(9.46) $\displaystyle\iota{\rm ch_{BC,\mathbf{Q}}}={\rm ch_{BC}}.$

###### Proof.

By [W20, Theorem 2], $\iota{\rm ch_{BC,\mathbf{Q}}}$ satisfies the uniqueness
conditions of Theorem 9.4, and so (9.46) holds. ∎

##  10\. Submersions and elliptic superconnections

The purpose of this Section is to state our main result in the case of a
submersion $p:M=X\times S\to S$, and to define infinite dimensional
superconnection forms on $S$, which will be used to establish this main
result.

This Section is organized as follows. In Subsection 10.1, we state our main
result, that involves
$\mathscr{F}\in\mathrm{D^{b}_{\mathrm{coh}}}\left(M\right)$.

In Subsection 10.2, using the results of Section 6, we show that in the proof,
$\mathscr{F}$ can be replaced by $\mathscr{E}\in\mathrm{B}\left(M\right)$. We
will then view $p_{*}\mathscr{E}$ as equipped with an antiholomorphic
superconnection $A^{p_{*}\mathscr{E}\prime\prime}$.

In Subsection 10.3, given a splitting $\mathscr{E}\simeq\mathscr{E}_{0}$, and
Hermitian metrics $g^{TX},g^{D}$ on $TX,D$, we obtain the adjoint
$A^{p_{*}\mathscr{E}_{0}\prime}$ of $A^{p_{*}\mathscr{E}_{0}\prime\prime}$.

In Subsection 10.4, we construct natural connections on $T_{\mathbf{R}}X$.

In Subsection 10.5, a Lichnerowicz formula is established for the curvature of
$A^{p_{*}\mathscr{E}_{0}}$, when $S$ is reduced to a point.

In Subsection 10.6, we give a Lichnerowicz formula for
$A^{p_{*}\mathscr{E}_{0},2}$ by taking a proper adiabatic limit of the
corresponding formula in the case of a single fiber. This curvature is a
second order elliptic differential operator along the fibers $X$.

Finally, in Subsection 10.7, by imitating the constructions of Section 8 in an
infinite-dimensional context, we obtain infinite-dimensional Chern character
forms
$\mathrm{ch}\left(A^{p_{*}\mathscr{E}_{0}\prime\prime},\omega^{X},g^{D}\right)$,
and we show that they have the same formal properties as the forms
$\mathrm{ch}\left(A^{E_{0}\prime\prime},h\right)$ that were considered in
Section 8. In particular, their class in Bott-Chern cohomology does not depend
on the metric data.

###  10.1. A theorem of Riemann-Roch-Grothendieck for submersions

Let $X,S$ be compact complex manifolds of dimension $n,n^{\prime}$. Put

(10.1) $M=X\times S.$

Then $M$ has dimension $m=n+n^{\prime}$. Let $p:M\to S,q:M\to X$ be the
projections. Then

(10.2) $TM=p^{*}TS\oplus q^{*}TX.$

Let $\mathscr{F}$ be an object in
$\mathrm{D^{b}_{\mathrm{coh}}}\left(M\right)$. As we saw in Subsection 3.4, by
a theorem by Grauert [Gr60, Theorem 10.4.6],
$Rp_{*}\mathscr{F}\in\mathrm{D^{b}_{\mathrm{coh}}}\left(S\right)$.

The purpose of the next sections is to prove a special case of our general
result.

######  Theorem 10.1.

The following identity holds:

(10.3)
$\mathrm{ch}_{\mathrm{BC}}\left(Rp_{*}\mathscr{F}\right)=p_{*}\left[q^{*}\mathrm{Td}_{\mathrm{BC}}\left(TX\right)\mathrm{ch}_{\mathrm{BC}}\left(\mathscr{F}\right)\right]\,\mathrm{in}\,H^{(=)}_{\mathrm{BC}}\left(S,\mathbf{R}\right).$

######  Remark 10.2.

We will exploit as much as possible the fact that $M$ is product. However, the
methods of [B13] can also be used to give a direct proof of Theorem 10.1 when
$p$ is an arbitrary holomorphic submersion.

###  10.2. Replacing $\mathscr{F}$ by $\mathscr{E}$

Let $\mathscr{F}\in\mathrm{D^{b}_{\mathrm{coh}}}\left(M\right)$. By Theorem
6.7, there exists
$\mathscr{E}=\left(E,A^{E\prime\prime}\right)\in\mathrm{B}\left(M\right)$ and
a quasi-isomorphism of $\mathcal{O}_{X}$-complexes
$\phi:\mathscr{E}\to\overline{\mathscr{F}}^{\infty}$, so that
$\mathscr{H}\mathscr{E}=\mathscr{H}\overline{\mathscr{F}}^{\infty}=\mathscr{H}\mathscr{F}$.
To prove our Theorem, we may as well assume that $\mathscr{F}=\mathscr{E}$.

We will use the notation of Sections 5 and 7. In particular $D$ denotes the
diagonal vector bundle on $M$ which is associated with $E$.

Note that $p_{*}\mathscr{E}$ is a
$\Omega^{0,\scriptsize\bullet}\left(S,\mathbf{C}\right)$-module, and
$A^{E\prime\prime}$ can be viewed as an antiholomorphic superconnection
$A^{p_{*}\mathscr{E}\prime\prime}$ on $p_{*}\mathscr{E}$. The difference with
respect to Subsections 5.1–5.3 is that $p_{*}\mathscr{E}$ is infinite-
dimensional.

As we saw in Subsection 3.1, a theorem of Grauert [GrR84, Theorem 10.4.6]
asserts that $Rp_{*}\mathscr{E}$ is an object in
$\mathrm{D^{b}_{\mathrm{coh}}}\left(S\right)$. Also by (3.11),

(10.4) $Rp_{*}\mathscr{E}=p_{*}\mathscr{E}.$

Therefore the $\mathcal{O}_{S}$-complex
$\left(p_{*}\mathscr{E},A^{p_{*}\mathscr{E}\prime\prime}\right)$ defines an
object in $\mathrm{D^{b}_{\mathrm{coh}}}\left(S\right)$. This result will be
given a direct proof in Theorem 11.6.

Let $i$ be the embedding of the fibers $X$ into $M$. By Propositions 5.11 and
6.10, the pull-back
$i^{*}_{b}\mathscr{E}=\left(i^{*}_{b}E,A^{i^{*}_{b}E\prime\prime}\right)$ is a
family of antiholomorphic superconnections along the fibers $X$, whose
corresponding associated element in
$\mathrm{D^{b}_{\mathrm{coh}}}\left(X\right)$ is just $Li^{*}\mathscr{E}$.

Let $\mathcal{D}$ be the diagonal bundle associated with $p_{*}\mathscr{E}$.
Then

(10.5) $\mathcal{D}=p_{*}i_{b}^{*}\mathscr{E},$

and the corresponding $A^{\mathcal{D}\prime\prime}$ (which is an analogue of
$v_{0}$ for $p_{*}\mathscr{E}$) is given by

(10.6) $A^{\mathcal{D}\prime\prime}=A^{i^{*}_{b}E\prime\prime}.$

The projection $P$ defined in Subsection 5.1 is given here by
$i^{*}_{b}:p_{*}\mathscr{E}\to\mathcal{D}=p_{*}i^{*}_{b}\mathscr{E}$.

By (10.2), we get

(10.7)
$\Lambda\left(\overline{T^{*}M}\right)=p^{*}\Lambda\left(\overline{T^{*}S}\right)\widehat{\otimes}q^{*}\Lambda\left(\overline{T^{*}X}\right).$

By (5.1), (10.7), we obtain

(10.8)
$E_{0}=p^{*}\Lambda\left(\overline{T^{*}S}\right)\widehat{\otimes}q^{*}\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D.$

Note that

(10.9)
$i^{*}_{b}E_{0}=q^{*}\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D.$

Over $X\times S$, we fix splittings as in (5.2), (5.3), so that

(10.10) $E\simeq E_{0}.$

These splittings induce corresponding splittings of $i_{b}^{*}E$, and we have
corresponding isomorphism,

(10.11) $i^{*}_{b}E\simeq i^{*}_{b}E_{0}.$

By (10.11), we get the non-canonical isomorphism

(10.12) $\mathcal{D}=\Omega^{0,\scriptsize\bullet}\left(X,D|_{X}\right).$

Put

(10.13)
$p_{*}\mathscr{E}_{0}=\Lambda\left(\overline{T^{*}S}\right)\widehat{\otimes}p_{*}i_{b}^{*}\mathscr{E}.$

By (10.5), (10.8), and (10.9), we have the identification252525It would be
more appropriate here to introduce the sub-filtration of $E$ associated with
$p^{*}\Lambda\left(\overline{T^{*}S}\right)$.,

(10.14) $p_{*}\mathscr{E}\simeq p_{*}\mathscr{E}_{0}.$

To keep in line with the previous notation, we will denote
$A^{p_{*}\mathscr{E}_{0}\prime\prime}$ the operator corresponding to
$A^{p_{*}\mathscr{E}\prime\prime}$ via the isomorphism (10.14).

Using (5.9), we get

(10.15) $A^{p_{*}\mathscr{E}_{0}\prime\prime}=\nabla^{D\prime\prime}+B.$

###  10.3. The adjoint of $A^{p_{*}\mathscr{E}_{0}\prime\prime}$

Since $M=X\times S$, then $\nabla^{D\prime\prime}$ splits as

(10.16)
$\nabla^{D\prime\prime}=\nabla^{D\prime\prime,X}+\nabla^{D\prime\prime,S},$

where the terms in the right-hand side of (10.16) differentiate along $X,S$.

Let $g^{TX}$ be a Hermitian metric on $TX$, and let $\omega^{X}$ denote the
corresponding Kähler $(1,1)$-form along the fibers $X$.262626To simplify the
presentation, we exploit as much as possible the product structure of
$M=X\times S$, to avoid the difficulties with the more general situation
considered in [B13]. Let $g^{T_{\mathbf{R}}X},g^{T^{*}_{\mathbf{R}}X}$ be the
induced metrics on $T_{\mathbf{R}}X,T^{*}_{\mathbf{R}}X$. We denote by
$\left\langle\,\right\rangle$ the corresponding scalar product on
$T_{\mathbf{R}}X$. If $J^{T_{\mathbf{R}}X}$ denotes the complex structure of
$T_{\mathbf{R}}X$, if $U,V\in T_{\mathbf{R}}X$, then272727Here, we choose the
sign conventions of Kobayashi-Nomizu [KN69, Chapter 9.5] and of [BGS88b,
BGS88c]. The opposite sign is also commonly used.

(10.17) $\omega^{X}\left(U,V\right)=\left\langle
U,J^{T_{\mathbf{R}}X}V\right\rangle.$

Let $g^{\Lambda\left(\overline{T^{*}X}\right)}$ denote the metric induced by
$g^{TX}$ on $\Lambda\left(\overline{T^{*}X}\right)$. Let $dx$ be the volume
form on $X$ which is induced by $g^{TX}$.

Let $g^{D}$ be a Hermitian metric on the $\mathbf{Z}$-graded vector bundle
$D$. Then $g^{D}$ defines a pure metric $h$ in $\mathscr{M}^{D}$.

We equip $\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D$ with the
metric $g^{\Lambda\left(\overline{T^{*}X}\right)}\widehat{\otimes}g^{D}$.

######  Definition 10.3.

If
$s,s^{\prime}\in\mathcal{D}=\Omega^{0,\scriptsize\bullet}\left(X,D|_{X}\right)$,
put

(10.18)
$\alpha\left(s,s^{\prime}\right)=\left(2\pi\right)^{-n}\int_{X}\left\langle
s,s^{\prime}\right\rangle_{g^{\Lambda\left(\overline{T^{*}X}\right)}\otimes
g^{D}}dx.$

Then $\alpha$ is a Hermitian product on $\mathcal{D}$, that defines a pure
metric in $\mathscr{M}^{\mathcal{D}}$.

Here we use the notation of Subsection 7.1, where the adjoint of an
antiholomorphic superconnection with respect to a Hermitian metric was
defined.

######  Definition 10.4.

Let $A^{p_{*}\mathscr{E}_{0}\prime}$ denote the adjoint of
$A^{p_{*}\mathscr{E}_{0}\prime\prime}$ with respect to $\alpha$.

Although $A^{E_{0}\prime\prime}$ and $A^{p_{*}\mathscr{E}_{0}\prime\prime}$
are essentially the same object, their adjoints $A^{E_{0}\prime}$ and
$A^{p_{*}\mathscr{E}_{0}\prime}$ are distinct. In particular, the metric
$g^{TX}$ plays no role in the definition of $A^{E_{0}\prime}$. Still, we will
show how to derive $A^{p_{*}\mathscr{E}_{0}\prime}$ from $A^{E_{0}\prime}$.

If $H$ is a smooth section of
$\Lambda\left(\overline{T^{*}M}\right)\widehat{\otimes}\mathrm{End}\left(D\right)$,
$H^{*}$ still denotes the adjoint of $H$ associated with the metric $h=g^{D}$,
as defined in Subsection 4.4.

Then $H$ can be viewed as a smooth section of

$\Lambda\left(\overline{T^{*}S}\right)\widehat{\otimes}\mathrm{End}\left(\Lambda\left(\overline{T^{*}X}\right)\right)\widehat{\otimes}\mathrm{End}\left(D\right).$

######  Definition 10.5.

Let $H^{*}_{\alpha}$ denote the adjoint of $H$ with respect to the metric
$\alpha$ on $\mathcal{D}$.

We use the conventions of Subsection 4.5, with $V=TX,g^{V^{*}}=g^{T^{*}X}$.
Then c is a map
$\Lambda\left(T^{*}_{\mathbf{C}}M\right)\to\Lambda\left(T^{*}_{\mathbf{C}}S\right)\widehat{\otimes}_{\mathbf{R}}c\left(T^{*}_{\mathbf{R}}X\right)$.
We extend c to a map

$\Lambda\left(T^{*}_{\mathbf{C}}M\right)\widehat{\otimes}\mathrm{End}\left(D\right)\to\Lambda\left(T^{*}_{\mathbf{C}}S\right)\widehat{\otimes}_{\mathbf{R}}c\left(T^{*}_{\mathbf{R}}X\right)\widehat{\otimes}_{\mathbf{R}}\mathrm{End}\left(D\right).$

Also we use the identification in (4.43),

(10.19)
$c\left(T^{*}_{\mathbf{R}}X\right)\otimes_{\mathbf{R}}\mathbf{C}=\mathrm{End}\left(\Lambda\left(\overline{T^{*}X}\right)\right).$

######  Proposition 10.6.

If $H$ is a smooth section of
$\Lambda\left(\overline{T^{*}M}\right)\widehat{\otimes}\mathrm{End}\left(D\right)$,
then

(10.20) $H^{*}_{\alpha}={}^{c}H^{*}.$

###### Proof.

We may as well assume that $D$ is trivial. We use the notation in (4.42). If
$f\in T^{*}X$, $\overline{f}$ and $i_{f_{*}}$ are adjoint to each other with
respect to the metric of $\Lambda\left(\overline{T^{*}X}\right)$. From (4.42),
we get (10.20). ∎

Recall that $\nabla^{D\prime}$ was defined after (7.14). As in (10.16), we
have the splitting

(10.21) $\nabla^{D\prime}=\nabla^{D\prime,X}+\nabla^{D\prime,S}.$

We can extend $\nabla^{D\prime,S}$ to smooth sections of
$\Lambda\left(T^{*}_{\mathbf{C}}S\right)\widehat{\otimes}\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D$.

Let $\nabla^{D\prime\prime,X*}_{\alpha}$ denote the formal adjoint of
$\nabla^{D\prime\prime,X}$ with respect to $\alpha$. Then
$\nabla^{D\prime\prime,X*}_{\alpha}$ is a classical Hermitian adjoint.

######  Proposition 10.7.

The following identity holds:

(10.22)
$A^{p_{*}\mathscr{E}_{0}\prime}=\nabla^{D\prime,S}+\nabla^{D\prime\prime,X*}_{\alpha}+{}^{c}B^{*}.$

###### Proof.

This is a consequence of (10.15), (10.16), and of Proposition 10.6. ∎

######  Definition 10.8.

Put

(10.23)
$A^{p_{*}\mathscr{E}_{0}}=A^{p_{*}\mathscr{E}_{0}\prime\prime}+A^{p_{*}\mathscr{E}_{0}\prime}.$

Then $A^{p_{*}\mathscr{E}_{0}}$ is a superconnection on
$p_{*}\mathscr{E}_{0}$.

Put

(10.24) $\nabla^{D,S}=\nabla^{D\prime\prime,S}+\nabla^{D\prime,S}.$

By (7.21), (10.15), and (10.22), we get

(10.25)
$A^{p_{*}\mathscr{E}_{0}}=\nabla^{D,S}+\nabla^{D\prime\prime,X}+\nabla^{D\prime\prime,X*}+{}^{c}C.$

Let $A^{\mathcal{D}\prime\prime*}$ denote the standard fiberwise $L_{2}$
adjoint of $A^{\mathcal{D}\prime\prime}$. We use the notation

(10.26)
$A^{\mathcal{D}}=A^{\mathcal{D}\prime\prime}+A^{\mathcal{D}\prime\prime*}.$

###  10.4. Some connections on $TX$

Put

(10.27)
$\beta=\frac{1}{2}\left(\overline{\partial}^{X}-\partial^{X}\right)i\omega^{X}.$

Then $\beta$ is a smooth section of
$\Lambda^{3}\left(T^{*}_{\mathbf{R}}X\right)$.

Let $\nabla^{T_{\mathbf{R}}X,\mathrm{LC}}$ be the Levi-Civita connection on
$T_{\mathbf{R}}X$, and let $\nabla^{TX}$ be the Chern connection on $TX$. Let
$\nabla^{T_{\mathbf{R}}X}$ be the connection on $T_{\mathbf{R}}X$ which is
induced by $\nabla^{TX}$. The connection
$\nabla^{T_{\mathbf{R}}X,\mathrm{LC}}$ preserves $TX$ if and only if $g^{TX}$
is Kähler, and in this case, it coincides with $\nabla^{T_{\mathbf{R}}X}$.

Let $\nabla^{\Lambda\left(\overline{T^{*}X}\right)}$ be the connection on
$\Lambda\left(\overline{T^{*}X}\right)$ which is induced by $\nabla^{TX}$.
This connection preserves the $\mathbf{Z}$-grading.

Let $\vartheta$ be the canonical $1$-form on $X$ with values in
$T_{\mathbf{R}}X$ that corresponds to the identity. Let $\tau$ denote the
torsion of $\nabla^{T_{\mathbf{R}}X}$. Then $\tau$ is the sum of a
$\left(2,0\right)$-form $\tau^{1,0}$ with values in $TX$, and of a
$\left(0,2\right)$-form $\tau^{0,1}$ with values in $\overline{TX}$.

Let $\left\langle\tau\wedge,\vartheta\right\rangle$ be the antisymmetrization
of the tensor $\left\langle\tau,\vartheta\right\rangle$. By [B89b, Proposition
2.1], or by (4.46), we get

(10.28) $\left\langle\tau\wedge_{,}\vartheta\right\rangle=-2\beta.$

Also equation (10.28) determines $\tau$. More precisely,

(10.29)
$\displaystyle\overline{\partial}^{X}i\omega^{X}=-\left\langle\tau^{0,1}\wedge,\vartheta^{1,0}\right\rangle,$
$\displaystyle\partial^{X}i\omega^{X}=\left\langle\tau^{1,0}\wedge,\vartheta^{0,1}\right\rangle.$

By (4.47), the connection $\nabla^{T_{\mathbf{R}}X,-1/2}$ on $T_{\mathbf{R}}X$
associated with $\nabla^{TX}$ is such that if $U,V\in T_{\mathbf{R}}X$, then

(10.30)
$\nabla^{T_{\mathbf{R}}X,-1/2}_{U}V=\nabla^{T_{\mathbf{R}}X}_{U}V-\frac{1}{2}\tau\left(U,V\right).$

As we saw at the end of Subsection 4.6, $\nabla^{T_{\mathbf{R}}X,-1/2}$
induces a connection $\nabla^{TX,-1/2}$ on $TX$, and the analogue of (4.59)
holds.

Here, we follow [B89b, Section 2 b)].

######  Definition 10.9.

Let $\overline{\nabla}^{T_{\mathbf{R}}X},\underline{\nabla}^{T_{\mathbf{R}}X}$
be the Euclidean connections on $T_{\mathbf{R}}X$ whose torsions
$\overline{\tau},\underline{\tau}$ are such that the tensors
$\left\langle\overline{\tau},\vartheta\right\rangle,\left\langle\underline{\tau},\vartheta\right\rangle$
are totally antisymmetric, and moreover,

(10.31)
$\displaystyle\left\langle\overline{\tau}\wedge_{,}\vartheta\right\rangle=2\beta,$
$\displaystyle\left\langle\underline{\tau}\wedge_{,}\vartheta\right\rangle=-2\beta,$

and let $\overline{R}^{T_{\mathbf{R}}X},\underline{R}^{T_{\mathbf{R}}X}$ be
their curvatures.

By [B89b, Proposition 2.5], the connection
$\overline{\nabla}^{T_{\mathbf{R}}X}$ preserves the complex structure, i.e.,
it induces a corresponding Hermitian connection $\overline{\nabla}^{TX}$ on
$TX$.

Let $A$ be the $1$-form on $X$ with values in antisymmetric sections of
$\mathrm{End}\left(T_{\mathbf{R}}X\right)$ such that if $U,V,W\in
T_{\mathbf{R}}X$, then

(10.32) $\left\langle
A\left(U\right)V,W\right\rangle=-\left\langle\tau\left(U,V\right),W\right\rangle+\left\langle\tau\left(U,W\right),V\right\rangle.$

From the properties of $\tau$ given before, we find that for $U\in
T_{\mathbf{R}}X$, $A\left(U\right)$ is a complex endomorphism of
$T_{\mathbf{R}}X$. By [B90, Proposition 2.5 and eq. (2.37)], we have the
identity

(10.33) $\overline{\nabla}^{TX}=\nabla^{TX}+A.$

For $t>0$, if $\omega^{X}$ is replaced by $\omega^{X}/t$, $\beta$ is replaced
by $\beta/t$. However, the above connections are unchanged. Also if
$\omega^{X}$ is closed, all the considered connections on $T_{\mathbf{R}}X$
coincide with $\nabla^{T_{\mathbf{R}}X}$.

