# Berndtsson-Lempert-Szőke Fields
associated to
Proper Holomorphic Families of Vector Bundles

Dror Varolin

###### Abstract.

Drawing on work of Berndtsson and of Lempert and Szőke, we define a complex
analytic structure for families of (possibly finite-dimensional) Hilbert
spaces that might not fit together to form a holomorphic vector bundle but
nevertheless have a reasonable definition of curvature that agrees with the
curvature of the Chern connection when the family of Hilbert spaces is locally
trivial. As one consequence, we obtain a new proof of a celebrated theorem of
Berndtsson on the curvature of direct images of semi-positively twisted
relative canonical bundles (i.e., adjoint bundles), and of its higher-rank
generalization by Liu-Yang.

Mathematics Subject Classification (2020 Database): 32L05

Dedicated to Bo Berndtsson and Laszlo Lempert

##  1\. Introduction

Let $p:X\to B$ be a proper holomorphic submersion of a complex manifold $X$ of
dimension $m+n$ onto a complex manifold $B$ of dimension $n$, and let $E\to X$
be a holomorphic vector bundle with smooth Hermitian metric $h$. We study the
family of Hilbert spaces ${\mathscr{H}}\to B$ whose fiber over $t\in B$ is

${\mathscr{H}}_{t}:=\left\\{f\in
H^{0}(X_{t},{\mathcal{O}}_{X_{t}}(K_{X_{t}}\otimes E|_{X_{t}}))\right\\},$

where $X_{t}:=p^{-1}(t)$, and whose Hermitian metric on the fiber
${\mathscr{H}}_{t}$ is

(1)
$(f_{1},f_{2})_{t}:=\int_{X_{t}}\sqrt{\text{\rm-1}}^{n^{2}}\left<f_{1}\wedge\bar{f}_{2},h\right>.$

The study of such families of Hilbert spaces was initiated in [B-2009] by
Berndtsson, who considered the case in which $E$ is a line bundle. Berndtsson
gave such families of Hilbert spaces a complex geometric structure by defining
notions of smooth and holomorphic sections of ${\mathscr{H}}\to B$, and he
noted that in general such structures do not yield vector bundles. (Some
natural examples are discussed in Paragraph 5.3.) However, even when not
locally trivial, ${\mathscr{H}}\to B$ resembles a holomorphic Hermitian vector
bundle in certain ways.

We propose a definition of the curvature endomorphism of ${\mathscr{H}}\to B$
(Definition 4.19). Our definition agrees with the usual definition if
${\mathscr{H}}\to B$ is locally trivial. In terms of this curvature, our first
result is the following theorem.

######  Theorem 1.

Let $(E,h)\to X\stackrel{{\scriptstyle p}}{{\to}}B$ be a proper Kähler family
of Hermitian holomorphic vector bundles and let $k\geq 1$ be an integer. If
the metric $h$ is $k$-non-negatively curved (resp. $k$-positively curved) in
the sense of Demailly then ${\mathscr{H}}\to B$ is $k$-non-negatively curved
(resp. $k$-positively curved) in the sense of Demailly.

Metrics are $1$-positively curved if and only if they are positively curved in
the sense of Griffiths, while metrics are $k$-positively curved for every $k$
if and only if they are positively curved in the sense of Nakano. Notions of
positivity are recalled in Paragraph 2.12.

We do not know if the hypotheses of Theorem 1 imply that ${\mathscr{H}}\to B$
is a holomorphic vector bundle. At present the main tool for local triviality
is the $L^{2}$ Extension package of Ohsawa-Takegoshi type. (See Paragraph
5.1.) If $E\to X$ has a metric with non-negative curvature in the sense of
Nakano then ${\mathscr{H}}\to B$ has the structure of a holomorphic vector
bundle. Under the hypotheses of the next two corollaries, one has such
metrics.

The first corollary of Theorem 1 is the following celebrated theorem of
Berndtsson.

######  Corollary 1.1 ([B-2009, Theorem 1.2]).

Let $(E,e^{-\varphi})\to X\stackrel{{\scriptstyle p}}{{\to}}B$ be a proper
Kähler family of holomorphic Hermitian line bundles. If the curvature of
$(E,e^{-\varphi})\to X$ is non-negative (resp. positive) then
${\mathscr{H}}\to B$ is a holomorphic vector bundle and the curvature of the
Chern connection of the $L^{2}$ metric (1) is non-negative (resp. positive) in
the sense of Nakano.

The next corollary is a direct consequence of Theorem 1, and local triviality
holds because Nakano positivity of the metric $h$ on fibers implies that for a
sufficiently plurisubharmonic function $\psi$ locally on the base $B$ the
metric $e^{-p^{*}\psi}h$ for $E\to X$ has Nakano-positive curvature.

######  Corollary 1.2.

Let $(E,h)\to X\stackrel{{\scriptstyle p}}{{\to}}B$ be a proper Kähler family
of holomorphic Hermitian vector bundles and let $k\geq 1$ be an integer.
Assume that for all $t\in B$ the metrics $h_{t}=h|_{X_{t}}$ are Nakano-
positive, and that the metric $h$ is $k$-non-negatively curved (resp.
$k$-positively curved) in the sense of Demailly. Then the holomorphic vector
bundle ${\mathscr{H}}\to B$ is $k$-non-negatively curved (resp. $k$-positively
curved) in the sense of Demailly.

Theorem 1 can be strengthened. Indeed, there is always a non-negative
contribution to the curvature of ${\mathscr{H}}\to B$ from the deformation-
theoretic properties of the proper holomorphic family $p:X\to B$, and this
contribution can be positive when the family is not locally trivial. That this
is so was already observed by Berndtsson in [B-2011], and later by Liu and
Yang in [LY-2014], who were the first to consider the case of higher rank $E$.
In all cases the extra positivity is a consequence of formulas for the
curvature of ${\mathscr{H}}\to B$. (Liu and Yang also obtained a number of
other equivalent curvature formulas and discussed several interesting
applications of these formulas.)

In [B-2011] and [LY-2014] the curvature formula is obtained under the
assumption of local triviality of ${\mathscr{H}}\to B$, which makes the
computation more manageable. In the non-locally trivial case the derivation of
the curvature formula is necessarily more complicated. To make its meaning
maximally clear, the curvature formula is obtained in a pair of theorems. To
state the first curvature theorem, we need some notation.

  1. i.

A _horizontal distribution_ for $p:X\to B$ is a (not necessarily integrable)
subbundle $\theta\subset T_{X}$ such that $dp|_{\theta}:\theta\to T_{B}$ is a
vector bundle isomorphism. Such distributions $\theta$ are in one-to-one
correspondence with _horizontal lifts_ , i.e., linear maps that assign to a
$(1,0)$-vector $\tau\in T^{1,0}_{B,t}$ a $(1,0)$-vector field
$\xi^{\theta}_{\tau}\in\Gamma(X_{t},T^{1,0}_{X}|_{X_{t}})$ such that
$dp\xi^{\theta}_{\tau}=\tau$. (See Paragraph 3.3.)

  2. ii.

Let $t\in B$ be fixed. For $\tau\in T^{1,0}_{B,t}$ we define

$\kappa^{\theta}_{\tau}:{\mathscr{H}}_{t}\to
H^{0}(X_{t},{\mathscr{C}}^{\infty}(\Lambda^{n-1,1}_{X_{t}}\otimes
E|_{X_{t}}))\quad\text{and}\quad\mathbf{I\\!I}^{\theta}_{\tau}:{\mathscr{H}}_{t}\to
H^{0}(X_{t},{\mathscr{C}}^{\infty}(\Lambda^{n,0}_{X_{t}}\otimes E|_{X_{t}}))$

by

$\kappa^{\theta}_{\tau}f:=\iota_{X_{t}}^{*}\bar{\partial}\xi^{\theta}_{\tau}\lrcorner
u\quad\text{and}\quad\mathbf{I\\!I}^{\theta}_{\tau}f:=P_{t}^{\perp}\iota_{X_{t}}^{*}(L^{1,0}_{\xi^{\theta}_{\tau}}u),$

where $u\in H^{0}(X,{\mathscr{C}}^{\infty}(\Lambda^{n,0}_{X}\otimes E))$ is
any section such that $\iota_{X_{t}}^{*}u=f$ and
$\iota_{X_{t}}^{*}L^{1,0}_{\xi^{\theta}_{\tau}}\bar{\partial}u\equiv 0$. Here
$P_{t}^{\perp}\\!:H^{0}(X_{t},L^{2}_{\ell oc}(K_{X_{t}}\otimes
E|_{X_{t}}))\to{\mathscr{H}}_{t}^{\perp}$ is the orthogonal projection and

$L^{1,0}_{\xi^{\theta}_{\sigma}}\alpha:=\nabla^{1,0}(\xi^{\theta}_{\sigma}\lrcorner\alpha)+\xi^{\theta}_{\sigma}\lrcorner\nabla^{1,0}\alpha,$

is the _twisted Lie derivative_ of an $E$-valued differential form $\alpha$
associated to the Chern connection $\nabla=\nabla^{1,0}+\bar{\partial}$ for
$(E,h)\to X$. (See paragraph 3.2.) As we will see later, these definitions are
independent of the choice of $u$.

######  Theorem 2.

Let $(E,h)\to X\stackrel{{\scriptstyle p}}{{\to}}B$ be a proper Kähler family
holomorphic Hermitian vector bundles. Then the curvature
$\Theta^{{\mathscr{H}}}$ of ${\mathscr{H}}\to B$ is given by

(2)
$(\Theta^{{\mathscr{H}}}_{\sigma\bar{\tau}}f_{1},f_{2})=\left(\Theta(h)_{\xi^{\theta}_{\sigma}\bar{\xi}^{\theta}_{\tau}}f_{1},f_{2}\right)-\int_{X_{t}}\\!\\!\\!\\!\sqrt{\text{\rm-1}}^{n^{2}}\\!\\!\left<\kappa^{\theta}_{\sigma}f_{1}\wedge\overline{\kappa^{\theta}_{\tau}f_{2}},h\right>-\int_{X_{t}}\\!\\!\\!\\!\sqrt{\text{\rm-1}}^{n^{2}}\\!\\!\left<\mathbf{I\\!I}^{\theta}_{\sigma}f_{1}\wedge\overline{\mathbf{I\\!I}^{\theta}_{\tau}f_{2}},h\right>,$

where $\theta$ is any horizontal distribution, $t\in B$, $\sigma,\tau\in
T^{1,0}_{B,t}$, $f_{1},f_{2}\in{\mathscr{H}}_{t}$, $\Theta(h)$ is the
curvature of $(E,h)\to X$ and $(\cdot,\cdot)$ is the inner product (1).

We emphasize that the curvature $\Theta^{{\mathscr{H}}}$ is independent of the
choice of $\theta$.

######  Remark 1.3.

Each of the three terms on the right hand side of Formula (2), while dependent
on the choice of horizontal distribution $\theta$, is independent of the way
in which the $(1,0)$-vectors $\sigma$ and $\tau$ are extended to
$(1,0)$-vector fields. We therefore adopt the inessential but simplifying
convention that these vectors are extended holomorphically, i.e.,
$\sigma,\tau\in{\mathcal{O}}(T^{1,0}_{B})_{t}$. $\diamond$

The precise definition of the curvature $\Theta^{{\mathscr{H}}}$ of
${\mathscr{H}}\to B$ requires some technology that we collect or develop in
Sections 2, 3 and 4. But with Theorem 2 stated we can explain the basic idea
behind the definition of $\Theta^{{\mathscr{H}}}$. Consider the family of
Hilbert spaces ${\mathscr{L}}\to B$ whose fiber over $t\in B$ is

${\mathscr{L}}_{t}:=\left\\{f\in H^{0}(X_{t},L^{2}_{\ell oc}(K_{X_{t}}\otimes
E|_{X_{t}}))\right\\},$

equipped with the same inner product (1). The curvature of ${\mathscr{L}}$ can
be defined because ${\mathscr{L}}$ is reasonably well-behaved: it is a smooth
Hilbert field in the sense of Lempert-Szőke.

Evidently the fibers of ${\mathscr{H}}\to B$ are subspaces of the fibers of
${\mathscr{L}}\to B$. Each horizontal distribution $\theta$ defines an
operator $\bar{\partial}^{\theta}$ on smooth sections of ${\mathscr{L}}$
(Definition 4.12). Remarkably, for every $\theta$ the operator
$\bar{\partial}^{\theta}$, when restricted to sections of ${\mathscr{H}}\to
B$, agrees with the $\bar{\partial}$-operator of ${\mathscr{H}}\to B$
introduced by Berndtsson, which we describe in slightly different form
(Definition 4.6). Therefore $({\mathscr{H}},\bar{\partial})$ is a ‘holomorphic
sub-field’ of $({\mathscr{L}},\bar{\partial}^{\theta})$.

######  Remark 1.4.

While Berndtsson’s $\bar{\partial}$-operator satisfies
$\bar{\partial}\bar{\partial}=0$, the operator $\bar{\partial}^{\theta}$
satisfies $\bar{\partial}^{\theta}\bar{\partial}^{\theta}=0$ if and only if
the horizontal distribution $\theta$ is Frobenius-integrable. Thus
$({\mathscr{L}},\bar{\partial}^{\theta})$ is a sort of ‘almost holomorphic’
Hilbert field. Like holomorphic Hilbert fields, almost holomorphic Hilbert
fields have at most one Chern connection, but the curvature of this Chern
connection is not a $(1,1)$-form if $\bar{\partial}\bar{\partial}\neq 0$.
$\diamond$

If ${\mathscr{H}}\to B$ is locally trivial then a Gauss-Griffiths-type formula
links the curvature of ${\mathscr{H}}$, the curvature of
$({\mathscr{L}},\bar{\partial}^{\theta})$ restricted to ${\mathscr{H}}$, and
the second fundamental form of
${\mathscr{H}}\subset({\mathscr{L}},\bar{\partial}^{\theta})$ (Proposition
2.33).

One can also define the second fundamental form in ${\mathscr{H}}$ even when
${\mathscr{H}}$ is not locally trivial. This definition yields the curvature
as an endomorphism of ${\mathscr{H}}$ viewed as a
${\mathscr{C}}^{\infty}_{B}$-module. Unlike the finite-rank case, such an
endomorphism is not automatically represented by a map of Hilbert fields. To
have such a map one must be able to extend a dense subset of every fiber
${\mathscr{H}}$ to sections of ${\mathscr{L}}$ with particular special
properties. The details are described abstractly, i.e., for general Hilbert
fields, in Paragraph 2.11, and then more concretely in Theorem 4.17 for
${\mathscr{H}}\subset{\mathscr{L}}$.

From the Gauss Formula perspective the sum of the first two terms on the right
hand side of the identity (2) is precisely the restriction to ${\mathscr{H}}$
of the curvature of $({\mathscr{L}},\bar{\partial}^{\theta})$ (Theorem 8.5),
while the last term on the right hand side of (2) is the negative of the
‘square of the second fundamental form’.

######  Remark 1.5.

By Ehresmann’s Theorem the properness of the holomorphic family $p:X\to B$
implies that the field of Hilbert spaces ${\mathscr{L}}\to B$ has a natural
structure of a smooth Hilbert bundle, i.e., it is locally trivial. The smooth
structure agrees with the notion of smooth structure introduced by Lempert and
Szőke in [LS-2014], as they point out.

Here we are more concerned with the complex structure, which is required for
any notion of Chern connection. From this perspective, we have not addressed
here the interesting question of whether for _some_ horizontal distribution
$\theta$ the field of Hilbert spaces ${\mathscr{L}}\to B$ has the structure of
an almost holomorphic vector bundle whose $\bar{\partial}$-operator is
$\bar{\partial}^{\theta}$.

Instead we circumvent this question by introducing the notion of Berndtsson-
Lempert-Szőke (BLS) fields. This notion combines Berndtsson’s complex
structures with the notion of smooth structure introduced by Lempert and
Szőke.

Our setup is general beyond the case of proper holomorphic families. One can
use the setup to give meaning to curvature in the setting of families of
complex manifolds-with-boundary, which have recently been considered by
Upadrashta [U-2021], and which significantly generalize some of the elegant
results of X. Wang [Wang-2017]. $\diamond$

Theorem 2 yields a proof of Corollary 1.2, without passing through Theorem 1.
To complete such a proof one must estimate the second and third terms on the
right hand side of the curvature formula (2). The third term can be estimated
using basic inequalities and a direct computation that is independent of the
horizontal distribution $\theta$, but the second term requires a good choice
of $\theta$, or equivalently, a good choice of almost holomorphic structure
for ${\mathscr{L}}\to B$. Such a choice is provided by the following theorem.

######  Theorem 3.

Let $(E,h)\to X\stackrel{{\scriptstyle p}}{{\to}}B$ be a proper Kähler family
of holomorphic vector bundles. Let $\omega$ be a relative Kähler form on $X$.
Then for each $t\in B$ there exists a smooth horizontal lift $\theta$ with the
following property: for every $f_{1},f_{2}\in{\mathscr{H}}_{t}$ one has

(3)
$-\int_{X_{t}}(\text{-}1)^{\tfrac{n^{2}}{2}}\left<\kappa^{\theta}_{\sigma}f_{1}\wedge\overline{\kappa^{\theta}_{\tau}f_{2}},h\right>=\left(\kappa^{\theta}_{\sigma}f_{1},\kappa^{\theta}_{\tau}f_{2}\right),$

where the right hand side is the inner product induced on $E$-valued
$(n-1,1)$-forms on $X_{t}$ by the metric $h$ and any Kähler metric on the
fiber $X_{t}$. Moreover, if $\dim H^{0,2}_{\bar{\partial}}(X_{t})$ is
independent of $t\in B$ then one can choose $\theta$ independent of $t$.

We emphasize that the horizontal lift of Theorem 3 is defined for the entire
family $X\to B$, but that, unless the Dolbeault numbers $\dim
H^{0,2}_{\bar{\partial}}(X_{t})$ are independent of $t\in B$, the identity (3)
only holds at one point $t\in B$.

The equality (3) of these two types of pairings of $(p,q)$-forms with $p+q\leq
n$ holds if one of these forms is primitive in the Hodge-Theoretic sense. This
equality is (a twisted version of) a special case of the so-called Hodge-
Riemann Bilinear Relations (Proposition 6.9). Theorem 3 is thus a consequence
of Theorem 7.3, which in turn relies heavily on the Hodge Theorem.

Theorem 3, while handy for a proof of Corollary 1.2 (see Paragraph 9.2) and of
interest in its own right, is not required in our proof of Theorem 1. In its
stead, one uses the Hodge Theorem in a different way to compute an exact
formula for the square of the second fundamental form, i.e., the third term on
the right hand side of (2). The statement is as follows.

######  Theorem 4.

Let $(E,h)\to X\stackrel{{\scriptstyle p}}{{\to}}B$ be a Kähler family of
holomorphic Hermitian vector bundles, let $\theta$ be a smooth horizontal
distribution, let $t\in B$ and let $\omega_{t}$ be a Kähler form on $X_{t}$.
Then for every $f_{1},f_{2}\in{\mathscr{H}}_{t}$ there exist smooth section
$u_{1},u_{2}\in H^{0}(X,{\mathscr{C}}^{\infty}(\Lambda^{n,0}_{X}\otimes E))$,
depending on $\theta$ and $\omega_{t}$ and satisfying
$\iota_{X_{t}}^{*}u_{i}=f_{i}$ and
$\iota_{X_{t}}^{*}L^{1,0}_{\xi^{\theta}_{\tau}}\bar{\partial}u_{i}=0$,
$i=1,2$, such that

$\displaystyle\int_{X_{t}}\\!\\!\\!\\!\sqrt{\text{\rm-1}}^{n^{2}}\\!\\!\left<\mathbf{I\\!I}^{\theta}_{\sigma}f_{1}\wedge\overline{\mathbf{I\\!I}^{\theta}_{\tau}f_{2}},h\right>=-\int_{X_{t}}\\!\\!\\!\\!\sqrt{\text{\rm-1}}^{n^{2}}\\!\\!\left<\kappa^{\theta}_{\sigma}f_{1}\wedge\overline{\kappa^{\theta}_{\tau}f_{2}},h\right>-\left(\xi^{\theta}_{\sigma}\lrcorner\bar{\partial}u_{1},\xi^{\theta}_{\tau}\lrcorner\bar{\partial}u_{2}\right)$
(4)
$\displaystyle\qquad-\left(\Lambda_{\omega_{t}}\iota_{X_{t}}^{*}\left(\xi^{\theta}_{\sigma}\lrcorner\sqrt{\text{\rm-1}}\Theta(h)\right)f_{1},\xi^{\theta}_{\tau}\lrcorner
u_{2}\right)-\left(\xi^{\theta}_{\sigma}\lrcorner
u_{1},\Lambda_{\omega_{t}}\iota_{X_{t}}^{*}\left(\xi^{\theta}_{\tau}\lrcorner\sqrt{\text{\rm-1}}\Theta(h)\right)f_{2}\right)$
$\displaystyle\qquad\qquad+\left(\left[\sqrt{\text{\rm-1}}\Theta(h_{t}),\Lambda_{\omega_{t}}\right](\xi^{\theta}_{\sigma}\lrcorner
u_{1}),(\xi^{\theta}_{\tau}\lrcorner u_{2})\right)$

for all $\sigma,\tau\in T^{1,0}_{B,t}$. The pairings $(\cdot,\cdot)$ in the
last four terms on the right hand side of (4) are the $L^{2}$ inner products
induced on $E|_{X_{t}}$-valued forms by the Hermitian metric $h_{t}$ and the
Kähler metric $\omega_{t}$.

The aforementioned use of the Hodge Theorem in the proof of Theorem 4 is
equivalent to a construction of Berndtsson, formalized here as Theorem 9.2.
With Theorem 9.2 in hand, the rest of the proof of Theorem 4 is a somewhat
involved integration-by-parts argument.

