$T_{\mathfrak{t}}(\overline{\xi})=T_{\mathfrak{t}}(\overline{\pi}^{\text{\rm
FS}},\overline{\mathcal{H}[-j]},\overline{\pi_{\ast}\mathcal{H}[-j]})\\\
+T_{\mathfrak{t}}(\overline{\pi}^{\text{\rm
FS}},\operatorname{\overline{cone}}(\overline{\mathcal{H}[-j]},\overline{\mathcal{F}}),\operatorname{\overline{cone}}(\overline{\pi_{\ast}\mathcal{H}[-j]},\overline{\pi_{\ast}\mathcal{F}})).$
(5.38)

It follows from [17, Thm. 2.27 (iv)] that the right hand side of this equality
does not depend on the choice of the auxiliary hermitian structures.

Finally, we consider the case when $\overline{\pi}$ has a metric different
from the Fubini-Study metric. Thus, let
$\overline{\xi}=(\overline{\pi},\overline{\mathcal{F}},\overline{\pi_{\ast}\mathcal{F}})$
and write $\overline{\xi}^{\prime}=(\overline{\pi}^{\text{\rm
FS}},\overline{\mathcal{F}},\overline{\pi_{\ast}\mathcal{F}})$. Then we put

$T_{\mathfrak{t}}(\overline{\xi})=T_{\mathfrak{t}}(\overline{\xi}^{\prime})+\overline{\pi}_{\flat}[\operatorname{ch}(\overline{F})\bullet\widetilde{\operatorname{Td}}_{m}(\overline{\pi},\overline{\pi}^{\text{\rm
FS}})].$ (5.39)

######  Definition 5.40.

Let $n$ and $\mathfrak{t}$ be as in Theorem 5.9. Then $T_{\mathfrak{t}}$ is
the assignment that to each relative metrized complex $\overline{\xi}$
associates $T_{\mathfrak{t}}(\overline{\xi})$ given by equations (5.37),
(5.38) and (5.39).

It remains to prove that $T_{\mathfrak{t}}$ satisfies axioms (i) to (iv).
Axiom (i) follows from the differential equations satisfied by the Bott-Chern
classes. Axiom (ii) follows from the functoriality of the canonical
resolution, the Chern forms and the Bott-Chern classes. Axiom (iii) follows
from the additivity of the canonical resolution and of the Chern character.
Finally Axiom (iv) follows from the multiplicativity of the Chern character.
This concludes the proof of Theorem 5.9. ∎

We finish this section showing the compatibility of analytic torsion classes
with the composition of projective bundles. Let $X$ be a smooth complex
variety. Consider the commutative diagram with cartesian square

$\textstyle{{\mathbb{P}}^{n_{1}}_{X}\underset{X}{\times}{\mathbb{P}}^{n_{2}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$$\scriptstyle{p}$$\textstyle{{\mathbb{P}}^{n_{1}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\textstyle{{\mathbb{P}}^{n_{2}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{2}}$$\textstyle{X}$

On $\pi_{1}$ and $\pi_{2}$ we introduce arbitrary hermitian structures and on
$p_{1}$ and $p_{2}$ the hermitian structures induced by the cartesian diagram.

######  Proposition 5.41.

Let $\overline{\mathcal{F}}$ be an object of
$\operatorname{\overline{\mathbf{D}}^{b}}({\mathbb{P}}^{n_{1}}_{X}\underset{X}{\times}{\mathbb{P}}^{n_{2}}_{X})$.
Put arbitrary hermitian structures on $(p_{1})_{\ast}\mathcal{F}$,
$(p_{2})_{\ast}\mathcal{F}$, and $p_{\ast}\mathcal{F}$. Then

$T(\overline{\pi}_{1})+(\overline{\pi}_{1})_{\flat}(T(\overline{p}_{1}))=T(\overline{\pi}_{2})+(\overline{\pi}_{2})_{\flat}(T(\overline{p}_{2})),$
(5.42)

where we are using the convention at the end of Definition 2.6.