By following [B89b, Section 2 b)] and [B13, Section 2.5], let us describe the
connection $\underline{\nabla}^{T_{\mathbf{R}}X}$ in more detail. Put

(10.34) $F=T^{*}X\oplus TX$

Then $F$ is a holomorphic Hermitian vector bundle on $X$. Let
$\nabla^{F}=\nabla^{T^{*}X}\oplus\nabla^{TX}$ denote the corresponding Chern
connection.

Let $b$ be the smooth section of
$\overline{T^{*}X}\otimes\mathrm{Hom}\left(TX,T^{*}X\right)$ such that if
$u,v\in TX$,then

(10.35)
$b\left(\overline{u}\right)v=i\partial^{X}\omega^{X}\left(\overline{u},v,\cdot\right).$

Put

(10.36)
$\mathfrak{b}=\left(g^{TX}\right)^{-1}\overline{b\left(g^{TX}\right)^{-1}}.$

Then $\mathfrak{b}$ is a section of
$T^{*}X\otimes\mathrm{Hom}\left(T^{*}X,TX\right)$.

Let $\underline{\nabla}^{F}$ be the connection on $F$,

(10.37) $\underline{\nabla}^{F}=\begin{bmatrix}\nabla^{T^{*}X}&b\\\
\mathfrak{b}&\nabla^{TX}\end{bmatrix}.$

Then $g^{\overline{TX}}$ identifies $\overline{TX}$ with $T^{*}X$, so that

(10.38) $F\simeq T_{\mathbf{C}}X=\overline{TX}\oplus TX.$

In [B89b, Theorem 2.9], [B13, Theorem 2.5.3], it was shown that via the
identification $F\simeq T_{\mathbf{C}}X$, the connection
$\underline{\nabla}^{F}$ is real and it induces the Euclidean connection
$\underline{\nabla}^{T_{\mathbf{R}}X}$ on $T_{\mathbf{R}}X$. If
$\underline{\nabla}^{T_{\mathbf{C}}X}$ denote the corresponding connection on
$T_{\mathbf{C}}X$, by (10.37), we get

(10.39)
$\underline{\nabla}^{T_{\mathbf{C}}X}=\begin{bmatrix}\nabla^{\overline{TX}}&\left(g^{\overline{TX}}\right)^{-1}b\\\
\left(g^{TX}\right)^{-1}\overline{b}&\nabla^{TX}\end{bmatrix}.$

By (10.39), we have the confirmation that
$\underline{\nabla}^{T_{\mathbf{R}}X}$ is unchanged when replacing
$\omega^{X}$ by $\omega^{X}/t$.

By (10.37), we get

(10.40)
$\underline{\nabla}^{F\prime\prime}=\begin{bmatrix}\nabla^{T^{*}X\prime\prime}&b\\\
0&\nabla^{TX\prime\prime}\end{bmatrix}.$

If $\overline{\partial}^{X}\partial^{X}i\omega^{X}=0$, by [B89b, Theorem 2.7],
[B13, Theorem 2.5.3], $\underline{\nabla}^{F\prime\prime}$ induces a new
holomorphic structure on $F$. We have the exact sequence of holomorphic vector
bundles,

(10.41) $0\to T^{*}X\to F\to TX\to 0.$

Equivalently, $\underline{\nabla}^{T_{\mathbf{C}}X\prime\prime}$ defines a
holomorphic structure on $T_{\mathbf{C}}X$, and the exact sequence (10.41) can
be rewritten in the form,

(10.42)
$0\to\left(\overline{TX},\nabla^{\overline{TX}\prime\prime}\right)\to\left(T_{\mathbf{C}}X,\underline{\nabla}^{T_{\mathbf{C}}X\prime\prime}\right)\to\left(TX,\nabla^{TX\prime\prime}\right)\to
0.$

In (10.42), $\nabla^{\overline{TX}\prime\prime}=\overline{\nabla^{TX\prime}}$
depends on the metric $g^{TX}$.

If $\overline{\partial}^{X}\partial^{X}i\omega^{X}=0$, then
$\underline{\nabla}^{T_{\mathbf{C}}X\prime\prime}$ is a holomorphic structure,
so that $\underline{R}^{T_{\mathbf{R}}X}$ is of type $(1,1)$. By [B89b,
Theorem 1.6], [B13, Theorem 2.5.6], under the same assumption, if $A,B,C,D\in
T_{\mathbf{R}}X$, then

(10.43)
$\left\langle\overline{R}^{T_{\mathbf{R}}X}\left(A,B\right)C,D\right\rangle=\left\langle\underline{R}^{T_{\mathbf{R}}X}\left(C,D\right)A,B\right\rangle.$

Equation (10.43) is compatible with the fact that
$\overline{R}^{T_{\mathbf{R}}X}$ takes its values in complex endomorphisms of
$T_{\mathbf{R}}X$.

Let $\overline{\nabla}^{\Lambda\left(\overline{T^{*}X}\right)}$ be the
connection on $\Lambda\left(\overline{T^{*}X}\right)$ induced by the unitary
connection $\overline{\nabla}^{TX}$ on $TX$.

Assume for the moment that $X$ is spin, or equivalently that the line bundle
$\det TX$ has a square root $\lambda$. Then $\lambda$ is a holomorphic
Hermitian line bundle. Let $\nabla^{\lambda}$ be the corresponding Chern
connection. Put

(10.44)
$S^{T_{\mathbf{R}}X}=\Lambda\left(\overline{T^{*}X}\right)\otimes\lambda^{-1}.$

Then $S^{T_{\mathbf{R}}X}$ is the Hermitian vector bundle of
$\left(T_{\mathbf{R}}X,\,g^{T_{\mathbf{R}}X}\right)$-spinors associated with
the underlying spin structure, the positive spinors corresponding to even
forms, and the negative spinors to odd forms. Also $S^{T_{\mathbf{R}}X}$
inherits a Hermitian connection $\nabla^{S^{T_{\mathbf{R}}X},\mathrm{LC}}$
corresponding to $\nabla^{T_{\mathbf{R}}X,\mathrm{LC}}$, that preserves its
$\mathbf{Z}_{2}$-grading. Let
$\nabla^{\Lambda\left(\overline{T^{*}X}\right),\mathrm{LC}}$ denote the
connection on $\Lambda\left(\overline{T^{*}X}\right)$ that is induced by
$\nabla^{S^{T_{\mathbf{R}}X}},\nabla^{\lambda}$. The connection
$\nabla^{\Lambda\left(\overline{T^{*}X}\right),\mathrm{LC}}$ preserves the
$\mathbf{Z}_{2}$-grading of $\Lambda\left(\overline{T^{*}X}\right)$.

The above considerations still make sense even if $X$ is not spin, since
locally, a square root $\lambda$ always exists.

By proceeding the way we did for $\nabla^{T_{\mathbf{R}}X,\mathrm{LC}}$, we
define the connection
$\overline{\nabla}_{s}^{\Lambda\left(\overline{T^{*}X}\right)}$ to be the
connection induced by $\overline{\nabla}^{T_{\mathbf{R}}X},\nabla^{\lambda}$
on $\Lambda\left(\overline{T^{*}X}\right)$.

Let $e_{1},\ldots,e_{2n}$ be an orthonormal basis of $T_{\mathbf{R}}X$, and
let $e^{1},\ldots,e^{2n}$ be the corresponding dual basis of
$T^{*}_{\mathbf{R}}X$. Let ${}^{c}A$ be the section of
$\Lambda\left(T^{*}_{\mathbf{R}}X\right)\widehat{\otimes}c\left(T_{\mathbf{R}}X\right)$
given by

(10.45) ${}^{c}A=\frac{1}{2}\left\langle
Ae_{i},e_{j}\right\rangle{}^{c}e_{i}{}^{c}e_{j}.$

By equation (10.33), we get

(10.46)
$\overline{\nabla}_{s}^{\Lambda\left(\overline{T^{*}X}\right)}=\nabla^{\Lambda\left(\overline{T^{*}X}\right)}+{}^{c}A.$

Observe that $i_{\cdot}\beta$ is a $1$-form with values in
$\Lambda^{2}\left(T^{*}_{\mathbf{R}}X\right)$, so that ${}^{c}i_{\cdot}\beta$
is a $1$-form with values in
$\mathrm{End}\left(\Lambda\left(\overline{T^{*}X}\right)\right)$. By [B89b,
eq. (2.34)] and by (10.28), we have the identity282828When $\omega^{X}$ is
replaced by $\omega^{X}/t$, $\beta$ is replaced by $\beta/t$. However, c now
depends on $t$. When suitably interpreted, (10.47) does not depend on the
scaling.

(10.47)
$\overline{\nabla}_{s}^{\Lambda\left(\overline{T^{*}X}\right)}=\nabla^{\Lambda\left(\overline{T^{*}X}\right),\mathrm{LC}}+{}^{c}\left(i_{\cdot}\beta\right).$

In general, the connections
$\overline{\nabla}_{s}^{\Lambda\left(\overline{T^{*}X}\right)}$ and
$\overline{\nabla}^{\Lambda\left(\overline{T^{*}X}\right)}$ do not coincide.
More precisely, we have the identity292929This point was clear in [B89b], and
was only stated implicitly in [B12, eq. (3.72)]. Still observe that these two
connections induce the same connection on
$\mathrm{End}\left(\Lambda\left(\overline{T^{*}X}\right)\right)$.

(10.48)
$\overline{\nabla}^{\Lambda\left(\overline{T^{*}X}\right)}=\overline{\nabla}_{s}^{\Lambda\left(\overline{T^{*}X}\right)}+\frac{1}{2}\mathrm{Tr}^{TX}\left[A\right].$

Let
$\nabla^{\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D,\mathrm{LC}}$
be the connection on $\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D$
that is induced by
$\nabla^{\Lambda\left(\overline{T^{*}X}\right),\mathrm{LC}},\nabla^{D}$. Let
$\overline{\nabla}^{\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D}$
denote the connection on
$\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D$ which is associated
with $\overline{\nabla}^{\Lambda\left(\overline{T^{*}X}\right)},\nabla^{D}$.

###  10.5. A Lichnerowicz formula for $A^{\mathcal{D},2}$

In this Subsection, we assume that $S$ is reduced to a point. Let
$D^{X,\mathrm{LC}}$ denote the classical Dirac operator acting on
$C^{\infty}\left(X,\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D\right)$.
Namely, if $e_{1},\ldots,e_{2n}$ is a basis $T_{\mathbf{R}}X$, and if
$e^{1},\ldots,e^{2n}$ is the corresponding dual basis of
$T^{*}_{\mathbf{R}}X$, then

(10.49)
$D^{X,\mathrm{LC}}={}^{\mathrm{c}}e^{i}\nabla^{\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D,\mathrm{LC}}_{e_{i}}.$

Now we assume that $e_{1},\ldots,e_{2n}$ is an orthonormal basis of
$T_{\mathbf{R}}X$. Let $K^{X}$ be the scalar curvature of $X$, and let $R^{D}$
be the curvature of $\nabla^{D}$. By Lichnerowicz formula, we get

(10.50)
$D^{X,\mathrm{LC},2}=-\frac{1}{2}\nabla_{e_{i}}^{\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D,\mathrm{LC},2}+\frac{1}{8}K^{X}+{}^{\mathrm{c}}\left(R^{D}+\frac{1}{2}\mathrm{Tr}\left[R^{TX}\right]\right).$

In the right-hand side of (10.50),
$\nabla_{e_{i}}^{\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D,\mathrm{LC},2}$
is our notation for the Bochner Laplacian. More precisely if
$e_{1},\ldots,e_{2n}$ is a locally defined orthonormal basis of
$T_{\mathbf{R}}X$, then

(10.51)
$\nabla_{e_{i}}^{\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D,\mathrm{LC},2}=\sum\nabla_{e_{i}}^{\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D,\mathrm{LC},2}-\nabla_{\sum\nabla^{T_{\mathbf{R}}X,\mathrm{LC}}_{e_{i}}e_{i}}^{\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D,\mathrm{LC}}$

Other Bochner Laplacians will be denoted in the same way, with the same
correction in the right-hand side.

######  Theorem 10.10.

The following identities hold:

$\displaystyle
A^{\mathcal{D}}=D^{X,\mathrm{LC}}+{}^{\mathrm{c}}\left(\beta+C\right),$
$\displaystyle
A^{\mathcal{D},2}=-\frac{1}{2}\left(\nabla^{\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D,\mathrm{LC}}_{e_{i}}+{}^{\mathrm{c}}\left(i_{e_{i}}\left(\beta+C\right)\right)\right)^{2}+\frac{K^{X}}{8}$
$\displaystyle+^{\mathrm{c}}\left(A^{E,2}+\frac{1}{2}\mathrm{Tr}\left[R^{TX}\right]-\overline{\partial}^{X}\partial^{X}i\omega^{X}\right)+\left({}^{\mathrm{c}}\left(\beta+C\right)\right)^{2}$
(10.52)
$\displaystyle+\frac{1}{2}\sum\left({}^{\mathrm{c}}\left(i_{e_{i}}\left(\beta+C\right)\right)\right)^{2}-{}^{\mathrm{c}}\left(\left(\beta+C\right)^{2}\right),$
$\displaystyle\left({}^{\mathrm{c}}\left(\beta+C\right)\right)^{2}+\frac{1}{2}\sum\left({}^{\mathrm{c}}\left(i_{e_{i}}\left(\beta+C\right)\right)\right)^{2}-^{\mathrm{c}}\left(\left(\beta+C\right)^{2}\right)$
$\displaystyle=\sum_{\begin{subarray}{c}i_{1<\ldots<i_{k}}\\\ k\geq
2\end{subarray}}\left(-1\right)^{k\left(k+1\right)/2}\frac{\left(1-k\right)}{2^{k/2}}{}^{\mathrm{c}}\left(\left(i_{e_{i_{1}}}\ldots
i_{e_{i_{k}}}\left(\beta+C\right)\right)^{2}\right).$

###### Proof.

We use [B89b, Theorems 1.1, 1.3, and 2.2] and (10.25). As explained in
Subsection 4.5, our normalization on the Clifford algebra is different from
the one [B89b]. Also $\beta+C$ is an odd section of
$\Lambda\left(T^{*}_{\mathbf{C}}X\right)\widehat{\otimes}\mathrm{End}\left(D\right)$,
while $\mathrm{End}\left(D\right)$ was absent in [B89b, Theorem 2.3]. Still,
for formal reasons, it is easy to check that the result of [B89b] remains
correct. The proof of our theorem is completed. ∎

######  Remark 10.11.

Observe that
$A^{\mathcal{D},2}=\left[A^{\mathcal{D}\prime\prime},A^{\mathcal{D}\prime}\right]$.
The standard consequences of Hodge theory hold true for $A^{\mathcal{D},2}$.

By (10.47), (10.48), we get

(10.53)
$-\frac{1}{2}\left(\nabla^{\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D,\mathrm{LC}}_{e_{i}}+{}^{\mathrm{c}}\left(i_{e_{i}}\left(\beta+C\right)\right)\right)^{2}\\\
=-\frac{1}{2}\left(\overline{\nabla}^{\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D}_{e_{i}}-\frac{1}{2}\mathrm{Tr}^{TX}\left[A\left(e_{i}\right)\right]+{}^{\mathrm{c}}\left(i_{e_{i}}C\right)\right)^{2}.$

Equation (10.53) makes clear that as it should be, the operator (10.53) is of
total degree $0$. Equations (10.10), (10.53) give a false sense of symmetry
between the roles of $\beta$ and $C$. In particular both c and $\beta$ do
depend on $g^{TX}$.

###  10.6. A Lichnerowicz formula for $A^{p_{*}\mathscr{E}_{0},2}$

Since $M=X\times S\to S$ is product, the connections on $TX$ defined in
Subsection 10.5 lift to $q^{*}TX$. Also the form $\beta$ in (10.27) lifts to
the form $q^{*}\beta$ on $M$.

Let $A^{\mathcal{D},\mathrm{LC}}$ denote the Levi-Civita superconnection in
the sense of [B86], [BGS88b, Section 2 a)],

(10.54) $A^{\mathcal{D},\mathrm{LC}}=\nabla^{D,S}+D^{X,\mathrm{LC}}.$

######  Theorem 10.12.

The following identities hold:

$\displaystyle
A^{p_{*}\mathscr{E}_{0}}=A^{\mathcal{D},\mathrm{\mathrm{LC}}}+{}^{\mathrm{c}}\left(q^{*}\beta+C\right),$
$\displaystyle
A^{p_{*}\mathscr{E}_{0},2}=-\frac{1}{2}\left(\nabla^{\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D,\mathrm{LC}}_{e_{i}}+{}^{\mathrm{c}}\left(i_{e_{i}}\left(q^{*}\beta+C\right)\right)\right)^{2}+\frac{K^{X}}{8}$
$\displaystyle+^{\mathrm{c}}\left(A^{E_{0},2}+\frac{1}{2}\mathrm{Tr}\left[R^{TX}\right]-q^{*}\overline{\partial}^{X}\partial^{X}i\omega^{X}\right)+\left({}^{\mathrm{c}}\left(q^{*}\beta+C\right)\right)^{2}$
(10.55)
$\displaystyle+\frac{1}{2}\sum\left({}^{\mathrm{c}}\left(i_{e_{i}}\left(q^{*}\beta+C\right)\right)\right)^{2}-{}^{\mathrm{c}}\left(\left(q^{*}\beta+C\right)^{2}\right),$
$\displaystyle\left({}^{\mathrm{c}}\left(q^{*}\beta+C\right)\right)^{2}+\frac{1}{2}\sum\left({}^{\mathrm{c}}\left(i_{e_{i}}\left(q^{*}\beta+C\right)\right)\right)^{2}-^{\mathrm{c}}\left(\left(q^{*}\beta+C\right)^{2}\right)$
$\displaystyle=\sum_{\begin{subarray}{c}i_{1<\ldots<i_{k}}\\\ k\geq
2\end{subarray}}\left(-1\right)^{k\left(k+1\right)/2}\frac{\left(1-k\right)}{2^{k/2}}{}^{\mathrm{c}}\left(\left(i_{e_{i_{1}}}\ldots
i_{e_{i_{k}}}\left(q^{*}\beta+C\right)\right)^{2}\right).$

###### Proof.

By (10.25), by the first identity in (10.10), and by (10.54), we get the first
identity in (10.12).

Let $U$ be a small open set in $S$. We will view $U$ as an open ball in
$\mathbf{C}^{n^{\prime}}$, and we replace $S\times X$ by $U\times X$. Given
$\epsilon>0$, we equip $T\mathbf{C}^{n^{\prime}}$ with the constant metric
$g^{T\mathbf{C}^{n^{\prime}}}$ which is Kähler, and $TM$ with the metric
$p^{*}\frac{g^{T\mathbf{C}^{n^{\prime}}}}{\epsilon}\oplus q^{*}g^{TX}$. Let
$\omega^{\mathbf{C}^{n^{\prime}}}$ be the Kähler form of
$\mathbf{C}^{n^{\prime}}$. If $f_{1},\ldots,f_{n^{\prime}}$ is an orthonormal
basis of $\mathbf{C}^{n^{\prime}}$, and if $f^{1},\ldots,f^{n^{\prime}}$ is
the corresponding dual basis, then

(10.56) $\omega^{\mathbf{C}^{n^{\prime}}}=-if^{\alpha}\overline{f}^{\alpha}.$

Also $\sqrt{\epsilon}f_{1},\ldots,\sqrt{\epsilon}f_{n^{\prime}}$ is an
orthonormal basis of $T\mathbf{C}^{n^{\prime}}$ for the metric
$g^{T\mathbf{C}^{n^{\prime}}}/\epsilon$.

Let $A^{\mathcal{D}}_{\epsilon}$ be the operator on $U\times X$, which is
analogue of the operator $A^{\mathcal{D}}$ considered in Theorem 10.10. Since
$\omega^{\mathbf{C}^{n^{\prime}}}$ is closed, the analogue of $\beta$ on
$U\times X$ is just $q^{*}\beta$. Using Theorem 10.10, we get a Lichnerowicz
formula for $A^{\mathcal{D},2}_{\epsilon}$. In this formula, c now depends on
$\epsilon$, and will instead be denoted ${}^{c_{\epsilon}}$. Also the adjoint
of $f^{\alpha}$ is now $\epsilon i_{f_{\alpha}}$. We will take a proper limit
of this formula as $\epsilon\to 0$.

Set

(10.57)
$\mathsf{A}^{\mathcal{D}}_{\epsilon}=e^{-i\omega^{\mathbf{C}^{n^{\prime}}}/\epsilon}A^{\mathcal{D}}_{\epsilon}e^{i\omega^{\mathbf{C}^{n^{\prime}}}/\epsilon}.$

The effect of the conjugation is to change $\epsilon
i_{\overline{f}_{\alpha}}$ into $\epsilon
i_{\overline{f}_{\alpha}}-f^{\alpha}$. In particular, as $\epsilon\to 0$,
after conjugation, we have the uniform convergence together with all the
derivatives,

(10.58)
${}^{c_{\epsilon}}\left(q^{*}\beta+C\right)\to{}^{c}\left(q^{*}\beta+C\right).$

By (10.49), by the first equation in (10.10), by (10.54), and by (10.58), we
find that as $\epsilon\to 0$,

(10.59) $\mathsf{A}^{\mathcal{D}}_{\epsilon}\to A^{p_{*}\mathscr{E}_{0}},$

in the sense that the coefficients of the considered differential operators
and their derivatives of any order converge uniformly over compact subsets.

Now we take the limit as $\epsilon\to 0$ of the analogue of the equation for
$\mathsf{A}^{\mathcal{D},2}_{\epsilon}$ one derives from (10.10). It is now
easy to take the limit as $\epsilon\to 0$ of our equation for
$\mathsf{A}^{\mathcal{D},2}_{\epsilon}$ and to obtain (10.12). The proof of
our theorem is completed. ∎

######  Remark 10.13.

Since the fibration $\pi:M\to S$ is product, none of the subtleties contained
in [B86, BGS88b, B12] appears in our formula.