Finally, Theorem 1 is a consequence of Theorems 2 and 4 because

$\displaystyle\left<p_{*}\left(\sqrt{\text{\rm-1}}^{n^{2}}\left<(\Theta(h)u_{1}\wedge\overline{u_{2}},h\right>\right),\sigma\wedge\bar{\tau}\right>$
$\displaystyle\quad=\left(\Theta(h)_{\xi^{\theta}_{\sigma}\bar{\xi}^{\theta}_{\tau}}f_{1},f_{2}\right)+\left(\Lambda_{\omega_{t}}\iota_{X_{t}}^{*}\left(\xi^{\theta}_{\sigma}\lrcorner\sqrt{\text{\rm-1}}\Theta(h)\right)f_{1},\xi^{\theta}_{\tau}\lrcorner
u_{2}\right)+\left(\xi^{\theta}_{\sigma}\lrcorner
u_{1},\Lambda_{\omega_{t}}\iota_{X_{t}}^{*}\left(\xi^{\theta}_{\tau}\lrcorner\sqrt{\text{\rm-1}}\Theta(h)\right)f_{2}\right)$
$\displaystyle\qquad\qquad-\left(\left[\sqrt{\text{\rm-1}}\Theta(h_{t}),\Lambda_{\omega_{t}}\right](\xi^{\theta}_{\sigma}\lrcorner
u_{1}),(\xi^{\theta}_{\tau}\lrcorner u_{2})\right)$

(Proposition 9.3). Indeed, together with Theorems 2 and 4, the latter identity
yields

(5)
$(\Theta^{{\mathscr{H}}}_{\sigma\bar{\tau}}f_{1},f_{2})=\left<p_{*}\left(\sqrt{\text{\rm-1}}^{n^{2}}\left<(\Theta(h)\wedge
u_{1})\wedge\overline{u_{2}},h\right>\right),\sigma\wedge\bar{\tau}\right>+\left(\xi^{\theta}_{\sigma}\lrcorner\bar{\partial}u_{1},\xi^{\theta}_{\tau}\lrcorner\bar{\partial}u_{2}\right).$

If ${\mathscr{H}}$ is locally trivial Formula (5) is due to Berndtsson
[B-2011, Section 2] if ${\rm Rank}(E)=1$ and to Liu-Yang [LY-2014, Theorem
1.6] in the higher rank case. The term

$\left(\xi^{\theta}_{\sigma}\lrcorner\bar{\partial}u_{1},\xi^{\theta}_{\tau}\lrcorner\bar{\partial}u_{2}\right)$

on the right hand side of (5) gives the extra positivity for the improvement
of Theorem 1 mentioned after the statement of Corollary 1.2. This term and the
term
$\sqrt{\text{\rm-1}}^{n^{2}}\\!\\!\left<\kappa^{\theta}_{\sigma}f_{1}\wedge\overline{\kappa^{\theta}_{\tau}f_{2}},h\right>$
are related to deformation theory of compact complex manifolds, and more
specifically to the Kodaira-Spencer map, which captures the first obstruction
to local triviality of holomorphic families. (See Section 7 for more detail.)

We end this introduction with an explanation one of our initial motivations.
If the family $X\to B$ is trivial then Corollary 1.1 has a considerably
simpler proof. In fact, this same simplified proof yields a significantly more
general result, stated as [B-2009, Theorem 1.1]. By our reading of [B-2009],
there is a problem implicitly stated in the second-to-last paragraph of the
introduction to [B-2009]: whether one can prove [B-2009, Theorem 1.2] (i.e.
Corollary 1.1 above) in a manner similar to the proof of [B-2009, Theorem
1.1]. Berndtsson states there that he was not “… able to find a natural
complex structure on the space of all (not necessarily holomorphic) $(n,0)$
forms, extending the complex structure on [${\mathscr{H}}$].” In fact, there
are infinitely many consistent ways to extend the complex structure of
${\mathscr{H}}$ to ${\mathscr{L}}$, each one corresponding to a choice of
horizontal lift $\theta$.

And aside from a solution of Berndtsson’s implicit problem, we obtain a
somewhat different geometric perspective. In the same penultimate paragraph of
the introduction of [B-2009] it is mentioned that the Kodaira-Spencer class of
$p:X\to B$ “plays somewhat the role of another second fundamental form,…”. It
seems that this informal explanation of the extra positivity as a second
fundamental form is not quite the same as the explanation coming from BLS
geometry. In BLS geometry the Kodaira-Spencer class affects both the curvature
of ${\mathscr{L}}$ and the second fundamental form of ${\mathscr{H}}$ in
${\mathscr{L}}$. In the curvature of ${\mathscr{L}}$ one sees the Hodge-
Riemann ‘square’

$-\int_{X_{t}}\\!\\!\\!\\!\sqrt{\text{\rm-1}}^{n^{2}}\\!\\!\left<\kappa^{\theta}_{\sigma}f_{1}\wedge\overline{\kappa^{\theta}_{\tau}f_{2}},h\right>$

of a certain representative $\kappa^{\theta}_{\sigma}f$ of the action on
$E$-valued $(n,0)$-forms of of the Kodaira-Spencer map of $p:X\to B$; a
representative that depends only on the choice of complex structure
$\bar{\partial}^{\theta}$ for ${\mathscr{L}}$, while in the second fundamental
form of ${\mathscr{H}}$ in ${\mathscr{L}}$ one sees the “difference of
squares”

$-\int_{X_{t}}\\!\\!\\!\\!\sqrt{\text{\rm-1}}^{n^{2}}\\!\\!\left<\kappa^{\theta}_{\sigma}f_{1}\wedge\overline{\kappa^{\theta}_{\tau}f_{2}},h\right>-\left(\xi^{\theta}_{\sigma}\lrcorner\bar{\partial}u_{1},\xi^{\theta}_{\tau}\lrcorner\bar{\partial}u_{2}\right)$

of two representatives of the Kodaira-Spencer map. The first of these is the
term appearing in the curvature of ${\mathscr{L}}$, and as already mentioned
it depends only on the choice of complex structure for ${\mathscr{L}}$. On the
other hand, the other representative

$\iota_{X_{t}}^{*}\xi^{\theta}_{\sigma}\lrcorner\bar{\partial}u=\kappa^{\theta}_{\sigma}f-\bar{\partial}\left(\iota_{X_{t}}^{*}(\xi^{\theta}_{\sigma}\lrcorner
u)\right)$

of the action of the Kodaira-Spencer map depends on how elements
$f\in{\mathscr{H}}_{t}$ are extended to sections $u$ of ${\mathscr{L}}$. Only
the term
$\left(\xi^{\theta}_{\sigma}\lrcorner\bar{\partial}u_{1},\xi^{\theta}_{\tau}\lrcorner\bar{\partial}u_{2}\right)$
survives in the formula (5) for $\Theta^{{\mathscr{H}}}$.

##  2\. Abstract Theory of Berndtsson-Lempert-Szőke Fields

In this section we discuss the general geometric structure with which
${\mathscr{H}}$ and ${\mathscr{L}}$ are eventually endowed. We start with a
slight reorganization and minor refinement of ideas introduced by Lempert and
Szőke in [LS-2014], and then we introduce definitions that are already
suggested or implied in [LS-2014]. For example, in [LS-2014] the introduction
of a $\bar{\partial}$-operator to define the holomorphic structure of a smooth
(qausi)-Hilbert field is not abstracted, but introduced in the ‘direct image’
constructions of [LS-2014, Part II] for the bundle of holomorphic sections.
And the discussion of subfields is not as explicit, but many of the elements
are hinted at in both [LS-2014] and [B-2009]. In fact, [B-2009] considers
subfields in the case of trivial families of complex manifolds.

###  2.1. Quasi-Hilbert fields

######  Definition 2.1.

A quasi-Hilbert field is a map $\pi:{\mathfrak{H}}\to B$ whose fibers are
topological vector spaces each of which is isomorphic to a Hilbert space.
$\diamond$

That is to say, for each $t\in B$ there is a Hilbert space $H_{t}$ such that
the fiber ${\mathfrak{H}}_{t}:=\pi^{-1}(t)$ is isomorphic, as a topological
vector space, to $H_{t}$, but ${\mathfrak{H}}_{t}$ does not have a canonical
inner product. The name is inspired by the fact that a linear isomorphism of
two Hilbert spaces is a quasi-isometry. We shall say that the Hilbert space
$H_{t}$ is quasi-isometric to ${\mathfrak{H}}_{t}$, even though the latter
does not have a canonical inner product.

######  Remark 2.2.

Even if the Hilbert space fibers of a quasi-Hilbert field are finite-
dimensional, one does not want all of them to be isomorphic to a single vector
space. Indeed, in our setting of a holomorphic family $E\to
X\stackrel{{\scriptstyle p}}{{\to}}B$ in which $p$ is proper, the field of
Hilbert spaces ${\mathscr{H}}\to B$ has constant fiber dimension if and only
if it is locally trivial. $\diamond$

######  Definition 2.3.

The dual of a quasi-Hilbert field ${\mathfrak{H}}\to B$ is the quasi-Hilbert
field ${\mathfrak{H}}^{*}\to B$ such that for each $t\in B$ the fiber
${\mathfrak{H}}_{t}^{*}$ is the topological dual of ${\mathfrak{H}}_{t}$,
i.e., the set of bounded linear functionals on ${\mathfrak{H}}_{t}$.
$\diamond$

By the Riesz Representation Theorem each fiber ${\mathfrak{H}}_{t}^{*}$ of the
dual quasi-Hilbert field ${\mathfrak{H}}^{*}$ is isometric to the dual fiber
${\mathfrak{H}}_{t}$. Therefore the Hilbert fields ${\mathfrak{H}}$ and
${\mathfrak{H}}^{*}$ are (quasi-)isomorphic quasi-Hilbert fields.

###  2.2. Smooth structures on quasi-Hilbert fields

We denote by $\Gamma(U,{\mathfrak{H}})$ the set of sections (i.e. right
inverses) of $\pi:{\mathfrak{H}}\to B$ over $U\subset B$.

######  Definition 2.4.

Let $B$ be a smooth manifold and let ${\mathfrak{H}}\to B$ be a quasi-Hilbert
field.

  1. i.

A smooth structure for a quasi-Hilbert field ${\mathfrak{H}}\to B$ is a sheaf
${\mathscr{C}}^{\infty}({\mathfrak{H}})\to B$ of
${\mathscr{C}}^{\infty}$-modules on $B$, whose elements are germs of sections
of $\pi:{\mathfrak{H}}\to B$, such that for each $t\in B$ the stalk
${\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}$ is mapped by the evaluation-
at-$t$ map to a dense subspace of the Hilbert space ${\mathfrak{H}}_{t}$. The
associated presheaf assigns to an open set $U\subset B$ a set of sections of
$\pi:\pi^{-1}(U)\to U$ denoted
$\Gamma(U,{\mathscr{C}}^{\infty}({\mathfrak{H}}))$, whose members are called
_smooth sections over $U$_.

  2. ii.

The smooth structure for ${\mathfrak{H}}^{*}\to B$ dual to the smooth
structure of ${\mathfrak{H}}\to B$ is the sheaf
${\mathscr{C}}^{\infty}({\mathfrak{H}}^{*})\to B$ associated to the presheaf
whose groups $\Gamma(U,{\mathscr{C}}^{\infty}({\mathfrak{H}}^{*}))$ are
defined as follows:
$\phi\in\Gamma(U,{\mathscr{C}}^{\infty}({\mathfrak{H}}^{*}))$ if and only if
for every $f\in\Gamma(U,{\mathscr{C}}^{\infty}({\mathfrak{H}}))$ the function
$\left<\phi,f\right>\in{\mathscr{C}}^{\infty}(U)$. $\diamond$

######  Proposition 2.5.

If a quasi-Hilbert field ${\mathfrak{H}}\to B$ with finite-dimensional fibers
admits a smooth structure then the function

$B\ni t\mapsto\dim{\mathfrak{H}}_{t}$

is lower semi-continuous.

###### Proof.

Since the smooth sections of ${\mathfrak{H}}$ restrict to dense subsets of the
fiber, one can find a basis for any ${\mathfrak{H}}_{t}$ whose elements are
smooth sections of ${\mathfrak{H}}$ evaluated at $t$. By the continuity
inherent in sheaves and the quasi-isometric invariance of linear independence,
at all $t^{\prime}$ sufficiently close to $t$ the restriction of these
sections to ${\mathfrak{H}}_{t^{\prime}}$ forms an independent set of vectors,
and hence $\dim{\mathfrak{H}}_{t^{\prime}}\geq\dim{\mathfrak{H}}_{t}$. ∎

###  2.3. Maps

We shall often encounter maps between (smooth) quasi-Hilbert fields.

######  Definition 2.6.

Let ${\mathfrak{H}}^{1},{\mathfrak{H}}^{2}\to B$ be quasi-Hilbert fields.

  1. a.

A map $F:{\mathfrak{H}}^{1}\to{\mathfrak{H}}^{2}$ is said to be a _map of
quasi-Hilbert fields_ if it is fiber-preserving, i.e.,

${{\mathfrak{H}}^{1}}$${{\mathfrak{H}}^{2}}$${B}$${B,}$$\scriptstyle{F}$$\scriptstyle{\rm
Id}$

and linear on fibers. We write $F\in{\rm
Lin}({\mathfrak{H}}^{1},{\mathfrak{H}}^{2})$.

(As is common in the study of linear transformations of Hilbert spaces, the
fiber restrictions need not be bounded or even everywhere defined.) If the
fiber restrictions are defined on dense subsets of the fibers we will say that
$F$ is _densely-defined_.

  2. b.

A map of quasi-Hilbert fields $F:{\mathfrak{H}}^{1}\to{\mathfrak{H}}^{2}$ is
called a _morphism_ if the restrictions to fibers are bounded. In this case we
write $F\in{\rm Hom}({\mathfrak{H}}^{1},{\mathfrak{H}}^{2})$.

  3. c.

Suppose moreover that ${\mathscr{C}}^{\infty}({\mathfrak{H}}_{1})$ and
${\mathscr{C}}^{\infty}({\mathfrak{H}}_{2})$ are smooth structures for
${\mathfrak{H}}_{1}$ and ${\mathfrak{H}}_{2}$ respectively. A map $F\in{\rm
Lin}({\mathfrak{H}}^{1},{\mathfrak{H}}^{2})$ is said to be _smooth_ if it maps
${\mathscr{C}}^{\infty}({\mathfrak{H}}^{1})$ to
${\mathscr{C}}^{\infty}({\mathfrak{H}}^{2})$, i.e., if for each $t\in B$ one
has $F{\mathfrak{f}}\in{\mathscr{C}}^{\infty}({\mathfrak{H}}^{2})_{t}$ for
every ${\mathfrak{f}}\in{\mathscr{C}}^{\infty}({\mathfrak{H}}^{1})_{t}$.

Thus a smooth map is densely-defined. However, a smooth map need not be a
morphism, and a morphism need not be smooth.

One often also encounters smooth-section maps.

######  Definition 2.7.

Let ${\mathfrak{H}}^{1},{\mathfrak{H}}^{2}\to B$ be quasi-Hilbert fields with
smooth structures ${\mathscr{C}}^{\infty}({\mathfrak{H}}^{1})$ and
${\mathscr{C}}^{\infty}({\mathfrak{H}}^{2})$.

  1. (1)

A map of sheaves
$\mathbf{L}:{\mathscr{C}}^{\infty}({\mathfrak{H}}^{1})\to{\mathscr{C}}^{\infty}({\mathfrak{H}}^{2})$
is said to be _${\mathscr{C}}^{\infty}$ -linear_ if

$\mathbf{L}(f{\mathfrak{f}})=f\mathbf{L}{\mathfrak{f}}$

for each $t\in B$, all
${\mathfrak{f}}\in{\mathscr{C}}^{\infty}({\mathfrak{H}}^{1})_{t}$ and all
$f\in{\mathscr{C}}^{\infty}_{B,t}$.

  2. (2)

A ${\mathscr{C}}^{\infty}$-linear $\mathbf{L}$ is said to be _tensorial at
$t\in B$_ if

${\mathfrak{f}}\in{\mathscr{C}}^{\infty}({\mathfrak{H}}^{1})_{t}\ \text{ and
}\ {\mathfrak{f}}(t)=0\quad\Rightarrow\quad(\mathbf{L}{\mathfrak{f}})(t)=0.$

We say $\mathbf{L}$ is _tensorial_ if $\mathbf{L}$ is tensorial at every point
of $B$.

Note that a tensorial ${\mathscr{C}}^{\infty}$-linear map determines a smooth
map of quasi-Hilbert fields in the sense of Definition 2.6.a. (The converse is
obvious.) Indeed, if $\mathbf{L}$ is a tensorial
${\mathscr{C}}^{\infty}$-linear map then one can define the map of quasi-
Hilbert fields $L$ by

$L{\mathfrak{f}}(t):=(\mathbf{L}{\mathfrak{f}})(t),\quad{\mathfrak{f}}\in{\mathscr{C}}^{\infty}({\mathfrak{H}})_{t},$

and the tensorial condition means $L$ is well-defined.

######  Remark 2.8.

If ${\mathscr{C}}^{\infty}({\mathfrak{H}}^{1})_{t}$ is generated, as a
${\mathscr{C}}^{\infty}_{B,t}$-module, by germs that do not vanish at $p$ then
every ${\mathscr{C}}^{\infty}$-linear map is tensorial at $t$. This is the
situation for smooth finite-rank vector bundles with their usual smooth
structure consisting of the sheaf of germs of smooth sections. $\diamond$

###  2.4. Connections and Curvature

######  Definition 2.9.

Let $\pi:{\mathfrak{H}}\to B$ be a quasi-Hilbert field with a smooth
structure.

  1. i.

A _connection_ for ${\mathfrak{H}}\to B$ is a map

$\nabla:\Gamma(B,{\mathscr{C}}^{\infty}(T_{B}))\times\Gamma(B,{\mathscr{C}}^{\infty}({\mathfrak{H}}))\in(\xi,{\mathfrak{f}})\mapsto\nabla_{\xi}{\mathfrak{f}}\in\Gamma(B,{\mathscr{C}}^{\infty}({\mathfrak{H}}))$

such that

$\nabla_{f\xi+\eta}=f\nabla_{\xi}+\nabla_{\eta}\quad\text{ and
}\nabla_{\xi}(f{\mathfrak{f}})=(\xi
f){\mathfrak{f}}+f\nabla_{\xi}{\mathfrak{f}}$

for all $f\in{\mathscr{C}}^{\infty}(B)$,
$\xi\in\Gamma(B,{\mathscr{C}}^{\infty}(T_{B}))$ and
${\mathfrak{f}}\in\Gamma(B,{\mathscr{C}}^{\infty}({\mathfrak{H}}))$.

  2. ii.

The map
$\Theta(\nabla):\Gamma(B,{\mathscr{C}}^{\infty}(\Lambda^{2}(T_{B})))\times\Gamma(B,{\mathscr{C}}^{\infty}({\mathfrak{H}}))\to\Gamma(B,{\mathscr{C}}^{\infty}({\mathfrak{H}}))$
defined by

$\left<\Theta(\nabla),(\xi\wedge\eta,{\mathfrak{f}})\right>:=\nabla_{\xi}\nabla_{\eta}{\mathfrak{f}}-\nabla_{\eta}\nabla_{\xi}{\mathfrak{f}}-\nabla_{[\xi,\eta]}{\mathfrak{f}}$

is called the _curvature_ of $\nabla$. $\diamond$

######  Definition 2.10.

Let ${\mathfrak{H}}\to B$ be a quasi-Hilbert field with smooth structure, and
let ${\mathfrak{h}}$ be a smooth metric for ${\mathfrak{H}}$. A connection
$\nabla$ for ${\mathfrak{H}}$ is said to be _compatible with the metric
${\mathfrak{h}}$_ if

$\left<d{\mathfrak{h}}({\mathfrak{f}}_{1},{\mathfrak{f}}_{2}),\xi\right>={\mathfrak{h}}(\nabla_{\xi}{\mathfrak{f}}_{1},{\mathfrak{f}}_{2})+{\mathfrak{h}}({\mathfrak{f}}_{1},\nabla_{\xi}{\mathfrak{f}}_{2})$

for all $t\in B$, all
${\mathfrak{f}}_{1},{\mathfrak{f}}_{2}\in{\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}$
and all $\xi\in T_{B,t}$. $\diamond$

One defines the covariant derivative of a smooth metric ${\mathfrak{h}}$ by

(6)
$\nabla{\mathfrak{h}}({\mathfrak{f}}_{1},{\mathfrak{f}}_{2}):=d({\mathfrak{h}}({\mathfrak{f}}_{1},{\mathfrak{f}}_{2}))-{\mathfrak{h}}(\nabla{\mathfrak{f}}_{1},{\mathfrak{f}}_{2})-{\mathfrak{h}}({\mathfrak{f}}_{1},\nabla{\mathfrak{f}}_{2}).$

In terms of this derivative, a connection $\nabla$ is compatible with a smooth
metric ${\mathfrak{h}}$ if and only if $\nabla{\mathfrak{h}}=0$.

The difference of two connections and the curvature of a connection are
${\mathscr{C}}^{\infty}$-linear. If the connections are metric compatible then
one can say more.

######  Proposition 2.11.

Let ${\mathfrak{H}}\to B$ be a quasi-Hilbert field with smooth structure, let
${\mathfrak{h}}$ be a smooth metric for ${\mathfrak{H}}$ and let
$\nabla,\nabla_{1},\nabla_{2}$ be ${\mathfrak{h}}$-compatible connections.
Then

$\nabla_{2}-\nabla_{1}\quad\text{and}\quad\Theta(\nabla)$

are tensorial.

