14)$$ \begin{align} {\langle}L_v\psi^k,L_v\psi^l{\rangle} = - \int_{X^{\prime}_s}{(\Box + 1)^{-1}(A_s \cup \psi^k)\cdot (A_{\overline{s}} \cup \psi^{\overline{l}})\, g\, dV}. \end{align} $$

(We write $\Box =\Box _{\partial }=\Box _{\bar \partial }$ when applied to $(n-1,1)$-forms.)

Now our main result Theorem 2.1 follows from equations (6.8), (6.10), (6.14) and the fact that $R^{\overline {l} k}_{i\overline {\jmath }}(s_0)=- \partial _{\overline {\jmath }} \partial _i H^{\overline {l} k}(s_0)$ in normal coordinates at a point $s_0 \in S$.