###  10.7. The elliptic superconnection forms

By equation (10.12), the operator $A^{p_{*}\mathscr{E}_{0},2}$ is elliptic
along the fibers $X$, so that $\exp\left(-A^{p_{*}\mathscr{E}_{0},2}\right)$
is a fiberwise trace class operator.

We will now imitate Definition 8.1 in an infinite dimensional context.

######  Definition 10.14.

Put

(10.60)
$\mathrm{ch}\left(A^{p_{*}\mathscr{E}_{0}\prime\prime},\omega^{X},g^{D}\right)=\varphi\mathrm{Tr_{s}}\left[\exp\left(-A^{p_{*}\mathscr{E}_{0},2}\right)\right].$

Then
$\mathrm{ch}\left(A^{p_{*}\mathscr{E}_{0}\prime\prime},\omega^{X},g^{D}\right)$
is a smooth even form on $S$.

Let $\mathscr{P}$ be the collection of parameters $\omega^{X},g^{D}$. Let
$d^{\mathscr{P}}$ be the de Rham operator on $\mathscr{P}$.

The metric $\alpha$ in (10.18) depends on $\omega^{X},g^{D}$. Then
$\alpha^{-1}d^{\mathscr{P}}\alpha$ is a $1$-form with values in $\alpha$-self-
adjoint endomorphisms.

Let $w_{1},\ldots,w_{n}$ be an orthonormal basis of $TX$ with respect to
$g^{TX}$. Using (10.18), and proceeding as in [BGS88c, eq. (1.109) and
Proposition 1.19], we get

(10.61)
$\alpha^{-1}d^{\mathscr{P}}\alpha=-{}^{\mathrm{c}}d^{\mathscr{P}}i\omega^{X}+\frac{1}{2}d^{\mathscr{P}}i\omega^{X}\left(w_{i},\overline{w}_{i}\right)+\left(g^{D}\right)^{-1}d^{\mathscr{P}}g^{D}.$

Now we have the obvious analogue of Theorem 8.2.

######  Theorem 10.15.

The form
$\mathrm{ch}\left(A^{p_{*}\mathscr{E}_{0}\prime\prime},\omega^{X},g^{D}\right)$
lies in $\Omega^{(=)}\left(S,\mathbf{R}\right)$, it is closed, and its Bott-
Chern cohomology class does not depend on $\omega^{X},g^{D}$, or on the
splitting of $E$. Also
$\varphi\mathrm{Tr_{s}}\left[\alpha^{-1}d^{\mathscr{P}}\alpha\exp\left(A^{p_{*}\mathscr{E}_{0},2}\right)\right]$
is a $1$-form on $\mathscr{P}$ with values in
$\Omega^{(=)}\left(S,\mathbf{R}\right)$, and moreover,

(10.62)
$d^{\mathscr{P}}\mathrm{ch}\left(A^{p_{*}\mathscr{E}_{0}\prime\prime},\omega^{X},g^{D}\right)\\\
=-\frac{\overline{\partial}^{S}\partial^{S}}{2i\pi}\varphi\mathrm{Tr_{s}}\left[\alpha^{-1}d^{\mathscr{P}}\alpha\exp\left(-A^{p_{*}\mathscr{E}_{0},2}\right)\right].$

###### Proof.

The proof of the fact that our forms are closed and lie in
$\Omega^{(=)}\left(S,\mathbf{R}\right)$ is formally the same as the proof of
Theorem 8.2. By proceeding as in the proof of Theorem 8.2, we get (10.62).

Let us now prove that the Bott-Chern class of
$\mathrm{ch}\left(A^{p_{*}\mathscr{E}_{0}\prime\prime},\omega^{X},g^{D}\right)$
does not depend on the splitting of $E$. Let
$\mathscr{E}^{\prime}_{0}=\left(E^{\prime}_{0},A^{E^{\prime}_{0}\prime\prime}\right)$
be another splitting of $E$. If $M^{\prime}=M\times\mathbf{P}^{1}$, and let
$p_{1},p_{2},p_{3}$ be the projections $M^{\prime}\to S,M^{\prime}\to
M,M^{\prime}\to\mathbf{P}^{1}$. Consider
$\mathscr{E}_{M^{\prime}}=p^{*}_{2,b}\mathscr{E}$, so that
$\mathscr{E}_{M^{\prime}}=\left(E_{M^{\prime}},A^{E_{M^{\prime}}\prime\prime}\right)$.
The diagonal bundle associated with $\mathscr{E}_{M^{\prime}}$ is
$p_{2}^{*}D$. Given $z\in\mathbf{P}^{1}$, let $j_{z}$ denote the embedding
$M\times\left\\{z\right\\}\to M^{\prime}$. For any $z\in\mathbf{P}^{1}$, then
$j_{z,b}^{*}\mathscr{E}_{M^{\prime}}=\mathscr{E}$. Let
$\mathscr{E}_{M^{\prime},0}=\left(E_{M^{\prime},0},A^{E_{M^{\prime},0}\prime\prime}\right)$
be a splitting of $\mathscr{E}_{M^{\prime}}$ such that
$j^{*}_{0,b}\mathscr{E}^{\prime}_{0}=\mathscr{E}_{0}$, and
$j^{*}_{\infty,b}\mathscr{E}_{M^{\prime},0}=\mathscr{E}^{\prime}_{0}$. Let $q$
be the projection $M^{\prime}\to S^{\prime}=S\times\mathbf{P}^{1}$. We equip
$p_{2}^{*}D$ with the metric $p_{2}^{*}g^{D}$. We will use our previous
results on the form
$\mathrm{ch}\left(A^{\mathscr{E}_{M^{\prime},0}\prime\prime},\omega^{X},p_{2}^{*}g^{D}\right)$.
The pull-backs of this form to $S\times 0$, $S\times\infty$ are just
$\mathrm{ch}\left(A^{p_{*}\mathscr{E}_{0}\prime\prime},\omega^{X},g^{D}\right),\mathrm{ch}\left(A^{p_{*}\mathscr{E}^{\prime}_{0}\prime\prime},\omega^{X},g^{D}\right)$.
Using the Poincaré-Lelong equation (2.25), we get

(10.63)
$\frac{\overline{\partial}^{S}\partial^{S}}{2i\pi}p_{1,*}\left[\mathrm{ch}\left(A^{q_{*}\mathscr{E}_{M^{\prime},0}\prime\prime},\omega^{X},p_{2}^{*}g^{D}\right)p^{*}_{3}\log\left(\left|z\right|^{2}\right)\right]\\\
=\mathrm{ch}\left(A^{p_{*}\mathscr{E}_{0}\prime\prime},\omega^{X},g^{D}\right)-\mathrm{ch}\left(A^{p_{*}\mathscr{E}^{\prime}_{0}\prime\prime},\omega^{X},g^{D}\right),$

from which we get the last statement in our theorem. ∎

######  Definition 10.16.

Let $\mathrm{ch}_{\mathrm{BC}}\left(A^{p_{*}\mathscr{E}\prime\prime}\right)\in
H_{\mathrm{BC}}^{(=)}\left(S,\mathbf{R}\right)$ be the common Bott-Chern
cohomology class of the forms
$\mathrm{ch}\left(A^{p_{*}\mathscr{E}_{0}\prime\prime},\omega^{X},g^{D}\right)$.

##  11\. Elliptic superconnection forms and direct images

The purpose of this Section is to prove that, with the notation of Section 10,
$\mathrm{ch}_{\mathrm{BC}}\left(A^{p_{*}\mathscr{E}\prime\prime}\right)$
coincides with $\mathrm{ch}_{\mathrm{BC}}\left(Rp_{*}\mathscr{E}\right)$.

This Section is organized as follows. In Subsection 11.1, we apply the
technique of spectral truncation of Subsection 8.10 to the infinite-
dimensional $p_{*}\mathscr{E}_{0}$. This will permit us to extend to
$p_{*}\mathscr{E}_{0}$ results we had established before only for ordinary
antiholomorphic superconnections.

In Subsection 11.2, we prove the critical result that there exists a classical
antiholomorphic superconnection $\underline{\mathscr{E}}$ on $S$ such that
$p_{*}\mathscr{E}$ and $\underline{\mathscr{E}}$ are quasi-isomorphic, the
main point being that $\underline{\mathscr{E}}$ is finite-dimensional.

In Subsection 11.3, we prove our main identity.

Finally, in Subsection 11.4, we briefly consider the case where
$\mathscr{H}Rp_{*}\mathscr{E}$ is locally free, in which case our main result
has a much simpler proof.

We make the same assumptions, and we use the same notation as in Section 10.

###  11.1. Spectral truncations in infinite dimensions

We will now proceed in infinite dimensions as in Subsection 8.10. Recall that
$A^{\mathcal{D}\prime\prime*}$ is the standard $L_{2}$-adjoint of
$A^{\mathcal{D}\prime\prime}$. Put

(11.1)
$\mathbb{A}^{p_{*}\mathscr{E}_{0}}=A^{p_{*}\mathscr{E}_{0}\prime\prime}+A^{\mathcal{D}\prime\prime*}.$

Equation (11.1) is the analogue of equation (8.72) for $\mathbb{A}^{E_{0}}$.

As in (8.73), we get

(11.2)
$\displaystyle\mathbb{A}^{p_{*}\mathscr{E}_{0},2}=\left[A^{p_{*}\mathscr{E}_{0}\prime\prime},A^{\mathcal{D}\prime\prime*}\right],$
$\displaystyle\left[A^{p_{*}\mathscr{E}_{0}\prime\prime},\mathbb{A}^{p_{*}\mathscr{E}_{0},2}\right]=0,\qquad\left[A^{\mathcal{D}\prime\prime*},\mathbb{A}^{p_{*}\mathscr{E}_{0},2}\right]=0.$

The operator $A^{\mathcal{D},2}$ being a second order elliptic operator along
the fibers $X$, it has discrete spectrum. As in (8.78), we get

(11.3)
$\mathrm{Sp}\,\mathbb{A}^{p_{*}\mathscr{E}_{0},2}=\mathrm{Sp}\,A^{\mathcal{D},2}.$

As in (8.80), for $a>0$, set

(11.4) $U_{a}=\left\\{s\in S,a\notin\mathrm{Sp}\,A^{\mathcal{D},2}\right\\}.$

Then $U_{a}$ is open in $S$.

######  Definition 11.1.

Over $U_{a}$, put

(11.5)
$P_{a,-}=\frac{1}{2i\pi}\int_{\begin{subarray}{c}\lambda\in\mathbf{C}\\\
\left|\lambda\right|=a\end{subarray}}\frac{d\lambda}{\lambda-\mathbb{A}^{p_{*}\mathscr{E}_{0},2}}.$

Then $P_{a,-}$ is a fiberwise smoothing projector acting on
$p_{*}\mathscr{E}_{0}$.

By (8.73), we get

(11.6)
$\displaystyle\left[A^{p_{*}\mathscr{E}_{0}\prime\prime},P_{a,-}\right]=0,$
$\displaystyle\left[A^{\mathcal{D}\prime\prime*},P_{a,-}\right]=0.$

Put

(11.7) $P_{a,+}=1-P_{a,-}.$

Then $P_{a,+}$ is also a projector.

As in (8.84), we get

(11.8) $P_{a,\pm}=P_{a,\pm}^{0}+P_{a,\pm}^{(\geq 1)}.$

Let $\mathcal{D}_{a,\pm}$ be the direct sum of the eigenspaces of
$A^{\mathcal{D},2}$ for eigenvalues $\lambda>a$ or $\lambda<a$. Then
$P^{0}_{a,\pm}$ is the orthogonal projectors
$\mathcal{D}\to\mathcal{D}_{a,\pm}$. Also the analogue of (8.85) holds. Also
$\mathcal{D}_{a,-}$ is a finite-dimensional vector bundle on $U_{a}$.

######  Definition 11.2.

Over $U_{a}$, put

(11.9) $p_{*}\mathscr{E}_{0,a,\pm}=P_{a,\pm}p_{*}\mathscr{E}_{0}.$

Then $p_{*}\mathscr{E}_{0,a,\pm}$ is a subbundle of $p_{*}\mathscr{E}_{0}$
which is also a $\Lambda\left(\overline{T^{*}S}\right)$-module. Let
$i_{a,\pm}$ be the corresponding embedding in $p_{*}\mathscr{E}_{0}$.

We use the notation of Subsection 4.1. We will establish an analogue of
Theorem 8.21. The main difference is that here, $p_{*}\mathscr{E}_{0}$ is
infinite-dimensional.

######  Theorem 11.3.

On $U_{a}$, we have a splitting of
$\Lambda\left(\overline{T^{*}S}\right)$-modules,

(11.10) $p_{*}\mathscr{E}_{0}=p_{*}\mathscr{E}_{0,a,+}\oplus
p_{*}\mathscr{E}_{0,a,-}.$

Moreover,

(11.11) $F^{p}p_{*}\mathscr{E}_{0,a,\pm}=p_{*}\mathscr{E}_{0,a,\pm}\cap
F^{p}p_{*}\mathscr{E}_{0}.$

Also $P_{a,\pm}$ induces a filtered isomorphism of
$\Lambda\left(\overline{T^{*}S}\right)$-modules,

(11.12)
$\Lambda\left(\overline{T^{*}S}\right)\widehat{\otimes}\mathcal{D}_{a,\pm}\simeq
p_{*}\mathscr{E}_{0,a,\pm}.$

As a $\Lambda\left(\overline{T^{*}S}\right)$-module,
$p_{*}\mathscr{E}_{0,a,-}$ verifies the conditions in (4.6), and the
associated diagonal bundle is just $\mathcal{D}_{a,-}$. Also it is equipped
with a splitting as in (5.2), (5.3).

Moreover, $A^{p_{*}\mathscr{E}_{0}\prime\prime}$ preserves the smooth sections
of $p_{*}\mathscr{E}_{0,a,\pm}$, and it induces an antiholomorphic
superconnection $A^{p_{*}\mathscr{E}_{0,a,\pm}\prime\prime}$ on
$p_{*}\mathscr{E}_{0,a,\pm}$.

On $U_{a}$, $P_{a,-}:\mathscr{E}_{0}\to\mathscr{E}_{0,a,-}$ and
$i_{a,-}=\mathscr{E}_{0,a,-}\to\mathscr{E}_{0}$ are quasi-isomorphisms of
$\mathcal{O}_{U_{a}}$-complexes, and $\mathscr{H}\mathscr{E}_{0,a,+}=0$.
Finally, for $s\in U_{a}$, the complex $\mathcal{D}_{a,+,s}$ is exact.

###### Proof.

The proof is essentially the same as the proof of Theorem 8.21. With respect
to that Theorem, here, only $D_{a,-}$ is finite-dimensional. Let us now prove
that if $s\in U_{a}$, $\mathcal{D}_{a,+}$ is exact. By proceeding as in the
proof of Proposition 8.22, i.e., using the analogue of (8.94), the part of
degree $0$ over $\Lambda\left(\overline{T^{*}S}\right)$ of
$A^{p_{*}\mathscr{E}}_{0,a,+}$ is just the restriction of
$A^{\mathcal{D}\prime\prime}$ to $\mathcal{D}_{a,+}$. By Hodge theory, the
complex $\mathcal{D}_{a,+,s}$ is exact. The proof of our theorem is completed.
∎

When considering another pair of metrics $g^{TX},g^{D}$ on $TX,D$, the obvious
analogue of Proposition 8.24 still holds. Details are left to the reader.

As a consequence of Theorem 11.3, we give another proof of a theorem of
Grauert [GrR84, Theorem 10.4.6].

######  Theorem 11.4.

The complex of $\mathcal{O}_{S}$-modules $p_{*}\mathscr{E}_{0}$ defines an
element in $\mathrm{D^{b}_{\mathrm{coh}}}\left(S\right)$. Moreover, the Leray
spectral sequence associated with the filtration of $p_{*}\mathscr{E}_{0}$ by
$\Lambda\left(\overline{T^{*}S}\right)$ degenerates at $E_{2}$.

###### Proof.

By Theorem 11.3, over $U_{a}$, $p_{*}\mathscr{E}_{0}$ and
$p_{*}\mathscr{E}_{0,a,-}$ are quasi-isomorphic. By Theorem 5.7,
$\mathscr{H}p_{*}\mathscr{E}_{0,a,-}$ is a coherent sheaf. This shows that
$p_{*}\mathscr{E}_{0}$ defines an object in
$\mathrm{D^{b}_{\mathrm{coh}}}\left(S\right)$. By Theorem 5.7, the spectral
sequence associated with $p_{*}\mathscr{E}_{0,a,-}$ degenerates at $E_{2}$.
Also on $D_{a,+}$, we have the identity

(11.13)
$1=\left[A^{\mathcal{D}\prime\prime},A^{\mathcal{D}\prime\prime*}\left[A^{\mathcal{D},2}\right]^{-1}\right].$

Equation (11.13) shows that on $p_{*}\mathscr{E}_{0,a,+}$, the spectral
sequence stops at $E_{1}$. This concludes the proof of our theorem. ∎

######  Remark 11.5.

In the context of the theory of determinant bundles of direct images, Theorems
11.3 and 11.4 can be used as a substitute of many arguments in [BGS88c,
Section 3], where non-projective Kähler manifolds are also considered.

###  11.2. The existence theorem

Recall that (10.13), (10.14) hold, and that
$A^{p_{*}\mathscr{E}_{0}\prime\prime}$ is given by (10.15).

Recall that if $\mathscr{F}\in\mathrm{D^{b}_{\mathrm{coh}}}\left(M\right)$,
$\mathscr{R}\mathscr{H}\mathscr{F}\in K\left(M\right)$ was defined in (8.62).

As we saw in Theorem 11.4, $p_{*}\mathscr{E}_{0}$ is an object in
$\mathrm{D^{b}_{\mathrm{coh}}}\left(S\right)$.

######  Theorem 11.6.

There exists a finite-dimensional antiholomorphic superconnection
$\underline{\mathscr{E}}=\left(\underline{E},A^{\underline{E}\prime\prime}\right)$
on $S$, and a morphism of
$\mathcal{O}^{\infty}_{S}\left(\Lambda\left(\overline{T^{*}S}\right)\right)$-modules
$\phi:\underline{\mathscr{E}}\to p_{*}\mathscr{E}_{0}$ which is a quasi-
isomorphism of $\mathcal{O}_{S}$-complexes, such that for any $s\in S$,
$\phi_{s}:\left(\underline{D},\underline{v}_{0}\right)_{s}\to\left(\mathcal{D},A^{\mathcal{D}\prime\prime}\right)_{s}$
is a quasi-isomorphism.

We have the identity

(11.14) $\underline{\mathscr{E}}\simeq Rp_{*}\mathscr{E}\
\mathrm{in}\,\mathrm{D^{b}_{\mathrm{coh}}}\left(S\right).$

In particular, we have the identity,

(11.15)
$\mathscr{R}\mathscr{H}\underline{\mathscr{E}}=p_{!}\left[\mathscr{R}\mathscr{H}\mathscr{E}\right]\
\mathrm{in}\ K\left(S\right).$

###### Proof.

By Proposition 6.12, there exists an object $\underline{\mathscr{E}}$ in
$\mathrm{B}\left(S\right)$ and a morphism of
$\mathcal{O}^{\infty}_{S}\left(\Lambda\left(\overline{T^{*}S}\right)\right)$-modules
$\phi:\underline{\mathscr{E}}\to p_{*}\mathscr{E}_{0}$, which is also a quasi-
isomorphism of $\mathcal{O}_{S}$-complexes. Using Theorem 11.3, over $U_{a}$,
if $\phi_{a}=P_{a,-}\phi$, $\phi_{a}$ induces a quasi-isomorphism
$\underline{\mathscr{E}}\to p_{*}\mathscr{E}_{0,a,-}$.

Over $U_{a}$, we now deal with finite dimensional antiholomorphic
superconnections. By Proposition 6.8, for $s\in U_{a}$, $\phi_{a}$ induces a
quasi-isomorphism $\underline{D}_{s}\to\mathcal{D}_{a,-,s}$. By Theorem 11.3,
for $s\in U_{a}$, the complex $\mathcal{D}_{a,+,s}$ is exact. Therefore, for
any $s\in U_{a}$, $\phi_{s}:\underline{D}_{s}\to\mathcal{D}_{s}$ is a quasi-
isomorphism.

Since $\phi$ is a quasi-isomorphism, we get (11.14), from which (11.15)
follows. The proof of our theorem is completed. ∎

######  Remark 11.7.

By equation (11.14) in Theorem 11.6, if $\underline{E}^{\prime}$ has the same
properties as $\underline{E}$, then

(11.16) $\underline{\mathscr{E}}\simeq\underline{\mathscr{E}}^{\prime}\
\mathrm{in}\ \mathrm{D^{b}_{\mathrm{coh}}}\left(S\right).$

Using Theorem 6.9 and (11.16), we deduce that there exists a quasi-isomorphism
$\psi:\underline{\mathscr{E}}\to\underline{\mathscr{E}}^{\prime}$. By
Proposition 6.8, this just says that if $\underline{D},\underline{D^{\prime}}$
are the corresponding diagonal bundles on $S$, for any $s\in S$,
$\psi_{s}:\underline{D}_{s}\to\underline{D}^{\prime}_{s}$ is a quasi-
isomorphism.

###  11.3. The Chern character of the direct image

We use the notation of Theorem 11.6. In particular $\underline{\mathscr{E}}$
is taken as in this Theorem.

######  Theorem 11.8.

The following identity holds:

(11.17)
$\mathrm{ch}_{\mathrm{BC}}\left(A^{p_{*}\mathscr{E}\prime\prime}\right)=\mathrm{ch}_{\mathrm{BC}}\left(\underline{\mathscr{E}}\right)\
\mathrm{in}\ H^{(=)}_{\mathrm{BC}}\left(S,\mathbf{R}\right).$

###### Proof.

The proof is an adaptation of the proof of Theorems 8.9 and 8.11. We will use
the notation in Theorem 11.6. For $t\in\mathbf{C}$, we form the cone
$\mathscr{C}_{t}=\mathrm{cone}\left(\underline{\mathscr{E}},p_{*}\mathscr{E}\right)$
associated with the morphism $t\phi$, with underlying infinite-dimensional
vector bundle $C_{t}$ on $S$. Note that $\phi$ induces a morphism
$\underline{D}\to\mathcal{D}$. The diagonal bundle $\mathbf{D}$ associated
with $\mathscr{C}_{t}$ is given
$\mathbf{D}_{t}=\mathrm{cone}\left(\underline{D},\mathcal{D}\right)$, where
the morphism is induced by $t\phi_{0}$. Also

(11.18)
$\mathbf{D}^{\scriptsize\bullet}_{t}=\underline{D}^{\scriptsize\bullet+1}\oplus\mathcal{D}^{\scriptsize\bullet}.$

By Theorem 11.6, for any $s\in S$, when $t\neq 0$, $\mathbf{D}_{t}$ is exact.