###### Proof.

Denoting either $\nabla_{2}-\nabla_{1}$ or $\Theta(\nabla)$ by $\mathbf{L}$,
one has
${\mathfrak{h}}(\mathbf{L}{\mathfrak{f}},{\mathfrak{g}})=-{\mathfrak{h}}({\mathfrak{f}},\mathbf{L}{\mathfrak{g}})$.
Therefore if ${\mathfrak{f}}\in{\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}$ and
${\mathfrak{f}}(t)=0$ then
${\mathfrak{h}}((\mathbf{L}{\mathfrak{f}})(t),{\mathfrak{g}}(t))=0$ for all
${\mathfrak{g}}\in{\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}$. Since
${\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}(t)$ is dense in
${\mathfrak{H}}_{t}$, $(\mathbf{L}{\mathfrak{f}})(t)=0$, i.e., $\mathbf{L}$ is
tensorial. ∎

###  2.5. Formulation with differential forms

Though it’s largely a matter of taste when the base manifold $B$ is finite-
dimensional (which is a standing assumption in the present article), one might
sometimes prefer to think of the connection as a total differential, rather
than talk about directional derivatives. To do so it is convenient to
introduce differential forms with values in ${\mathfrak{H}}$.

######  Definition 2.12.

Let ${\mathfrak{H}}\to B$ be a smooth quasi-Hilbert field. An
${\mathfrak{H}}$-valued differential $k$-form is a section ${\mathfrak{b}}$ of
$\Lambda^{k}_{B}\otimes{\mathfrak{H}}\to B$. If ${\mathfrak{H}}\to B$ is a
smooth quasi-Hilbert field then we say that ${\mathfrak{b}}$ is _smooth_ if
for each $k$-tuple of vector fields
$\tau_{1},...,\tau_{k}\in\Gamma(B,{\mathscr{C}}^{\infty}(T_{B}))$ the section
$\left<\alpha,\tau_{1}\wedge\cdots\wedge\tau_{k}\right>$ of ${\mathfrak{H}}\to
B$ obtained by contracting in the factor $\Lambda^{k}_{B}$ lies in
$\Gamma(B,{\mathscr{C}}^{\infty}({\mathfrak{H}}))$. We denote by
$\Gamma(B,{\mathscr{C}}^{\infty}(\Lambda^{k}_{B}\otimes{\mathfrak{H}}))$ the
collection of smooth ${\mathfrak{H}}$-valued $k$-forms.

######  Remark 2.13.

It may very well be that the fibers of ${\mathfrak{H}}\to B$ are infinite-
dimensional. In this case, one often sees the analytic completion
$\hat{\otimes}$. Here we assume our ${\mathfrak{H}}$-valued $k$-forms can be
written locally as the form
$\alpha^{1}\otimes{\mathfrak{f}}_{1}+\cdots+\alpha^{m_{k}}\otimes{\mathfrak{f}}_{m_{k}}$,
where $m=\dim_{{\mathbb{C}}}B$ and $m_{k}=\binom{m}{k}$. The
${\mathfrak{H}}$-valued $k$-form is smooth precisely when the local forms
$\alpha^{i}$ and the local sections ${\mathfrak{f}}_{i}$ are smooth.
$\diamond$

######  Remark 2.14.

Note that since the set of smooth sections restricts to dense subsets of the
fibers, the same is automatically true of the set of smooth forms. $\diamond$

From the ‘forms’ point of view a connection $\nabla$ can be seen as a map

$\nabla:\Gamma(B,{\mathscr{C}}^{\infty}({\mathfrak{H}}))\to\Gamma(B,{\mathscr{C}}^{\infty}(\Lambda^{1}_{B}\otimes{\mathfrak{H}})),$

and we can extend $\nabla$ to higher twisted exterior derivatives

$\nabla:\Gamma(B,{\mathscr{C}}^{\infty}(\Lambda^{k}_{B}\otimes{\mathfrak{H}}))\to\Gamma(B,{\mathscr{C}}^{\infty}(\Lambda^{k+1}_{B}\otimes{\mathfrak{H}}))$

in the usual way. A connection is not ${\mathscr{C}}^{\infty}(B)$-linear, but
the difference of two connections is. And if the difference
$\nabla_{2}-\nabla_{1}$ is also tensorial then

$\nabla_{2}-\nabla_{1}\in\Gamma(B,{\mathscr{C}}^{\infty}(\Lambda^{1}_{B}\otimes{\rm
Lin}({\mathfrak{H}},{\mathfrak{H}}))).$

Similarly, if the curvature $\nabla\nabla$ of a connection $\nabla$ is
tensorial then it can be identified with a $2$-form with values in ${\rm
Lin}({\mathfrak{H}},{\mathfrak{H}})$:

$\Theta(\nabla)\in\Gamma(B,{\mathscr{C}}^{\infty}(\Lambda^{2}_{B}\otimes{\rm
Lin}({\mathfrak{H}},{\mathfrak{H}}))).$

As noted in Proposition 2.11, this is the case for metric connections.

###  2.6. Smooth metrics

######  Definition 2.15.

Let ${\mathfrak{H}}\to B$ be a quasi-Hilbert field.

  1. i.

A metric ${\mathfrak{h}}$ for ${\mathfrak{H}}\to B$ is an assignment to each
fiber ${\mathfrak{H}}_{t}$ of a complete inner product ${\mathfrak{h}}_{t}$ so
that $({\mathfrak{H}}_{t},{\mathfrak{h}}_{t})$ and ${\mathfrak{H}}_{t}$ are
quasi-isometric for all $t\in B$.

  2. ii.

If ${\mathscr{C}}^{\infty}({\mathfrak{H}})\to B$ is a smooth structure for
${\mathfrak{H}}\to B$ then the metric ${\mathfrak{h}}$ is _smooth_ if

${\mathfrak{h}}({\mathfrak{f}}_{1},{\mathfrak{f}}_{2})\in{\mathscr{C}}^{\infty}(U)$

for any smooth sections
${\mathfrak{f}}_{1},{\mathfrak{f}}_{2}\in\Gamma(U,{\mathscr{C}}^{\infty}({\mathfrak{H}}))$.
$\diamond$

###  2.7. Smooth (quasi-)Hilbert fields

######  Definition 2.16.

  1. (1)

A _smooth quasi-Hilbert field_ is a quasi-Hilbert field with a smooth
structure that admits a connection.

  2. (2)

A smooth quasi-Hilbert field ${\mathfrak{H}}\to
B,{\mathscr{C}}^{\infty}({\mathfrak{H}})$ over $B$ is said to be a _smooth
Hilbert field_ if it has a connection $\mathring{\nabla}$, called a _reference
connection_ , and a smooth metric ${\mathfrak{h}}$ with the following
property. For each $t\in B$, $\xi\in T_{B,t}$ and
${\mathfrak{f}}\in{\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}$ there is a
constant $C=C(t,\xi,{\mathfrak{f}})$ such that

(7)
$\left|(\mathring{\nabla}_{\xi}{\mathfrak{h}})({\mathfrak{f}},{\mathfrak{g}}))(t)\right|^{2}\leq
C{\mathfrak{h}}({\mathfrak{g}},{\mathfrak{g}})(t).$

for each ${\mathfrak{g}}\in{\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}$.
$\diamond$

######  Proposition 2.17.

A smooth Hilbert field $({\mathfrak{H}},{\mathfrak{h}})$ has a metric
compatible connection.

###### Proof.

Let $\mathring{\nabla}$ be a reference connection. Any other connection is
related to $\mathring{\nabla}$ by some ${\mathscr{C}}^{\infty}$-linear map of
sheaves
$\mathbf{A}:{\mathscr{C}}^{\infty}({\mathfrak{H}})\to{\mathscr{C}}^{\infty}(T^{*}_{B}\otimes{\mathfrak{H}})$,
i.e., it is of the form $\nabla^{\mathbf{A}}:=\mathring{\nabla}+\mathbf{A}$.
To find a metric compatible connection, one needs to find $\mathbf{A}$
satisfying $\nabla^{\mathbf{A}}{\mathfrak{h}}=0$ (c.f. definition (6)), or
equivalently

$\mathring{\nabla}{\mathfrak{h}}({\mathfrak{f}},{\mathfrak{g}})={\mathfrak{h}}(\mathbf{A}{\mathfrak{f}},{\mathfrak{g}})+{\mathfrak{h}}({\mathfrak{f}},\mathbf{A}{\mathfrak{g}}).$

First, by (7) we see that for each
${\mathfrak{f}}\in{\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}$ there exists
$\mathbf{A}{\mathfrak{f}}\in{\mathscr{C}}^{\infty}(T^{*}_{B}\otimes{\mathfrak{H}})_{t}$
such that

$\tfrac{1}{2}\mathring{\nabla}{\mathfrak{h}}({\mathfrak{f}},{\mathfrak{g}})={\mathfrak{h}}(\mathbf{A}{\mathfrak{f}},{\mathfrak{g}}).$

Since
$\overline{\mathring{\nabla}{\mathfrak{h}}({\mathfrak{f}},{\mathfrak{g}})}=\mathring{\nabla}{\mathfrak{h}}({\mathfrak{g}},{\mathfrak{f}})$,
we see that if ${\mathfrak{g}}\in{\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}$
then

$\left|(\mathring{\nabla}_{\xi}{\mathfrak{h}})({\mathfrak{f}},{\mathfrak{g}}))(t)\right|^{2}\lesssim{\mathfrak{h}}({\mathfrak{f}},{\mathfrak{f}})(t)$

for every ${\mathfrak{f}}\in{\rm Domain}(\mathbf{A})$. Thus
${\mathfrak{g}}\in{\rm Domain}(\mathbf{A}^{\dagger})$, the Hilbert space
adjoint of $\mathbf{A}$, and moreover

${\mathfrak{h}}(\mathbf{A}{\mathfrak{f}},{\mathfrak{g}})={\mathfrak{h}}({\mathfrak{f}},\mathbf{A}^{\dagger}{\mathfrak{g}})$

for all
${\mathfrak{f}},{\mathfrak{g}}\in{\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}$.
And we also have

${\mathfrak{h}}(\mathbf{A}{\mathfrak{f}},{\mathfrak{g}})=\mathring{\nabla}{\mathfrak{h}}({\mathfrak{f}},{\mathfrak{g}})=\tfrac{1}{2}\overline{\mathring{\nabla}{\mathfrak{h}}({\mathfrak{g}},{\mathfrak{f}})}=\tfrac{1}{2}\overline{{\mathfrak{h}}(\mathbf{A}{\mathfrak{g}},{\mathfrak{f}})}={\mathfrak{h}}({\mathfrak{f}},\mathbf{A}{\mathfrak{g}}),$

so that in fact $\mathbf{A}$ is self-adjoint on its domain
${\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}$. Therefore

${\mathfrak{h}}(\mathbf{A}{\mathfrak{f}},{\mathfrak{g}})+{\mathfrak{h}}({\mathfrak{f}},\mathbf{A}{\mathfrak{g}})={\mathfrak{h}}(2\mathbf{A}{\mathfrak{f}},{\mathfrak{g}})=\mathring{\nabla}{\mathfrak{h}}({\mathfrak{f}},{\mathfrak{g}}),$

and the proof is complete. ∎

######  Remark 2.18.

Note that, conversely, the estimates (7) on $\mathring{\nabla}$ and
${\mathfrak{h}}$ are necessary for the existence of $\mathbf{A}$. Moreover,
the ${\mathfrak{h}}$-Hermitian operator $\mathbf{A}$ must be tensorial. There
may be other solutions; but all of them must be of the form
$\mathbf{A}+\mathbf{B}$ such that $\mathbf{B}^{\dagger}=-\mathbf{B}$, with
appropriately defined domains which we will not discuss here. $\diamond$

###  2.8. Holomorphic Quasi-Hilbert fields and BLS fields

Since we are interested in the situation in which the base is a complex
manifold, we consider the obvious complexification of the notion of
connection: In Definition 2.9 one allows the functions
$f\in{\mathscr{C}}^{\infty}(B)$ to be complex valued, and for a complex vector
field $\xi\in\Gamma(B,{\mathscr{C}}^{\infty}(T_{B}\otimes{\mathbb{C}}))$ one
defines

$\nabla_{\xi}:=\nabla_{{\rm Re\ }\xi}+\sqrt{\text{\rm-1}}\nabla_{{\rm Im\
}\xi}.$

These notions, which just use the natural complex structures of the targets,
only require $B$ to be a smooth manifold. But if $B$ is a complex manifold (or
more generally an almost complex manifold) then there is a splitting
$T_{B}\otimes{\mathbb{C}}=T^{1,0}_{B}\oplus T^{0,1}_{B}$. Thus one has a
decomposition

$\nabla=\nabla^{1,0}+\nabla^{0,1}$

induced by the splitting

$\Gamma(B,{\mathscr{C}}^{\infty}(\Lambda^{k+1}_{B}\otimes{\mathfrak{H}}))=\Gamma(B,{\mathscr{C}}^{\infty}(\Lambda^{1,0}_{B}\wedge\Lambda^{k}_{B}\otimes{\mathfrak{H}}))\oplus\Gamma(B,{\mathscr{C}}^{\infty}(\Lambda^{0,1}_{B}\wedge\Lambda^{k}_{B}\otimes{\mathfrak{H}})),$

and one can talk about ${\mathfrak{H}}$-valued $(p,q)$-forms as sections of
$\Lambda^{p,q}_{B}\otimes{\mathfrak{H}}\to B$. Such forms are smooth if they
are smooth as ${\mathfrak{H}}$-valued $(p+q)$-forms. For $0\leq
p,q\leq\dim_{{\mathbb{C}}}B$ the set of smooth ${\mathfrak{H}}$-valued
$(p,q)$-forms
$\Gamma(B,\Lambda^{p,q}_{B}\otimes{\mathfrak{H}})\cap\Gamma(B,{\mathscr{C}}^{\infty}(\Lambda^{p+q}_{B}\otimes{\mathfrak{H}}))$
shall be denoted
$\Gamma(B,{\mathscr{C}}^{\infty}(\Lambda^{p,q}_{B}\otimes{\mathfrak{H}}))$.

######  Definition 2.19.

Let $B$ be an almost complex manifold and let ${\mathfrak{H}}\to B$ be a
smooth quasi-Hilbert field.

  1. i.

An almost complex structure for ${\mathfrak{H}}\to B$ is a family of linear
maps of sheaves

$\bar{\partial}:{\mathscr{C}}^{\infty}(\Lambda^{p,q}_{B}\otimes{\mathfrak{H}})\to{\mathscr{C}}^{\infty}(\Lambda^{p,q+1}_{B}\otimes{\mathfrak{H}}),\quad
0\leq p,q\leq n,$

compatible with the $\bar{\partial}$-operator on $B$ in the sense that

$\bar{\partial}(f\wedge{\mathfrak{b}})=\bar{\partial}f\wedge{\mathfrak{b}}+(-1)^{p^{\prime}+q^{\prime}}f\wedge\bar{\partial}{\mathfrak{b}},\quad
f\in{\mathscr{C}}^{\infty}(\Lambda^{p^{\prime},q^{\prime}}_{B}).$

The almost complex structure is _involutive_ or _complex_ or _integrable_ if
$\bar{\partial}\bar{\partial}=0$.

  2. ii.

A smooth section ${\mathfrak{f}}$ of the smooth quasi-Hilbert field
${\mathfrak{H}}\to B$ with almost complex structure $\bar{\partial}$ is said
to be _holomorphic_ on an open subset $U\subset B$ if
$\bar{\partial}{\mathfrak{f}}=0$ on $U$. The set of holomorphic sections on
$U$ is denoted $\Gamma(U,{\mathcal{O}}({\mathfrak{H}}))$.

  3. iii.

An _(almost) holomorphic quasi-Hilbert field_ 111Since we are now working with
complex connections, when we talk about a smooth Hilbert field, in Definition
2.16.b we replace the vectors $\xi\in T_{B,t}$ with vectors vectors $\xi\in
T_{B,t}\otimes{\mathbb{C}}$. is a smooth quasi-Hilbert field with an (almost)
complex structure.

  4. iv.

A _Berndtsson-Lempert-Szőke_ (BLS) field is an almost holomorphic smooth
Hilbert field admitting a reference connection $\mathring{\nabla}$ such that
$\mathring{\nabla}^{0,1}=\bar{\partial}$.

  5. v.

A BLS field is said to be _integrable_ if its almost complex structure is
integrable.

######  Remark 2.20.

If ${\mathfrak{H}}$ is an almost holomorphic smooth Hilbert field then
${\mathfrak{H}}$ is BLS if and only if for any reference connection
$\mathring{\nabla}$ the map of sheaves
$\mathring{\nabla}^{0,1}-\bar{\partial}$ is tensorial and the associated map
of Hilbert fields is a morphism. $\diamond$

######  Proposition 2.21.

On every BLS field $({\mathfrak{H}},{\mathfrak{h}},\bar{\partial})\to B$ there
exists a unique connection $\nabla$, called the _BLS Chern connection_ , or
simply the _Chern connection_ , such that

$\nabla{\mathfrak{h}}=0\quad\text{and}\quad\nabla^{0,1}=\bar{\partial}.$

###### Proof.

Let $\mathring{\nabla}=\mathring{\nabla}^{1,0}+\bar{\partial}$ be the
reference connection of the underlying quasi-Hilbert field. We need to find a
linear map of sheaves
$\mathbf{A}:{\mathscr{C}}^{\infty}({\mathfrak{H}})\to{\mathscr{C}}^{\infty}(T^{*1,0}_{B}\otimes{\mathfrak{H}})$
such that the connection $\nabla^{\mathbf{A}}:=\mathring{\nabla}+\mathbf{A}$
is metric compatible, i.e., satisfies $\nabla_{\mathbf{A}}{\mathfrak{h}}=0$.
Our assumption is that for each $\xi\in T_{B,t}\otimes{\mathbb{C}}$ and each
${\mathfrak{f}}\in{\mathscr{C}}^{\infty}({\mathfrak{H}})$ the linear
functional $(\mathring{\nabla}_{\xi}{\mathfrak{h}})({\mathfrak{f}},\cdot)$ is
bounded. In particular, if we take $\xi\in T^{1,0}_{B,t}$, we find that the
linear functional
$(\mathring{\nabla}_{\xi}{\mathfrak{h}})({\mathfrak{f}},\cdot)$ defined by

$(\mathring{\nabla}_{\xi}{\mathfrak{h}})({\mathfrak{f}},{\mathfrak{g}})=\partial_{\xi}({\mathfrak{h}}({\mathfrak{f}},{\mathfrak{g}}))-{\mathfrak{h}}(\mathring{\nabla}^{1,0}_{\xi}{\mathfrak{f}},{\mathfrak{g}})-{\mathfrak{h}}({\mathfrak{f}},\bar{\partial}_{\bar{\xi}}{\mathfrak{g}})$

is bounded. Hence there exists a linear map of sheaves
$\mathbf{A}:{\mathscr{C}}^{\infty}({\mathfrak{H}})\to{\mathscr{C}}^{\infty}(T^{*1,0}_{B}\otimes{\mathfrak{H}})$
such that

$\partial_{\zeta}({\mathfrak{h}}({\mathfrak{f}},{\mathfrak{g}}))-{\mathfrak{h}}(\mathring{\nabla}^{1,0}_{\zeta}{\mathfrak{f}},{\mathfrak{g}})-{\mathfrak{h}}({\mathfrak{f}},\bar{\partial}_{\bar{\zeta}}{\mathfrak{g}})={\mathfrak{h}}(\left<\mathbf{A},\zeta\right>{\mathfrak{f}},{\mathfrak{g}})\quad\text{
for all }\zeta\in T^{1,0}_{B,t}.$

Letting $\nabla:=\mathring{\nabla}^{1,0}+\mathbf{A}+\bar{\partial}$, for
$T_{B,t}\ni\xi=\eta+\bar{\eta}\in T_{B,t}^{1,0}\oplus
T_{B,t}^{0,1}=T_{B,t}\otimes{\mathbb{C}}$ and
${\mathfrak{f}},{\mathfrak{g}}\in{\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}$
we compute that

$\displaystyle\nabla_{\xi}{\mathfrak{h}}({\mathfrak{f}},{\mathfrak{g}})$
$\displaystyle=$ $\displaystyle
d_{\xi}({\mathfrak{h}}({\mathfrak{f}},{\mathfrak{g}}))-{\mathfrak{h}}(\nabla_{\xi}{\mathfrak{f}},{\mathfrak{g}})-{\mathfrak{h}}({\mathfrak{f}},\nabla_{\xi}{\mathfrak{g}})$
$\displaystyle=$
$\displaystyle\partial_{\zeta}({\mathfrak{h}}({\mathfrak{f}},{\mathfrak{g}}))+\bar{\partial}_{\bar{\zeta}}({\mathfrak{h}}({\mathfrak{f}},{\mathfrak{g}}))-{\mathfrak{h}}(\nabla^{1,0}_{\zeta}{\mathfrak{f}},{\mathfrak{g}})-{\mathfrak{h}}(\bar{\partial}_{\bar{\zeta}}{\mathfrak{f}},{\mathfrak{g}})-{\mathfrak{h}}({\mathfrak{f}},\nabla^{1,0}_{\zeta}{\mathfrak{g}})-{\mathfrak{h}}({\mathfrak{f}},\bar{\partial}_{\bar{\zeta}}{\mathfrak{g}})$
$\displaystyle=$
$\displaystyle\partial_{\zeta}({\mathfrak{h}}({\mathfrak{f}},{\mathfrak{g}}))-{\mathfrak{h}}(\nabla^{1,0}_{\zeta}{\mathfrak{f}},{\mathfrak{g}})-{\mathfrak{h}}({\mathfrak{f}},\bar{\partial}_{\bar{\zeta}}{\mathfrak{g}})+\overline{\partial_{\zeta}({\mathfrak{h}}({\mathfrak{g}},{\mathfrak{f}}))-{\mathfrak{h}}(\nabla^{1,0}_{\zeta}{\mathfrak{g}},{\mathfrak{f}})-{\mathfrak{h}}({\mathfrak{g}},\bar{\partial}_{\bar{\zeta}}{\mathfrak{f}})}$
$\displaystyle=$
$\displaystyle\partial_{\zeta}({\mathfrak{h}}({\mathfrak{f}},{\mathfrak{g}}))-{\mathfrak{h}}(\mathring{\nabla}^{1,0}_{\zeta}{\mathfrak{f}},{\mathfrak{g}})-{\mathfrak{h}}({\mathfrak{f}},\bar{\partial}_{\bar{\zeta}}{\mathfrak{g}})-{\mathfrak{h}}\left(\left<\mathbf{A},\zeta\right>{\mathfrak{f}},{\mathfrak{g}}\right)$
$\displaystyle\qquad+\overline{\partial_{\zeta}({\mathfrak{h}}({\mathfrak{g}},{\mathfrak{f}}))-{\mathfrak{h}}(\nabla^{1,0}_{\zeta}{\mathfrak{g}},{\mathfrak{f}})-{\mathfrak{h}}({\mathfrak{g}},\bar{\partial}_{\bar{\zeta}}{\mathfrak{f}})-{\mathfrak{h}}\left(\left<\mathbf{A},\zeta\right>{\mathfrak{g}},{\mathfrak{f}}\right)}$
$\displaystyle=0.$

Therefore $\nabla:=\mathring{\nabla}+\mathbf{A}$ is the Chern connection.
Moreover, we have shown that

(8)
${\mathfrak{h}}(\nabla^{1,0}{\mathfrak{b}}_{1},{\mathfrak{b}}_{2})=\partial({\mathfrak{h}}({\mathfrak{b}}_{1},{\mathfrak{b}}_{2}))-{\mathfrak{h}}({\mathfrak{b}}_{1},\bar{\partial}{\mathfrak{b}}_{2})$

on all smooth ${\mathfrak{H}}$-valued forms. Since the right hand side does
not depend on the connection, we see that the Chern connection is unique. ∎

######  Remark 2.22.

In view of the uniqueness of the Chern connection, we shall often refer to the
BLS field $({\mathfrak{H}},{\mathfrak{h}})$ and omit explicit reference to the
connection. Often we even omit reference to the metric when the latter is
clear from the context. $\diamond$

######  Proposition 2.23.

The curvature $\Theta({\mathfrak{h}})$ of the Chern connection of a BLS field
$({\mathfrak{H}},{\mathfrak{h}},\bar{\partial})\to B$ is a smooth ${\rm
Lin}({\mathfrak{H}},{\mathfrak{H}})$-valued $2$-form that is
${\mathfrak{h}}$-anti-Hermitian. Moreover, if $\bar{\partial}\bar{\partial}=0$
then the curvature is a ${\rm Lin}({\mathfrak{H}},{\mathfrak{H}})$-valued
$(1,1)$-form.