###### Proof.

By the anomaly formulas (Proposition 5.10), if equation (5.42) holds for a
particular choice of hermitian structures on $\mathcal{F}$,
$(p_{1})_{\ast}\mathcal{F}$, $(p_{2})_{\ast}\mathcal{F}$, and
$p_{\ast}\mathcal{F}$, then it holds for any other choice. Let

$\overline{\mathcal{F}}_{2}\dashrightarrow\overline{\mathcal{F}}_{1}\dashrightarrow\overline{\mathcal{F}}_{0}\dashrightarrow$

be a distinguished triangle and put hermitian structures on the direct images
as before. Then Proposition 5.12 implies that, if equation (5.42) holds for
two of them, then it also holds for the third. Since the objects of the form
$\mathcal{G}(k,l):=p^{\ast}\mathcal{G}\otimes
p_{1}^{\ast}\mathcal{O}(k)\otimes p_{2}^{\ast}\mathcal{O}(l)$ are a generating
class of
$\operatorname{\mathbf{D}^{b}}({\mathbb{P}}^{n_{1}}_{X}\underset{X}{\times}{\mathbb{P}}^{n_{2}}_{X})$,
the previous discussion shows that it is enough to prove the case
$\mathcal{F}=\mathcal{G}(k,l)$, with the hermitian structure of $\mathcal{F}$
induced by a hermitian structure of $\mathcal{G}$ and the Fubini-Study metric
on $\mathcal{O}(k)$ and $\mathcal{O}(l)$, and the hermitian structures on the
direct images defined as in (5.3). In this case the result follows easily from
the functoriality and the projection formula. ∎

##  6 Compatible analytic torsion classes

In this section we study the compatibility between analytic torsion classes
for closed immersions and analytic torsion classes for projective spaces. It
turns out that, once the compatibility between the diagonal embedding of
${\mathbb{P}}^{n}$ into ${\mathbb{P}}^{n}\times{\mathbb{P}}^{n}$ and the
second projection of ${\mathbb{P}}^{n}\times{\mathbb{P}}^{n}$ onto
${\mathbb{P}}^{n}$ is established, then all the other possible compatibilities
follow. Essentially this observation can be traced back to [15].

Let $n$, $V$, $\overline{V}$ and ${\mathbb{P}}^{n}(V)$ be as in the previous
section. We consider the diagram

$\textstyle{{\mathbb{P}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\Delta\
\ \
}$$\textstyle{{\mathbb{P}}^{n}\times{\mathbb{P}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$$\textstyle{{\mathbb{P}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{{\mathbb{P}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\textstyle{\operatorname{Spec}{\mathbb{C}}\
.}$

On ${\mathbb{P}}^{n}$ we have the tautological short exact sequence

$0\rightarrow\mathcal{O}(-1)\rightarrow V\rightarrow Q\rightarrow 0\ .$

This induces on ${\mathbb{P}}^{n}\times{\mathbb{P}}^{n}$ the exact sequence

$0\rightarrow p_{2}^{\ast}\mathcal{O}(-1)\rightarrow V\rightarrow
p_{2}^{\ast}Q\rightarrow 0\ .$

By composition with the injection $p_{1}^{\ast}\mathcal{O}(-1)\hookrightarrow
V$, we obtain a morphism $p_{1}^{\ast}\mathcal{O}(-1)\to p_{2}^{\ast}Q,$ hence
a section of $p_{2}^{\ast}Q\otimes p_{1}^{\ast}\mathcal{O}(1)$. The zero locus
of this section is the image of the diagonal. Moreover, the associated Koszul
complex is quasi-isomorphic to $\Delta_{\ast}\mathcal{O}_{{\mathbb{P}}^{n}}$.
That is, the sequence

$0\rightarrow\Lambda^{n}(p_{2}^{\ast}Q^{\vee})\otimes
p_{1}^{\ast}\mathcal{O}_{{\mathbb{P}}^{n}}(-n)\rightarrow\dots{\\\
}\dots\rightarrow\Lambda^{1}(p_{2}^{\ast}Q^{\vee})\otimes
p_{1}^{\ast}\mathcal{O}_{{\mathbb{P}}^{n}}(-1)\rightarrow\mathcal{O}_{{\mathbb{P}}^{n}\times{\mathbb{P}}^{n}}\rightarrow\Delta_{\ast}\mathcal{O}_{{\mathbb{P}}^{n}}\rightarrow
0$ (6.1)

is exact.