We fix splittings of $E,\underline{E}$ as in (5.2), (5.3). In the sequel, we
may as well replace $\underline{\mathscr{E}},\mathscr{E}$ by
$\underline{\mathscr{E}}_{0},\mathscr{E}_{0}$. In particular $\mathscr{C}_{t}$
also splits and can be replaced by $\mathscr{C}_{t,0}$.

Let $g^{\underline{D}}$ be a Hermitian metric on $\underline{D}$. Let
$g^{\mathbf{D}}$ denote the direct sum Hermitian metric on $\mathbf{D}$
associated with $g^{\underline{D}},g^{\mathcal{D}}$. We equip $C_{t,0}$ with
the pure metric $g^{\mathbf{D}}$. Let $\square^{\mathbf{D}}_{t}$ denote the
corresponding fiberwise Hodge Laplacian acting on $\mathbf{D}$. Since for any
$s\in S$, $\mathbf{D}_{t}$ is exact, $\square^{\mathbf{D}}_{t,s}$ is
invertible, and it has a uniform positive lower bound over $S$. Its heat
kernel is again fiberwise trace class.

Let $A^{C_{t,0}\prime\prime}$ denote the antiholomorphic superconnection on
$C_{t}$ that is associated with
$A^{\underline{E}_{0}\prime\prime},A^{p_{*}\mathscr{E}_{0}\prime\prime},t\phi$.

Given $t\in\mathbf{C}$, we can still define the Chern form
$\mathrm{ch}\left(A^{C_{t,0}\prime\prime},\omega^{X},g^{D},g^{\underline{D}}\right)$,
for which the analogue of Theorem 10.15 holds, with a similar proof. Let
$\mathrm{ch}_{\mathrm{BC}}\left(A^{C_{t,0}\prime\prime}\right)\in
H_{\mathrm{BC}}^{(=)}\left(S,\mathbf{R}\right)$ denote the corresponding
common Bott-Chern cohomology class.

For $T>0$, let $g_{T}^{D},g_{T}^{\underline{D}}$ be the metrics on
$D,\underline{D}$ constructed as in (8.35). Let $g^{\mathbf{D}}_{T}$ be the
metric on $\mathbf{D}$ that is associated with the metrics
$\omega^{X}/T,g^{D}_{T},g^{\underline{D}}_{T}T^{-n-1}$. If $N^{\mathbf{D}}$ is
the number operator of $\mathbf{D}$, then

(11.19) $g^{\mathbf{D}}_{T}=T^{-n}g^{\mathbf{D}}T^{N^{\mathbf{D}}}.$

The proof of (11.19) is elementary. The extra factor $T^{-n}$ comes from the
contribution of the volume form on $X$ to the Hermitian product on
$\mathcal{D}$. This normalizing factor does not contribute to the computation
of the adjoint $A^{C_{t,0}\prime}_{T}$ of $A^{C_{t,0}\prime\prime}$. By
(11.19), if $\square^{\mathbf{D}}_{t,T}$ denotes the associated fiberwise
Laplacian, then

(11.20) $\square^{\mathbf{D}}_{t,T}=T\square^{\mathbf{D}}_{t}.$

We claim that if $t\neq 0$, as $T\to+\infty$,

(11.21)
$\mathrm{ch}\left(A^{C_{t,0}\prime\prime},\omega^{X}/T,g_{T}^{D},g_{T}^{\underline{D}}T^{-n-1}\right)\to
0.$

The proof is an infinite-dimensional analogue of equation (8.40) in Theorem
8.9, with $HD=0$ which can be established by the methods of [BGV92, Theorems
9.19 and 9.23], [B97, Theorems 9.5 and 9.6].

By (11.21), we deduce that for $t\neq 0$,

(11.22) $\mathrm{ch}_{\mathrm{BC}}\left(A^{C_{t,0}\prime\prime}\right)=0.$

By taking $t\to 0$ in (11.22), we get

(11.23) $\mathrm{ch}_{\mathrm{BC}}\left(A^{C_{0,0}\prime\prime}\right)=0.$

As in (8.56), we have the identity

(11.24)
$\mathrm{ch}_{\mathrm{BC}}\left(A^{C_{0,0}\prime\prime}\right)=\mathrm{ch}_{\mathrm{BC}}\left(A^{p_{*}\mathscr{E}_{0}\prime\prime}\right)-\mathrm{ch}_{\mathrm{BC}}\left(\underline{\mathscr{E}}\right).$

By (11.23), (11.24), we get (11.17). The proof of our theorem is completed. ∎

######  Remark 11.9.

By Theorem 8.12 and Remark 11.7, we already know that
$\mathrm{ch}_{\mathrm{BC}}\left(\underline{\mathscr{E}}\right)$ does not
depend on the choice of $\underline{E}$. Theorem 11.8 gives a direct proof of
this fact.

######  Theorem 11.10.

The following identity holds:

(11.25)
$\mathrm{ch}_{\mathrm{BC}}\left(A^{p_{*}\mathscr{E}\prime\prime}\right)=\mathrm{ch}_{\mathrm{BC}}\left(Rp_{*}\
\mathscr{E}\right)\ \mathrm{in}\
H^{(=)}_{\mathrm{BC}}\left(S,\mathbf{R}\right).$

###### Proof.

Using equation (11.14) in Theorem 11.6 and Theorem 11.8, we get (11.25). ∎

###  11.4. The case where $\mathscr{H}Rp_{*}\mathscr{E}$ is locally free

By Theorem 8.16, we know that

(11.26)
$\mathrm{ch}_{\mathrm{BC}}\left(\underline{\mathscr{E}}\right)=\mathrm{ch}_{\mathrm{BC}}\left(\mathscr{R}\mathscr{H}\underline{\mathscr{E}}\right)\
\mathrm{in}\ H^{(=)}_{\mathrm{BC}}\left(S,\mathbf{R}\right).$

Also $\mathscr{H}Rp_{*}\mathscr{E}$ is a $\mathbf{Z}$-graded coherent sheaf on
$S$. Using equation (11.17) in Theorem 11.10 and (11.26), we get

(11.27)
$\mathrm{ch}_{\mathrm{BC}}\left(A^{p_{*}\mathscr{E}\prime\prime}\right)=\mathrm{ch}_{\mathrm{BC}}\left(\mathscr{H}Rp_{*}\mathscr{E}\right)\
\mathrm{in}\ H^{(=)}_{\mathrm{BC}}\left(S,\mathbf{R}\right).$

When $\mathscr{H}Rp_{*}\mathscr{E}$ is locally free, there is a simpler proof
of Theorem 11.10 not relying on Theorem 11.6. In this case, the same arguments
as in Subsection 8.5 show that the cohomology of the complex
$\left(\mathcal{D},A^{\mathcal{D}\prime\prime}\right)$ is a finite-dimensional
holomorphic vector bundle $H$ on $S$. Using fiberwise Hodge theory, we can
identify $H$ with the corresponding fiberwise harmonic sections $\mathcal{H}$.
Let $g^{H}$ denote the corresponding Hermitian metric on $H$.

Under the above assumption, a strict analogue of Theorem 8.9 can be proved.
More precisely, by proceeding as in Berline-Getzler-Vergne [BGV92, Theorems
9.19 and 9.23], [B97, Theorems 9.5 and 9.6], and using the obvious analogue of
(11.19), one can show that as $T\to+\infty$, we have the uniform convergence
of forms on $S$,

(11.28)
$\mathrm{ch}\left(A^{p_{*}\mathscr{E}_{0}\prime\prime},\omega^{X}/T,g_{T}^{D}\right)\to\mathrm{ch}\left(H,g^{H}\right).$

By (11.28), we deduce that

(11.29)
$\mathrm{ch}_{\mathrm{BC}}\left(A^{p_{*}\mathscr{E}\prime\prime}\right)=\mathrm{ch}_{\mathrm{BC}}\left(H\right)\
\mathrm{in}\ H^{(=)}_{\mathrm{BC}}\left(S,\mathbf{R}\right).$

Moreover,

(11.30) $\mathscr{H}Rp_{*}\mathscr{E}=H\,\mathrm{in}\,K\left(S\right).$

Using (11.29), (11.30), we reobtain Theorem 11.10.

The above strategy was used in Bismut [B13, Theorems 4.10.4 and 4.11.2] in the
case where $D$ is a holomorphic vector bundle on $M$, and
$\left(E,A^{E\prime\prime}\right)=\left(D,\nabla^{D\prime\prime}\right)$, in
order to establish Theorem 11.10.

##  12\. A proof of Theorem 10.1 when
$\overline{\partial}^{X}\partial^{X}\omega^{X}=0$.

We make the same assumptions as in Section 10. We will prove Theorem 10.1
under the assumption that $\overline{\partial}^{X}\partial^{X}\omega^{X}=0$,
by extending [B13, Theorem 5.1.2]. Under this assumption, by the results of
Subsections 2.4 and 10.4, $\underline{R}^{T_{\mathbf{R}}X}$ is a $(1,1)$ form,
so that
$\widehat{A}\left(T_{\mathbf{R}}X,\underline{\nabla}^{T_{\mathbf{R}}X}\right)$
is a closed form in $\Omega^{(=)}\left(X,\mathbf{R}\right)$.

######  Theorem 12.1.

If $\overline{\partial}^{X}\partial^{X}\omega^{X}=0$, as $t\to 0$, we have the
uniform convergence of forms on $S$,

(12.1)
$\mathrm{ch}\left(A^{p_{*}\mathscr{E}_{0}\prime\prime},\omega^{X}/t,g^{D}\right)\\\
\to
p_{*}\left[q^{*}\left[\widehat{A}\left(T_{\mathbf{R}}X,\underline{\nabla}^{T_{\mathbf{R}}X}\right)e^{c_{1}\left(TX,g^{TX}\right)/2}\right]\mathrm{ch}\left(A^{E_{0}\prime\prime},g^{D}\right)\right].$

In particular, if $\omega^{X}$ is closed, as $t\to 0$,

(12.2)
$\mathrm{ch}\left(A^{p_{*}\mathscr{E}_{0}\prime\prime},\omega^{X}/t,g^{D}\right)\to
p_{*}\left[q^{*}\mathrm{Td}\left(TX,g^{TX}\right)\mathrm{ch}\left(A^{E_{0}\prime\prime},g^{D}\right)\right].$

If $\overline{\partial}^{X}\partial^{X}\omega^{X}=0$, then

(12.3)
$\mathrm{ch}_{\mathrm{BC}}\left(Rp_{*}\mathscr{E}\right)=p_{*}\left[q^{*}\mathrm{Td}_{\mathrm{BC}}\left(TX\right)\mathrm{ch}_{\mathrm{BC}}\left(\mathscr{E}\right)\right]\,\mathrm{in}\,H^{(=)}_{\mathrm{BC}}\left(S,\mathbf{R}\right).$

###### Proof.

In the proof, we will assume that
$\overline{\partial}^{X}\partial^{X}\omega^{X}=0$. Also, we fix $s\in S$. The
analysis will be done in the fiber $X\times\left\\{s\right\\}$.

Given $t>0,x,x^{\prime}\in X$, let $P_{t}\left(x,x^{\prime}\right)$ denote the
smooth kernel for $\exp\left(-A^{p_{*}\mathscr{E}_{0},2}_{t}\right)$ along the
fiber $X\times\\{s\\}$ with respect to the volume
$dx^{\prime}/\left(2\pi\right)^{n}$ 303030In the proof, the parameter $s$ will
be usually omitted.. Then

(12.4)
$\mathrm{ch}\left(A^{p_{*}\mathscr{E}_{0}\prime\prime},\omega^{X}/t,g^{D}\right)=\varphi\int_{X}\mathrm{Tr_{s}}\left[P_{t}\left(x,x\right)\right]\frac{dx}{\left(2\pi\right)^{n}}.$

To calculate the asymptotics of (12.4) as $t\to 0$, we will use the local
index theoretic methods of [B86, BGS88c, B12].

We denote by ${}^{\mathrm{c}_{t}}$ the map c associated with the metric
$g^{TX}/t$ on $TX$, that induces the metric $tg^{T^{*}X}$ on $T^{*}X$. By
(4.42), if $f\in T^{*}X$, if $f_{*}\in\overline{TX}$ corresponds to $f$ by the
metric $g^{TX}$, then

(12.5) $\displaystyle c_{t}\left(\overline{f}\right)=\overline{f}\wedge,$
$\displaystyle c_{t}\left(f\right)=-i_{tf_{*}}.$

Let $A_{t}^{p_{*}\mathscr{E}_{0}}$ denote the superconnection
$A^{p_{*}\mathscr{E}_{0}}$ associated with $\omega^{X}/t,g^{D}$. In the right-
hand side of the equation for $A^{p_{*}\mathscr{E}_{0},2}_{t}$ in (10.12), by
(10.53), in the first row of the right-hand side, the first term is now

(12.6)
$-\frac{1}{2}t\left(\overline{\nabla}_{e_{i}}^{\Lambda\left(\overline{T^{*}X}\right)\otimes
D}-\frac{1}{2}\mathrm{Tr}^{TX}\left[A\left(e_{i}\right)\right]+{}^{\mathrm{c}_{t}}\left(i_{e_{i}}C\right)\right)^{2}.$

The second term is also scaled by the factor $t$. The term containing
$\overline{\partial}^{X}\partial^{X}i\omega^{X}$ is absent in the right-hand
side. Also the $e_{i}$ are replaced by $\sqrt{t}e_{i}$.

We will compute the limit as $t\to 0$ of
$\mathrm{Tr_{s}}\left[P_{t}\left(x,x\right)\right]$. We fix $x\in X$. We take
geodesic coordinates on $X$ near $x$ with respect to the metric $g^{TX}$. This
way, for $\epsilon>0$ small enough, for any $x\in X$, we identify the open
ball $B^{T_{\mathbf{R},x}X}\left(0,\epsilon\right)$ in $T_{\mathbf{R},x}X$ to
a corresponding open geodesic ball $B^{X}\left(x,\epsilon\right)$ in $X$.
Given $Y\in T_{\mathbf{R},x}X$, along the geodesic
$u\in\mathbf{R}\to\exp_{x}\left(uY\right)\in X$, we trivialize $TX,D$ by
parallel transport with respect to the connections
$\overline{\nabla}^{TX},\nabla^{D}$. Also we trivialize
$\Lambda\left(\overline{T^{*}X}\right)$ via the connection
$\overline{\nabla}^{\Lambda\left(\overline{T^{*}X}\right)}$. If
$w_{1},\ldots,w_{n}$ is an orthonormal basis of $T_{x}X$, it can also be
viewed as an orthonormal trivialization of $TX$ near $x$. In this
trivialization, we have the identity

(12.7) $\omega^{X}=-iw^{i}\overline{w}^{i}.$

If $e_{1},\ldots,e_{2n}$ is an orthonormal basis of $T_{\mathbf{R},x}X$, via
the parallel transport with respect to $\overline{\nabla}^{T_{\mathbf{R}}X}$
along the above geodesics, we obtain an orthonormal basis of $T_{\mathbf{R}}X$
near $x$.

Using the above trivialization, near $x$, the operator
$A^{p_{*}\mathscr{E}_{0},2}_{t}$ acts on

$H_{x,\epsilon}=C^{\infty}\left(B^{T_{\mathbf{R},x}\left(0,\epsilon\right)},\Lambda\left(T^{*}_{\mathbf{C},s}S\right)\widehat{\otimes}\Lambda\left(\overline{T^{*}_{x}X}\right)\widehat{\otimes}D_{s,x}\right).$

A first standard step is to construct an elliptic operator acting on

$H_{x}=C^{\infty}\left(T_{\mathbf{R},x}X,\Lambda\left(T^{*}_{\mathbf{C},s}S\right)\widehat{\otimes}\Lambda\left(\overline{T^{*}_{x}X}\right)\widehat{\otimes}D_{s,x}\right),$

that coincides with $A^{p_{*}\mathscr{E}_{0},2}_{t}$ on
$B^{T_{\mathbf{R},x}X}\left(0,\epsilon/2\right)$, the point being that the
asymptotics of $\mathrm{Tr_{s}}\left[P_{t}\left(x,x\right)\right]$ as $t\to 0$
can be computed using this new operator. This step being standard, we will
skip it, and pretend that $A^{p_{*}\mathscr{E}_{0},2}_{t}$ acts on $H_{x}$,
and that $P_{t}\left(Y,Y^{\prime}\right)$ is now a kernel on
$T_{\mathbf{R},x}X$.

For $a>0,u\in H_{x}$, put

(12.8) $F_{a}u\left(Y\right)=u\left(aY\right).$

Set

(12.9) $L_{t,x}=F_{\sqrt{t}}A^{p_{*}\mathscr{E}_{0},2}_{t}F_{1/\sqrt{t}}.$

Let $Q_{t,x}\left(Y,Y^{\prime}\right),Y,Y^{\prime}\in T_{\mathbf{R},x}X$ be
the smooth kernel for $\exp\left(-L_{t,x}\right)$ with respect to the volume
form $\frac{dY^{\prime}}{\left(2\pi\right)^{n}}$. Clearly

(12.10) $P_{t}\left(x,x\right)=t^{-n}Q_{t,x}\left(0,0\right).$

Among the monomials in the $\overline{w}^{i},i_{\overline{w}_{j}}$, observe
that up to obvious permutations, the only monomial whose supertrace on
$\Lambda\left(\overline{T^{*}_{x}X}\right)$ is nonzero is
$\prod_{i=1}^{n}\overline{w}^{i}i_{\overline{w}_{i}}$, and moreover

(12.11)
$\mathrm{Tr_{s}}^{\Lambda\left(\overline{T^{*}_{x}X}\right)}\left[\prod_{i=1}^{n}\overline{w}^{i}i_{\overline{w}_{i}}\right]=\left(-1\right)^{n}.$

If $I$ is a strictly ordered multi-index with values in $1,\ldots,n$, put
$\overline{w}^{I}=\prod_{i\in
I}\overline{w}^{i},i_{\overline{w}_{I}}=\prod_{i\in I}i_{\overline{w}_{i}}$.
Any $T\in\mathrm{End}\left(\Lambda\left(\overline{T^{*}_{x}X}\right)\right)$
can be written uniquely as a linear combination of operators of the form
$\overline{w}^{I}i_{\overline{w}_{J}}$. Put

(12.12) $T_{t}=e^{-i\omega^{X}_{x}/t}Te^{i\omega^{X}_{x}/t}.$

Conjugation by the form $e^{-i\omega_{x}^{X}/t}$ leaves unchanged the
operators $\overline{w}^{i}\wedge$ and replaces the operators
$i_{t\overline{w}_{i}}$ by $i_{t\overline{w}_{i}}-w^{i}\wedge$. Let
$T_{t}^{\max}\in\mathbf{C}$ be the coefficient of
$\left(-i\right)^{n}\prod_{1}^{n}\overline{w}^{i}w^{i}$ in the obvious
expansion of $T_{t}$. By (12.11), (12.12), we get

(12.13)
$\mathrm{Tr_{s}}^{\Lambda\left(\overline{T^{*}_{x}X}\right)}\left[T\right]=\frac{t^{n}}{i^{n}}T_{t}^{\max}.$

If $T$ is instead an element of
$\mathrm{End}\left(\Lambda\left(\overline{T_{x}^{*}X}\right)\right)\widehat{\otimes}\mathrm{End}\left(D_{s,x}\right)$,
we still define $T_{t}$ as in (12.12). Equation (12.13) is replaced by

(12.14)
$\mathrm{Tr_{s}}^{\Lambda\left(\overline{T^{*}_{x}X}\right)\widehat{\otimes}D_{s,x}}\left[T\right]=\frac{t^{n}}{i^{n}}\mathrm{Tr_{s}}^{D_{s,x}}\left[T_{t}\right]^{\max}.$

When viewing $w^{i},\overline{w}^{i}$ as differential forms on $X$ at $x$,
then $\left(-i\right)^{n}\prod_{1}^{n}\overline{w}^{i}w^{i}$ is just the
canonical volume $2n$-form.

In $L_{t,x}$, no matrix operator of the form $w^{i}\wedge$ appears. Put

(12.15) $M_{t,x}=e^{-i\omega_{x}^{X}/t}L_{t,x}e^{i\omega_{x}^{X}/t}.$

The above conjugation will play the role of Getzler rescaling [G86] in local
index theory.

Let $R_{t,x}\left(Y,Y^{\prime}\right),Y,Y^{\prime}\in T_{\mathbf{R},x}X$ be
the smooth kernel for $\exp\left(-M_{t,x}\right)$ with respect to the volume
$\frac{dY^{\prime}}{\left(2\pi\right)^{n}}$. By (12.10), (12.14), we get

(12.16)
$\mathrm{Tr_{s}}\left[P_{t}\left(x,x\right)\right]=\frac{1}{i^{n}}\mathrm{Tr_{s}}^{D_{s,x}}\left[R_{t,x}\left(0,0\right)\right]^{\max}.$

To compute the asymptotics of (12.16) as $t\to 0$, we use the second equation
in (10.12) for $A^{p_{*}\mathscr{E}_{0},2}_{t}$ while implementing the above
trivializations and conjugations. We will not write the conjugations
explicitly.