###### Proof.

By Proposition 2.11 $\Theta({\mathfrak{h}})$ is tensorial. If
${\mathfrak{b}}_{1}$ and ${\mathfrak{b}}_{2}$ are smooth $k$-forms then

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
dd{\mathfrak{h}}({\mathfrak{b}}_{1},{\mathfrak{b}}_{2})=d\left({\mathfrak{h}}(\nabla{\mathfrak{b}}_{1},{\mathfrak{b}}_{2})+(-1)^{k}{\mathfrak{h}}({\mathfrak{b}}_{1},\nabla{\mathfrak{b}}_{2})\right)$
$\displaystyle=$
$\displaystyle{\mathfrak{h}}(\Theta({\mathfrak{h}}){\mathfrak{b}}_{1},{\mathfrak{b}}_{2})+(-1)^{k+1}{\mathfrak{h}}(\nabla{\mathfrak{b}}_{1},\nabla{\mathfrak{b}}_{2})+(-1)^{k}{\mathfrak{h}}(\nabla{\mathfrak{b}}_{1},\nabla{\mathfrak{b}}_{2})+{\mathfrak{h}}({\mathfrak{b}}_{1},\Theta({\mathfrak{h}}){\mathfrak{b}}_{2})$
$\displaystyle=$
$\displaystyle{\mathfrak{h}}(\Theta({\mathfrak{h}}){\mathfrak{b}}_{1},{\mathfrak{b}}_{2})+{\mathfrak{h}}({\mathfrak{b}}_{1},\Theta({\mathfrak{h}}){\mathfrak{b}}_{2}),$

so every metric-compatible connection has anti-Hermitian curvature. In
particular, the $(0,2)$-component of a metric-compatible connection is the
Hermitian conjugate of the $(2,0)$-component, so if also
$\bar{\partial}\bar{\partial}=0$ then $\nabla^{1,0}\nabla^{1,0}=0$. ∎

###  2.9. Smooth (Quasi-)Hilbert Subfields

######  Definition 2.24.

Let $({\mathfrak{L}},{\mathfrak{h}}^{{\mathfrak{L}}})\to B$ be a smooth
Hilbert field.

  1. i.

A smooth Hilbert subfield of
$({\mathfrak{L}},{\mathfrak{h}}^{{\mathfrak{L}}})$ is a smooth Hilbert field
$({\mathfrak{H}},{\mathfrak{h}}^{{\mathfrak{H}}})$ such that

    1. a)

${\mathfrak{H}}_{t}$ is a closed subspace of ${\mathfrak{L}}_{t}$ for every
$t\in B$,

    2. b)

${\mathscr{C}}^{\infty}({\mathfrak{H}})=\\{{\mathfrak{f}}\in{\mathscr{C}}^{\infty}({\mathfrak{L}}))\
;\ {\mathfrak{f}}(t)\in{\mathfrak{H}}_{t}\text{ for all }t\in{\rm
Domain}({\mathfrak{f}})\\}$, and

    3. c)

${\mathfrak{h}}^{{\mathfrak{L}}}|_{{\mathfrak{H}}}={\mathfrak{h}}^{{\mathfrak{H}}}$.

  2. ii.

To any smooth Hilbert subfield ${\mathfrak{H}}\subset{\mathfrak{L}}$ there is
associated the _orthogonal projector_ : the unique morphism
$P=P_{{\mathfrak{H}}}:{\mathfrak{L}}\to{\mathfrak{H}}$ defined by fiberwise
orthogonal projection.

  3. iii.

We say that the smooth Hilbert subfield ${\mathfrak{H}}\subset{\mathfrak{L}}$
is _regular_ if $P_{{\mathfrak{H}}}:{\mathfrak{L}}\to{\mathfrak{H}}$ is
smooth.

######  Remark 2.25.

A smooth quasi-Hilbert subfield ${\mathfrak{H}}\subset{\mathfrak{L}}$ is a
smooth quasi-Hilbert field that satisfies (i.a) and (i.b) of Definition 2.24.
$\diamond$

######  Remark 2.26.

To any smooth Hilbert subfield ${\mathfrak{H}}\subset{\mathfrak{L}}$ one can
associate its orthogonal complement ${\mathfrak{H}}^{\perp}$. In general
${\mathfrak{H}}^{\perp}$ need not be a smooth Hilbert subfield of
${\mathfrak{L}}$, but this is certainly the case if ${\mathfrak{H}}$ is
regular. $\diamond$

Let ${\mathfrak{L}}\to B$ be a smooth (quasi-)Hilbert and
${\mathfrak{H}}\subset{\mathfrak{L}}$ a smooth (quasi-)Hilbert subfield.
Suppose both fields are equipped with connections $\nabla^{{\mathfrak{L}}}$
and $\nabla^{{\mathfrak{H}}}$. One can consider the map

$\mathbf{N}^{{\mathfrak{L}}/{\mathfrak{H}}}:=\nabla^{{\mathfrak{L}}}|_{{\mathfrak{H}}}-\nabla^{{\mathfrak{H}}}:{\mathscr{C}}^{\infty}({\mathfrak{H}})\to{\mathscr{C}}^{\infty}(\Lambda^{1}_{B}\otimes{\mathfrak{L}}),$

which depends of course on the choice of connections. One computes that

$\mathbf{N}^{{\mathfrak{L}}/{\mathfrak{H}}}(f{\mathfrak{f}})=\nabla^{{\mathfrak{L}}}(f{\mathfrak{f}})-\nabla^{{\mathfrak{H}}}(f{\mathfrak{f}})=df\otimes{\mathfrak{f}}+f\nabla^{{\mathfrak{L}}}{\mathfrak{f}}-(df\otimes{\mathfrak{f}}+f\nabla^{{\mathfrak{H}}}{\mathfrak{f}})=f\mathbf{N}^{{\mathfrak{L}}/{\mathfrak{H}}}{\mathfrak{f}}\quad\text{
for all }{\mathfrak{f}}\in{\mathscr{C}}^{\infty}({\mathfrak{H}}),$

which shows that $\mathbf{N}^{{\mathfrak{L}}/{\mathfrak{H}}}$ is a linear map
of sheaves.

######  Definition 2.27.

The linear map of sheaves $\mathbf{N}^{{\mathfrak{L}}/{\mathfrak{H}}}$ is
called the _second fundamental map_ of the (quasi)-Hilbert fields with
connections.

######  Proposition 2.28.

If the connections $\nabla^{{\mathfrak{L}}}$ and $\nabla^{{\mathfrak{H}}}$ are
metric-compatible then the map $\mathbf{N}^{{\mathfrak{L}}/{\mathfrak{H}}}$ is
tensorial.

###### Proof.

If
${\mathfrak{f}},{\mathfrak{g}}\in{\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}$
and ${\mathfrak{f}}(t)=0$ then

${\mathfrak{h}}(\mathbf{N}^{{\mathfrak{L}}/{\mathfrak{H}}}{\mathfrak{f}},{\mathfrak{g}})={\mathfrak{h}}((\nabla^{{\mathfrak{L}}}-\nabla^{{\mathfrak{H}}}){\mathfrak{f}},{\mathfrak{g}})={\mathfrak{h}}({\mathfrak{f}},(\nabla^{{\mathfrak{L}}}-\nabla^{{\mathfrak{H}}}){\mathfrak{g}}),$

and the right hand side vanishes at $t$. Therefore by density
$(\mathbf{N}^{{\mathfrak{L}}/{\mathfrak{H}}}{\mathfrak{f}})(t)=0$. ∎

Therefore one has a densely defined map of Hilbert fields
$\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}:{\mathfrak{H}}\to\Lambda^{1}_{B}\otimes{\mathfrak{L}}$
such that

$\mathbf{N}^{{\mathfrak{L}}/{\mathfrak{H}}}{\mathfrak{f}}=\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}{\mathfrak{f}}\quad\text{for
all }{\mathfrak{f}}\in{\mathscr{C}}^{\infty}({\mathfrak{H}}).$

######  Definition 2.29.

The map of Hilbert fields $\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}$ is
called the _second fundamental form_ of the Hilbert fields with metric-
compatible connections $({\mathfrak{L}},\nabla^{{\mathfrak{L}}})$ and
$({\mathfrak{H}},\nabla^{{\mathfrak{H}}})$.

If ${\mathfrak{H}}\subset{\mathfrak{L}}$ is a regular Hilbert subfield then
any connection $\nabla^{{\mathfrak{L}}}$ for ${\mathfrak{L}}$ defines a
connection
$\nabla^{{\mathfrak{H}}}:=P_{{\mathfrak{H}}}\nabla^{{\mathfrak{L}}}$. In this
case,
$\mathbf{N}^{{\mathfrak{L}}/{\mathfrak{H}}}=P_{{\mathfrak{H}}}^{\perp}\nabla^{{\mathfrak{L}}}$.
And if $\nabla^{{\mathfrak{L}}}$ is metric-compatible then for any
${\mathfrak{f}},{\mathfrak{g}}\in{\mathscr{C}}^{\infty}({\mathfrak{H}})$ one
has

${\mathfrak{h}}(\nabla^{{\mathfrak{H}}}{\mathfrak{f}},{\mathfrak{g}})={\mathfrak{h}}(\nabla^{{\mathfrak{L}}}{\mathfrak{f}},{\mathfrak{g}})=d({\mathfrak{h}}({\mathfrak{f}},{\mathfrak{g}}))-{\mathfrak{h}}({\mathfrak{f}},\nabla^{{\mathfrak{L}}}{\mathfrak{g}})=d({\mathfrak{h}}({\mathfrak{f}},{\mathfrak{g}}))-{\mathfrak{h}}({\mathfrak{f}},\nabla^{{\mathfrak{H}}}{\mathfrak{g}}),$

so that $\nabla^{{\mathfrak{H}}}$ is also metric compatible. Therefore
$\mathbf{N}^{{\mathfrak{L}}/{\mathfrak{H}}}$ is tensorial, and one has a
second fundamental form $\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}$,
which moreover takes its values in
$\Lambda^{1}_{B}\otimes{\mathfrak{H}}^{\perp}$.

######  Remark 2.30.

In the case of our main examples ${\mathscr{H}}\subset{\mathscr{L}}$ discussed
in the introduction, if the Hilbert field ${\mathscr{H}}$ is smooth then it is
a regular smooth Hilbert subfield of ${\mathscr{L}}$. This fact is a special
case of a result of Kodaira and Spencer, to the effect that smoothness of the
harmonic projections is equivalent to constancy of the dimensions of the
harmonic spaces. The latter in turn means that the harmonic spaces fit
together to form a vector bundle, and in the case of harmonic $(p,0)$-forms,
which are automatically holomorphic, this vector bundle is even holomorphic.
Thus for ${\mathscr{H}}$, which is the special case $p=n$, Kodaira-Spencer
tells us that if ${\mathscr{H}}$ is smooth then ${\mathscr{H}}$ is a
holomorphic vector bundle, and then the projection $P_{{\mathscr{H}}}$ is
smooth. $\diamond$

###  2.10. BLS Subfields and the Gauss-Griffiths Formula

######  Definition 2.31.

Let ${\mathfrak{L}}\to B,\bar{\partial}^{{\mathfrak{L}}}$ be an almost
holomorphic quasi-Hilbert field.

  1. i.

An almost holomorphic quasi-Hilbert field ${\mathfrak{H}}\to
B,\bar{\partial}^{{\mathfrak{H}}}$ is said to be a _subfield_ of
${\mathfrak{L}}$ if

    1. a)

${\mathfrak{H}}_{t}$ is a closed subspace of ${\mathfrak{L}}_{t}$ for every
$t\in B$,

    2. b)

${\mathscr{C}}^{\infty}({\mathfrak{H}})=\\{{\mathfrak{f}}\in{\mathscr{C}}^{\infty}({\mathfrak{L}}))\
;\ {\mathfrak{f}}(t)\in{\mathfrak{H}}_{t}\text{ for all }t\in{\rm
Domain}({\mathfrak{f}})\\}$, and

    3. c)

$\bar{\partial}^{{\mathfrak{L}}}|_{{\mathscr{C}}^{\infty}({\mathfrak{H}})}=\bar{\partial}^{{\mathfrak{H}}}$.

  2. ii.

A subfield ${\mathfrak{H}}\to B$ of ${\mathfrak{L}}\to B$ is said to be a _BLS
subfield_ of ${\mathfrak{L}}$ if
$P_{{\mathfrak{H}}}{\mathscr{C}}^{\infty}({\mathfrak{L}})\subset{\mathscr{C}}^{\infty}({\mathfrak{H}})$.

######  Remark.

The inclusions
$P_{{\mathfrak{H}}}{\mathscr{C}}^{\infty}({\mathfrak{L}})\subset{\mathscr{C}}^{\infty}({\mathfrak{H}})\subset{\mathscr{C}}^{\infty}({\mathfrak{L}})$
imply that ${\mathfrak{H}}\subset{\mathfrak{L}}$ is a regular subfield and
that
$P_{{\mathfrak{H}}}{\mathscr{C}}^{\infty}({\mathfrak{L}})={\mathscr{C}}^{\infty}({\mathfrak{H}})$.
In particular, a smooth section of ${\mathfrak{L}}$ that takes all its values
in ${\mathfrak{H}}$ is a smooth section of ${\mathfrak{H}}$.

Conversely, if ${\mathfrak{L}}$ is a BLS field and
${\mathfrak{H}}\subset{\mathfrak{L}}$ is a regular subfield such that the
subspace $\\{(P_{{\mathfrak{H}}}{\mathfrak{f}})(p)\ ;\
{\mathfrak{f}}\in{\mathscr{C}}^{\infty}({\mathfrak{L}})_{p}\\}\subset{\mathfrak{H}}_{p}$
is dense for every $p\in B$ then ${\mathfrak{H}}\subset{\mathfrak{L}}$ is a
BLS subfield. $\diamond$

######  Proposition 2.32.

If ${\mathfrak{H}}\subset{\mathfrak{L}}$ is a BLS subfield then the BLS Chern
connection for ${\mathfrak{H}}\to B$ is
$\nabla^{{\mathfrak{H}}}=P_{{\mathfrak{H}}}\nabla^{{\mathfrak{L}}}|_{{\mathfrak{H}}}$,
where $\nabla^{{\mathfrak{L}}}|_{{\mathfrak{H}}}$ denotes the restriction of
the connection $\nabla^{{\mathfrak{L}}}$ to sections of ${\mathfrak{H}}$.

###### Proof.

If ${\mathfrak{f}}_{1},{\mathfrak{f}}_{2}\in\Gamma(B,{\mathfrak{H}})$ then

$\displaystyle\partial{\mathfrak{h}}^{{\mathfrak{H}}}({\mathfrak{f}}_{1},{\mathfrak{f}}_{2})-{\mathfrak{h}}^{{\mathfrak{H}}}({\mathfrak{f}}_{1},\bar{\partial}^{{\mathfrak{H}}}{\mathfrak{f}}_{2})$
$\displaystyle=$
$\displaystyle\partial{\mathfrak{h}}^{{\mathfrak{L}}}({\mathfrak{f}}_{1},{\mathfrak{f}}_{2})-{\mathfrak{h}}^{{\mathfrak{L}}}({\mathfrak{f}}_{1},\bar{\partial}^{{\mathfrak{L}}}{\mathfrak{f}}_{2})$
$\displaystyle=$
$\displaystyle{\mathfrak{h}}^{{\mathfrak{L}}}(\nabla^{{\mathfrak{L}}1,0}{\mathfrak{f}}_{1},{\mathfrak{f}}_{2})={\mathfrak{h}}^{{\mathfrak{L}}}(P_{{\mathfrak{H}}}\nabla^{{\mathfrak{L}}1,0}{\mathfrak{f}}_{1},{\mathfrak{f}}_{2})={\mathfrak{h}}^{{\mathfrak{H}}}(P_{{\mathfrak{H}}}\nabla^{{\mathfrak{L}}1,0}{\mathfrak{f}}_{1},{\mathfrak{f}}_{2}),$

and since (8) uniquely determines the Chern connection, the result follows. ∎

By property (c) of Definition 2.31 one also sees that

$\mathbf{N}^{{\mathfrak{L}}/{\mathfrak{H}}}=P^{\perp}_{{\mathfrak{H}}}\nabla^{{\mathfrak{L}}1,0}|_{{\mathfrak{H}}}.$

Consequently the second fundamental map
$\mathbf{N}^{{\mathfrak{L}}/{\mathfrak{H}}}$ of a BLS subfield
${\mathfrak{H}}\subset{\mathfrak{L}}$ is tensorial, and its associated second
fundamental form $\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}$ takes values
in $\Lambda^{1,0}_{B}\otimes{\mathfrak{H}}^{\perp}$.

######  Proposition 2.33 (Gauss-Griffiths Formula).

Let ${\mathfrak{L}}$ be a BLS field and let
${\mathfrak{H}}\subset{\mathfrak{L}}$ be a BLS subfield.

  1. a.

For any smooth sections
${\mathfrak{f}}_{1},{\mathfrak{f}}_{2}\in\Gamma(B,{\mathscr{C}}^{\infty}({\mathfrak{H}}))$
and any $(1,0)$-vectors $\xi,\eta\in T^{1,0}_{B,t}$, $t\in B$ one has

(9)
${\mathfrak{h}}(\Theta(\nabla^{{\mathfrak{L}}})(\xi,\bar{\eta}){\mathfrak{f}}_{1},{\mathfrak{f}}_{2})={\mathfrak{h}}(\Theta(\nabla^{{\mathfrak{H}}})(\xi,\bar{\eta}){\mathfrak{f}}_{1},{\mathfrak{f}}_{2})+{\mathfrak{h}}(\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}(\xi){\mathfrak{f}}_{1},\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}(\eta){\mathfrak{f}}_{2})$

  2. b.