On $T_{{\mathbb{P}}^{n}}$ and $T_{{\mathbb{P}}^{n}\times{\mathbb{P}}^{n}}$ we
consider the Fubini-Study metrics. We denote by $\overline{\Delta}$ and
$\overline{p}_{2}$ the morphisms of
$\operatorname{\overline{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$ determined by
these metrics. As in [17, Ex. 5.7], we have that
$\overline{p_{2}}\circ\overline{\Delta}=\overline{\operatorname{id}}_{{\mathbb{P}}^{n}}$,
where $T_{\overline{\operatorname{id}_{{\mathbb{P}}^{n}}}}=\overline{0}$. The
Fubini-Study metric on $\mathcal{O}(-1)$ and the metric induced by the
tautological exact sequence on $Q$ induce a metric $\overline{K}(\Delta)$ on
the Koszul complex. This is a hermitian structure on
$\Delta_{\ast}\mathcal{O}_{{\mathbb{P}}^{n}}$. Finally on
$\mathcal{O}_{{\mathbb{P}}^{n}}$ we consider the trivial metric. This is a
hermitian structure on $(p_{2})_{\ast}K(\Delta)$.

Fix a real additive genus $S$ and denote by $T_{S}$ the theory of analytic
torsion classes for closed immersions that is compatible with the projection
formula and transitive, associated to $S$ (Theorem 3.11). Moreover, fix a
family of real numbers $\mathfrak{t}=\\{t_{nk}\ |\ n\geq 0,\ -n\leq k\leq
0\\}$ and denote $T_{\mathfrak{t}}$ the theory of generalized analytic torsion
classes for projective spaces associated to this family.

Compatible analytic torsion classes for closed immersions and for projective
spaces should combine to provide analytic torsion classes for arbitrary
projective morphisms, and these classes should be transitive. The transitivity
condition for the composition
$\operatorname{id}_{{\mathbb{P}}^{n}}=p_{2}\circ\Delta$ should give us

$0=T(\overline{\operatorname{id}}_{{\mathbb{P}}^{n}},\overline{\mathcal{O}}_{{\mathbb{P}}^{n}},\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})=T_{\mathfrak{t}}(\overline{p_{2}},\overline{K}(\Delta),\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})+(\overline{p}_{2})_{\flat}(T_{S}(\overline{\Delta},\overline{\mathcal{O}}_{{\mathbb{P}}^{n}},\overline{K}(\Delta))).$

In general we define

######  Definition 6.2.

The theories of analytic torsion classes $T_{S}$ and $T_{\mathfrak{t}}$ are
called _compatible_ if

$T_{\mathfrak{t}}(\overline{p_{2}},\overline{K}(\Delta),\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})+(\overline{p}_{2})_{\flat}(T_{S}(\overline{\Delta},\overline{\mathcal{O}}_{{\mathbb{P}}^{n}},\overline{K}(\Delta)))=0.$
(6.3)

######  Theorem 6.4.

Let $S$ be a real additive genus. Then there exists a unique family of real
numbers $\mathfrak{t}=\\{t_{n,k}\ |\ n\geq 0,\ -n\leq k\leq 0\\}$ such that
the theories of analytic torsion classes $T_{S}$ and $T_{\mathfrak{t}}$ are
compatible. The theory $T_{\mathfrak{t}}$ will also be denoted $T_{S}$.

###### Proof.

The first step is to make explicit equation (6.3) in terms of the main
characteristic numbers $\mathfrak{t}$. To this end, first observe that, since
the exact sequence

$0\to T_{p_{2}}\to T_{{\mathbb{P}}^{n}\times{\mathbb{P}}^{n}}\to
p_{2}^{\ast}T_{{\mathbb{P}}^{n}}\to 0$ (6.5)

is split and the hermitian metric on
$T_{{\mathbb{P}}^{n}\times{\mathbb{P}}^{n}}$ is the orthogonal direct sum
metric, $\overline{p_{2}}=\pi^{\ast}_{1}(\overline{\pi}^{\text{\rm FS}})$.
Next, we denote by $\overline{K}(\Delta)_{i}$ the component of degree $i$ of
the Koszul complex, and we define