We view $\overline{R}^{TX}_{x}$ as a section of
$\Lambda^{2}\left(T^{*}_{\mathbf{R},x}X\right)\otimes\mathrm{End}\left(T_{x}X\right)$.
Let $\overline{\Gamma}^{TX}$ be the connection form for
$\overline{\nabla}^{TX}$ in the above parallel transport trivialization. It is
well-known that for $Y\in T_{\mathbf{R},x}X$,

(12.17)
$\overline{\Gamma}^{TX}_{Y}=\frac{1}{2}\overline{R}_{x}^{TX}\left(Y,\cdot\right)+\mathcal{O}\left(\left|Y\right|^{2}\right).$

By (12.17), we deduce that if
$\overline{\Gamma}^{\Lambda\left(\overline{T^{*}X}\right)}$ is the
corresponding connection form for
$\overline{\nabla}^{\Lambda\left(\overline{T^{*}X}\right)}$, then

(12.18)
$\overline{\Gamma}^{\Lambda\left(\overline{T^{*}X}\right)}_{Y}=-\frac{1}{2}\left\langle\overline{R}_{x}^{TX}\left(Y,\cdot\right)\overline{w}_{i},w_{j}\right\rangle\overline{w}^{i}i_{\overline{w}^{j}}+\mathcal{O}\left(\left|Y\right|^{2}\right),$

with $\mathcal{O}\left(\left|Y\right|^{2}\right)$ still being linear
combinations of the $\overline{w}^{i}i_{\overline{w}_{j}}$.

Let $\Gamma^{D}$ be the connection form in the considered trivialization of
$D$. Then

(12.19)
$-F_{\sqrt{t}}\frac{1}{2}t\left(\overline{\nabla}_{e_{i}}^{\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D}-\frac{1}{2}\mathrm{Tr}^{TX}\left[A\left(e_{i}\right)\right]+{}^{\mathrm{c}_{t}}i_{e_{i}}C\right)^{2}F_{1/\sqrt{t}}\\\
=-\frac{1}{2}\Biggl{(}\nabla_{e_{i}\left(\sqrt{t}Y\right)}+\sqrt{t}\left(\overline{\Gamma}^{\Lambda\left(\overline{T^{*}X}\right)}+\Gamma^{D}\right)_{\sqrt{t}Y}\left(e_{i}(\sqrt{t}Y)\right)\\\
-\frac{\sqrt{t}}{2}\mathrm{Tr}^{TX}\left[A\left(e_{i}\right)\right]_{{}_{\sqrt{t}Y}}+\sqrt{t}{}^{c_{t}}i_{e_{i}}C_{\sqrt{t}Y}\Biggr{)}^{2}.$

We conjugate (12.19) by $e^{-i\omega_{x}^{X}/t}$. Using (12.18), we find that
as $t\to 0$,

(12.20)
$\nabla_{e_{i}\left(\sqrt{t}Y\right)}+\sqrt{t}\left(\overline{\Gamma}^{\Lambda\left(\overline{T^{*}X}\right)}+\Gamma^{D}\right)_{\sqrt{t}Y}\left(e_{i}(\sqrt{t}Y)\right)\\\
\to\nabla_{e_{i}}+\frac{1}{2}\left\langle\overline{R}^{TX}_{x}\left(Y,e_{i}\right)\overline{w}_{i},w_{j}\right\rangle\overline{w}^{i}w^{j}.$

Note that ${}^{c_{t}}B=B$, and ${}^{c_{t}}B^{*}$ is obtained from $B^{*}$ by
replacing the exterior variables $w^{i}$ by $-i_{t\overline{w}_{i}}$. By the
considerations that follow (12.11), when conjugating $-i_{t\overline{w}_{i}}$
by $e^{-i\omega^{X}_{x}/t}$, we obtain $w^{i}-i_{t\overline{w}_{i}}$, so that
as $t\to 0$, after conjugation,

(12.21) ${}^{c_{t}}i_{e_{i}}C_{\sqrt{t}Y}\to i_{e_{i}}C_{x}.$

By (12.21), we deduce that

(12.22) $\sqrt{t}{}^{c_{t}}i_{e_{i}}C_{\sqrt{t}Y}\to 0.$

By (12.20), (12.22), we find that after conjugation, as $t\to 0$,

(12.23)
$-\frac{1}{2}t\left(\overline{\nabla}_{e_{i}}^{\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D}-\frac{1}{2}\mathrm{Tr}^{TX}\left[A\left(e_{i}\right)\right]+{}^{\mathrm{c}_{t}}i_{e_{i}}C\right)^{2}\\\
\to-\frac{1}{2}\left(\nabla_{e_{i}}+\frac{1}{2}\left\langle\overline{R}_{x}^{TX}\left(Y,e_{i}\right)\overline{w}_{i},w_{j}\right\rangle\overline{w}^{i}w^{j}\right)^{2}.$

We will use the fact that $\overline{\partial}^{X}\partial^{X}\omega^{X}=0$,
and also (10.43). Recall that $w^{i},\overline{w}^{j}$ are considered as
standard differential forms on $X$ at $x$. Then (12.23) can be rewritten in
the form

(12.24)
$-\frac{1}{2}t\left(\overline{\nabla}_{e_{i}}^{\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}D}-\frac{1}{2}\mathrm{Tr}^{TX}\left[A\left(e_{i}\right)\right]+{}^{\mathrm{c}_{t}}i_{e_{i}}C\right)^{2}\\\
\to-\frac{1}{2}\left(\nabla_{e_{i}}+\frac{1}{2}\left\langle\underline{R}_{x}^{TX}Y,e_{i}\right\rangle\right)^{2}.$

Consider now the remaining terms in the right-hand side of (10.12), while
still using the fact that $\overline{\partial}^{X}\partial^{X}\omega^{X}=0$.
By proceeding as before, we find that after conjugation,

(12.25)
${}^{\mathrm{c}_{t}}\left(A^{E_{0},2}+\frac{1}{2}\mathrm{Tr}\left[R^{TX}\right]\right)_{\sqrt{t}Y}\to\left(A^{E_{0},2}+\frac{1}{2}\mathrm{Tr}\left[R^{TX}\right]\right)_{s,x}.$

To obtain the behavior as $t\to 0$ of the remaining terms in the right-hand
side of the second equation in (10.12), we will use the third identity with
$\beta$ replaced by $\beta/t$, c by ${}^{c_{t}}$, and $e_{i}$ by
$\sqrt{t}e_{i}$. After conjugation, for $k\geq 3$, the limit of the
corresponding terms vanishes. Since $\beta$ is a $3$-form,
$i_{e_{i_{1}}}i_{e_{i_{2}}}\beta$ is a $1$-form, whose square vanishes.
Ultimately, after conjugation, the limit of all the terms vanishes.

Put

(12.26)
$M_{0,s,x}=-\frac{1}{2}\left(\nabla_{e_{i}}+\frac{1}{2}\left\langle\underline{R}_{x}^{TX}Y,e_{i}\right\rangle\right)^{2}+A^{E_{0},2}_{s,x}+\frac{1}{2}\mathrm{Tr}\left[R^{TX}_{x}\right].$

By (10.12), (12.24), (12.25), and by the considerations that follow, we see
that as $t\to 0$,

(12.27) $M_{t,x}\to M_{0,s,x},$

in the sense that we have a uniform convergence of the coefficients of the
considered operators together with all their derivatives over compact subsets
of $T_{\mathbf{R},x}X$.

Let $R_{0,s,x}\left(Y,Y^{\prime}\right),Y,Y^{\prime}\in T_{\mathbf{R},x}X$
denote the smooth kernel associated with the operator
$\exp\left(-M_{0,s,x}\right)$ with respect to
$\frac{dY^{\prime}}{\left(2\pi\right)^{n}}$. By proceeding as in [B86, Theorem
4.12], [BGV92, Theorem 10.23] using (12.27), as $t\to 0$, we have the uniform
convergence,

(12.28) $R_{t,x}\left(0,0\right)\to R_{0,s,x}\left(0,0\right).$

Note that $\mathrm{Tr_{s}}^{D_{s,x}}\left[R_{0,x}\left(0,0\right)\right]$
takes its values in
$\Lambda\left(T^{*}_{\mathbf{C}}S\right)\widehat{\otimes}\Lambda\left(T^{*}_{\mathbf{C}}X\right)$,
to which the full operator $\varphi$ now applies. Let
$\left\\{\varphi\mathrm{Tr}^{D_{s,x}}\left[Q_{0,s,x}\left(0,0\right)\right]\right\\}^{\max}\in\Lambda\left(T^{*}_{\mathbf{C}}S\right)$
be the coefficient of the volume form $dx$ viewed as a $2n$ form on
$T_{\mathbf{R},x}X$. Using (12.16), (12.28), as $t\to 0$, we have the uniform
convergence over $X$,

(12.29)
$\varphi\left(2\pi\right)^{-n}\mathrm{Tr_{s}}\left[P_{t}\left(x,x\right)\right]\to\left\\{\varphi\mathrm{Tr}^{D_{s,x}}\left[R_{0,s,x}\left(0,0\right)\right]\right\\}^{\max}.$

In the right-hand side of (12.27) one recognizes the classical harmonic
oscillator in index theory [G86], [BGV92, Proposition 4.19]. Using Mehler’s
formula [GlJ87, Theorem 1.5.10] and (8.1), we get

(12.30)
$\varphi\mathrm{Tr}^{D_{s,x}}\left[R_{0,s,x}\left(0,0\right)\right]=q^{*}\left[\widehat{A}\left(T_{\mathbf{R}}X,\underline{\nabla}^{T_{\mathbf{R}}X}\right)e^{c_{1}\left(TX,g^{TX}\right)/2}\right]\mathrm{ch}\left(A^{E_{0}\prime\prime},g^{D}\right).$

Using (12.4), (12.29), (12.30), and dominated convergence, we get (12.1). If
$\omega^{X}$ is closed,
$\overline{\nabla}^{T_{\mathbf{R}}X}=\nabla^{T_{\mathbf{R}}X}$, so that (12.1)
gives (12.2).

If $\omega^{X}$ is closed, (12.3) follows from (12.2). We will now prove
(12.3) when we only assume that
$\overline{\partial}^{X}\partial^{X}\omega^{X}=0$. Recall that if
$\alpha\in\Omega^{(=)}\left(X,\mathbf{R}\right)$ is closed,
$\left\\{\alpha\right\\}$ denotes its Bott-Chern cohomology class in
$H^{(=)}_{\mathrm{BC}}\left(X,\mathbf{R}\right)$. By Theorem 11.10 and by
(12.1), we get

(12.31)
$\mathrm{ch}_{\mathrm{BC}}\left(Rp_{*}\mathscr{E}\right)=p_{*}\left[q^{*}\left\\{\widehat{A}\left(T_{\mathbf{R}}X,\underline{\nabla}^{T_{\mathbf{R}}X}\right)\right\\}q^{*}e^{c_{1,\mathrm{BC}}\left(TX\right)}\mathrm{ch}_{\mathrm{BC}}\left(\mathscr{E}\right)\right].$

Now we proceed as in [B13, Theorem 5.2.1]. We have the equivalent exact
sequence of holomorphic vector bundles on $X$ in (10.41), (10.42). Using the
results of Bismut-Gillet-Soulé [BGS88a, Theorem 1.29] on Bott-Chern classes,
from the exact sequence in (10.41), using the notation in (2.27), we get

(12.32)
$\left\\{\widehat{A}\left(T_{\mathbf{R}}X,\underline{\nabla}^{T_{\mathbf{R}}X}\right)\right\\}=\widehat{A}_{\mathrm{BC}}\left(TX\right).$

By (2.28), we obtain

(12.33)
$\mathrm{Td}_{\mathrm{BC}}\left(TX\right)=\widehat{A}_{\mathrm{BC}}\left(TX\right)e^{c_{1,\mathrm{BC}}\left(TX\right)/2}.$

By (12.31)–(12.33), we get (12.3). The proof of our theorem is completed. ∎

######  Remark 12.2.

It is not possible to deform geometrically the exact sequence (10.42) by
scaling $\omega^{X}$. However, on the exact sequence (10.41), the deformation
arguments of [BGS88a] can still be used. More precisely, that if
$d\in\mathbf{C}^{*}$, replacing $\omega^{X}$ by $d\omega^{X}$ in the
construction of $\underline{\nabla}^{F\prime\prime}$ in (10.40) does not
change the Bott-Chern class, which ultimately allows us to make $d=0$, which
is equivalent to what we did in the proof of Theorem 12.1. Similar ideas will
be used in a more sophisticate context in the rest of the paper.

##  13\. The hypoelliptic superconnections

The purpose of this Section is to construct an antiholomorphic superconnection
over $S$ with fiberwise hypoelliptic curvature. This superconnection is a
deformation of the elliptic superconnections of Section 10. Our constructions
extend what was done in [B13, Chapter 6] for holomorphic vector bundles on $M$
to the case of antiholomorphic superconnections $A^{E\prime\prime}$.

This Section is organized as follows. In Subsection 13.1, we introduce the
total space $\mathcal{X}$ of an extra copy $\widehat{TX}$ of $TX$, and also
the antiholomorphic superconnection
$\mathscr{N}_{X}=\pi_{*}\mathscr{C}^{\mathcal{X}}$ on $X$.

In Subsection 13.2, given a Hermitian metric $g^{\widehat{TX}}$ on
$\widehat{TX}$, we obtain a corresponding splitting of $\mathscr{N}_{X}$.

In Subsection 13.3, we introduce the total space
$\underline{\pi}:\mathcal{M}\to M$ of $q^{*}\widehat{TX}$, and we construct a
natural infinite-dimensional antiholomorphic superconnection
$\mathscr{E}_{M}=\mathscr{E}\widehat{\otimes}_{b}\underline{\pi}_{*}\mathscr{C}^{M}$
over $M$.

In Subsection 13.4, we introduce the superconnection
$\mathcal{A}^{\prime\prime}$ that corresponds to $A^{E_{M}\prime\prime}$ via a
natural splitting of $E_{M}$.

In Subsection 13.5, given metrics $g^{\widehat{TX}},g^{D}$ on
$\widehat{TX},D$, we construct a nonpositive Hermitian form on the diagonal
bundle associated with $E_{M}$, and the corresponding adjoint superconnection
$\mathcal{A}^{\prime}$ of $\mathcal{A^{\prime\prime}}$.

In Subsection 13.6, if $\underline{q}:\mathcal{M}\to X$ is the obvious
projection, given a holomorphic section $z$ of $\underline{q}^{*}TX$ on
$\mathcal{M}$, we construct an antiholomorphic superconnection
$A^{E_{M}\prime\prime}_{z}$ on
$\Lambda\left(T^{*}M\right)\widehat{\otimes}\mathscr{E}_{M}$.

In Subsection 13.7, still using the above splittings, we obtain a
superconnection $\mathcal{A}^{\prime\prime}_{Z}$ and its companion
$\mathcal{A}^{\prime}_{Z}$, which also depends on the choice of a Kähler form
$\omega^{X}$ on $X$.

In Subsection 13.8, we construct a Hermitian form $\epsilon_{X}$ along the
fibers $\mathcal{X}$.

In Subsection 13.9, we compute the proper adjoint of $\mathcal{A}_{Z}$.

In Subsection 13.10, we give a Lichnerowicz formula for the curvature
$\mathcal{A}_{Z}^{2}$.

In Subsection 13.11, we specialize our constructions with $z$ is the canonical
section $y$, in which case $\mathcal{A}_{Z}$ is self-adjoint, and its
curvature is fiberwise hypoelliptic.

In Subsection 13.12, we study the behavior of $\mathcal{A}_{Y}$ under the
scaling of $g^{\widehat{TX}}$.

Finally, in Subsection 13.13, we give the arguments which show that when
replacing $g^{\widehat{TX}}$ by $b^{4}g^{\widehat{TX}}$, if
$g^{TX}=g^{\widehat{TX}}$, as $b\to 0$, the hypoelliptic superconnection
$\mathcal{A}_{Y,b}$ is a deformation of $A^{p_{*}\mathscr{E}_{0}}$.

In this Section, we make the same assumptions as in Section 10, and we use the
corresponding notation.

###  13.1. The total space of $\widehat{TX}$

Let $X$ be a compact complex manifold. Let $\pi:\mathcal{X}\to X$ be the total
space of $TX$, the fiber being denoted $\widehat{TX}$ to distinguish it from
the usual tangent bundle $TX$, so that $TX$ and $\widehat{TX}$ are canonically
isomorphic. The conjugate bundle to $\widehat{TX}$ will be denoted
$\overline{\widehat{TX}}$ and its dual $\overline{\widehat{T^{*}X}}$. The real
bundle associated with $\widehat{TX}$ is denoted by
$\widehat{T_{\mathbf{R}}X}$.

We have the exact sequences of holomorphic vector bundles on $\mathcal{X}$,

(13.1) $\displaystyle 0\to\pi^{*}\widehat{TX}\to T\mathcal{X}\to\pi^{*}TX\to
0,$ $\displaystyle 0\to\pi^{*}T^{*}X\to
T^{*}\mathcal{X}\to\pi^{*}\widehat{T^{*}X}\to 0.$

Recall that the trivial antiholomorphic superconnection $\mathscr{C}^{X}$ was
defined in (5.12). We define $\mathscr{C}^{\mathcal{X}}$ on $\mathcal{X}$ in
the same way. Then

(13.2) $\mathscr{C}^{\mathcal{X}}=\pi^{*}_{b}\mathscr{C}^{X}.$

Put

(13.3) $\mathscr{N}_{X}=\pi_{*}\mathscr{C}^{\mathcal{X}}.$

Then $\mathscr{N}_{X}$ is a $\Lambda\left(\overline{T^{*}X}\right)$-module.
Let $N_{X}$ be the infinite-dimensional vector bundle associated with
$\mathscr{N}_{X}$. The operator $\overline{\partial}^{\mathcal{X}}$ can be
viewed as an antiholomorphic superconnection $A^{N_{X}\prime\prime}$ on
$N_{X}$, so that $\mathscr{N}_{X}=\left(N_{X},A^{N_{X}\prime\prime}\right)$.

Put

(13.4)
$\mathbf{I}=C^{\infty}\left(\widehat{TX},\pi^{*}\Lambda\left(\overline{\widehat{T^{*}X}}\right)\right).$

Then $\mathbf{I}$ is the diagonal vector bundle associated with $N_{X}$. Also
$\mathbf{I}^{0}$ is the vector bundle on $X$ of smooth complex functions along
the fibre $\widehat{TX}$, and

(13.5)
$\mathbf{I}=\Lambda\left(\overline{\widehat{T^{*}X}}\right)\otimes\mathbf{I}^{0}.$

###  13.2. A Hermitian metric on $\widehat{TX}$

Here, we follow [B08, Section 3.12] and [B13, Section 6.1]. The constructions
given in these references will permit us to construct a splittings of
$\mathscr{N}_{X}$.

Let $g^{\widehat{TX}}$ be a Hermitian metric on $\widehat{TX}$, let
$\nabla^{\widehat{TX}}$ be the holomorphic Hermitian connection on
$\left(\widehat{TX},g^{\widehat{TX}}\right)$, and let $R^{\widehat{TX}}$ be
its curvature. Let $\nabla^{\Lambda\left(\overline{\widehat{T^{*}X}}\right)}$
be the induced connection on
$\Lambda\left(\overline{\widehat{T^{*}X}}\right)$.

Let $T^{H}\mathcal{X}\subset T\mathcal{X}$ be the horizontal subbundle
associated with the connection $\nabla^{\widehat{TX}}$. We have the
identification of smooth vector bundles on $\mathcal{X}$,

(13.6) $T^{H}\mathcal{X}=\pi^{*}TX,$

and also identification of smooth vector bundles on $\mathcal{X}$,

(13.7) $T\mathcal{X}=T^{H}\mathcal{X}\oplus\pi^{*}\widehat{TX}.$

By (13.6), (13.7), we get the smooth identification,

(13.8) $T\mathcal{X}=\pi^{*}\left(TX\oplus\widehat{TX}\right).$

By (13.8), we get the identification of smooth vector bundles,

(13.9)
$\Lambda\left(\overline{T^{*}\mathcal{X}}\right)=\pi^{*}\left(\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}\Lambda\left(\overline{\widehat{T^{*}X}}\right)\right).$

By (13.9), we get the identification of smooth vector bundles on $X$,

(13.10)
$N_{X}=\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}\mathbf{I}.$

By (13.9), we have the identity

(13.11)
$\Omega^{0,\scriptsize\bullet}\left(\mathcal{X},\mathbf{C}\right)=\Omega^{0,\scriptsize\bullet}\left(X,\mathbf{I}^{\scriptsize\bullet}\right).$

Let $\mathbf{I}^{\mathrm{c}}$ denote the vector bundle of elements of
$\mathbf{I}$ which have compact support. Then $\mathbf{I}^{\mathrm{c}}$
inherits a $L_{2}$ Hermitian metric $g^{\mathbf{I}^{\mathrm{c}}}$ from the
metric $g^{\widehat{TX}}$, such that if $r,t\in\mathbf{I}^{\mathrm{c}}$, then

(13.12) $\left\langle
r,t\right\rangle_{g^{\mathbf{I}^{\mathrm{c}}}}=\int_{\widehat{T_{\mathbf{R}}X}}\left\langle
r,t\right\rangle_{g^{\Lambda\left(\overline{\widehat{T^{*}X}}\right)}}\frac{dY}{\left(2\pi\right)^{n}}.$

If $U\in T_{\mathbf{R}}X$, let $U^{H}\in T_{\mathbf{R}}^{H}\mathcal{X}$ denote
the horizontal lift of $U$.

If $U\in T_{\mathbf{R}}X$, if $s$ is a smooth section of $\mathbf{I}$ on $X$,
set

(13.13)
$\nabla^{\mathbf{I}}_{U}s=\nabla^{\Lambda\left(\overline{\widehat{T^{*}X}}\right)}_{U^{H}}s.$

Then $\nabla^{\mathbf{I}}$ induces a unitary connection
$\nabla^{\mathbf{I}^{\mathrm{c}}}$ on $\mathbf{I}^{\mathrm{c}}$ that preserves
its $\mathbf{Z}$-grading. Let $\nabla^{\mathbf{I}^{0}}$ be the restriction of
$\nabla^{\mathbf{I}}$ to $\mathbf{I}^{0}$. By (13.5), the connection
$\nabla^{\mathbf{I}}$ is just the connection on $\mathbf{I}$ that is induced
by
$\nabla^{\Lambda\left(\overline{\widehat{T^{*}X}}\right)},\nabla^{\mathbf{I}^{0}}$.

Let $\overline{\partial}^{V}$ denote the $\overline{\partial}$-operator acting
along the fibers $\widehat{TX}$ of $\pi$. Using (13.11), we see that
$\nabla^{\mathbf{I}\prime\prime}$ and $\overline{\partial}^{V}$ act on
$\Omega^{0,\scriptsize\bullet}\left(\mathcal{X},\mathbf{C}\right)$.