If moreover the quadratic form

${\mathbb{G}}_{\xi\bar{\eta}}(f_{1},f_{2}):={\mathfrak{h}}(\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}(\xi)f_{1},\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}(\eta)f_{2})$

defines a bounded conjugate-linear functional
${\mathbb{G}}_{\xi,\bar{\eta}}(f,\cdot)$ for every $t\in B$, every
$f\in{\mathfrak{H}}_{t}$ and every pair of $(1,0)$-vectors $\xi,\eta\in
T^{1,0}_{B,t}$, then

(10)
$\Theta(\nabla^{{\mathfrak{L}}})|_{{\mathfrak{H}}}=\Theta(\nabla^{{\mathfrak{H}}})-(\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}})^{\dagger}\wedge\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}.$

###### Proof.

Fix
${\mathfrak{f}}_{1},{\mathfrak{f}}_{2}\in\Gamma(B,{\mathscr{C}}^{\infty}({\mathfrak{H}}))$.
Then by orthogonality

$\displaystyle{\mathfrak{h}}(\Theta(\nabla^{{\mathfrak{L}}}){\mathfrak{f}}_{1},{\mathfrak{f}}_{2})$
$\displaystyle=$
$\displaystyle{\mathfrak{h}}\left(((P_{\mathfrak{H}}\nabla^{{\mathfrak{L}}1,0}+P^{\perp}_{{\mathfrak{H}}}\nabla^{{\mathfrak{L}}1,0})\bar{\partial}+\bar{\partial}(P_{\mathfrak{H}}\nabla^{{\mathfrak{L}}1,0}+P^{\perp}_{{\mathfrak{H}}}\nabla^{{\mathfrak{L}}1,0})){\mathfrak{f}}_{1},{\mathfrak{f}}_{2}\right)$
$\displaystyle=$
$\displaystyle{\mathfrak{h}}((\nabla^{{\mathfrak{H}}1,0}\bar{\partial}+\bar{\partial}\nabla^{{\mathfrak{H}}1,0}){\mathfrak{f}}_{1},{\mathfrak{f}}_{2})+{\mathfrak{h}}(\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}\bar{\partial}+\bar{\partial}\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}){\mathfrak{f}}_{1},{\mathfrak{f}}_{2})$
$\displaystyle=$
$\displaystyle{\mathfrak{h}}((\nabla^{{\mathfrak{H}}1,0}\bar{\partial}+\bar{\partial}\nabla^{{\mathfrak{H}}1,0}){\mathfrak{f}}_{1},{\mathfrak{f}}_{2})+{\mathfrak{h}}(\bar{\partial}(\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}{\mathfrak{f}}_{1}),{\mathfrak{f}}_{2}).$

But

$\bar{\partial}{\mathfrak{h}}(\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}{\mathfrak{f}}_{1},{\mathfrak{f}}_{2})={\mathfrak{h}}(\bar{\partial}(\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}{\mathfrak{f}}_{1}),{\mathfrak{f}}_{2})-{\mathfrak{h}}(\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}{\mathfrak{f}}_{1},\nabla^{{\mathfrak{L}}1,0}{\mathfrak{f}}_{2})={\mathfrak{h}}(\bar{\partial}(\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}{\mathfrak{f}}_{1}),{\mathfrak{f}}_{2})-{\mathfrak{h}}(\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}{\mathfrak{f}}_{1},P_{{\mathfrak{H}}}^{\perp}\nabla^{{\mathfrak{L}}1,0}{\mathfrak{f}}_{2}),$

and since
${\mathfrak{h}}(\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}{\mathfrak{f}}_{1},{\mathfrak{f}}_{2})=0$,

${\mathfrak{h}}(\bar{\partial}\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}{\mathfrak{f}}_{1},{\mathfrak{f}}_{2})={\mathfrak{h}}(\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}{\mathfrak{f}}_{1},\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}{\mathfrak{f}}_{2}).$

Thus Formula (9) is proved.

To obtain Formula (10) from Formula (9) we need to show that the domain of the
Hilbert space adjoint of the densely defined operator
$\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}(\eta)$ contains the image of
$\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}(\xi)$. Now, the domain of
$\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}(\eta)$ consists of all
$\gamma\in{\mathfrak{H}}_{t}^{\perp}$ such that

$\left|{\mathfrak{h}}(\gamma,\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}(\eta)g)\right|^{2}\leq
C_{\gamma,\eta}{\mathfrak{h}}(g,g)\quad\text{for all
}g\in({\mathscr{C}}^{\infty}({\mathfrak{H}})_{t})(t).$

Since by hypothesis

$\left|{\mathbb{G}}_{\xi\bar{\eta}}(f,g)\right|^{2}=\left|{\mathfrak{h}}(\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}(\xi)f,\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}(\eta)g)\right|^{2}\lesssim{\mathfrak{h}}(g,g),$

we see that for each $f\in{\mathfrak{H}}_{t}$

$\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}(\xi)f\in{\rm
Domain}(\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}(\eta)^{\dagger})\quad\text{and}\quad{\mathfrak{h}}(\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}(\eta)^{\dagger}\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}(\xi)f,g)={\mathfrak{h}}(\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}(\xi)f,\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}(\eta)g).$

Therefore for each $f\in{\mathfrak{H}}_{t}$

$\left<(\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}})^{\dagger}\wedge\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}f,\xi\wedge\bar{\eta}\right>=-\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}(\eta)^{\dagger}\mathbf{I\\!I}^{{\mathfrak{L}}/{\mathfrak{H}}}(\xi)f,$

which shows that (9) implies (10). The proof is complete. ∎

######  Remark 2.34.

Note that if ${\mathfrak{H}}\to B$ has an _integrable_ structure, i.e.,
$\bar{\partial}^{{\mathfrak{H}}}\bar{\partial}^{{\mathfrak{H}}}=0$, then the
right hand side of (10) is a $(1,1)$-form. Thus, even if
$\bar{\partial}^{{\mathfrak{L}}}\bar{\partial}^{{\mathfrak{L}}}\neq 0$, the
restriction of the curvature of ${\mathfrak{L}}$ to an integrable BLS
subbundle is a $(1,1)$-form.

In general, the complex structure of ${\mathfrak{H}}$ need not be integrable.
In that case, Proposition 2.33 computes only the $(1,1)$-part of the curvature
operator of ${\mathfrak{H}}$. It is not hard to extend Proposition 2.33 to
compute the full curvature form of ${\mathfrak{H}}$. The part of the formula
coming from the form ${\mathbb{G}}$ remains of type $(1,1)$, and the type
$(2,0)$ and $(0,2)$ parts come from the curvature form of ${\mathfrak{L}}$.

In our application, in which ${\mathfrak{H}}$ is the Hilbert field
${\mathscr{H}}$ defined in the introduction, the complex structure of the
subfield is always integrable. $\diamond$

######  Definition 2.35.

Let ${\mathfrak{L}}$ be a BLS field and let
${\mathfrak{H}}\subset{\mathfrak{L}}$ be a BLS subfield.

  1. a.

The quadratic form ${\mathbb{G}}$ defined in Proposition 2.33.b is called the
_Gauss-Griffiths form_ of the BLS subfield
${\mathfrak{H}}\subset{\mathfrak{L}}$.

  2. b.

If the Gauss form satisfies the continuity condition of Proposition 2.33.b
then we say that ${\mathfrak{H}}\subset{\mathfrak{L}}$ is a _strongly BLS
subfield_.

###  2.11. iBLS Fields

If ${\mathfrak{L}}$ is a BLS field and ${\mathfrak{H}}$ is a BLS subfield then
Formula (10) is an identity of Hilbert field maps. Equivalently, Formula (9)
depends only on the value of the sections
${\mathfrak{f}}_{1},{\mathfrak{f}}_{2}$ at a point, and not on higher
derivatives of these sections, provided of course that
${\mathfrak{f}}_{1},{\mathfrak{f}}_{2}\in{\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}$.
Similarly, the second fundamental form of a BLS subfield is a pointwise
object.

If ${\mathfrak{L}}\to B$ is a BLS field and
${\mathfrak{H}}\subset{\mathfrak{L}}$ is a Hilbert subfield that is not
necessarily BLS, one might try to use the Gauss-Griffiths Formula (10) to
define the curvature of ${\mathfrak{H}}\to B$. There are several issues that
arise:

  1. 1.

For the resulting curvature of ${\mathfrak{H}}$ to be a densely-defined
operator, one must know that the set of germs of smooth sections of
${\mathfrak{L}}$ at $t\in B$ whose value at $t$ lies in ${\mathfrak{H}}_{t}$
is dense in ${\mathfrak{H}}_{t}$.

  2. 2.

One must define the second fundamental form of ${\mathfrak{H}}$ in
${\mathfrak{L}}$ in a manner that does not require a connection for
${\mathfrak{H}}$.

  3. 3.

If one wishes to use Formula (10) to define the curvature of ${\mathfrak{H}}$
as a tensorial operator then the second fundamental form defined in 2 must be
tensorial, as must the wedge product of the second fundamental form and its
Hilbert space adjoint, i.e., the object corresponding to the second quantity
on the right hand side of (2.33) must be well-defined.

  4. 4.

Ideally the curvature of ${\mathfrak{H}}$ should be independent of the ambient
BLS field ${\mathfrak{L}}$.

We shall introduce a geometric theory of iBLS fields that resolves the first
three issues. Unfortunately, we are unable at present to resolve the fourth
issue at the abstract level. However, in the concrete setting of Berndtsson’s
iBLS subfield ${\mathscr{H}}$, the large collection of ambient BLS fields
${\mathscr{L}}^{\theta}$ that we define (in fact, we could not come up with
any other examples of ambient BLS fields for ${\mathscr{H}}$) all yield the
same curvature operator.

#### Substalk bundles

For each $t\in B$ let

(11)
$\tilde{\mathfrak{H}}_{{\mathfrak{L}},t}:=\\{{\mathfrak{f}}\in{\mathscr{C}}^{\infty}({\mathfrak{L}})_{t}\text{
and
}{\mathfrak{f}}(t)\in{\mathfrak{H}}_{t}\\}\subset{\mathscr{C}}^{\infty}({\mathfrak{L}})_{t}.$

The space $\tilde{\mathfrak{H}}_{{\mathfrak{L}},t}$ is a ${\mathbb{C}}$-vector
subspace of the stalk ${\mathscr{C}}^{\infty}({\mathfrak{L}})_{t}$. Define the
family $\tilde{\mathfrak{H}}_{{\mathfrak{L}}}\to B$ whose fiber over $t$ is
$\tilde{\mathfrak{H}}_{{\mathfrak{L}},t}$.

######  Remark.

The ‘family’ notation is only introduced for convenience; we give this family
no additional structure. In particular, we note that
$\tilde{\mathfrak{H}}_{{\mathfrak{L}}}$ is not a sheaf, since in general there
is no notion of a smooth local section of
$\tilde{\mathfrak{H}}_{{\mathfrak{L}}}$.

However, we _can_ define the notion of a smooth section of
$\tilde{\mathfrak{H}}_{{\mathfrak{L}}}$ over $U\subset B$; it is a section
${\mathfrak{f}}\in\Gamma(U,{\mathscr{C}}^{\infty}({\mathfrak{L}}))$ such that
${\mathfrak{f}}(t)\in{\mathfrak{H}}_{t}$ for all $t\in U$. $\diamond$

######  Definition 2.36.

Let ${\mathfrak{L}}\to B$ be a smooth quasi-Hilbert field. We say that
${\mathfrak{H}}\subset{\mathfrak{L}}$ is a _bona fide_ quasi-Hilbert subfield
if for each $t\in B$ the space
$\tilde{\mathfrak{H}}_{{\mathfrak{L}},t}(t):=\\{{\mathfrak{f}}(t)\ ;\
{\mathfrak{f}}\in\tilde{\mathfrak{H}}_{{\mathfrak{L}},t}\\}$ is a dense
subspace of ${\mathfrak{H}}_{t}$.

Bona fide subfields resolve Issue 1 above.

######  Definition 2.37.

Let ${\mathfrak{L}}\to B$ be a BLS field and let
${\mathfrak{H}}\subset{\mathfrak{L}}$ be a (not necessarily smooth) Hilbert
subfield.

  1. (1)

A substalk bundle ${\mathcal{S}}$ of ${\mathscr{C}}^{\infty}({\mathfrak{L}})$
is a choice of ${\mathbb{C}}$-vector subspace
${\mathcal{S}}_{t}\subset{\mathscr{C}}^{\infty}({\mathfrak{L}})_{t}$ for each
$t\in B$; we write
${\mathcal{S}}\subset\tilde{\mathfrak{H}}_{{\mathfrak{L}}}$. We say that
${\mathcal{S}}$ is a _tuning_ of ${\mathfrak{H}}$ (in ${\mathfrak{L}}$), or
that $({\mathfrak{H}},{\mathcal{S}})$ is a _tuned_ Hilbert subfield.

  2. (2)

The family $\tilde{\mathfrak{H}}_{{\mathfrak{L}}}\to B$ of subspaces of the
stalks of ${\mathscr{C}}^{\infty}({\mathfrak{L}})$ defined by (11) is called
the maximal substalk bundle of ${\mathfrak{H}}$.

  3. (3)

A substalk bundle $\Sigma\subset\tilde{\mathfrak{H}}_{{\mathfrak{L}}}$ is said
to _formalize ${\mathfrak{H}}$_ if

    1. (a)

$\Sigma_{t}\subset\tilde{\mathfrak{H}}_{{\mathfrak{L}},t}$ is a dense subspace
for every $t$, and

    2. (b)

for every $t\in B$, if ${\mathfrak{f}}\in\Sigma_{t}$ and ${\mathfrak{f}}(t)=0$
then $(\nabla^{{\mathfrak{L}}}{\mathfrak{f}})(t)\in{\mathfrak{H}}_{t}$.

In this case we say that $\Sigma$ is a _formal_ tuning, or that
$({\mathfrak{H}},\Sigma)$ is formally tuned.

######  Example 2.38.

If ${\mathfrak{H}}$ is a BLS subfield of a BLS field ${\mathfrak{L}}$ then the
tuning ${\mathscr{C}}^{\infty}({\mathfrak{H}})$ of ${\mathfrak{H}}$ is formal.
In fact, Property i of Definition 2.37.c holds because density is part of the
BLS hypothesis, and the proof of Proposition 2.28 gives Property ii.
$\diamond$

As Example 2.38 suggests, tunings of ${\mathfrak{H}}$ in ${\mathfrak{L}}$ are
meant to be ‘non-integrable’ replacements for the non-existent smooth
structure ${\mathscr{C}}^{\infty}({\mathfrak{H}})$ of ${\mathfrak{H}}$.

#### Infinitesimal Second Fundamental Form

######  Definition 2.39.

Let ${\mathfrak{L}}$ be a BLS field and let $({\mathfrak{H}},{\mathcal{S}})$
be a tuned quasi-Hilbert subfield. The map

$\mathbf{N}^{{\mathfrak{L}}/({\mathfrak{H}},{\mathcal{S}})}:=P_{{\mathfrak{H}}}^{\perp}\nabla^{{\mathfrak{L}}}|_{{\mathcal{S}}}$

is called the second fundamental map (of $({\mathfrak{H}},{\mathcal{S}})$ in
${\mathfrak{L}}$).

In some sense, $\mathbf{N}^{{\mathfrak{L}}/({\mathfrak{H}},{\mathcal{S}})}$ is
${\mathscr{C}}^{\infty}_{B}$-linear. Indeed, for
$f\in{\mathscr{C}}^{\infty}_{B,t}$ and ${\mathfrak{f}}\in{\mathcal{S}}$ one
has

$P^{\perp}_{{\mathfrak{H}}}\nabla(f{\mathfrak{f}})=P^{\perp}_{{\mathfrak{H}}}(df\otimes{\mathfrak{f}}+f\nabla{\mathfrak{f}})=fP^{\perp}_{{\mathfrak{H}}}\nabla{\mathfrak{f}}.$

However, the projection $P^{\perp}_{{\mathfrak{H}}}$ need not be smooth, so
one has to understand this linearity in an appropriate sense.

Obviously
$(\mathbf{N}^{{\mathfrak{L}}/({\mathfrak{H}},\tilde{\mathfrak{H}}_{{\mathfrak{L}}})}{\mathfrak{f}})(t)\in{\mathfrak{H}}_{t}^{\perp}$.
However, $\tilde{\mathfrak{H}}_{{\mathfrak{L}}}$ is _never_ a formal tuning of
${\mathfrak{H}}$, so the map
$\mathbf{N}^{{\mathfrak{L}}/({\mathfrak{H}},\tilde{\mathfrak{H}}_{{\mathfrak{L}}})}$
is never tensorial222We are slightly abusing language here, since there is no
sheaf $\tilde{\mathfrak{H}}_{{\mathfrak{L}}}$ whose stalk at every $t\in B$ is
$\tilde{\mathfrak{H}}_{{\mathfrak{L}},t}$. However,
$\tilde{\mathfrak{H}}_{{\mathfrak{L}},t}$ is a
${\mathscr{C}}^{\infty}_{B,t}$-module, so ${\mathscr{C}}^{\infty}$-linear
means ${\mathscr{C}}^{\infty}_{B,t}$-linear, and by _tensorial_ we mean that
if ${\mathfrak{f}}\in\tilde{\mathfrak{H}}_{{\mathfrak{L}},t}$ and
${\mathfrak{f}}(t)=0$ then
$(\mathbf{N}^{{\mathfrak{L}}/{\mathfrak{H}}}{\mathfrak{f}})(t)=0$.; not even
if ${\mathfrak{H}}\subset{\mathfrak{L}}$ is a BLS subfield. The latter claim
might seem confusing, in view of Proposition 2.28. However, the second
fundamental map $\mathbf{N}^{{\mathfrak{L}}/{\mathfrak{H}}}$ in Proposition
2.28 corresponds here to the map
$\mathbf{N}^{{\mathfrak{L}}/({\mathfrak{H}},{\mathscr{C}}^{\infty}({\mathfrak{h}}))}$,
and there is no analogue of the sheaf ${\mathscr{C}}^{\infty}({\mathfrak{H}})$
in the present context, since we are not assuming that ${\mathfrak{H}}$ is a
smooth quasi-Hilbert field.

######  Proposition 2.40.

If ${\mathfrak{L}}$ is a BLS field and $({\mathfrak{H}},\Sigma)$ is a formally
tuned Hilbert subfield then there exists a $(1,0)$-form
$\mathbf{I\\!I}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)}$ with values in ${\rm
Lin}(\tilde{\mathfrak{H}}_{{\mathfrak{L}}},{\mathfrak{H}}^{\perp})$ such that

(12)
$(\mathbf{N}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)}{\mathfrak{f}})(t)=\mathbf{I\\!I}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)}(t){\mathfrak{f}}(t)$

for all $t\in B$ and all ${\mathfrak{f}}\in\Sigma_{t}$.

###### Proof.

The formality of $\Sigma$ implies that the linear map of sheaves
$\mathbf{N}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)}$ is formal, and thus
$\mathbf{I\\!I}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)}(t)$ is well-defined
by (12). ∎

######  Remark 2.41.

The name _formal_ is thus revealed to be a play on words: if
$({\mathfrak{H}},\Sigma)$ is formal then its second fundamental map
$\mathbf{N}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)}$ is tensorial, and hence
defined by a _form_. $\diamond$

#### Infinitesimal (almost) holomorphic subfield

######  Definition 2.42.

Let ${\mathfrak{L}}\to B$ be a BLS field, let
${\mathfrak{H}}\subset{\mathfrak{L}}$ be a quasi-Hilbert subfield and let
${\mathcal{S}}$ be a tuning of ${\mathfrak{H}}$, i.e., a substalk bundle of
$\tilde{\mathfrak{H}}_{{\mathfrak{L}}}$. We say that
$({\mathfrak{H}},{\mathcal{S}})$ is an _infinitesimally almost holomorphic
subfield_ of ${\mathfrak{L}}$ if for each $t\in B$, each $\tau\in
T^{1,0}_{B,t}$ and each ${\mathfrak{f}}\in\tilde{\mathcal{S}}_{t}$ one has

$\left<\bar{\partial}^{{\mathfrak{L}}}{\mathfrak{f}},\bar{\tau}\right>(t)\in{\mathfrak{H}}_{t}.$

If moreover

$\left<\bar{\partial}^{{\mathfrak{L}}}\bar{\partial}^{{\mathfrak{L}}}{\mathfrak{f}},\bar{\sigma}\wedge\bar{\tau}\right>(t)=0$

for each $t\in B$, all $\sigma,\tau\in T^{1,0}_{B,t}$ and all
${\mathfrak{f}}\in\tilde{\mathcal{S}}_{t}$ then we say that
$({\mathfrak{H}},{\mathcal{S}})$ is an _infinitesimally holomorphic subfield_
of ${\mathfrak{L}}$.

######  Remark 2.43.

In practice, given a Hilbert subfield ${\mathfrak{H}}$ of a BLS field
${\mathfrak{L}}$, one finds that there are always sections
${\mathfrak{f}}\in\tilde{\mathfrak{H}}_{{\mathfrak{L}},t}$ such that
$(\bar{\partial}^{{\mathfrak{L}}}{\mathfrak{f}})(t)\not\in
T^{0,1}_{B,t}\otimes{\mathfrak{H}}_{t}$. Thus any useful notion of almost
holomorphic subfield must involve a choice of tuning. $\diamond$

If the quasi-Hilbert subfield ${\mathfrak{H}}\subset{\mathfrak{L}}$ is
moreover infinitesimally almost holomorphic, which is the case of primary
interest for us, then one trivially has the following proposition.

######  Proposition 2.44.

Let ${\mathfrak{L}}$ be a BLS field and let
$({\mathfrak{H}},{\mathcal{S}})\subset{\mathfrak{L}}$ be an infinitesimally
almost holomorphic subfield. Then

$\mathbf{N}^{{\mathfrak{L}}/({\mathfrak{H}},{\mathcal{S}})}=P^{\perp}_{{\mathfrak{H}}}\nabla^{{\mathfrak{L}}1,0}|_{{\mathcal{S}}}.$

#### iBLS fields

Propositions 2.40 and 2.44 take care of the first aspect of issue number 3
discussed on page 3.. The second issue, namely that of showing that
$(\mathbf{N}^{{\mathfrak{L}}/{\mathfrak{H}}})^{\dagger}\wedge\mathbf{N}^{{\mathfrak{L}}/{\mathfrak{H}}}$
is well-defined and tensorial, is better handled by considering the quadratic
form (10) rather than the formula (9) in Proposition 2.33.