$\overline{(p_{2})_{\ast}K(\Delta)_{i}}=\begin{cases}\overline{\mathcal{O}}_{{\mathbb{P}}^{n}},&\text{for
}i=0,\\\ \overline{0},&\text{for }i>0.\end{cases}$

Finally using Corollary 5.15, functoriality and the compatibility with the
projection formula, we derive

$\displaystyle
T_{\mathfrak{t}}(\overline{p_{2}},\overline{K}(\Delta),\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})$
$\displaystyle=\sum_{i=0}^{n}(-1)^{i}T_{\mathfrak{t}}(\overline{p_{2}},\overline{K}(\Delta)_{i},\overline{(p_{2})_{\ast}K(\Delta)_{i}})$
$\displaystyle=\sum_{i=0}^{n}(-1)^{i}T_{\mathfrak{t}}(\pi_{1}^{\ast}\overline{\xi}_{n}(-i)\otimes\Lambda^{i}\overline{Q}^{\vee})$
$\displaystyle=\sum_{i=0}^{n}(-1)^{i}t_{n,-i}\operatorname{ch}(\Lambda^{i}\overline{Q}^{\vee}).$

Thus, the second and last step is to solve the equation

$\sum_{i=0}^{n}(-1)^{i}t_{n,-i}\operatorname{ch}(\Lambda^{i}\overline{Q}^{\vee})=-(p_{2})_{\ast}(T_{S}(\overline{\Delta},\overline{\mathcal{O}}_{{\mathbb{P}}^{n}},\overline{K}(\Delta))\bullet\operatorname{Td}(\overline{p_{2}})).$
(6.6)

Since the left hand side of equation (6.6) is closed, in order to be able to
solve this equation we have to show that the right hand side is also closed.
We compute

$\displaystyle\operatorname{d}_{\mathcal{D}}(p_{2})_{\ast}$
$\displaystyle(T_{S}(\overline{\Delta},\overline{\mathcal{O}}_{{\mathbb{P}}^{n}},\overline{K}(\Delta))\bullet\operatorname{Td}(\overline{p_{2}}))$
$\displaystyle=(p_{2})_{\ast}\left(\sum_{i=0}^{n}(-1)^{i}\operatorname{ch}(\overline{K}(\Delta)_{i})\operatorname{Td}(\overline{p}_{2})-\Delta_{\ast}(\operatorname{ch}(\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})\operatorname{Td}(\overline{\Delta}))\operatorname{Td}(\overline{p}_{2})\right)$
$\displaystyle=(p_{2})_{\ast}\left(\sum_{i=0}^{n}(-1)^{i}p_{2}^{\ast}(\operatorname{ch}(\Lambda^{i}\overline{Q}^{\vee}))p_{1}^{\ast}(\operatorname{ch}(\overline{\mathcal{O}}(-i)))\operatorname{Td}(\overline{p}_{2})\right)-1$
$\displaystyle=\sum_{i=0}^{n}(-1)^{i}\operatorname{ch}(\Lambda^{i}\overline{Q}^{\vee})(p_{2})_{\ast}\left(p_{1}^{\ast}(\operatorname{ch}(\overline{\mathcal{O}}(-i)))\operatorname{Td}(\overline{p}_{2})\right)-1$
$\displaystyle=\sum_{i=0}^{n}(-1)^{i}\operatorname{ch}(\Lambda^{i}\overline{Q}^{\vee})\pi_{1}^{\ast}\pi_{\ast}\left(\operatorname{ch}(\overline{\mathcal{O}}(-i))\operatorname{Td}(\overline{\pi})\right)-1$
$\displaystyle=1-1=0.$

In the first equality we have used the differential equation of $T_{S}$. In
the second one we have used the definition of the Koszul complex, the equation
$\operatorname{ch}(\overline{\mathcal{O}}_{{\mathbb{P}}^{n}})=1$ and the fact
that, by the choice of hermitian structures on $T_{\overline{\Delta}}$ and
$T_{\overline{p}_{2}}$ we have
$\operatorname{Td}(\overline{\Delta})\bullet\Delta^{\ast}(\operatorname{Td}(\overline{p}_{2}))=1$.
The third equality is the projection formula and the fourth is base change for
cohomology. For the last equality we have used equation (5.22).