Let $\widehat{y}$ be the tautological section of $\pi^{*}\widehat{TX}$ on
$\mathcal{X}$, let $\overline{\widehat{y}}$ be the conjugate section of
$\pi^{*}\overline{\widehat{TX}}$, and let
$\widehat{Y}=\widehat{y}+\overline{\widehat{y}}$ be the tautological section
of $\pi^{*}\widehat{T_{\mathbf{R}}X}$, so that
$\left|\widehat{Y}\right|^{2}_{g^{\widehat{T_{\mathbf{R}}X}}}=2\left|\widehat{y}\right|^{2}_{g^{\widehat{TX}}}$.

Let $w_{1},\ldots,w_{n}$ be a basis of $TX$, let $w^{1},\ldots,w^{n}$ be the
corresponding dual basis of $T^{*}X$. We denote with a hat the corresponding
objects associated with $\widehat{TX},\widehat{T^{*}X}$. We have results
established in [BGS88b, Theorem 2.8], [B13, Propositions 6.1.2, 6.1.4, and
6.1.6].

######  Proposition 13.1.

The following identities hold:

(13.14) $\displaystyle\nabla^{\mathbf{I}\prime\prime,2}=0,\
\nabla^{\mathbf{I}\prime,2}=0,$
$\displaystyle\nabla^{\mathbf{I},2}=-\nabla^{V}_{R^{\widehat{TX}}\widehat{Y}}-\left\langle
R^{\widehat{TX}}\overline{\widehat{w}}_{i},\overline{\widehat{w}}^{j}\right\rangle\overline{\widehat{w}}^{i}i_{\overline{\widehat{w}}_{j}}.$

Also

(13.15)
$\displaystyle\left[\nabla^{\mathbf{I}},\overline{\partial}^{V}\right]=0,$
$\displaystyle\left[\nabla^{\mathbf{I}},\overline{\partial}^{V*}\right]=0.$

Finally, we have the identity of operators acting on
$\Omega^{0,\scriptsize\bullet}\left(\mathcal{X},\mathbf{C}\right)$,

(13.16)
$\overline{\partial}^{\mathcal{X}}=\nabla^{\mathbf{I}\prime\prime}+\overline{\partial}^{V}.$

######  Remark 13.2.

The decomposition (13.16) can be viewed as a special case of equation (5.6)
when $\mathscr{E}$ is replaced by $\mathscr{N}_{X}$. In (13.16),
$\overline{\partial}^{\mathcal{X}}$ does not depend on $g^{\widehat{TX}}$, but
the splitting in the right-hand depends on $g^{\widehat{TX}}$.

By the above, it follows that
$\left(\mathbf{I}^{\mathrm{c}},\nabla^{\mathbf{I}^{\mathrm{c}}\prime\prime}\right)$
is a holomorphic Hermitian vector bundle313131Here, ‘holomorphic vector
bundle’ is taken in the naive sense.. The holomorphic structure depends
explicitly on the metric $g^{\widehat{TX}}$.

###  13.3. The antiholomorphic superconnection $\mathscr{E}_{M}$

We make the same assumptions as in Subsections 10.1, 10.2, and 13.1. Put

(13.17) $\mathcal{M}=S\times\mathcal{X}.$

Let $\underline{\pi}$ be the projection $\mathcal{M}\to M$, let
$\underline{p},\underline{q}$ be the obvious maps $\mathcal{M}\to
S,\mathcal{M}\to X$.

As in (13.2), we have

(13.18) $\mathscr{C}^{\mathcal{M}}=\underline{\pi}^{*}_{b}\mathscr{C}^{M}.$

Put

(13.19) $\mathscr{N}_{M}=\underline{\pi}_{*}\mathscr{C}^{\mathcal{M}}.$

Then

(13.20) $\mathscr{N}_{M}=\underline{q}^{*}_{b}\mathscr{N}_{X}.$

Let $N_{M}$ be the infinite-dimensional vector bundle on $M$ that is
associated with $\mathscr{N}_{M}$.

The associated diagonal vector bundle is just $q^{*}\mathbf{I}$.

Let $\mathscr{E}=\left(E,A^{E\prime\prime}\right)$ be an antiholomorphic
superconnection on $M$. Put

(13.21) $\mathscr{E}^{\mathcal{M}}=\underline{\pi}^{*}_{b}\mathscr{E}.$

Then
$\mathscr{E}^{\mathcal{M}}=\left(E^{\mathcal{M}},A^{E^{\mathcal{M}}\prime\prime}\right)$
is an antiholomorphic superconnection on $\mathcal{M}$, and the associated
diagonal bundle is just $\underline{\pi}^{*}D$. Also
$\mathscr{E}^{\mathcal{M}}$ is a
$\Lambda\left(\overline{T^{*}\mathcal{M}}\right)$-module.

Put

(13.22) $\mathscr{E}_{M}=\underline{\pi}_{*}\mathscr{E}^{\mathcal{M}}.$

Recall that the tensor product $\widehat{\otimes}_{b}$ was defined in (5.77).
Then

(13.23) $\mathscr{E}_{M}=\mathscr{E}\widehat{\otimes}_{b}\mathscr{N}_{M}.$

Observe that $\mathscr{E}_{M}=\left(E_{M},A^{E_{M}\prime\prime}\right)$ is an
antiholomorphic superconnection on $M$, with an infinite-dimensional $E_{M}$,
and the associated diagonal bundle is $D\widehat{\otimes}q^{*}\mathbf{I}$.

###  13.4. A metric description of $A^{E_{M}\prime\prime}$

We fix a splitting of $E$ as in (5.2), (5.3), so that $E\simeq E_{0}$.

We proceed as in Subsections 13.2 and 13.3. We fix a metric $g^{\widehat{TX}}$
on the vector bundle $\widehat{TX}$ over $X$. This induces a corresponding
splitting of $\mathscr{N}_{X}$, and also of $\mathscr{N}_{M}$.

The corresponding vector bundles $E_{0},N_{M,0}$ are given by

(13.24) $\displaystyle
E_{0}=\Lambda\left(\overline{T^{*}M}\right)\widehat{\otimes}D,$ $\displaystyle
N_{M,0}=\Lambda\left(\overline{T^{*}M}\right)\widehat{\otimes}q^{*}\mathbf{I}.$

The above two splittings induce a corresponding splitting of
$\mathscr{E}_{M}$, so that

(13.25)
$E_{M,0}=\Lambda\left(\overline{T^{*}M}\right)\widehat{\otimes}D\widehat{\otimes}q^{*}\mathbf{I}.$

Equation (13.25) can also be written in the form

(13.26) $E_{M,0}=E_{0}\widehat{\otimes}q^{*}\mathbf{I}.$

######  Definition 13.3.

Let $\mathcal{A}^{\prime\prime}$ be the antiholomorphic superconnection on
$E_{M,0}$ that corresponds to $A^{E_{M}\prime\prime}$.

We write $A^{E_{0}\prime\prime}$ as in (5.9). In particular, $D$ is equipped
with an antiholomorphic connection $\nabla^{D\prime\prime}$.

As we saw in Subsection 13.2, $\mathbf{I}$ is equipped with a holomorphic
structure $\nabla^{\mathbf{I}\prime\prime}$.

######  Definition 13.4.

Let $\nabla^{D\widehat{\otimes}q^{*}\mathbf{I}\prime\prime}$ be the
antiholomorphic connection on $D\widehat{\otimes}q^{*}\mathbf{I}$ associated
with $\nabla^{D\prime\prime},\nabla^{\mathbf{I}\prime\prime}$.

######  Proposition 13.5.

The following identity holds:

(13.27)
$\mathcal{A}^{\prime\prime}=\nabla^{D\widehat{\otimes}q^{*}\mathbf{I}\prime\prime}+B+\overline{\partial}^{V}.$

###### Proof.

Our proposition follows from equations (5.9), (13.16). ∎

###  13.5. The adjoint of $\mathcal{A}^{\prime\prime}$

Let $g^{D}$ be a Hermitian metric on $D$. Recall that $g^{\widehat{TX}}$
induces a Hermitian metric on $\mathbf{I}^{\mathrm{c}}$. Therefore
$D\widehat{\otimes}q^{*}\mathbf{I}^{\mathrm{c}}$ is equipped with a Hermitian
metric $g^{D\widehat{\otimes}q^{*}\mathbf{I}^{\mathrm{c}}}$.

We are now in a situation formally similar to what was done in a finite-
dimensional context in Section 7, and in an infinite-dimensional context in
[BGS88b], in [B13, Chapter 4], and in Section 10, with $M,S$ replaced by
$\mathcal{M},M$, and $p$ replaced by $\underline{\pi}$. We briefly explain the
construction of the adjoint of $\mathcal{A}^{\prime\prime}$.

Let
$s,s^{\prime}\in\Omega\left(M,D\widehat{\otimes}q^{*}\mathbf{I}^{\mathrm{c}}\right)$.
We can write $s,s^{\prime}$ in the form

(13.28) $\displaystyle s=\sum\alpha_{i}r_{i},$ $\displaystyle
s^{\prime}=\sum\beta_{j}t_{j},$

with $\alpha_{i},\beta_{j}\in\Omega\left(M,\mathbf{C}\right)$ and
$r_{i},t_{j}\in
C^{\infty}\left(M,D\widehat{\otimes}q^{*}\mathbf{I}^{\mathrm{c}}\right)$. As
in (7.10), (7.12), put

(13.29)
$\theta_{g^{D\widehat{\otimes}q^{*}\mathbf{I}^{\mathrm{c}}}}\left(s,s^{\prime}\right)=\frac{i^{m}}{\left(2\pi\right)^{m}}\sum\int_{M}\widetilde{\alpha}_{i}\wedge\overline{\beta}_{j}\left\langle
r_{i},t_{j}\right\rangle_{g^{D\widehat{\otimes}q^{*}\mathbf{I}^{\mathrm{c}}}}.$

######  Definition 13.6.

Let $\mathcal{A}^{\prime}$ be the formal adjoint of
$\mathcal{A}^{\prime\prime}$ with respect to
$\theta_{g^{D\widehat{\otimes}q^{*}\mathbf{I}^{\mathrm{c}}}}$.

Then $\mathcal{A}^{\prime}$ is exactly the adjoint of the antiholomorphic
superconnection $\mathcal{A}^{\prime\prime}$ in the sense of Definition 7.4
with respect to the pure metric
$g^{D\widehat{\otimes}q^{*}\mathbf{I}^{\mathrm{c}}}$.

Recall that the holomorphic connection $\nabla^{D\prime}$ on $D$ was defined
in (7.15). Since $g^{D}$ is a pure metric in $\mathscr{M}^{D}$,
$\nabla^{D\prime}$ is just a standard holomorphic connection.

######  Definition 13.7.

Let $\nabla^{D\widehat{\otimes}q^{*}\mathbf{I}\prime}$ be the holomorphic
connection on $D\widehat{\otimes}q^{*}\mathbf{I}$ induced by
$\nabla^{D\prime},\nabla^{\mathbf{I}\prime}$.

We use the notation in (7.17). Let $\overline{\partial}^{V*}$ denote the
formal adjoint of $\overline{\partial}^{V}$ with respect to the metric
$g^{\mathbf{I}^{\mathrm{c}}}$.

######  Proposition 13.8.

The following identity holds:

(13.30)
$\mathcal{A}^{\prime}=\nabla^{D\widehat{\otimes}q^{*}\mathbf{I}\prime}+B^{*}+\overline{\partial}^{V*}.$

###### Proof.

This is a consequence of Proposition 13.5. ∎

Then $\mathcal{A}^{\prime\prime},\mathcal{A}^{\prime}$ act as differential
operators on $\Omega\left(M,D\widehat{\otimes}q^{*}\mathbf{I}\right)$, and

(13.31) $\displaystyle\mathcal{A}^{\prime\prime 2}=0,$
$\displaystyle\mathcal{A}^{\prime 2}=0.$

###  13.6. The antiholomorphic superconnection $A^{E_{M}\prime\prime}_{z}$

Recall that $A^{E^{\mathcal{M}}\prime\prime},A^{E_{M}\prime\prime}$ are the
same operator, that acts on
$C^{\infty}\left(\mathcal{M},E^{\mathcal{M}}\right)=C^{\infty}\left(M,E_{M}\right)$.
We extend these operators to operators acting on
$C^{\infty}\left(\mathcal{M},\underline{\pi}^{*}\Lambda\left(T^{*}M\right)\widehat{\otimes}E^{\mathcal{M}}\right)=C^{\infty}\left(M,\Lambda\left(T^{*}M\right)\widehat{\otimes}E_{M}\right)$,
that still verify Leibniz rule with respect to multiplication by smooth forms
in $\underline{\pi}^{*}\Lambda\left(T^{*}M\right),\Lambda\left(T^{*}M\right)$.

Let $z$ be a holomorphic section of $\underline{q}^{*}TX$ on $\mathcal{M}$.
Since $q^{*}TX\subset TM$, the contraction operator $i_{z}$ acts on the above
vector spaces.

######  Definition 13.9.

Put

(13.32) $\displaystyle
A^{E^{\mathcal{M}}\prime\prime}_{z}=A^{E^{\mathcal{M}}\prime\prime}+i_{z},$
$\displaystyle A^{E_{M}\prime\prime}_{z}=A^{E_{M}\prime\prime}+i_{z}.$

Both operators have total degree $+1$. Since $z$ is holomorphic, we get

(13.33) $\displaystyle A^{E^{\mathcal{M}}\prime\prime,2}_{z}=0,$
$\displaystyle A^{E_{M}\prime\prime,2}_{z}=0.$

Also $A^{E^{\mathcal{M}}\prime\prime}_{z}$ is an antiholomorphic
superconnection on
$\underline{\pi}^{*}\Lambda\left(T^{*}M\right)\widehat{\otimes}\mathscr{E}^{\mathcal{M}}$,
and $A^{E_{M}\prime\prime}_{z}$ is an antiholomorphic superconnection on
$\Lambda\left(T^{*}M\right)\widehat{\otimes}\mathscr{E}_{M}$. These two
operators are in fact the same operator. In the sequel, we will deal mostly
with the second one.

###  13.7. The superconnection $\mathcal{A}_{Z}$

We now make the same assumptions as in Subsections 13.4 and 13.6.

As in Subsection 10.3, let $g^{TX}$ be a Hermitian metric on $TX$ with Kähler
form $\omega^{X}$. The exponential $e^{-i\omega^{X}}$ is taken in the exterior
algebra $\Lambda\left(T^{*}_{\mathbf{C}}X\right)$.

Let $z$ denote a holomorphic section of $\underline{q}^{*}TX$ on
$\mathcal{M}$. Put

(13.34) $Z=z+\overline{z}.$

Then $Z$ is a smooth section of $\underline{q}^{*}T_{\mathbf{R}}X$. Let
$\overline{z}_{*}\in\underline{q}^{*}T^{*}X$ correspond to
$\overline{z}\in\underline{q}^{*}\overline{TX}$ by the metric $g^{TX}$. By
(10.7), $\overline{z}_{*}$ can also be viewed as a section of $T^{*}M$.

Note that $TX\subset TM$, and so the contraction operators
$i_{z},i_{\overline{z}}$ act as operators of degree $1,-1$ on
$\Lambda\left(T^{*}_{\mathbf{C}}M\right)$, and so they act on
$\Omega\left(M,D\widehat{\otimes}q^{*}\mathbf{I}\right)$.

Now we follow [B13, Section 6.3].

######  Definition 13.10.

Put

(13.35)
$\displaystyle\mathcal{A}^{\prime\prime}_{Z}=\mathcal{A}^{\prime\prime}+i_{z},$
$\displaystyle\mathcal{A}^{\prime}_{Z}=e^{q^{*}i\omega^{X}}\left(\mathcal{A}^{\prime}+i_{\overline{z}}\right)e^{-q^{*}i\omega^{X}},\quad\mathcal{A}_{Z}=\mathcal{A}^{\prime\prime}_{Z}+\mathcal{A}^{\prime}_{Z}.$

Since $z$ is holomorphic,

(13.36) $\displaystyle\mathcal{A}_{Z}^{\prime\prime 2}=0,$
$\displaystyle\mathcal{A}^{\prime 2}_{Z}=0.$

By (13.35), we get

(13.37)
$\mathcal{A}^{\prime}_{Z}=\mathcal{A}^{\prime}-q^{*}\partial^{X}i\omega^{X}+i_{\overline{z}}+\overline{z}_{*}\wedge.$

Also

(13.38)
$\mathcal{A}^{2}_{Z}=\left[\mathcal{A}^{\prime\prime}_{Z},\mathcal{A}^{\prime}_{Z}\right].$

Let $\nabla^{D\widehat{\otimes}q^{*}\mathbf{I}}$ be the connection on
$D\widehat{\otimes}q^{*}\mathbf{I}$ which is induced by
$\nabla^{D},\nabla^{\mathbf{I}}$. Recall that $C$ was defined in (7.21). By
(13.27), (13.30), (13.35), and (13.37), we get

$\displaystyle\mathcal{A}^{\prime\prime}_{Z}=\nabla^{D\widehat{\otimes}q^{*}\mathbf{I}\prime\prime}+B+\overline{\partial}^{V}+i_{z},$
(13.39)
$\displaystyle\mathcal{A}^{\prime}_{Z}=\nabla^{D\widehat{\otimes}q^{*}\mathbf{I}\prime}+B^{*}-q^{*}\partial^{X}i\omega^{X}+\overline{\partial}^{V*}+i_{\overline{z}}+\overline{z}_{*}\wedge,$
$\displaystyle\mathcal{A}_{Z}=\nabla^{D\widehat{\otimes}q^{*}\mathbf{I}}+C-q^{*}\partial^{X}i\omega^{X}+\overline{\partial}^{V}+i_{z}+\overline{\partial}^{V*}+i_{\overline{z}}+\overline{z}_{*}\wedge.$

We denote by $L$ the operator of multiplication by $\omega^{X}$ and by
$\Lambda$ its adjoint with respect to $g^{TX}$. Let $w_{1}\ldots,w_{n}$ be an
orthonormal basis of $\left(TX,g^{TX}\right)$. Then

(13.40) $\displaystyle L=-iw^{i}\wedge\overline{w}^{i},$
$\displaystyle\Lambda=ii_{\overline{w}_{i}}i_{w_{i}}.$

Conjugation by $\exp\left(i\Lambda\right)$ changes $w^{i},\overline{w}^{i}$
into $w^{i}-i_{\overline{w}_{i}},\overline{w}^{i}+i_{w_{i}}$.

Put

$\displaystyle\mathcal{B}^{\prime\prime}_{Z}=\exp\left(i\Lambda\right)\mathcal{A}^{\prime\prime}_{Z}\exp\left(-i\Lambda\right),$
(13.41)
$\displaystyle\mathcal{B}^{\prime}_{Z}=\exp\left(i\Lambda\right)\mathcal{A}^{\prime}_{Z}\exp\left(-i\Lambda\right),$
$\displaystyle\mathcal{B}_{Z}=\exp\left(i\Lambda\right)\mathcal{A}_{Z}\exp\left(-i\Lambda\right).$

Then

(13.42)
$\mathcal{B}_{Z}=\mathcal{B}^{\prime\prime}_{Z}+\mathcal{B}^{\prime}_{Z}.$

Set

(13.43)
$\mathcal{E}=\exp\left(i\Lambda\right)\left(\nabla^{D\widehat{\otimes}q^{*}\mathbf{I}}+C-q^{*}\partial^{X}i\omega^{X}\right)\exp\left(-i\Lambda\right).$

By (13.7), and by the considerations that follow (13.40), we get

(13.44)
$\mathcal{B}_{Z}=\mathcal{E}+\overline{\partial}^{V}+i_{z}+\overline{\partial}^{V*}+\overline{z}_{*}.$

We still count the degree in $\Lambda\left(T^{*}_{\mathbf{C}}M\right)$ as the
difference of the antiholomorphic and of the holomorphic degree. Then
$A^{\prime\prime}_{Z},B^{\prime\prime}_{Z}$ are operators of total degree $1$,
and $A^{\prime}_{Z},B^{\prime}_{Z}$ are operators of total degree $-1$.

Observe that the above operators can be considered as standard differential
operators on $\mathcal{M}$.

###  13.8. Two Hermitian forms

Here we follow [B13, Sections 6.5 and 6.6]. We will slightly modify the
construction of the Hermitian form
$\theta_{g^{D\widehat{\otimes}q^{*}\mathbf{I}}}$.

Set

(13.45)
$\mathbb{F}=q^{*}\left(\Lambda\left(T^{*}_{\mathbf{C}}X\right)\widehat{\otimes}\Lambda\left(\overline{\widehat{T^{*}X}}\right)\right)\widehat{\otimes}D.$

Then $\mathbb{F}$ is a vector bundle on $M$. Recall that
$\Lambda\left(T^{*}_{\mathbf{C}}X\right)$ is $\mathbf{Z}$-graded by the
difference of the antiholomorphic and the holomorphic degrees. Also
$\Lambda\left(\overline{\widehat{T^{*}X}}\right)$ and $D$ are
$\mathbf{Z}$-graded. Then $\mathbb{F}$ inherits a corresponding
$\mathbf{Z}$-grading. Similarly $\Lambda\left(T^{*}_{\mathbf{C}}S\right)$
inherits a similar grading.

As in [B13, Section 6.5], we will use the identification

(13.46)
$\Lambda\left(\widehat{T^{*}_{\mathbf{C}}X}\right)=\Lambda\left(T^{*}X\right)\widehat{\otimes}\Lambda\left(\overline{\widehat{T^{*}X}}\right).$

By (13.45), (13.46), we get

(13.47)
$\mathbb{F}=q^{*}\left(\Lambda\left(\overline{T^{*}X}\right)\widehat{\otimes}\Lambda\left(\widehat{T^{*}_{\mathbf{C}}X}\right)\right)\widehat{\otimes}D.$

Let $\sigma$ be the involution of $\mathcal{X}$ given by
$\left(x,\widehat{y}\right)\to\left(x,-\widehat{y}\right)$. Then $\sigma$
induces an involution of $\mathcal{M}$.