######  Definition 2.45.

Let ${\mathfrak{L}}$ be a BLS field, let ${\mathfrak{H}}\subset{\mathfrak{L}}$
be a bona fide quasi-Hilbert subfield and let $\Sigma$ be a formal tuning of
${\mathfrak{H}}$. The _Gauss-Griffiths form_ of
$({\mathfrak{L}},{\mathfrak{H}},\Sigma)$ is the sesquilinear form
${\mathbb{G}}:\Sigma\times\Sigma\to{\mathscr{C}}^{\infty}(\Lambda^{2}_{B})$
defined by

${\mathbb{G}}({\mathfrak{f}},{\mathfrak{g}}):={\mathfrak{h}}\left(\mathbf{\mathbf{I\\!I}}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)}{\mathfrak{f}},\mathbf{\mathbf{I\\!I}}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)}{\mathfrak{g}}\right).$

Note that the Gauss-Griffiths form applied to ${\mathfrak{f}},{\mathfrak{g}}$
is a complex $2$-form, and its contraction with two vectors $\sigma,\tau\in
T_{B}\otimes{\mathbb{C}}$ is given by

${\mathbb{G}}_{\sigma\bar{\tau}}({\mathfrak{f}},{\mathfrak{g}})={\mathfrak{h}}\left(\mathbf{\mathbf{I\\!I}}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)}(\sigma){\mathfrak{f}},\mathbf{\mathbf{I\\!I}}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)}(\tau){\mathfrak{g}}\right).$

Just as we required the continuity (7) of $\mathring{\nabla}{\mathfrak{h}}$ in
the setting of smooth Hilbert fields in order to guarantee that such Hilbert
fields have a metric-compatible connection (c.f. the proof of Proposition
2.17), if we want the curvature of a bona-fide, formally tuned quasi-Hilbert
subfield of a BLS field to be given by a map of sheaves then we must require
the analogous continuity of the Gauss-Griffiths form.

Finally, we can define _infinitesimal BLS_ fields, or iBLS fields, as follows.

######  Definition 2.46.

Let ${\mathfrak{H}}\to B$ be a quasi-Hilbert field.

  1. i.

An iBLS field is a triple $({\mathfrak{H}},{\mathfrak{L}},\Sigma)$ where
${\mathfrak{L}}\to B$ is a BLS field, called the _ambient_ BLS field (for
${\mathfrak{H}}$), and $\Sigma\subset\tilde{\mathfrak{H}}_{{\mathfrak{L}}}$ is
a substalk bundle, called the _formalizing_ substalk bundle, such that

    1. (a)

${\mathfrak{H}}\subset{\mathfrak{L}}$ is a bona fide subfield, i.e.,
$\tilde{\mathfrak{H}}_{{\mathfrak{L}},t}(t)$ is dense in ${\mathfrak{H}}_{t}$
for every $t\in B$,

    2. (b)

$({\mathfrak{H}},\Sigma)$ is an infinitesimally almost holomorphic Hilbert
subfield of ${\mathfrak{L}}$, and

    3. (c)

$({\mathfrak{H}},\Sigma)\subset{\mathfrak{L}}$ is formally tuned.

  2. ii.

If in addition to (a), (b) and (c) one has that

    1. (d)

for each $t\in B$, every $\sigma,\tau\in T_{B,t}\otimes{\mathbb{C}}$ and each
$f\in\Sigma_{t}(t)$ the conjugate-linear functional
${\mathbb{G}}_{\sigma\bar{\tau}}({\mathfrak{f}}(t),\cdot)_{t}$ is bounded on
$\Sigma_{t}(t)$ then we say that $({\mathfrak{H}},{\mathfrak{L}},\Sigma)$ is
_strongly iBLS_.

  3. iii.

And if moreover $({\mathfrak{H}},\Sigma)$ is an infinitesimally holomorphic
subfield of ${\mathfrak{L}}$ then we shall say that
$({\mathfrak{H}},{\mathfrak{L}},\Sigma)$ is an _integrable_ (strongly) iBLS
field.

We shall often abuse notation by repressing the triple
$({\mathfrak{H}},{\mathfrak{L}},\Sigma)$ and referring only to
${\mathfrak{H}}$ as an iBLS field when the rest of the data is clear from the
context.

######  Proposition 2.47.

Let $({\mathfrak{H}},{\mathfrak{L}},\Sigma)$ be an iBLS field. Then there
exists a unique linear map of substalk bundles
$\mathbf{A}:\Sigma\to\Lambda^{2}_{B}\otimes\Sigma$ such that

${\mathbb{G}}_{\sigma\bar{\tau}}({\mathfrak{f}},{\mathfrak{g}})={\mathfrak{h}}\left(\left<\mathbf{A},\sigma\wedge\bar{\tau}\right>{\mathfrak{f}},{\mathfrak{g}}\right)=-{\mathfrak{h}}\left({\mathfrak{f}},\left<\mathbf{A},\sigma\wedge\bar{\tau}\right>{\mathfrak{g}}\right).$

###### Proof.

The map $\mathbf{A}$ defined by the Riesz Representation Theorem satisfies the
first equality. Since
$\overline{{\mathbb{G}}_{\sigma\bar{\tau}}({\mathfrak{f}},{\mathfrak{g}})}={\mathbb{G}}_{\tau\bar{\sigma}}({\mathfrak{g}},{\mathfrak{f}})$,
we have

${\mathbb{G}}_{\sigma\bar{\tau}}({\mathfrak{f}},{\mathfrak{g}})=\overline{{\mathbb{G}}_{\tau\bar{\sigma}}({\mathfrak{g}},{\mathfrak{f}})}=\overline{{\mathfrak{h}}\left(\left<\mathbf{A},\bar{\tau}\wedge\sigma\right>{\mathfrak{g}},{\mathfrak{f}}\right)}=-\overline{{\mathfrak{h}}\left(\left<\mathbf{A},\sigma\wedge\bar{\tau}\right>{\mathfrak{g}},{\mathfrak{f}}\right)}=-{\mathfrak{h}}\left({\mathfrak{f}},\left<\mathbf{A},\sigma\wedge\bar{\tau}\right>{\mathfrak{g}}\right).$

The proof is complete. ∎

In analogy with the Gauss-Griffiths formula, we define the operator
$\mathbf{I\\!I}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)})^{\dagger}\wedge\mathbf{I\\!I}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)}$
by

$\displaystyle\left<\mathbf{A},\sigma\wedge\bar{\tau}\right>$
$\displaystyle=:$
$\displaystyle\left<(\mathbf{I\\!I}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)})^{\dagger}\wedge\mathbf{I\\!I}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)},\sigma\wedge\bar{\tau}\right>$
$\displaystyle=$
$\displaystyle\left<(\mathbf{I\\!I}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)})^{\dagger},\bar{\tau}\right>\left<\mathbf{I\\!I}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)},\sigma\right>-\left<\mathbf{I\\!I}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)},\sigma\right>\left<(\mathbf{I\\!I}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)})^{\dagger},\bar{\tau}\right>.$

######  Definition 2.48.

The _curvature_ of an iBLS field $({\mathfrak{H}},{\mathfrak{L}},\Sigma)$ is
the $2$-form

$\Theta^{\text{(rel
${\mathfrak{L}}$)}}({\mathfrak{H}}):=\Theta(\nabla^{{\mathfrak{L}}})|_{{\mathfrak{H}}}+(\mathbf{I\\!I}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)})^{\dagger}\wedge\mathbf{I\\!I}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)}.$

Equivalently, for any $t\in B$

${\mathfrak{h}}(\Theta^{\text{(rel
${\mathfrak{L}}$)}}({\mathfrak{H}})(\xi,\eta){\mathfrak{f}}_{1},{\mathfrak{f}}_{2})={\mathfrak{h}}(\Theta(\nabla^{{\mathfrak{L}}})(\xi,\eta){\mathfrak{f}}_{1},{\mathfrak{f}}_{2})-{\mathfrak{h}}(\mathbf{I\\!I}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)}(\xi){\mathfrak{f}}_{1},\mathbf{I\\!I}^{{\mathfrak{L}}/({\mathfrak{H}},\Sigma)}(\eta){\mathfrak{f}}_{2})$

for all
${\mathfrak{f}}_{1},{\mathfrak{f}}_{2}\in{\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}$
and all $(1,0)$-vectors $\xi,\eta\in{\mathbb{C}}\otimes T_{B_{t}}$.

(We omit reference to $\Sigma$ in $\Theta^{(\text{rel ${\mathfrak{L}}$})}$,
but the curvature does depend on the choice of $\Sigma$.)

######  Proposition 2.49.

If ${\mathfrak{H}}$ is an iBLS subfield of a BLS field ${\mathfrak{L}}$ and
${\mathfrak{H}}$ is integrable, i.e.,
$(\bar{\partial}^{2}{\mathfrak{f}})(t)=0$ for every $f\in\Sigma_{t}$, then
$\Theta(\nabla^{{\mathfrak{L}}})|_{{\mathfrak{H}}}$ has $(1,1)$-coefficients.
More precisely, for each $f_{1},f_{2}\in\Sigma_{t}(t)$ and each $\xi,\eta\in
T^{1,0}_{B,t}$ one has
${\mathfrak{h}}\left(\Theta(\nabla^{{\mathfrak{L}}})(\xi,\eta)f_{1},f_{2}\right)={\mathfrak{h}}\left(\Theta(\nabla^{{\mathfrak{L}}})(\bar{\xi},\bar{\eta})f_{1},f_{2}\right)=0$.
Consequently, if ${\mathfrak{H}}$ is integrable then $\Theta^{({\rm rel}\
{\mathfrak{L}})}({\mathfrak{H}})$ has $(1,1)$-coefficients, i.e.,

$\Theta^{({\rm rel}\ {\mathfrak{L}})}({\mathfrak{H}})(\xi,\eta)=\Theta^{({\rm
rel}\ {\mathfrak{L}})}({\mathfrak{H}})(\bar{\xi},\bar{\eta})=0$

for all $\xi,\eta\in T^{1,0}_{B,t}$.

###### Proof.

The statement about $\Theta^{({\rm rel}\ {\mathfrak{L}})}({\mathfrak{H}})$
follows from the statement about
$\Theta(\nabla^{{\mathfrak{L}}})|_{{\mathfrak{H}}}$ and the fact that
$\mathbf{I\\!I}^{{\mathscr{L}}/{\mathfrak{H}}}$ has $(1,0)$-form coefficients.
Finally, since $\Theta(\nabla^{{\mathfrak{L}}})$ is anti-Hermitian, it
suffices to show that $\Theta(\nabla^{{\mathfrak{L}}})^{0,2}{\mathfrak{f}}=0$
for any ${\mathfrak{f}}\in{\mathfrak{H}}$. The latter holds because
$\Theta(\nabla^{{\mathfrak{L}}})^{0,2}{\mathfrak{f}}=\bar{\partial}^{{\mathfrak{H}}}\bar{\partial}^{{\mathfrak{H}}}{\mathfrak{f}}=0$
for all ${\mathfrak{f}}\in{\mathfrak{H}}$. ∎

A definition of the curvature of an iBLS subfield that depends heavily on the
ambient BLS field is rather undesirable, but we have not found a reasonable
way to overcome this issue. On the other hand, the iBLS field
${\mathscr{H}}\to B$, which is our main concern, has the interesting property
that it has a huge number of natural ambient BLS fields
${\mathscr{L}}^{\theta}$, where $\theta$ is the parameter ‘enumerating’ these
BLS fields, such that

  1. (i)

${\mathscr{C}}^{\infty}({\mathscr{L}}^{\theta})$ is independent of $\theta$,

  2. (ii)

the formal tuning $\Sigma$ of ${\mathscr{H}}\subset{\mathscr{L}}^{\theta}$ is
independent of $\theta$, and

  3. (iii)

the curvature $\Theta^{{\rm rel}\ {\mathscr{L}}^{\theta}}({\mathscr{H}})$ is
independent of $\theta$.

In fact, the only dependence on the parameter $\theta$ is through the
$\bar{\partial}$-operator $\bar{\partial}^{\theta}$. We will use these
properties to define the curvature of the iBLS field ${\mathscr{H}}\to B$.

###  2.12. Positivity

All of the definitions of positivity of holomorphic Hermitian vector bundles
have direct analogues in the setting of BLS fields and iBLS fields, with one
important change. Since one wants curvature to measure an infinitesimal
quantity related to local variation, it is natural to evaluate the curvature
only on vectors that can be continued to a local section. For a BLS field
$({\mathfrak{H}},{\mathfrak{h}})\to B$ the curvature $\Theta({\mathfrak{h}})$
is evaluated at the fiber ${\mathfrak{H}}_{t}$ only on vectors in
${\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}(t)$, while for iBLS fields
$({\mathfrak{H}},{\mathfrak{L}},\Sigma)$ over $B$ the curvature
$\Theta^{(\text{rel }{\mathfrak{L}})}({\mathfrak{h}})$ is evaluated at the
fiber ${\mathfrak{H}}_{t}$ only on vectors in $\Sigma_{t}(t)$. For simplicity
we shall state the definitions of positivity only for BLS fields. For iBLS
fields, simply replace $\Theta({\mathfrak{h}})$ by $\Theta^{(\text{rel
}{\mathfrak{L}})}({\mathfrak{H}})$ and
${\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}(t)$ by $\Sigma_{t}(t)$.

Let ${\mathfrak{H}}\to B,{\mathfrak{h}}$ be a BLS field. Using the metric
${\mathfrak{h}}$, one defines Hermitian forms $\\{\ ,\
\\}_{{\mathfrak{h}},\Theta({\mathfrak{h}})}$ on the fibers
$T^{1,0}_{B,t}\otimes{\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}(t)$ by

$\left\\{\tau_{1}\otimes{\mathfrak{f}}_{1},\tau_{2}\otimes{\mathfrak{f}}_{2}\right\\}_{{\mathfrak{h}},\Theta({\mathfrak{h}})}:={\mathfrak{h}}(\Theta({\mathfrak{h}})_{\tau_{1},\bar{\tau}_{2}}{\mathfrak{f}}_{1},{\mathfrak{f}}_{2})$

for indecomposable tensors in
$T^{1,0}_{B,t}\otimes{\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}(t)$, and
extending sesqui-linearly on each fiber.

######  Definition 2.50.

Let ${\mathfrak{H}}\to B$ be a BLS field with smooth Hermitian metric
${\mathfrak{h}}$, and fix a smooth Hermitian metric $g$ on $B$.

  1. (i)

We say that ${\mathfrak{h}}$ has positive curvature in the sense of Griffiths
at a point $t\in B$ if there exists $c>0$ such that

$\left\\{\tau\otimes{\mathfrak{f}},\tau\otimes{\mathfrak{f}}\right\\}_{{\mathfrak{h}},\Theta({\mathfrak{h}})}\geq
c{\mathfrak{h}}({\mathfrak{f}},{\mathfrak{f}})g(\tau,\tau)$

for all $\tau\otimes{\mathfrak{f}}\in
T^{1,0}_{B,t}\otimes{\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}(t)$.

  2. (ii)

We say that ${\mathfrak{h}}$ has positive curvature in the sense of Nakano at
a point $t\in B$ if there exists $c>0$ such that

$\left\\{\sum_{j=1}^{n}\tau_{j}\otimes{\mathfrak{f}}_{j},\sum_{k=1}^{n}\tau_{k}\otimes{\mathfrak{f}}_{k}\right\\}_{{\mathfrak{h}},\Theta({\mathfrak{h}})}\geq
c\sum_{j=1}^{n}{\mathfrak{h}}({\mathfrak{f}}_{j},{\mathfrak{f}}_{j})g(\tau_{j},\tau_{j})$

for all
$\tau_{1}\otimes{\mathfrak{f}}_{1},...,\tau_{n}\otimes{\mathfrak{f}}_{n}\in
T^{1,0}_{B,t}\otimes{\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}(t)$, where
$n=\min\left({\rm dim}_{{\mathbb{C}}}B,{\rm
Rank}({\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}(t))\right)$.

  3. (iii)

Let $k$ be an integer between $1$ and $\min\left({\rm
dim}_{{\mathbb{C}}}B,{\rm
Rank}({\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}(t))\right)$. We say that
${\mathfrak{h}}$ has $k$-positive curvature in the sense of Demailly at a
point $t\in B$ if there exists $c>0$ such that

$\left\\{\sum_{j=1}^{k}\tau_{j}\otimes{\mathfrak{f}}_{j},\sum_{\ell=1}^{k}\tau_{\ell}\otimes{\mathfrak{f}}_{\ell}\right\\}_{{\mathfrak{h}},\Theta({\mathfrak{h}})}\geq
c\sum_{j=1}^{k}{\mathfrak{h}}({\mathfrak{f}}_{j},{\mathfrak{f}}_{j})g(\tau_{j},\tau_{j})$

for all
$\tau_{1}\otimes{\mathfrak{f}}_{1},...,\tau_{k}\otimes{\mathfrak{f}}_{k}\in
T^{1,0}_{B,t}\otimes{\mathscr{C}}^{\infty}({\mathfrak{H}})_{t}(t)$ $\diamond$

Clearly these notions of positivity do not depend on the choice of the metric
$g$. Note furthermore that $1$-positivity is just Griffiths positivity, that
$n$-positivity is just Nakano positivity, and that $m$-positivity implies
$k$-positivity if $m\geq k$.

There are also corresponding notions of nonnegative curvature (one takes $c$
nonnegative) and negative curvature (one reverses the inequalities and takes
$c<0$).

##  3\. Relative Canonical Bundle, Lie Derivatives and Horizontal Lifts

The present section introduces the basic definitions and tools needed to to
define our Hilbert fields of interest and to carry out computation of the
various geometric objects on these Hilbert fields; objects that were discussed
more abstractly in the previous section.

###  3.1. Families of twisted canonical sections

Fix a proper holomorphic family of vector bundles $(E,h)\to
X\stackrel{{\scriptstyle p}}{{\to}}B$. In the Introduction we defined the
Hilbert field ${\mathscr{H}}\to B$, whose fibers and metric are respectively

${\mathscr{H}}_{t}:=H^{0}(X_{t},{\mathcal{O}}_{X_{t}}(K_{X_{t}}\otimes
E))\quad\text{and}\quad(f_{1},f_{2})_{t}:=(-1)^{\frac{n^{2}}{2}}\int_{X_{t}}\left<f_{1}\wedge\bar{f}_{2},h\right>.$

In Section 4 we shall present the iBLS structures for ${\mathscr{H}}\to B$. To
do so, one must have a way to vary $E$-valued canonical sections on $X_{t}$
with respect to $t$. Following Berndtsson, we define the sections of
${\mathscr{H}}\to B$ to be fiberwise-holomorphic sections of $K_{X/B}\otimes
E\to X$, where $K_{X/B}$ denotes the relative canonical bundle. Since $p:X\to
B$ is a submersion, there are two equivalent ways of defining the relative
canonical bundle. The first is as a quotient

$K_{X/B}:=K_{X}\otimes p^{*}K_{B}^{*}$

of the canonical bundle of $X$ by the pullback of the canonical bundle of $B$,
and the second is by equivalence classes of $E$-valued $(n,0)$-forms.

####  3.1.1. Representation of twisted relative canonical sections as twisted
canonical sections

Given a point $t\in B$ and a coordinate system $s=(s^{1},...,s^{m})$ in an
open set $U$ containing $t$, the restricted projection map $p:X_{U}\to U$,
where $X_{U}:=p^{-1}(U)$, can be expressed in terms of coordinates as
$p=(p^{1},...,p^{m})$ for global holomorphic functions
$p^{i}\in{\mathcal{O}}(X_{U})$. Since $p$ is a submersion, i.e.,
$dp^{1}\wedge...\wedge dp^{m}$ is nowhere-zero on $X_{U}$, any section $F$ of
$K_{X}|_{X_{t}}\to X_{t}$ can be written uniquely as $F=f\wedge
dp^{1}\wedge...\wedge dp^{m}$ for some section $f$ of $K_{X_{t}}\to X_{t}$.
This first perspective on the relative canonical bundle is well-suited to
extension of canonical sections from a the fiber $X_{t}$ to $X$, in the sense
of Ohsawa-Takegoshi. But for the purpose of differentiating sections of
${\mathscr{H}}\to B$ the ‘canonical sections’ approach has drawbacks.
Therefore, we will not be using this point of view except when we discuss the
question of local triviality of ${\mathscr{H}}\to B$ in Section 5.

####  3.1.2. Representation by equivalence classes of relative top-forms

Consider the holomorphic vector bundle $\Lambda^{n,0}_{X}\to X$ whose local
sections are $(n,0)$-forms on $X$, and the subbundle $Z(p)^{n,0}\to X$ whose
fiber at $x\in X$ is defined by

$Z(p)^{n,0}_{x}=\left\\{f\in\Lambda^{n,0}_{X,x}\ ;\
\left<f,\xi_{1}\wedge...\wedge\xi_{n}\right>=0\text{ for all
}\xi_{1}\wedge...\wedge\xi_{n}\in\Lambda^{n}({\rm Kernel}(dp(x)))\right\\}.$

Thus $Z(p)^{n,0}\to X$ is the annihilator of $\Lambda^{n}T^{1,0}_{X/B}$, where
$T^{1,0}_{X/B}:={\rm kernel}(dp)$ is the so-called _vertical $(1,0)$-tangent
bundle_ of $p:X\to B$. As $p$ is a submersion, $T^{1,0}_{X/B}$ is a
holomorphic vector bundle of rank $n$, and thus $\Lambda^{n}T^{1,0}_{X/B}$ has
rank $1$. Therefore there is a line bundle $L$ such that

$0\to Z(p)^{n,0}\to\Lambda^{n,0}_{X}\to L\to 0$

is an exact sequence of vector bundles, and the line bundle $L$ is precisely
$K_{X/B}$. From this perspective a section of $K_{X/B}\otimes E\to X$ is an
equivalence class of sections of $\Lambda_{X}^{n,0}\otimes E\to X$, where two
sections are equivalent if their difference is a section of $Z(p)^{n,0}\otimes
E\to X$. We note also that a section $u$ of $\Lambda_{X}^{n,0}\otimes E\to X$
is a section of $Z(p)^{n,0}\otimes E\to X$ if and only if
$\iota_{X_{t}}^{*}u=0$ for every $t\in B$.