Both sides of equation (6.6) are closed and defined up to boundaries, hence
this is an equation in cohomology classes. The tautological exact sequence
induces exact sequences

$0\to\Lambda^{k}Q^{\vee}\to\Lambda^{k}V^{\vee}\to\Lambda^{k-1}Q^{\vee}\otimes\mathcal{O}(1)\to
0,$

that give us equations

$\operatorname{ch}(\Lambda^{k}Q^{\vee})=\binom{n+1}{k}-\operatorname{ch}(\Lambda^{k-1}Q^{\vee})\operatorname{ch}(\mathcal{O}(1)).$

Hence

$\operatorname{ch}(\Lambda^{k}Q^{\vee})=\sum^{k}_{i=0}(-1)^{i}\binom{n+1}{k-i}\operatorname{ch}(\mathcal{O}(i)).$

Since the classes $\operatorname{ch}(\mathcal{O}(i))$, $i=0,\dots,n$, form a
basis of
$\bigoplus_{p}H^{2p}_{\mathcal{D}}({\mathbb{P}}^{n},{\mathbb{R}}(p))$, the
same is true for the classes $\operatorname{ch}(\Lambda^{i}Q^{\vee})$,
$i=0,\dots,n$. Therefore, if $\mathbf{1}_{1}\in
H^{1}_{\mathcal{D}}({\mathbb{P}}^{n},{\mathbb{R}}(1))$ is the class
represented by the constant function $1$, the classes
$\mathbf{1}_{1}\bullet\operatorname{ch}(\Lambda^{i}Q^{\vee})$, $i=0,\dots,n$
form a basis of
$\bigoplus_{p=1}^{n+1}H^{2p-1}_{\mathcal{D}}({\mathbb{P}}^{n},{\mathbb{R}}(p))$.
This implies that equation (6.6) has a unique solution. ∎

######  Remark 6.7.

Given a theory $T$ of analytic torsion classes for projective spaces, obtained
from an arbitrary choice of characteristic numbers, in general, it does not
exist an additive genus such that the associated theory of singular Bott-Chern
classes is compatible with $T$. It would be interesting to characterize the
collections of characteristic numbers that arise from Theorem 6.4.

By definition, compatible analytic torsion classes for closed immersions and
projective spaces satisfy a compatibility condition for the trivial vector
bundle and the diagonal embedding. When adding the functoriality and the
projection formula, we obtain compatibility relations for arbitrary sections
of the trivial projective bundle and arbitrary objects.

Let $X$ be a smooth complex variety, let $\pi\colon{\mathbb{P}}_{X}^{n}\to X$
be the projective space over $X$ and let $s\colon X\to{\mathbb{P}}_{X}^{n}$ be
a section. Choose any hermitian structure on $T_{\pi}$. Since we have an
isomorphism $T_{s}\dashrightarrow s^{\ast}T_{\pi}[-1]$, this hermitian
structure induces a hermitian structure on $s$. Denote by $\overline{\pi}$ and
$\overline{s}$ the corresponding morphisms in
$\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. With this choice
of hermitian structures, we have

$\overline{\pi}\circ\overline{s}=(\pi\circ
s,\operatorname{\overline{cone}}(s^{\ast}T_{\overline{\pi}}[-1],s^{\ast}T_{\overline{\pi}}[-1]))=(\operatorname{id}_{X},\overline{0}),$

because the cone of the identity is meager.