Let $\sigma^{*}$ denote the obvious action of $\sigma$ on
$\Lambda\left(\widehat{T^{*}_{\mathbf{C}}X}\right)$. Namely, if
$N^{\Lambda\left(\widehat{T^{*}_{\mathbf{C}}X}\right)}$ is the number operator
of $\Lambda\left(\widehat{T^{*}_{\mathbf{C}}X}\right)$, then $\sigma^{*}$ acts
like
$\left(-1\right)^{N^{\Lambda\left(\widehat{T^{*}_{\mathbf{C}}X}\right)}}$.
Also, we make $\sigma^{*}$ act trivially on $D$. Then $\sigma^{*}$ acts on
$\mathbb{F}$ in (13.47).

We fix one fiber $X$ of the projection $p$, and the corresponding fiber
$\mathcal{X}$ of $\underline{p}$. Note that

(13.48)
$\Omega\left(X,D|_{X}\widehat{\otimes}\mathbf{I}\right)=C^{\infty}\left(\mathcal{X},\underline{\pi}^{*}\mathbb{F}|_{\mathcal{X}}\right).$

Let $dv_{\mathcal{X}}$ denote the volume form on $\mathcal{X}$ which is
associated with the metrics $g^{TX},g^{\widehat{TX}}$. If $s,s^{\prime}\in
C^{\infty,\mathrm{c}}\left(\mathcal{X},\underline{\pi}^{*}\mathbb{F}|_{\mathcal{X}}\right)$,
set

(13.49) $\left\langle
s,s^{\prime}\right\rangle_{L_{2}}=\left(2\pi\right)^{-2n}\int_{\mathcal{X}}\left\langle
s,s^{\prime}\right\rangle_{g^{\mathbb{F}}}dv_{\mathcal{X}}.$

######  Definition 13.11.

If $s,s^{\prime}\in
C^{\infty,\mathrm{c}}\left(\mathcal{X},\underline{\pi}^{*}\mathbb{F}|_{\mathcal{X}}\right)$,
put

(13.50)
$\eta_{X}\left(s,s^{\prime}\right)=\left\langle\sigma^{*}s,s^{\prime}\right\rangle_{L_{2}}.$

Then $\eta_{X}$ is a Hermitian form on
$C^{\infty,\mathrm{c}}\left(\mathcal{X},\underline{\pi}^{*}\mathbb{F}|_{\mathcal{X}}\right)$.
Note that $\eta_{X}$ is non-degenerate, it is positive on $\sigma$-invariant
forms, and negative on $\sigma$-anti-invariant forms. Also the
$C^{\infty,\mathrm{c}}\left(\mathcal{X},\underline{\pi}^{*}\mathbb{F}^{i}|_{\mathcal{X}}\right)$
are mutually orthogonal with respect to $\left\langle\,\right\rangle_{L_{2}}$
and $\eta_{X}$.

Let $\underline{\sigma}^{*}$ be the restriction of $\sigma^{*}$ to
$\Lambda\left(\overline{\widehat{T^{*}X}}\right)$. We make
$\underline{\sigma}^{*}$ act trivially on
$\Lambda\left(T^{*}_{\mathbf{C}}X\right)$ and on $D$. By (13.45),
$\underline{\sigma}^{*}$ also acts on $\mathbb{F}$. The difference with
$\sigma^{*}$ is that $\underline{\sigma}^{*}$ acts trivially on
$\Lambda\left(T^{*}X\right)$. Then $\underline{\sigma}^{*}$ acts on
$D|_{X}\widehat{\otimes}\mathbf{I}$. Let
$s,s^{\prime}\in\Omega\left(X,D|_{X}\widehat{\otimes}\mathbf{I}^{\mathrm{c}}\right)$.
Then $s,s^{\prime}$ can be written in the form

(13.51) $\displaystyle s=\sum\alpha_{i}r_{i},$ $\displaystyle
s^{\prime}=\sum\beta_{j}t_{j},$

with $\alpha_{i},\beta_{j}\in\Omega\left(X,\mathbf{C}\right)$, and
$r_{i},t_{j}\in
C^{\infty}\left(X,D|_{X}\widehat{\otimes}\mathbf{I}^{\mathrm{c}}\right)$.

If $\alpha\in\Lambda\left(T^{*}_{\mathbf{C}}X\right)$, we define
$\widetilde{\alpha}$ as in (4.33). Also we use the same notation as in
Subsection 8.2. As we observed in that Subsection, multiplication by
$i\omega^{X}$ is a self-adjoint operator with respect to the form $\theta$ in
(7.1). Also the exponential $e^{-i\omega^{X}}$ is taken in
$\Lambda\left(T^{*}_{\mathbf{C}}X\right)$.

######  Definition 13.12.

Put

(13.52)
$\epsilon_{X}\left(s,s^{\prime}\right)=\frac{i^{n}}{\left(2\pi\right)^{n}}\int_{X}\widetilde{\alpha}_{i}\wedge\overline{e^{-i\omega^{X}}\beta}_{j}\left\langle\underline{\sigma}^{*}r,t\right\rangle_{g^{D|_{X}\widehat{\otimes}\mathbf{I}^{\mathrm{c}}}}.$

Equation (13.52) can be rewritten in the form,

(13.53)
$\epsilon_{X}\left(s,s^{\prime}\right)=\frac{i^{n}}{\left(2\pi\right)^{n}}\int_{\mathcal{X}}\widetilde{\alpha}_{i}\wedge\overline{e^{-i\omega^{X}}\beta}_{j}\left\langle\underline{\sigma}^{*}r,t\right\rangle_{g^{D|_{X}\widehat{\otimes}\Lambda\left(\overline{\widehat{T^{*}X}}\right)}}\frac{dY}{\left(2\pi\right)^{n}}.$

By [B13, Section 6.5 and Theorem 6.6.1], $\epsilon_{X}$ is a Hermitian form on
$C^{\infty,\mathrm{c}}\left(\mathcal{X},\underline{\pi}^{*}\mathbb{F}|_{\mathcal{X}}\right)=\Omega\left(X,\mathbf{I}^{\mathrm{c}}\widehat{\otimes}D|_{X}\right)$.
As explained in [B13, Section 6.6], if instead, we replace integration over
$X$ by integration over $M$, while replacing
$\frac{i^{n}}{\left(2\pi\right)^{n}}$ by
$\frac{i^{m}}{\left(2\pi\right)^{m}}$, we obtain a Hermitian form
$\epsilon_{M}$ on
$\Omega\left(M,D\widehat{\otimes}q^{*}\mathbf{I}^{\mathrm{c}}\right)$.

######  Remark 13.13.

As explained in Subsection 8.2, the term $e^{-i\omega^{X}}$ in (13.52) can be
interpreted as a generalized metric on the trivial line $\mathbf{C}$ over $M$.
Equivalently, $e^{-i\omega^{X}}g^{D}|_{X}$ can be viewed as a generalized
metric on $D|_{X}$.

By [B13, eq. (6.5.10) and Theorem 6.6.1], if

$s,s^{\prime}\in\Omega\left(X,D|_{X}\widehat{\otimes}q^{*}\mathbf{I}^{\mathrm{c}}\right)=C^{\infty}\left(\mathcal{X},\underline{\pi}^{*}\mathbb{F}|_{\mathcal{X}}\right),$

we get

(13.54)
$\epsilon_{X}\left(s,s^{\prime}\right)=\eta_{X}\left(e^{i\Lambda}s,e^{i\Lambda}s^{\prime}\right).$

Then $\epsilon_{X}$ is also a non-degenerate Hermitian form, which can also be
proved directly. By (13.54), or by direct inspection, the
$C^{\infty,\mathrm{c}}\left(\mathcal{X},\underline{\pi}^{*}\mathbb{F}^{i}|_{\mathcal{X}}\right)$
are mutually orthogonal with respect to $\epsilon_{X}$.

We will denote with an upper ${\dagger}$ the adjoint of an operator with
respect to $\epsilon_{X}$ or $\eta_{X}$, the corresponding Hermitian form
being explicitly mentioned.

We have the following result established in [B13, Proposition 6.5.1 and
Theorem 6.6.1].

######  Proposition 13.14.

If adjoints are taken with respect to $\epsilon_{X}$, if $e\in TX,f\in
T^{*}X$, then

(13.55) $\displaystyle i_{e}^{{\dagger}}=-\overline{e}_{*}\wedge-
i_{\overline{e}},$ $\displaystyle i_{\overline{e}}^{{\dagger}}=e_{*}\wedge-
i_{e},$ $\displaystyle\left(f\wedge\right)^{{\dagger}}=-\overline{f}\wedge,$
$\displaystyle\left(\overline{f}\wedge\right)^{{\dagger}}=-f\wedge.$

If $e\in TX,E=e+\overline{e}\in T_{\mathbf{R}}X$, then $e_{*}\wedge-
i_{E},\overline{e}_{*}\wedge+i_{E}$ are skew-adjoint with respect to
$\epsilon_{X}$.

###  13.9. Self-adjointness of $\mathcal{A}_{Z},\mathcal{B}_{Z}$

Note that $p_{*}\left(\Lambda\left(T^{*}M\right)\widehat{\otimes}E_{M}\right)$
is a $\Lambda\left(T_{\mathbf{C}}^{*}S\right)$-module.

We will use the conventions of Subsection 10.3 when taking adjoints of
antiholomorphic superconnections over the base $S$. These adjoints will be
taken with respect to the Hermitian forms $\epsilon_{X}$ or $\eta_{X}$, and
will be denoted with a †. The adjoints of $\mathcal{A}_{Z}$ and of its
components $\mathcal{A}^{\prime\prime}_{Z},\mathcal{A}^{\prime}_{Z}$ are taken
with respect to $\epsilon_{X}$, and the adjoints of $\mathcal{B}_{Z}$ and of
its components $\mathcal{B}^{\prime\prime}_{Z},\mathcal{B}^{\prime}_{Z}$ are
taken with respect to $\eta_{X}$. In the case of $\mathcal{A}_{Z}$ or of its
components, the adjoint could be taken as well with respect to the standard
Hermitian form $\epsilon_{M}$.

Let $z_{-}$ be the section of $\underline{q}^{*}TX$ given by

(13.56) $z_{-}\left(x,Y\right)=-z\left(x,-Y\right).$

Then $z_{-}$ is still holomorphic. Let $Z_{-}$ be the corresponding section of
$\underline{q}^{*}T_{\mathbf{R}}X$.

######  Theorem 13.15.

The operators $\mathcal{A}^{\prime\prime}_{Z},\mathcal{B}^{\prime\prime}_{Z}$
are of total degree $1$, and the operators
$\mathcal{A}^{\prime}_{Z},\mathcal{B}^{\prime}_{Z}$ are of total degree $-1$.
Moreover, we have the identities

(13.57)
$\displaystyle\mathcal{A}^{\prime\prime{\dagger}}_{Z}=\mathcal{A}^{\prime}_{Z_{-}},$
$\displaystyle\mathcal{B}^{\prime\prime{\dagger}}_{Z}=\mathcal{B}^{\prime}_{Z_{-}},$
$\displaystyle\mathcal{A}^{{\dagger}}_{Z}=\mathcal{A}_{Z_{-}},$
$\displaystyle\mathcal{B}^{{\dagger}}_{Z}=\mathcal{B}_{Z_{-}}.$

###### Proof.

The proof is the same as in [B13, Theorem 6.5.2]. For $\mathcal{A}_{Z}$ and
its components, it is an easy consequence of (13.52) and of Proposition 13.14.
∎

######  Remark 13.16.

When considering $\mathcal{A}_{Z}$ and its components, we may take adjoints
with respect to $\epsilon_{M}$, and the above results would still hold.
Adjoints with respect to $\epsilon_{M}$ are simply classical adjoints. Using
integration along the fiber $p_{*}$ to evaluate $\epsilon_{M}$ is enough to
explain the coincidence of the adjoints.

###  13.10. A formula for the curvature of $\mathcal{A}_{Z}$

When identifying $TX$ and $\widehat{TX}$, we denote by $\widehat{g}^{TX}$ the
metric on $TX$ corresponding to $g^{\widehat{TX}}$, and by
$\widehat{\nabla}^{TX}$ the connection on $TX$ that corresponds to
$\nabla^{\widehat{TX}}$. Equivalently, $\widehat{\nabla}^{TX}$ is the Chern
connection on $\left(TX,\widehat{g}^{TX}\right)$. Let $\widehat{\tau}$ be the
torsion of $\widehat{\nabla}^{TX}$. Then $\widehat{\tau}$ has the same
properties as $\tau$. 323232The torsion $\widehat{\tau}$ takes its values in
$TX$ and not in $\widehat{TX}$.

Recall that $Z=z+\overline{z}$ is a smooth section on $\mathcal{M}$ of
$\underline{q}^{*}T_{\mathbf{R}}X$. Let $\rho:\mathcal{M}\to\mathcal{X}$ be
the obvious projection. We identify $Z$ with its horizontal lift $Z^{H}$ in
$\rho^{*}T^{H}_{\mathbf{R}}\mathcal{X}\subset T_{\mathbf{R}}\mathcal{M}$.

We denote by $\widehat{\nabla}^{T_{\mathbf{R}}X,H}Z$ the section on
$\mathcal{M}$ of $\underline{\pi}^{*}\left(T^{*}_{\mathbf{R}}M\otimes
q^{*}T_{\mathbf{R}}X\right)$ that is obtained by taking the covariant
derivative of $Z$ with respect to the pull-back of
$\widehat{\nabla}^{T_{\mathbf{R}}X}$ along horizontal directions in
$\mathcal{M}$. Other horizontal covariant derivatives will be denoted in a
similar way.

Let $\Delta^{V}_{g^{\widehat{TX}}}$ be the Laplacian along the fibers
$\widehat{TX}$ with respect to the metric $g^{\widehat{TX}}$.

In this Subsection, ${\dagger}$ refers to the adjoint with respect to the form
$\epsilon_{X}$ in Definition 13.12. By Proposition 13.14 and by (13.56), we
get

(13.58) $i_{z_{-}}^{{\dagger}}=\overline{z}_{*}\wedge+i_{\overline{z}}.$

Now we follow Subsection 4.6. For $a\in\mathbf{R}$, we introduce the
connection $\widehat{\nabla}^{TX,a}$ on $TX$ associated with
$\widehat{\nabla}^{TX}$333333In [B13], the connection $\nabla^{TX}$ on $TX$
was used instead. This accounts for minor differences with respect to [B13]..
If $\widehat{\nabla}^{\Lambda\left(T^{*}_{\mathbf{R}}X\right),a}$ is the
induced connection on $\Lambda\left(T^{*}_{\mathbf{R}}X\right)$, by (4.49), if
$U\in T_{\mathbf{R}}X$, then

(13.59)
$\widehat{\nabla}^{\Lambda\left(T^{*}_{\mathbf{R}}X\right),a}_{U}=\widehat{\nabla}_{U}^{\Lambda\left(T^{*}_{\mathbf{R}}X\right)}-ai_{\widehat{\tau}\left(U,\cdot\right)}.$

The connection $\widehat{\nabla}^{\Lambda\left(T^{*}_{\mathbf{R}}X\right),a}$
preserves the complex type of the forms. In particular, by (4.52), we get

(13.60)
$d^{X}=\widehat{\nabla}_{\mathrm{a}}^{\Lambda\left(T^{*}_{\mathbf{R}}X\right),-1/2},$

and (13.60) splits into formulas for $\overline{\partial}^{X},\partial^{X}$.
Equation (13.60) can be extended to an equation for $d^{M}$,

(13.61)
$d^{M}=\widehat{\nabla}_{\mathrm{a}}^{\Lambda\left(T^{*}_{\mathbf{R}}X\right),-1/2}.$

In the right-hand side of (13.61), exterior differentiation on $S$ is done
implicitly, and does not require the choice of a connection on $TS$.

Recall that $\mathbb{F}$ was defined in (13.45). Let
$\widehat{\nabla}^{\mathbb{F}}$ be the connection on $\mathbb{F}$ induced by
$\widehat{\nabla}^{\Lambda\left(T^{*}_{\mathbf{C}}X\right)},\nabla^{D},\nabla^{\Lambda\left(\overline{\widehat{T^{*}X}}\right)}$.

For $a\in\mathbf{R}$, let $\widehat{\nabla}^{\mathbb{F},a}$ be the connection
on $\mathbb{F}$ induced by
$\widehat{\nabla}^{\Lambda\left(T^{*}_{\mathbf{C}}X\right),a},\nabla^{D},\nabla^{\Lambda\left(\overline{\widehat{T^{*}X}}\right)}$.
By (13.59), we get

(13.62)
$\widehat{\nabla}^{\mathbb{F},a}_{U}=\widehat{\nabla}^{\mathbb{F}}_{U}-ai_{\widehat{\tau}\left(U,\cdot\right)}.$

By (4.51), we obtain

(13.63)
$\widehat{\nabla}^{\mathbb{F},-1/2}_{\mathrm{a}}=\widehat{\nabla}^{\mathbb{F}}_{\mathrm{a}}+i_{\widehat{\tau}}.$

As explained after (13.61), equation (13.63) can be viewed as an identity of
operators acting on
$\Omega\left(M,q^{*}\Lambda\left(\overline{\widehat{T^{*}X}}\right)\widehat{\otimes}D\right)$.
Equation (13.63) can be rewritten in the form

(13.64)
$\widehat{\nabla}^{\mathbb{F},-1/2}_{\mathrm{a}}=\nabla^{q^{*}\Lambda\left(\overline{\widehat{T^{*}X}}\right)\widehat{\otimes}D},$

where the right-hand side is simply the natural extension of a connection on
the vector bundle
$q^{*}\Lambda\left(\overline{\widehat{T^{*}X}}\right)\widehat{\otimes}D$ to an
operator acting on
$\Omega\left(M,q^{*}\Lambda\left(\overline{\widehat{T^{*}X}}\right)\widehat{\otimes}D\right)$.

######  Definition 13.17.

Let ${}^{0}\widehat{\nabla}^{\mathbb{F}}$ be the fibrewise connection on
$\mathbb{F}$ such that if $U\in T_{\mathbf{R}}X$, $U=u+\overline{u},u\in TX$,

(13.65)
${}^{0}\widehat{\nabla}^{\mathbb{F}}_{U}=\widehat{\nabla}^{\mathbb{F},-1}_{U}-i_{u}q^{*}\partial^{X}i\omega^{X}.$

Put

(13.66)
$\mathbf{F}=q^{*}\Lambda\left(T^{*}_{\mathbf{C}}X\right)\widehat{\otimes}D\widehat{\otimes}q^{*}\mathbf{I}.$

By (13.5), (13.45), we get

(13.67) $\mathbf{F}=\mathbb{F}\widehat{\otimes}q^{*}\mathbf{I}^{0}.$

Equation (13.67) can be rewritten in the form,

(13.68)
$\mathbf{F}=C^{\infty}\left(\widehat{T_{\mathbf{R}}X},\underline{\pi}^{*}\mathbb{F}|_{\widehat{T_{\mathbf{R}}X}}\right).$

Let $\widehat{\nabla}^{\mathbf{F}}$ be the connection on $\mathbf{F}$
associated with
$\widehat{\nabla}^{\Lambda\left(T^{*}_{\mathbf{C}}X\right)},\nabla^{D},\nabla^{\mathbf{I}}$.
Equivalently, $\widehat{\nabla}^{\mathbf{F}}$ is the connection on
$\mathbf{F}$ induced by
$\widehat{\nabla}^{\mathbb{F}},\nabla^{\mathbf{I}^{0}}$. More generally, when
replacing $\mathbb{F}$ by $\mathbf{F}$, we obtain corresponding objects
associated with $\mathbf{F}$.

As in Subsection 4.6, $\widehat{\nabla}^{\mathbf{F},a}_{\mathrm{a}}$ denotes
the antisymmetrization of $\widehat{\nabla}^{\mathbf{F},a}$. By (13.7),
(13.64) can be rewritten in the form

$\displaystyle\mathcal{A}^{\prime\prime}_{Z}=\widehat{\nabla}^{\mathbf{F},-1/2\prime\prime}_{\mathrm{a}}+B+\overline{\partial}^{V}+i_{z},$
(13.69)
$\displaystyle\mathcal{A}^{\prime}_{Z}=\widehat{\nabla}^{\mathbf{F},-1/2\prime}_{\mathrm{a}}+B^{*}-q^{*}\partial^{X}i\omega^{X}+\overline{\partial}^{V*}+i_{\overline{z}}+\overline{z}_{*}\wedge,$
$\displaystyle\mathcal{A}_{Z}=\widehat{\nabla}^{\mathbf{F},-1/2}_{\mathrm{a}}+C-q^{*}\partial^{X}i\omega^{X}+\overline{\partial}^{V}+i_{z}+\overline{\partial}^{V*}+i_{\overline{z}}+\overline{z}_{*}\wedge.$

Recall that the curvature $A^{E_{0},2}$ was defined in Subsection 7.2. It is a
section of degree $0$ of
$\Lambda\left(T^{*}_{\mathbf{C}}M\right)\widehat{\otimes}\mathrm{End}\left(D\right)$.

Let $w_{1},\ldots,w_{n}$ be a basis of $TX$. The corresponding basis of
$\widehat{TX}$ is denoted $\widehat{w}_{1},\ldots,\widehat{w}_{n}$. In the
sequel, we assume that $\widehat{w}_{1},\ldots,\widehat{w}_{n}$ is an
orthonormal basis of $\left(\widehat{TX},g^{\widehat{TX}}\right)$. As before,
if $e\in T_{\mathbf{R}}X$, $e_{*}\in T^{*}_{\mathbf{R}}X$ corresponds to $e$
via the metric $g^{T_{\mathbf{R}}X}$.

We establish an extension of [B13, Theorem 6.8.1].