######  Proposition 3.1.

Let $p:X\to B$ be a proper holomorphic submersion of complex manifolds. For
each $t\in B$ there is an isomorphism

$\iota_{t}:\Gamma(X_{t},(K_{X/B}\otimes
E)|_{X_{t}})\to\Gamma(X_{t},K_{X_{t}}\otimes E|_{X_{t}})$

such that, with $\iota_{X_{t}}:X_{t}\hookrightarrow X$ denoting the natural
inclusion of the fiber,

$\iota_{t}(f)=\iota_{X_{t}}^{*}u$

for any $(n,0)$-form $u$ representing a section $f$ of $K_{X/B}\otimes E$ over
an open subset $U$ of $t$.

######  Remark 3.2.

In computation one often works locally on the base $B$, in some coordinate
chart $U$ with local coordinates $s=(s^{1},...,s^{m})$. In terms of these
coordinates the projection $p$ of the holomorphic family has the expression
$p=(p^{1},...,p^{m})$. The functions $p^{i}$ are of course constant on the
fibers of $p$, and their differentials $dp^{i}$ and $d\bar{p}^{i}$ are global
forms on $p^{-1}(U)$. Every section of $Z(p)^{n,0}|_{U}$ is of the form
$dp^{1}\wedge v_{1}+...+dp^{m}\wedge v_{m}$ for some smooth $E$-valued
$(n-1,0)$-forms
$v_{1},...,v_{m}\in\Gamma(p^{-1}(U),{\mathscr{C}}^{\infty}_{X}(\Lambda^{n-1,0}_{X}\otimes
E))$. Thus two sections $u_{1},u_{2}$ of $\Lambda^{n,0}_{X}\otimes E|_{U}$
define the same $E$-valued relative canonical section over $U$ if and only if
there exist
$v_{1},...,v_{m}\in\Gamma(p^{-1}(U),{\mathscr{C}}^{\infty}_{X}(\Lambda^{n-1,0}_{X}\otimes
E))$ such that

$u_{2}-u_{1}=dp^{1}\wedge v_{1}+\cdots+dp^{m}\wedge v_{m},$

or equivalently $u_{1}\wedge dp^{1}\wedge\cdots\wedge dp^{m}=u_{2}\wedge
dp^{1}\wedge\cdots\wedge dp^{m}$. $\diamond$

###  3.2. Lie derivatives of twisted forms

We shall need to compute derivatives of integrals of the form (1). The usual
tool for taking the derivative ‘through the integral’ is the Lie derivative.
When the objects being differentiated have natural action by diffeomorphisms,
the Lie derivative has a very natural definition, namely differentiation of
the pullback by the flow of the vector field along which one is computing
derivatives. For example, differential forms, vector fields and the tensors
one can construct from these behave well with respect to this definition.

In the twisted setting, i.e., when one looks at differential forms with values
in a vector bundle, one can still make use of the classical Lie derivative by
working in locally trivial open sets, but many formulas become local and
keeping track of these local formulas can be cumbersome. It is clearer and
more efficient to identify certain combinations of terms with a global
meaning, and to work with those combinations of terms as the basic objects.

######  Definition 3.3.

Let $X$ be a complex manifold, let $E\to X$ be a holomorphic vector bundle
with connection $\nabla$, let $u$ be an $E$-valued $(p,q)$-form and let $\xi$
be a complex vector field on $X$, i.e., a section of
$T_{X}\otimes{\mathbb{C}}\to X$.

  1. i.

The Lie derivative along $\xi$, which acts on complex-valued forms, is defined
by

$L_{\xi}:=L_{{\rm Re\ }\xi}+\sqrt{\text{\rm-1}}L_{{\rm Im\ }\xi},$

where $L$ on the right-hand side denotes the real Lie derivative of real
vector fields.

  2. ii.

The twisted Lie derivative of an $E$-valued $(p,q)$-form $u$ along $\xi$ is

$L^{\nabla}_{\xi}u=L_{\xi}u:=\nabla(\xi\lrcorner u)+\xi\lrcorner\nabla u,$

######  Remark 3.4.

In the untwisted case, i.e., when the underlying vector bundle $E$ is the
trivial line bundle and the connection $\nabla$ is the usual exterior
derivative, Definition 3.3.ii agrees with the aforementioned definition of Lie
derivative because of the formula

$L_{\xi}u=d(\xi\lrcorner u)+\xi\lrcorner(du)$

of H. Cartan. $\diamond$

Via the splitting $\nabla=\nabla^{1,0}+\nabla^{0,1}$ one can define the
$(1,0)$\- and $(0,1)$-Lie derivative of a $(p,q)$-form $u$ with values in a
holomorphic vector bundle along a complex vector field $\xi$ by

$L^{\nabla 1,0}_{\xi}u:=\nabla^{1,0}(\xi\lrcorner
u)+\xi\lrcorner\nabla^{1,0}u\quad\text{and}\quad L^{\nabla
0,1}_{\xi}u:=\nabla^{0,1}(\xi\lrcorner u)+\xi\lrcorner\nabla^{0,1}u.$

When the connection $\nabla$ is clear from the context, we shall write
$L^{1,0}_{\xi}$ instead of $L^{\nabla 1,0}_{\xi}$ (and similarly for $(0,1)$)
but we shall retain the notation $L^{\nabla}$ for the full twisted Lie
derivative for both clarity and emphasis.

######  Remark.

Note that $L^{1,0}_{\xi}$ (resp. $L^{0,1}_{\xi}$) preserves the bi-degree
$(p,q)$ if and only if $\xi$ is a $(1,0)$ (resp. $(0,1)$) vector field or
$q=0$ (resp. $p=0$). $\diamond$

######  Remark.

One can also define
$L^{0,1}_{\xi}u:=\xi\lrcorner\bar{\partial}u+\bar{\partial}(\xi\lrcorner u)$.
In the present article these two meanings of $L^{0,1}_{\xi}$ do not conflict
because we only use connections $\nabla$ satisfying
$\nabla^{0,1}=\bar{\partial}$. $\diamond$

######  Proposition 3.5.

Let $X$ be a Kähler manifold of dimension $n+m$, let $Z\subset X$ be an
$n$-dimensional submanifold, let $E\to X$ be a holomorphic vector bundle with
Hermitian metric $h$ and its Chern connection, and let $u,u^{\prime}$ be
$E$-valued $(n,0)$-forms on $X$. If $\xi$ is a $(1,0)$-vector field on $X$
then

$\iota_{Z}^{*}L_{\xi}\left<u\wedge\bar{u}^{\prime},h\right>=\iota_{Z}^{*}\left<\left(L^{1,0}_{\xi}u\right)\wedge\bar{u}^{\prime},h\right>+\iota_{Z}^{*}\left<u\wedge\overline{L^{0,1}_{\bar{\xi}}u^{\prime}},h\right>$

and

$\iota_{Z}^{*}L_{\bar{\xi}}\left<u\wedge\bar{u}^{\prime},h\right>=\iota_{Z}^{*}\left<\left(L^{0,1}_{\bar{\xi}}u\right)\wedge\bar{u}^{\prime},h\right>+\iota_{Z}^{*}\left<u\wedge\overline{L^{1,0}_{\xi}u^{\prime}},h\right>.$

###### Proof.

We compute that

$\displaystyle L_{\xi}\left<u\wedge\bar{u}^{\prime},h\right>$ $\displaystyle=$
$\displaystyle
d\left(\xi\lrcorner\left<u\wedge\bar{u}^{\prime},h\right>\right)+\xi\lrcorner
d\left<u\wedge\bar{u}^{\prime},h\right>$ $\displaystyle=$ $\displaystyle
d\left(\left<(\xi\lrcorner
u)\wedge\bar{u}^{\prime},h\right>+(-1)^{n}\left<u\wedge\overline{(\bar{\xi}\lrcorner
u^{\prime})},h\right>\right)+\xi\lrcorner\left(\left<\nabla
u\wedge\bar{u}^{\prime},h\right>+(-1)^{n}\left<u\wedge\overline{\nabla
u^{\prime}},h\right>\right)$ $\displaystyle=$
$\displaystyle\left<\nabla(\xi\lrcorner
u)\wedge\bar{u}^{\prime},h\right>+(-1)^{n-1}\left<(\xi\lrcorner
u)\wedge\overline{\nabla u^{\prime}},h\right>+(-1)^{n}\left<\nabla
u\wedge\overline{(\bar{\xi}\lrcorner u^{\prime})},h\right>$
$\displaystyle+\left<u\wedge\overline{\nabla(\bar{\xi}\lrcorner
u^{\prime})},h\right>+\left<(\xi\lrcorner(\nabla
u))\wedge\bar{u}^{\prime},h\right>+(-1)^{n+1}\left<\nabla
u\wedge\overline{(\bar{\xi}\lrcorner u^{\prime})},h\right>$
$\displaystyle+(-1)^{n}\left<\xi\lrcorner u\wedge\overline{\nabla
u^{\prime}},h\right>+\left<u,\overline{(\bar{\xi}\lrcorner(\nabla
u^{\prime}))},h\right>$ $\displaystyle=$
$\displaystyle\left<\nabla(\xi\lrcorner
u)\wedge\bar{u}^{\prime},h\right>+\left<u\wedge\overline{\nabla(\bar{\xi}\lrcorner
u^{\prime})},h\right>+\left<(\xi\lrcorner(\nabla
u))\wedge\bar{u}^{\prime},h\right>+\left<u,\overline{(\bar{\xi}\lrcorner(\nabla
u^{\prime}))},h\right>$ $\displaystyle=$
$\displaystyle\left<\nabla(\xi\lrcorner
u)\wedge\bar{u}^{\prime},h\right>+\left<(\xi\lrcorner(\nabla
u))\wedge\bar{u}^{\prime},h\right>+\left<u\wedge\overline{(\bar{\xi}\lrcorner(\nabla
u^{\prime}))},h\right>,$

where the last equality holds because $\bar{\xi}\lrcorner u^{\prime}=0$.
Therefore

$\displaystyle\iota_{Z}^{*}L_{\xi}\left<u\wedge\bar{u}^{\prime},h\right>$
$\displaystyle=$ $\displaystyle\iota_{Z}^{*}\left<\nabla(\xi\lrcorner
u)\wedge\bar{u}^{\prime},h\right>+\iota_{Z}^{*}\left<(\xi\lrcorner(\nabla
u))\wedge\bar{u}^{\prime},h\right>+\iota_{Z}^{*}\left<u,\overline{(\bar{\xi}\lrcorner(\nabla
u^{\prime}))},h\right>$ $\displaystyle=$
$\displaystyle\iota_{Z}^{*}\left<\nabla^{1,0}(\xi\lrcorner
u)\wedge\bar{u}^{\prime},h\right>+\iota_{Z}^{*}\left<(\xi\lrcorner(\nabla^{1,0}u))\wedge\bar{u}^{\prime},h\right>+\iota_{Z}^{*}\left<u,\overline{(\bar{\xi}\lrcorner(\bar{\partial}u^{\prime}))},h\right>,$

the last equality being a consequence of the fact that upon restriction to $Z$
all of the forms whose bidegree is not $(n,n)$ must vanish. ∎

######  Lemma 3.6.

Let $p:X\to B$ be a proper holomorphic family, let $E\to X,h$ be a Hermitian
holomorphic vector bundle equipped with its Chern connection and let $u$ and
$u^{\prime}$ be $E$-valued $(n,0)$-forms on $X$. If $\tau$ is a $(1,0)$-vector
field on $B$ and $\xi_{\tau}$ is a $(1,0)$-vector field on $X$ such that
$dp\xi_{\tau}=\tau$ then

$\tau\int_{X_{t}}\left<u\wedge\bar{u}^{\prime},h\right>=\int_{X_{t}}\left<L^{1,0}_{\xi_{\tau}}u\wedge\bar{u}^{\prime},h\right>+\int_{X_{t}}\left<u\wedge\overline{L^{0,1}_{\bar{\xi}_{\tau}}u^{\prime}},h\right>,$

where $X_{t}:=p^{-1}(t)$.

###### Proof.

Write $\tau=\tau_{1}+\sqrt{\text{\rm-1}}\tau_{2}$ in terms of its real and
imaginary parts, and similarly write
$\xi_{\tau}=\xi_{1}+\sqrt{\text{\rm-1}}\xi_{2}$. Since $p$ is holomorphic,
$dp\xi_{i}=\tau_{i}$. Then

$\displaystyle\tau_{i}\int_{X_{t}}\left<u\wedge\bar{u}^{\prime},h\right>$
$\displaystyle=$ $\displaystyle\left.\frac{\partial}{\partial
r}\right|_{r=0}\int_{\Phi^{r}_{\xi_{i}}(X_{t})}\left<u\wedge\bar{u}^{\prime},h\right>=\left.\frac{\partial}{\partial
r}\right|_{r=0}\int_{X_{t}}\left(\Phi^{r}_{\xi_{i}}\right)^{*}\left<u\wedge\bar{u}^{\prime},h\right>$
$\displaystyle=$
$\displaystyle\int_{X_{t}}L_{\xi_{i}}\left<u\wedge\bar{u}^{\prime},h\right>,$

so the lemma follows from Definition 3.3 and Proposition 3.5. ∎

Similarly one has the following lemma.

######  Lemma 3.7.

Let $X$, $Z$, $E$ and $h$ be as in Proposition 3.5, let $\alpha$ and $\beta$
be $E$-valued forms of bidegree $(n,1)$ and $(n-1,0)$ repectively. If $\xi$ is
a $(1,0)$-vector field on $X$ then

$L^{1,0}_{\xi}\left<\alpha\wedge\bar{\beta},h\right>=\left<(L^{1,0}_{\xi}\alpha)\wedge\bar{\beta},h\right>+\left<\alpha\wedge\overline{L^{0,1}_{\bar{\xi}}\beta},h\right>,$

and hence

$\int_{Z}L^{1,0}_{\xi}\left<\alpha\wedge\bar{\beta},h\right>=\int_{Z}\left<(L^{1,0}_{\xi}\alpha)\wedge\bar{\beta},h\right>+\int_{Z}\left<\alpha\wedge\overline{L^{0,1}_{\bar{\xi}}\beta},h\right>.$

###### Proof.

We compute that

$\displaystyle L^{1,0}_{\xi}\left<\alpha\wedge\bar{\beta},h\right>$
$\displaystyle=$
$\displaystyle\xi\lrcorner\partial\left<\alpha\wedge\bar{\beta},h\right>+\partial\left(\xi\lrcorner\left<\alpha\wedge\bar{\beta},h\right>\right)$
$\displaystyle=$
$\displaystyle\xi\lrcorner\left<\nabla^{1,0}\alpha\wedge\bar{\beta},h\right>+(-1)^{n+1}\xi\lrcorner\left<\alpha\wedge\overline{\bar{\partial}\beta},h\right>+\partial\left<\xi\lrcorner\alpha\wedge\bar{\beta},h\right>$
$\displaystyle=$
$\displaystyle\left<(\xi\lrcorner\nabla^{1,0}\alpha)\wedge\bar{\beta},h\right>+(-1)^{n+1}\left<(\xi\lrcorner\alpha)\wedge\overline{\bar{\partial}\beta},h\right>+\left<\alpha\wedge\overline{\bar{\xi}\lrcorner\bar{\partial}\beta},h\right>$
$\displaystyle+\left<\nabla^{1,0}(\xi\lrcorner\alpha)\wedge\bar{\beta},h\right>+(-1)^{n}\left<\xi\lrcorner\alpha\wedge\overline{\bar{\partial}\beta},h\right>$
$\displaystyle=$
$\displaystyle\left<(\xi\lrcorner\nabla^{1,0}\alpha)\wedge\bar{\beta},h\right>+\left<\nabla^{1,0}(\xi\lrcorner\alpha)\wedge\bar{\beta},h\right>+\left<\alpha\wedge\overline{\bar{\xi}\lrcorner\bar{\partial}\beta},h\right>$
$\displaystyle=$
$\displaystyle\left<(L^{0,1}_{\xi}\alpha)\wedge\bar{\beta},h\right>+\left<\alpha\wedge\overline{L^{0,1}_{\bar{\xi}}\beta},h\right>,$

and the proof is completed by integrating over $Z$. ∎

######  Proposition 3.8.

Let $X$, $Z$, $E$ and $h$ be as in Proposition 3.5, let $u$ and $u^{\prime}$
be $E$-valued $(n,0)$-forms on $X$ and let $\gamma$ be an $E$-valued
$(n-1,1)$-form on $X$. If $\xi$ is a $(1,0)$-vector field on $X$ then

(13)
$L^{1,0}_{\bar{\xi}}\left<\gamma\wedge\bar{u},h\right>=\left<L^{1,0}_{\bar{\xi}}\gamma\wedge\bar{u},h\right>+\left<\gamma\wedge\overline{L^{0,1}_{\xi}u},h\right>$

and

(14)
$L^{0,1}_{\xi}\left<u\wedge\bar{u}^{\prime},h\right>=\left<(L^{0,1}_{\xi}u)\wedge\bar{u}^{\prime},h\right>.$

In particular,

(15)
$\iota_{Z}^{*}L_{\bar{\xi}}\left<\gamma\wedge\bar{u},h\right>=\iota_{Z}^{*}\left<L^{1,0}_{\bar{\xi}}\gamma\wedge\bar{u},h\right>+\iota_{Z}^{*}\left<\gamma\wedge\overline{L^{0,1}_{\xi}u},h\right>$

###### Proof.

As in the proof of Proposition 3.5 we have

$L_{\bar{\xi}}\left<\gamma\wedge\bar{u},h\right>=\left<\nabla(\bar{\xi}\lrcorner\gamma)\wedge\bar{u},h\right>+\left<(\bar{\xi}\lrcorner(\nabla\gamma))\wedge\bar{u},h\right>+\left<\gamma\wedge\overline{\nabla(\xi\lrcorner
u)},h\right>+\left<\gamma\wedge\overline{(\xi\lrcorner(\nabla u))},h\right>$

and

$L_{\xi}\left<u\wedge\bar{u}^{\prime},h\right>=\left<\nabla(\xi\lrcorner
u)\wedge\bar{u}^{\prime},h\right>+\left<(\xi\lrcorner(\nabla
u))\wedge\bar{u}^{\prime},h\right>+\left<u\wedge\overline{\nabla(\bar{\xi}\lrcorner
u^{\prime})},h\right>+\left<u\wedge\overline{(\bar{\xi}\lrcorner(\nabla
u^{\prime}))},h\right>.$

Taking the $(n,n)$-components of both sides of the first identity yields (13),
and taking the $(n-1,n+1)$-parts of the second identity yields (14). Finally,
since $L^{0,1}_{\bar{\xi}}\left<\gamma\wedge\bar{u},h\right>$ is of bidegree
$(n-1,n+1)$, (15) follows from (13) ∎

######  Lemma 3.9.

Let $(E,h)\to X\stackrel{{\scriptstyle p}}{{\to}}B$ be a holomorphic family of
Hermitian holomorphic vector bundles, let $\tau$ be a holomorphic vector field
on $B$, let $\xi_{\tau}$ be a lift of $\tau$ to $X$, and let $u$ be an
$E$-valued $(n,0)$-form on $X$. Then with respect to the Chern connection of
$(E,h)$

$\bar{\partial}\iota_{X_{t}}^{*}\left(L^{1,0}_{\xi_{\tau}}u\right)=\iota_{X_{t}}^{*}\left(-(\xi_{\tau}\lrcorner\Theta(h))u-\nabla^{1,0}((\bar{\partial}\xi_{\tau})\lrcorner
u)+L^{1,0}_{\xi_{\tau}}\bar{\partial}u\right),$

where the $(n,1)$-form $\iota_{X_{t}}^{*}((\xi_{\tau}\lrcorner\Theta(h))u)$ is
defined by

$\bar{\zeta}\lrcorner\left(\iota_{X_{t}}^{*}\left((\xi_{\tau}\lrcorner\Theta(h))u\right)\right):=\iota_{X_{t}}^{*}\left(\Theta(h)_{\xi_{\tau}\bar{\zeta}}u\right).$

###### Proof.

One has

$\displaystyle\bar{\partial}_{X_{t}}\iota_{X_{t}}^{*}L^{1,0}_{\xi_{\tau}}u$
$\displaystyle=$
$\displaystyle\iota_{X_{t}}^{*}\bar{\partial}L^{1,0}_{\xi_{\tau}}u=\iota_{X_{t}}^{*}\left(\bar{\partial}\nabla^{1,0}(\xi_{\tau}\lrcorner
u)+\bar{\partial}(\xi_{\tau}\lrcorner\nabla^{1,0}u)\right)$ $\displaystyle=$
$\displaystyle\iota_{X_{t}}^{*}\left(\bar{\partial}\nabla^{1,0}(\xi_{\tau}\lrcorner
u)-\xi_{\tau}\lrcorner\bar{\partial}\nabla^{1,0}u+(\bar{\partial}\xi_{\tau})\lrcorner\nabla^{1,0}u\right)$
$\displaystyle=$
$\displaystyle\iota_{X_{t}}^{*}\left(\Theta(h)(\xi_{\tau}\lrcorner
u)-\nabla^{1,0}\bar{\partial}(\xi_{\tau}\lrcorner
u)-(\xi_{\tau}\lrcorner\Theta(h)u)+\xi_{\tau}\lrcorner(\nabla^{1,0}\bar{\partial}u)+(\bar{\partial}\xi_{\tau})\lrcorner\nabla^{1,0}u\right)$
$\displaystyle=$
$\displaystyle\iota_{X_{t}}^{*}\left(-(\xi_{\tau}\lrcorner\Theta(h))u-\left(\nabla^{1,0}\bar{\partial}(\xi_{\tau}\lrcorner
u)-\xi_{\tau}\lrcorner(\nabla^{1,0}\bar{\partial}u)\right)+(\bar{\partial}\xi_{\tau})\lrcorner\nabla^{1,0}u\right)$
$\displaystyle=$
$\displaystyle\iota_{X_{t}}^{*}\left(-(\xi_{\tau}\lrcorner\Theta(h))u-\nabla^{1,0}((\bar{\partial}\xi_{\tau})\lrcorner
u)+\nabla^{1,0}(\xi_{\tau}\lrcorner\bar{\partial}u)+\xi_{\tau}\lrcorner\nabla^{1,0}\bar{\partial}u+(\bar{\partial}\xi_{\tau})\lrcorner\nabla^{1,0}u\right)$
$\displaystyle=$
$\displaystyle\iota_{X_{t}}^{*}\left(-(\xi_{\tau}\lrcorner\Theta(h))u-\nabla^{1,0}((\bar{\partial}\xi_{\tau})\lrcorner
u)+L^{1,0}_{\xi_{\tau}}\bar{\partial}u+(\bar{\partial}\xi_{\tau})\lrcorner\nabla^{1,0}u\right)$
$\displaystyle=$
$\displaystyle\iota_{X_{t}}^{*}\left(-(\xi_{\tau}\lrcorner\Theta(h))u-\nabla^{1,0}((\bar{\partial}\xi_{\tau})\lrcorner
u)+L^{1,0}_{\xi_{\tau}}\bar{\partial}u\right).$

Indeed, $(\bar{\partial}\xi_{\tau})\lrcorner\nabla^{1,0}u=0$, because
$\bar{\partial}\xi_{\tau}$ is vertical, and $\nabla^{1,0}u$ has bidegree
$(n+1,0)$. ∎

######  Proposition 3.10.

Let $(E,h)\to X\stackrel{{\scriptstyle p}}{{\to}}B$ be a holomorphic family of
holomorphic Hermitian vector bundles, let $t\in B$ and let $\tau\in
H^{0}(B,{\mathcal{O}}(T^{1,0}_{B}))$. Let $\xi_{\tau}$ be a lift of $\tau$,
i.e., a $(1,0)$-vector field $\xi_{\tau}$ on $X$ such that
$dp\xi_{\tau}=\tau$. Then with respect to the Chern connection for $(E,h)$,
one has the following statements.

  1. a.

If $u,\tilde{u}\in H^{0}(X,{\mathscr{C}}^{\infty}(\Lambda^{n,0}_{X}\otimes
E))$ agree on the fibers of $X\to B$, i.e., they represent the same smooth
section $[u]=[\tilde{u}]$ of $K_{X/B}\otimes E\to X$, then

$\iota^{*}_{X_{t}}L^{1,0}_{\xi_{\tau}}u=\iota^{*}_{X_{t}}L^{1,0}_{\xi_{\tau}}\tilde{u}\quad\text{and}\quad\iota^{*}_{X_{t}}L^{1,0}_{\xi_{\tau}}\bar{\partial}u=\iota^{*}_{X_{t}}L^{1,0}_{\xi_{\tau}}\bar{\partial}\tilde{u}.$

  2. b.

If moreover $\iota^{*}_{X_{t}}u\in{\mathscr{H}}_{t}$ then

$\iota^{*}_{X_{t}}L^{1,0}_{\tilde{\xi}_{\tau}}\bar{\partial}u=\iota^{*}_{X_{t}}L^{1,0}_{\xi_{\tau}}\bar{\partial}u$

for any other lift $\tilde{\xi}_{\tau}$ of $\tau$.

  3. c.

Finally, if $\iota_{X_{t}}^{*}u=0$ and
$\iota_{X_{t}}^{*}L^{1,0}_{\xi_{\tau}}\bar{\partial}u=0$ then
$\iota_{X_{t}}^{*}L^{1,0}_{\xi_{\tau}}u\in{\mathscr{H}}_{t}$.

###### Proof.

Since $v:=\tilde{\xi}_{\tau}-\xi_{\tau}$ is tangent to the fibers of $p$,

$\iota^{*}_{X_{t}}L^{1,0}_{\tilde{\xi}_{\tau}}\bar{\partial}u_{1}-\iota^{*}_{X_{t}}L^{1,0}_{\xi_{\tau}}\bar{\partial}u_{1}=\iota_{X_{t}}^{*}L^{1,0}_{v}\bar{\partial}u=L^{1,0}_{v_{t}}\bar{\partial}\iota_{X_{t}}^{*}u=0,$

where $v_{t}:=v|_{X_{t}}$. Thus (b) is proved.

To prove (a) it suffices to show that if $u\in
H^{0}(X,{\mathscr{C}}^{\infty}(\Lambda^{n,0}_{X}\otimes E))$ restricts to $0$
on fibers then $L^{1,0}_{\xi_{\tau}}u$ and
$L^{1,0}_{\xi_{\tau}}\bar{\partial}u$ restrict to $0$ on fibers. Now, using
coordinates $s=(s^{1},...,s^{m})$ on the base, in terms of which we write
$p=(p^{1},...,p^{m})$ and $\tau=\tau^{\mu}\frac{\partial}{\partial s^{\mu}}$
(see Remark 3.2), the section $u$ vanishes on fibers if and only if locally in
the base one has

$u=dp^{\mu}\wedge v_{\mu}$

for some $E$-valued $(n-1,0)$-forms $v_{1},...,v_{m}$. But then

$L^{1,0}_{\xi_{\tau}}u=L^{1,0}_{\xi_{\tau}}dp^{\mu}\wedge
v_{\mu}=(p^{*}\partial\tau^{\mu}\wedge
v_{\mu})+dp^{\mu}\wedge\iota_{X_{t}}^{*}L^{1,0}_{\xi_{\tau}}v_{\mu}=dp^{\mu}\wedge\left(p^{*}\left(\tfrac{\partial\tau^{\nu}}{\partial
s^{\mu}}\right)v_{\nu}+L^{1,0}_{\xi_{\tau}}v_{\mu}\right)$

and

$-L^{1,0}_{\xi_{\tau}}\bar{\partial}u=L^{1,0}_{\xi_{\tau}}dp^{\mu}\wedge\bar{\partial}v_{\mu}=(p^{*}\partial\tau^{\mu}\wedge\bar{\partial}v_{\mu})+dp^{\mu}\wedge\iota_{X_{t}}^{*}L^{1,0}_{\xi_{\tau}}\bar{\partial}v_{\mu}=dp^{\mu}\wedge\left(p^{*}\left(\tfrac{\partial\tau^{\nu}}{\partial
s^{\mu}}\right)\bar{\partial}v_{\nu}+L^{1,0}_{\xi_{\tau}}\bar{\partial}v_{\mu}\right),$

so that $\iota_{X_{t}}^{*}L^{1,0}_{\xi_{\tau}}u=0$ and
$\iota_{X_{t}}^{*}L^{1,0}_{\xi_{\tau}}\bar{\partial}u=0$ for all $t\in B$.

Finally, if $\tau$ is a holomorphic vector field, $\bar{\partial}\xi_{\tau}$
is a $(0,1)$-form with values in the vertical bundle $T^{1,0}_{X/B}$.
Therefore if $\iota_{X_{t}}^{*}u=0$ then
$\iota_{X_{t}}^{*}\left(\nabla^{1,0}(\bar{\partial}\xi_{\tau})\lrcorner
u)\right)=0$ and $\iota_{X_{t}}^{*}(\xi_{\tau}\lrcorner\Theta(h))u$. If also
$\iota_{X_{t}}^{*}L^{1,0}_{\xi_{\tau}}\bar{\partial}u=0$ then by Lemma 3.9
$\bar{\partial}\iota_{X_{t}}^{*}\left(L^{1,0}_{\xi_{\tau}}u\right)=0$, so (c)
is proved. ∎

###  3.3. Horizontal lifts

In differentiating sections sections of ${\mathscr{H}}\to B$ or the ambient
$L^{2}$ bundle ${\mathscr{L}}\to B$ one must choose _horizontal_ lifts of
$(1,0)$-vector fields from $B$ to $X$.

######  Definition 3.11.

A horizontal distribution for the smooth family $p:X\to B$ is a distribution
$H\subset T_{X}$ such that $dp|_{H}:H\to T_{B}$ is an isomorphism of vector
bundles. Equivalently, there is a morphism of vector bundles $\theta:T_{B}\to
T_{X}$ such that $dp\theta(\tau)=\tau$. $\diamond$

We shall abuse notation and write $\theta$ instead of $H$ for the horizontal
distribution. Thus $\theta$ is both a subbundle of $T_{X}$ and a right inverse
to $dp:T_{X}\to T_{B}$.

In our situation the total space $X$ and base manifold $B$ are complex
manifolds and $p:X\to B$ is holomorphic, i.e., $dpJ=Jdp$, where $J$ denotes
the almost complex structures. Regardless of whether $J$ is integrable, there
is a map that assigns to any horizontal distribution $\theta\subset T_{X}$ a
unique horizontal $(1,0)$-distribution $\theta^{1,0}\subset T^{1,0}_{X}$,
defined as $\theta^{1,0}=s^{1,0}\theta$, where

$s^{1,0}:T_{X}\ni\xi\mapsto\tfrac{1}{2}(\xi-\sqrt{\text{\rm-1}}J\xi)\in
T^{1,0}_{X}.$

The correspondence $\theta\leftrightarrow\theta^{1,0}$ is bijective, and since
$p:X\to B$ is holomorphic,

$dp\theta^{1,0}\cong T^{1,0}_{B}\quad\text{and}\quad dp\theta^{0,1}\cong
T^{0,1}_{B},$

where $\theta^{0,1}:=\overline{\theta^{1,0}}$.

To slightly shrink the sizes of many already-too-large formulas, we shall
write

$\xi^{\theta}_{\tau}:=\theta^{1,0}(\tau)\quad\text{and}\quad\bar{\xi}^{\theta}_{\tau}:=\theta^{0,1}(\bar{\tau})=\overline{\theta^{1,0}(\tau)},\qquad\tau\in
T^{1,0}_{B}.$

######  Proposition 3.12.

Let $p:X\to B$ be a holomorphic submersion and let $\theta$ be a horizontal
distribution for $p$. Then $\theta$ is integrable if and only if

(16)
$\left[\xi^{\theta}_{\sigma},\xi^{\theta}_{\tau}\right]=\xi^{\theta}_{[\sigma,\tau]}$

for all local vector fields $\sigma,\tau$ on $B$.

###### Proof.

If (16) holds then the Lie brackets of vector fields tangent to $\theta$ are
tangent to $\theta$, i.e., $\theta$ is integrable.

Conversely, suppose $\theta$ is integrable. Then
$\left[\xi^{\theta}_{\sigma},\xi^{\theta}_{\tau}\right]=\xi^{\theta}_{\eta}$
for some vector field $\eta$ on $B$, and we must show that
$\eta=[\sigma,\tau]$. By the Inverse Function Theorem each point $P\in X$ has
a local coordinate system $(t,x)$ such that $p(t,x)=t$ is a local coordinate
at $p(P)$. In these coordinates, if
$\sigma=\sigma^{\mu}\frac{\partial}{\partial t^{\mu}}$ and
$\tau=\tau^{\nu}\frac{\partial}{\partial t^{\nu}}$ then

$\xi^{\theta}_{\sigma}=\sigma^{\mu}\frac{\partial}{\partial
t^{\mu}}+\alpha^{j}\frac{\partial}{\partial
x^{j}}\quad\text{and}\quad\xi^{\theta}_{\tau}=\tau^{\nu}\frac{\partial}{\partial
t^{\nu}}+\beta^{k}\frac{\partial}{\partial x^{k}}$

and therefore

$\left[\xi^{\theta}_{\sigma},\xi^{\theta}_{\tau}\right]=[\sigma,\theta]^{\mu}\frac{\partial}{\partial
t^{\mu}}+\gamma^{i}\frac{\partial}{\partial x^{i}}.$

Thus

$\eta=dp\left(\left[\xi^{\theta}_{\sigma},\xi^{\theta}_{\tau}\right]\right)=[\sigma,\tau],$

as claimed. ∎

##  4\. Hilbert Fields of Sections

In this section we discuss the structure of the Hilbert fields
${\mathscr{H}}\to B$ and ${\mathscr{L}}\to B$ associated to a holomorphic
family of vector bundles $(E,h)\to X\stackrel{{\scriptstyle p}}{{\to}}B$ that
we defined in the introduction.

###  4.1. Berndtsson’s Hilbert field

######  Definition 4.1.

The Hilbert field ${\mathscr{H}}\to B$ with its $L^{2}$-inner product (1) is
called the _Berndtsson Hilbert field of the family $(E,h)\to
X\stackrel{{\scriptstyle p}}{{\to}}B$_. $\diamond$

#### Smooth and holomorphic sections of ${\mathscr{H}}\to B$

We begin with Berndtsson’s definition of sections of ${\mathscr{H}}\to B$.
However, we phrase the definition in terms of sections of the relative
canonical bundle, and for the moment repress the fact that these sections are
equivalence classes of twisted $(n,0)$-forms.

######  Definition 4.2.

A section ${\mathfrak{f}}$ of ${\mathscr{H}}\to B$ is a section
$f\in\Gamma(X,K_{X/B}\otimes E)$ such that for each $t\in B$
$\iota_{t}f\in{\mathscr{H}}_{t}=H^{0}(X_{t},{\mathcal{O}}(K_{X_{t}}\otimes
E|_{X_{t}}))$ (c.f. Proposition 3.1). We write

${\mathfrak{f}}\in\Gamma(B,{\mathscr{H}}).$

We also use the notation

${\mathfrak{f}}={\mathfrak{i}}(f)\quad\text{ and }\quad
f={\mathfrak{a}}({\mathfrak{f}}),$

and say that ${\mathfrak{f}}\in\Gamma(B,{\mathscr{H}})$ is induced by $f$, and
that $f\in\Gamma(X,K_{X/B}\otimes E)$ is associated to ${\mathfrak{f}}$. We
say ${\mathfrak{f}}:B\to{\mathscr{H}}$ is holomorphic (resp.
${\mathscr{C}}^{r}$, measurable , etc.) if ${\mathfrak{a}}({\mathfrak{f}})$ is
holomorphic (resp. ${\mathscr{C}}^{r}$, measurable , etc.). We denote by
$\Gamma(U,{\mathscr{C}}^{\infty}({\mathscr{H}}))$ and
$\Gamma(U,{\mathcal{O}}({\mathscr{H}}))$ the set of smooth and holomorphic
sections over an open set $U\subset B$. Finally, we let
${\mathscr{C}}^{\infty}({\mathscr{H}})$ and ${\mathcal{O}}({\mathscr{H}})$
denote the sheaves of germs of smooth and holomorphic sections of
${\mathscr{H}}\to B$ respectively. $\diamond$

######  Remark 4.3.

We emphasize that there may well exist $t\in B$ and $f\in{\mathscr{H}}_{t}$
such that there is no _smooth_ (or even continuous) section ${\mathfrak{f}}$
of ${\mathscr{H}}\to B$ satisfying ${\mathfrak{f}}(t)=f$. This is the case
precisely when ${\mathscr{H}}\to B$ is not locally trivial near $t\in B$.
$\diamond$

Unfortunately the sheaf ${\mathscr{C}}^{\infty}({\mathscr{H}})$ does not
always define a smooth structure on ${\mathscr{H}}$. The next proposition
characterizes precisely when this is the case.

######  Proposition 4.4.

The sheaf ${\mathscr{C}}^{\infty}({\mathscr{H}})$ is a smooth structure for
the (quasi-)Hilbert field ${\mathscr{H}}$, in the sense of Definition 2.4.i,
if and only if ${\mathscr{H}}$ is a holomorphic vector bundle.

###### Proof.

If ${\mathscr{H}}\to B$ is locally trivial then the smooth sections clearly
form a dense subset on every fiber; in fact, every element of every fiber is
the evaluation of a smooth (and even holomorphic) section. Conversely, if
${\mathscr{H}}\to B$ has a smooth structure then
$t\mapsto\dim{\mathscr{H}}_{t}$ is lower semi-continuous by Proposition 2.5.
On the other hand, a standard argument using Montel’s Theorem shows that
$t\mapsto\dim{\mathscr{H}}_{t}$ is upper semi-continuous. Therefore the rank
of the fibers of ${\mathscr{H}}\to B$ is constant, and thus ${\mathscr{H}}\to
B$ is a holomorphic vector bundle. ∎

#### Berndtsson’s $\bar{\partial}$ operator for ${\mathscr{H}}\to B$

Let us now turn to the definition of the $\bar{\partial}$ operator for
${\mathscr{H}}\to B$. We begin with the following simple but important fact.

######  Proposition 4.5.

Let $u$ be an $E$-valued $(n,0)$-form on the total space of the family $X\to
B$, let $t\in B$, and suppose that $\iota_{X_{t}}^{*}u\in{\mathscr{H}}_{t}$.
Then for any lifts $\xi^{1}_{\tau}$ and $\xi^{2}_{\tau}$ of the $(1,0)$-vector
$\tau\in T^{1,0}_{B,t}$ one has

$\bar{\xi}^{1}_{\tau}\lrcorner\bar{\partial}u=\bar{\xi}^{2}_{\tau}\lrcorner\bar{\partial}u$

as $E$-valued $(n,0)$-forms on $X$ along $X_{t}$.

###### Proof.

The proposition is equivalent to the requirement that
$\bar{\eta}\lrcorner\bar{\partial}u=0$ for any $\eta\in T^{1,0}_{X_{t}}$.
Since $\eta$ is tangent to $X_{t}$, $\bar{\eta}\lrcorner\bar{\partial}u=0$ if
and only if $\bar{\eta}\lrcorner\bar{\partial}\iota_{X_{t}}^{*}u=0$, and the
latter holds because $\iota_{X_{t}}^{*}u$ is holomorphic. ∎

By Proposition 4.5 if ${\mathfrak{f}}\in\Gamma(B,{\mathscr{H}})$ then the
associated section $f:={\mathfrak{a}}({\mathfrak{f}})$ satisfies

(17) $\bar{\partial}f(\bar{\zeta})=0\quad\text{ for all }\zeta\in
T^{1,0}_{X/B},$

where $T^{1,0}_{X/B}:={\rm kernel}(dp)\cap T^{1,0}_{X}\subset T^{1,0}_{X}$ is
the vertical holomorphic tangent bundle. One has an exact sequence of anti-
holomorphic vector bundles

$0\to T^{0,1}_{X/B}\to T^{0,1}_{X}\to p^{*}T^{0,1}_{B}\to 0,$

and therefore $\bar{\partial}f$ induces a unique section
$\bar{\eth}f\in\Gamma(X,p^{*}T^{*0,1}_{B}\otimes K_{X/B}\otimes E)$ defined by

$\bar{\eth}f((x,\bar{\tau}))=\bar{\partial}f(x)(\bar{\xi}_{\tau})\quad\text{
for any }\xi_{\tau}\in T^{1,0}_{X,x}\text{ such that }dp(x)\xi_{\tau}=\tau\in
T^{1,0}_{B,p(x)}.$

We therefore obtain the following definition, which is equivalent to the
definition of $\bar{\partial}$ for ${\mathscr{H}}\to B$ given in [B-2009].

######  Definition 4.6 ($\bar{\partial}$-operator for ${\mathscr{H}}$).

The operator

$\bar{\partial}:\Gamma(B,{\mathscr{C}}^{\infty}({\mathscr{H}}))\to\Gamma(B,{\mathscr{C}}^{\infty}(\Lambda^{0,1}_{B}\otimes{\mathscr{H}}))$

defined by

$\bar{\partial}{\mathfrak{f}}:={\mathfrak{i}}(\bar{\eth}{\mathfrak{a}}({\mathfrak{f}}))$

is the $\bar{\partial}$-operator for ${\mathscr{H}}\to B$.

######  Remark 4.7.

Evidently ${\mathfrak{f}}\in\Gamma(B,{\mathcal{O}}({\mathscr{H}}))$ in the
sense of Definition 4.2 if and only if $\bar{\partial}{\mathfrak{f}}=0$, i.e.,
if and only if ${\mathfrak{f}}\in\Gamma(B,{\mathcal{O}}({\mathscr{H}}))$ in
the sense of Definition 2.19. $\diamond$

######  Remark 4.8.

For higher-bidegree forms ${\mathfrak{b}}$, expressed in terms of a
$\bar{\partial}$-closed local frame $\alpha^{1},...,\alpha^{N}$ for
$\Lambda^{p,q}_{B}$ as ${\mathfrak{b}}=\alpha^{i}\otimes{\mathfrak{b}}_{i}$,
the operator $\bar{\partial}$ is given by

$\bar{\partial}{\mathfrak{b}}=\alpha^{i}\otimes\bar{\partial}{\mathfrak{b}}_{i},$

and therefore $\bar{\partial}\bar{\partial}=0$. Thus ${\mathscr{H}}\to B$ is a
holomorphic Hilbert field. $\diamond$

######  Remark 4.9.

At first glance, the definition of the $\bar{\partial}$ operator for
${\mathscr{H}}$ appearing in [B-2009] might look somewhat different than the
definition we presented here. Here we have used the representation of sections
of ${\mathscr{H}}$ as relative canonical sections $f\in
p_{*}{\mathcal{O}}_{X}(K_{X/B}\otimes E)_{t}$, whereas in Berndtsson’s
approach one chooses an $E$-valued $(n,0)$-form $u\in
p_{*}{\mathcal{O}}_{X}(\Lambda^{n,0}_{X}\otimes E)$ representing $f$. Since