######  Theorem 13.18.

The following identity holds:

(13.70)
$\mathcal{A}^{2}_{Z}=\left[\overline{\partial}^{V}+i_{z},\partial^{V{\dagger}}+i_{z_{-}}^{{\dagger}}\right]+\left[{}\widehat{\nabla}^{\mathbf{F},-1/2\prime\prime}_{\mathrm{a}}+B,i_{z_{-}}^{{\dagger}}\right]\\\
+\left[\widehat{\nabla}^{\mathbf{F},-1/2\prime}_{\mathrm{a}}+B^{*}-q^{*}\partial^{X}i\omega^{X},i_{z}\right]-q^{*}\overline{\partial}^{X}\partial^{X}i\omega^{X}\\\
-\nabla^{V}_{R^{\widehat{TX}}\widehat{Y}}-\left\langle
R^{\widehat{TX}}\overline{\widehat{w}}_{i},\overline{\widehat{w}}^{j}\right\rangle\overline{\widehat{w}}^{i}i_{\overline{\widehat{w}}_{j}}+A^{E_{0},2}.$

Equation (13.70) can be rewritten in the form,

(13.71)
$\mathcal{A}^{2}_{Z}=\left[\overline{\partial}^{V}+i_{z},\partial^{V{\dagger}}+i_{z_{-}}^{{\dagger}}\right]+{}^{0}\widehat{\nabla}^{\mathbf{F}}_{Z}+i_{Z}C-q^{*}\overline{\partial}^{X}\partial^{X}i\omega^{X}\\\
-\nabla^{V}_{R^{\widehat{TX}}\widehat{Y}}-\left\langle
R^{\widehat{TX}}\overline{\widehat{w}}_{i},\overline{\widehat{w}}^{j}\right\rangle\overline{\widehat{w}}^{i}i_{\overline{\widehat{w}}_{j}}+A^{E_{0},2}+i_{\widehat{\nabla}^{T_{\mathbf{R}}X,H}Z}+\nabla^{TX\prime\prime,H}\overline{z}_{*}.$

Moreover,

(13.72)
$\left[\overline{\partial}^{V}+i_{z},\partial^{V{\dagger}}+i_{z_{-}}^{{\dagger}}\right]=\frac{1}{2}\left(-\Delta^{V}_{g^{\widehat{TX}}}+\left|Z\right|^{2}_{g^{TX}}\right)\\\
+\overline{\widehat{w}}^{i}\wedge\left(i_{\nabla_{\overline{\widehat{w}}_{i}}\overline{z}}+\nabla_{\overline{\widehat{w}}_{i}}\overline{z}_{*}\right)-i_{\overline{\widehat{w}}_{i}}i_{\nabla_{\widehat{w}_{i}}z}.$

###### Proof.

Since $\widehat{\sigma}^{*}$ commutes with $\overline{\partial}^{V}$, we get

(13.73) $\overline{\partial}^{V*}=\overline{\partial}^{V{\dagger}}.$

Equation (13.70) follows from Proposition 13.1 and from equations (13.38),
(13.58), (13.10), and (13.73). Equation (13.71) is an easy consequence. The
only mysterious point is the that in ${}^{0}\widehat{\nabla}^{\mathbf{F}}$,
the connection $\widehat{\nabla}^{\mathbf{F},-1}$ appears, while there was
$\widehat{\nabla}^{\mathbf{F},-1/2}$ in equation (13.70). But this is entirely
explained in (4.56)–(4.58).

The proof of (13.72) is left to the reader, which completes the proof of our
theorem. ∎

###  13.11. The hypoelliptic curvature

Among the sections $z$, we will distinguish the tautological section $y$ of
$\underline{q}^{*}TX$ on $\mathcal{M}$ corresponding to the tautological
section $\widehat{y}$ of $\widehat{TX}$. Note that $y,\widehat{y}$ already
appeared in Subsection 13.2. The section $y$ is such that

(13.74) $y_{-}=y.$

We use the notation

(13.75) $Y=y+\overline{y}.$

By (13.10), we get

$\displaystyle\mathcal{A}^{\prime\prime}_{Y}=\widehat{\nabla}^{\mathbf{F},-1/2\prime\prime}_{\mathrm{a}}+B+\overline{\partial}^{V}+i_{y},$
(13.76)
$\displaystyle\mathcal{A}^{\prime}_{Y}=\widehat{\nabla}^{\mathbf{F},-1/2\prime}_{\mathrm{a}}+B^{*}-q^{*}\partial^{X}i\omega^{X}+\overline{\partial}^{V*}+i_{\overline{y}}+\overline{y}_{*}\wedge,$
$\displaystyle\mathcal{A}_{Y}=\widehat{\nabla}^{\mathbf{F},-1/2}_{\mathrm{a}}+C-q^{*}\partial^{X}i\omega^{X}+\overline{\partial}^{V}+i_{y}+\overline{\partial}^{V*}+i_{\overline{y}}+\overline{y}_{*}\wedge.$

By Theorem 13.15, we get

(13.77)
$\displaystyle\mathcal{A}^{\prime\prime{\dagger}}_{Y}=\mathcal{A}^{\prime}_{Y},$
$\displaystyle\mathcal{A}_{Y}^{{\dagger}}=\mathcal{A}_{Y}.$

Put

(13.78) $\gamma=\nabla^{TX}-\widehat{\nabla}^{TX}.$

Then $\gamma$ is a section of $T^{*}X\otimes\mathrm{End}\left(TX\right)$. If
$u\in TX$, then $\overline{\gamma u_{*}}$ is a $(1,1)$-form on $X$.

######  Definition 13.19.

Let ${}^{1}\widehat{\nabla}^{\mathbb{F}}$ be the fiberwise connection on
$\mathbb{F}$, which is such that if $u\in TX,U=u+\overline{u}\in
T_{\mathbf{R}}X$, then

(13.79)
${}^{1}\widehat{\nabla}^{\mathbb{F}}_{U}={}^{0}\widehat{\nabla}^{\mathbb{F}}_{U}+\overline{\gamma
u_{*}}.$

As before, we may as well replace $\mathbb{F}$ by $\mathbf{F}$ and obtain a
fiberwise connection ${}^{1}\widehat{\nabla}^{\mathbf{F}}$ on $\mathbf{F}$.

######  Theorem 13.20.

The following identity holds:

(13.80)
$\mathcal{A}^{2}_{Y}=\left[\overline{\partial}^{V}+i_{y},\partial^{V{\dagger}}+i_{y}^{{\dagger}}\right]+\left[{}\widehat{\nabla}^{\mathbf{F},-1/2\prime\prime}_{\mathrm{a}}+B,i_{y}^{{\dagger}}\right]\\\
+\left[\widehat{\nabla}^{\mathbf{F},-1/2\prime}_{\mathrm{a}}+B^{*}-q^{*}\partial^{X}i\omega^{X},i_{y}\right]-q^{*}\overline{\partial}^{X}\partial^{X}i\omega^{X}\\\
-\nabla^{V}_{R^{\widehat{TX}}\widehat{Y}}-\left\langle
R^{\widehat{TX}}\overline{\widehat{w}}_{i},\overline{\widehat{w}}^{j}\right\rangle\overline{\widehat{w}}^{i}i_{\overline{\widehat{w}}_{j}}+A^{E_{0},2}.$

Equation (13.80) can be rewritten in the form

(13.81)
$\mathcal{A}^{2}_{Y}=\left[\overline{\partial}^{V}+i_{y},\partial^{V{\dagger}}+i_{y}^{{\dagger}}\right]+{}^{1}\widehat{\nabla}^{\mathbf{F}}_{Y}+i_{Y}C-q^{*}\overline{\partial}^{X}\partial^{X}i\omega^{X}\\\
-\nabla^{V}_{R^{\widehat{TX}}\widehat{Y}}-\left\langle
R^{\widehat{TX}}\overline{\widehat{w}}_{i},\overline{\widehat{w}}^{j}\right\rangle\overline{\widehat{w}}^{i}i_{\overline{\widehat{w}}_{j}}+A^{E_{0},2}.$

Moreover,

(13.82)
$\left[\overline{\partial}^{V}+i_{y},\partial^{V{\dagger}}+i_{y}^{{\dagger}}\right]=\frac{1}{2}\left(-\Delta^{V}_{g^{\widehat{TX}}}+\left|Y\right|^{2}_{g^{TX}}\right)\\\
+\overline{\widehat{w}}^{i}\wedge\left(i_{\overline{w}_{i}}+\overline{w}_{i*}\right)-i_{\overline{\widehat{w}}_{i}}i_{w_{i}}.$

###### Proof.

This is a consequence of Theorem 13.18. In the proof of (13.81), we have
exploited the fact that $\widehat{\nabla}^{T_{\mathbf{R}}X,H}Y=0$, and the
related equation

(13.83) $\nabla^{TX\prime\prime,H}\overline{y}_{*}=\overline{\gamma y}_{*}.$

∎

By Theorem 13.18 and by Hörmander [H67], $\mathcal{A}^{2}_{Y}$ is a
hypoelliptic operator along the fibers $\mathcal{X}$. More precisely,
$\mathcal{A}^{2}_{Y}$ is a hypoelliptic Laplacian in the sense of [B05, B08,
B12].

######  Remark 13.21.

In (13.81), we may as well replace ${}^{1}\widehat{\nabla}^{\mathbf{F}}_{Y}$
by ${}^{1}\widehat{\nabla}^{\mathbb{F}}_{Y^{H}}$.

###  13.12. Scaling the metric $g^{\widehat{TX}}$

Here we follow [B13, Section 7.1]. For $b>0$, when $g^{\widehat{TX}}$ is
replaced by $b^{4}g^{\widehat{TX}}$, let
$\epsilon_{X,b},\epsilon_{M,b},\mathcal{A}^{\prime}_{Y,b},\mathcal{A}_{Y,b}$
be the analogues of
$\epsilon_{X},\epsilon_{M},\mathcal{A}^{\prime}_{Y},\mathcal{A}_{Y}$.

######  Definition 13.22.

For $a>0$, let $\delta_{a}$ be the dilation of $\mathcal{M}$:
$\left(x,Y\right)\to\left(x,aY\right)$. For $a>0$, let
$\underline{\delta}^{*}_{a}$ be the action of $\delta_{a}$ on smooth sections
of $\Lambda\left(\overline{\widehat{T^{*}X}}\right)$. The action of
$\underline{\delta}^{*}_{a}$ extends to $\mathbb{F}$.

Also $y/b^{2}$ is a holomorphic section on $\mathcal{M}$ of
$\underline{q}^{*}TX$. One verifies easily that

(13.84)
$A_{Y/b^{2}}=\underline{\delta}^{*}_{1/b^{2}}\mathcal{A}_{Y,b}\underline{\delta}^{*-1}_{1/b^{2}}.$

This identity is obvious because $\delta^{*}_{1/b^{2}}$ maps the metric
$b^{4}g^{\widehat{TX}}$ to $g^{\widehat{TX}}$, and $Y$ to $Y/b^{2}$.

Now we follow [B13, Section 6.9]. For $a>0$, put

(13.85) $K_{a}s\left(x,Y\right)=s\left(x,aY\right).$

######  Definition 13.23.

Put

(13.86) $\displaystyle\mathcal{C}_{Y,b}=K_{b}\mathcal{A}_{Y/b^{2}}K_{b}^{-1},$
$\displaystyle\mathcal{D}_{Y,b}=K_{b}\mathcal{B}_{Y/b^{2}}K_{b}^{-1}.$

By (13.7), (13.86), we get

(13.87)
$\mathcal{D}_{Y,b}=\exp\left(i\Lambda\right)\mathcal{C}_{Y,b}\exp\left(-i\Lambda\right).$

By (13.7), (13.44), we get

(13.88)
$\displaystyle\mathcal{C}_{Y,b}=\nabla^{D\widehat{\otimes}q^{*}\mathbf{I}}+C-q^{*}\partial^{X}i\omega^{X}+\frac{1}{b}\left(\overline{\partial}^{V}+i_{y}+\overline{\partial}^{V*}+i_{\overline{y}}+\overline{y}_{*}\right),$
$\displaystyle\mathcal{D}_{Y,b}=\mathcal{E}+\frac{1}{b}\left(\overline{\partial}^{V}+i_{y}+\overline{\partial}^{V*}+\overline{y}_{*}\right).$

###  13.13. Hypoelliptic and elliptic superconnections

In this Subsection, we assume that $\widehat{g}^{TX}=g^{TX}$. Let
$w_{1},\ldots,w_{n}$ be an orthonormal basis of $TX$ with respect to $g^{TX}$,
let $\widehat{w}_{1},\ldots,\widehat{w}_{n}$ be the corresponding basis of
$\widehat{TX}$. We denote the dual basis in the usual way.

We use the identification in (13.46). Set

(13.89) $\displaystyle\widehat{L}=-iw^{j}\wedge\overline{\widehat{w}}^{j},$
$\displaystyle\widehat{\Lambda}=ii_{\overline{\widehat{w}}_{j}}i_{w_{j}}.$

In (13.89), $\widehat{L}$ is just the multiplication operator by the fibrewise
Kähler form $\widehat{\omega}^{\mathcal{X},V}$. As in [B13, eq. (6.9.8)], we
get

(13.90)
$\left[\overline{\partial}^{V}+i_{y},\overline{\partial}^{V*}+\overline{y}_{*}\wedge\right]=\frac{1}{2}\left(-\Delta^{V}_{g^{TX}}+\left|Y\right|_{g^{TX}}^{2}\right)-i\left(\widehat{L}-\widehat{\Lambda}\right).$

The right-hand side of (13.90) is a harmonic oscillator.

Let
$\mathcal{S}^{(0,\scriptsize\bullet)}\left(\widehat{TX},\pi^{*}\Lambda\left(T^{*}X\right)\right)$
be the Schwartz space of rapidly decreasing sections of
$\pi^{*}\left(\Lambda\left(\overline{\widehat{T^{*}X}}\right)\widehat{\otimes}\Lambda\left(T^{*}X\right)\right)$
along the fibre $\widehat{TX}$. The operator in (13.90) is essentially self-
adjoint on
$\mathcal{S}^{(0,\scriptsize\bullet)}\left(\widehat{TX},\pi^{*}\Lambda\left(T^{*}X\right)\right)$.
By [B90, Proposition 1.5 and Theorem 1.6]343434In [B90], the operator
$\overline{\partial}^{V}+i_{\sqrt{-1}y}$ and its adjoint were considered
instead, which explains the minor discrepancies with respect to what is done
here., [B13, Section 6.9], its spectrum is $\mathbf{N}$, and its kernel is
$1$-dimensional and spanned by

(13.91)
$\delta=\exp\left(i\widehat{\omega}^{\mathcal{X},V}-\left|Y\right|^{2}_{g^{TX}}/2\right).$

Recall that $dY$ is the volume form along the fibers $\widehat{TX}$ with
respect to $g^{\widehat{TX}}$. Note that

(13.92) $\displaystyle\sigma^{*}\delta=\delta,$
$\displaystyle\left(2\pi\right)^{-n}\int_{\widehat{TX}}\left|\delta\right|^{2}dY=1.$

Let $P$ be the $L_{2}$ orthogonal projection operator on vector space spanned
by $\delta$.

We embed $\mathcal{D}=\Omega^{0,\scriptsize\bullet}\left(X,D|_{X}\right)$ into
$\Omega\left(X,D|_{X}\widehat{\otimes}\mathbf{I}\right)$ by the embedding
$\alpha\to\underline{\pi}^{*}\alpha\wedge\delta$, so that $\mathcal{D}$ is
just the $L_{2}$ kernel of
$\overline{\partial}^{V}+i_{y}+\overline{\partial}^{V*}+\overline{y}_{*}\wedge$.
More generally, we embed $p_{*}\mathscr{E}_{0}$ into
$p_{*}\left[\mathscr{E}_{0}\widehat{\otimes}q^{*}\left(\Lambda\left(T^{*}X\right)\widehat{\otimes}\mathbf{I}\right)\right]$
by the same formula.

Recall that the elliptic superconnection $A^{p_{*}\mathscr{E}_{0}}$ was
defined in Definition 10.8, and that the differential operator $\mathcal{E}$
was defined in (13.43). Now we give an extension of [B13, Theorem 6.9.2].

######  Theorem 13.24.

The following identity holds:

(13.93) $P\mathcal{E}P=A^{p_{*}\mathscr{E}_{0}}.$

###### Proof.

Observe that (13.93) is an identity of differential operators over $M$, which
we will prove by local methods on $M$. We will use the considerations that
follow (13.40) to evaluate $\mathcal{E}$.

Let $\nabla^{\Lambda\left(T^{*}X\right)}$ be the connection induced by
$\nabla^{TX}$ on $\Lambda\left(T^{*}X\right)$. Since
$g^{TX}=\widehat{g}^{TX}$, we get

(13.94)
$\nabla^{\Lambda\left(T^{*}X\right)\widehat{\otimes}\mathbf{I}}\delta=0.$

Recall that $C=B+B^{*}$. As explained after equation (13.40), conjugation by
$\exp\left(i\Lambda\right)$ changes $\overline{w}^{i}$ into
$\overline{w}^{i}+i_{w_{i}}$. Recall that $\delta$ in (13.91) is of total
degree $0$. The effect of the composition of the $i_{w_{j}}$ is to map
$\delta$ into forms of total positive degree353535Recall that the total degree
is the difference of the antiholomorphic and of the holomorphic degree., which
are orthogonal to $\delta$. Using (13.94) and proceeding as in the proof of
[B13, Theorem 6.9.2], we find that

(13.95)
$P\exp\left(i\Lambda\right)\left(\nabla^{D\widehat{\otimes}q^{*}\mathbf{I}\prime\prime}+B\right)\exp\left(-i\Lambda\right)P=\nabla^{D\prime\prime}+B.$

In [B13], the proof of (13.95) uses equations (13.63), (13.11).

Using (10.15), we can rewrite (13.95) in the form

(13.96)
$P\exp\left(i\Lambda\right)\left(\nabla^{D\widehat{\otimes}q^{*}\mathbf{I}\prime\prime}+B\right)\exp\left(-i\Lambda\right)P=A^{p_{*}\mathscr{E}_{0}\prime\prime}.$

Conjugation by $\exp\left(i\Lambda\right)$ changes $w^{i}$ into
$w^{i}-i_{\overline{w}_{i}}$. By proceeding again as in [B13, Theorem 6.9.2],
we get

(13.97)
$P\exp\left(i\Lambda\right)\left(\nabla^{D\widehat{\otimes}q^{*}\mathbf{I}\prime}+B^{*}-q^{*}\partial^{X}i\omega^{X}\right)\exp\left(-i\Lambda\right)P\\\
=\nabla^{D\prime,S}+\nabla^{D\prime\prime,X*}_{\alpha}+{}^{c}B^{*}.$

By (10.22), we can rewrite (13.97) in the form,

(13.98)
$P\exp\left(i\Lambda\right)\left(\nabla^{D\widehat{\otimes}q^{*}\mathbf{I}\prime}+B^{*}-q^{*}\partial^{X}i\omega^{X}\right)\exp\left(-i\Lambda\right)P=A^{p_{*}\mathscr{E}_{0}\prime}.$

By (13.96), (13.98), we get (13.93). The proof of our theorem is completed. ∎

######  Remark 13.25.

Recall that $\mathcal{D}_{Y,b}$ was defined in (13.87), and that it splits as

(13.99)
$\mathcal{D}_{Y,b}=\mathcal{D}^{\prime\prime}_{Y,b}+\mathcal{D}^{\prime}_{Y,b}.$

By (13.57), when taking adjoints with respect to $\eta_{X}$, then

(13.100)
$\mathcal{D}^{\prime\prime{\dagger}}_{Y,b}=\mathcal{D}^{\prime}_{Y,b}.$

By (13.50), (13.92), on $\Omega^{0,\scriptsize\bullet}\left(X,D|_{X}\right)$,
$\eta_{X}$ restrict to the Hermitian product
$\left\langle\,\right\rangle_{L_{2}}$. Since the contribution of
$\nabla^{D\widehat{\otimes}q^{*}\mathbf{I}\prime\prime}+B$ to $P\mathcal{E}P$
is $A^{p_{*}\mathscr{E}_{0}\prime\prime}$, the contribution of its adjoint
$\nabla^{D\widehat{\otimes}q^{*}\mathbf{I}\prime}+B^{*}-q^{*}\partial^{X}i\omega^{X}$
has to be $A^{p_{*}\mathscr{E}_{0}\prime}$. This explains some aspects of the
proof of Theorem 13.24.

##  14\. The hypoelliptic superconnection forms

The purpose of this Section is to construct the superconnection forms
associated with the hypoelliptic superconnections of Section 13, and to prove
that their class in Bott-Chern cohomology coincides with the class of the
elliptic superconnection forms.

This Section is organized as follows. In Subsection 14.1, we construct the
hypoelliptic superconnection forms, that depend on metrics
$g^{TX},g^{\widehat{TX}},g^{D}$.

In Subsection 14.2, when $g^{TX}=\widehat{g}^{TX}$, we show that when
replacing $g^{\widehat{TX}}$ by $b^{4}g^{\widehat{TX}}$, as $b\to 0$, the
hypoelliptic superconnection forms converge to the elliptic superconnection
forms, which gives the main result of this Section.

In this Section, we make the same assumptions as in Sections 10 and 13, and we
use the corresponding notation.

###  14.1. The hypoelliptic forms

We will say that two metrics $g^{\widehat{TX}},g^{\widehat{TX}\prime}$ lie in
the same projective class if there exists a constant $c>0$ such that
$g^{\widehat{TX}\prime}=cg^{\widehat{TX}}$. Let $\mathbf{p}$ be such a
projective class. Once we fix $g^{\widehat{TX}}\in\mathbf{p}$, then
$\mathbf{p}\simeq\mathbf{R}_{+}^{*}$.

We fix $\mathbf{p}$. Let $\mathscr{Q}_{\mathbf{p}}$ be the collection of
parameters $\omega^{X},g^{D},g^{\widehat{TX}}$, with
$g^{\widehat{TX}}\in\mathbf{p}$. We denote by $d^{\mathscr{Q}_{\mathbf{p}}}$
the de Rham operator on $\mathscr{Q}_{\mathbf{p}}$. A key fact is that when
$g^{\widehat{TX}}\in\mathbf{p}$, then $\nabla^{\mathbf{I}}$ does not depend on
$g^{\widehat{TX}}$.

By [BL08, Section 3.3], the operator $\exp\left(-\mathcal{A}^{2}_{Y}\right)$
is trace class, and is given by a smooth kernel along the fibers $\mathcal{X}$
that is rapidly decreasing along the fibre $\widehat{TX}$ together with its
derivatives, and this uniformly over $S$.

######  Definition 14.1.

Put

(14.1)
$\mathrm{ch}\left(\mathcal{A}^{\prime\prime}_{Y},\omega^{X},g^{D},g^{\widehat{TX}}\right)=\varphi\mathrm{Tr_{s}}\left[\exp\left(-\mathcal{A}^{2}_{Y}\right)\right].$