######  Proposition 6.8.

Let $S$ be a real additive genus. Let $T_{S}$ denote both, the theory of
analytic torsion classes for closed immersions determined by $S$, and the
theory of analytic torsion classes for projective spaces compatible with it.
Let $\overline{\mathcal{F}}$ be an object of
$\operatorname{\overline{\mathbf{D}}^{b}}(X)$. Put a hermitian structure on
$s_{\ast}\mathcal{F}$. Then

$T_{S}(\overline{\pi},\overline{s_{\ast}\mathcal{F}},\overline{\mathcal{F}})+\overline{\pi}_{\flat}(T_{S}(\overline{s},\overline{\mathcal{F}},\overline{s_{\ast}\mathcal{F}}))=0.$
(6.9)

###### Proof.

By the anomaly formulas Proposition 3.17 and Proposition 5.10, if equation
(6.9) holds for a particular choice of hermitian structure on
$s_{\ast}\mathcal{F}$ then it holds for any other choice. Therefore we can
assume that the hermitian structure on $s_{\ast}\mathcal{F}$ is given by
$\overline{K}(s)\otimes\pi^{\ast}\overline{\mathcal{F}}$, where
$\overline{K}(s)$ is the Koszul complex associated to the section $s$. By the
projection formulas, if (6.9) holds for the trivial bundle $\mathcal{O}_{X}$
then it holds for arbitrary objects of
$\operatorname{\overline{\mathbf{D}}^{b}}(X)$.

We now prove that, if equation (6.9) holds for a particular choice of
hermitian structure $\overline{\pi}$, then it holds for any other choice.
Thus, assume that equation (6.9) is satisfied for $\overline{\pi}$ and
$\overline{s}$. Let $\overline{\pi}^{\prime}$ be another choice of hermitian
structure on $\pi$ and $\overline{s}^{\prime}$ be the hermitian structure
induced on $s$. On one hand, we have

$T_{S}(\overline{\pi}^{\prime},\overline{K}(s),\overline{\mathcal{O}}_{X})=T_{S}(\overline{\pi},\overline{K}(s),\overline{\mathcal{O}}_{X})+\pi_{\ast}\left(\operatorname{ch}(\overline{K}(s)\bullet\widetilde{\operatorname{Td}}_{m}(\overline{\pi}^{\prime},\overline{\pi})\bullet\operatorname{Td}(\overline{\pi}^{\prime}))\right).$
(6.10)

On the other hand, we have

$\displaystyle
T_{S}(\overline{s}^{\prime},\overline{\mathcal{F}},\overline{s_{\ast}\mathcal{F}})\bullet\operatorname{Td}(\overline{\pi}^{\prime})$
(6.11)
$\displaystyle\phantom{A}=\left(T_{S}(\overline{s},\overline{\mathcal{F}},\overline{s_{\ast}\mathcal{F}})+s_{\ast}(\widetilde{\operatorname{Td}}_{m}(\overline{s}^{\prime},\overline{s})\operatorname{Td}(\overline{s}^{\prime}))\right)\bullet\left(\operatorname{Td}(\overline{\pi})-\operatorname{d}_{\mathcal{D}}(\widetilde{\operatorname{Td}}_{m}(\overline{\pi}^{\prime},\overline{\pi})\bullet\operatorname{Td}(\overline{\pi}^{\prime}))\right)$
$\displaystyle\phantom{A}=T_{S}(\overline{s},\overline{\mathcal{F}},\overline{s_{\ast}\mathcal{F}})\bullet\operatorname{Td}(\overline{\pi})+s_{\ast}(\widetilde{\operatorname{Td}}_{m}(\overline{s}^{\prime},\overline{s})\operatorname{Td}(\overline{s}^{\prime}))\bullet\operatorname{Td}(\overline{\pi}^{\prime})$
$\displaystyle\phantom{AAAAAAAAAAAAAAA}-T_{S}(\overline{s},\overline{\mathcal{F}},\overline{s_{\ast}\mathcal{F}})\bullet\operatorname{d}_{\mathcal{D}}(\widetilde{\operatorname{Td}}_{m}(\overline{\pi}^{\prime},\overline{\pi})\bullet\operatorname{Td}(\overline{\pi}^{\prime}))$

In the group
$\bigoplus_{p}\widetilde{D}_{D}^{2p-1}({\mathbb{P}}^{n}_{X},N_{s},p)$ we have

$T_{S}(\overline{s},\overline{\mathcal{F}},\overline{s_{\ast}\mathcal{F}})\bullet\operatorname{d}_{\mathcal{D}}(\widetilde{\operatorname{Td}}_{m}(\overline{\pi}^{\prime},\overline{\pi})\bullet\operatorname{Td}(\overline{\pi}^{\prime}))\\\
=\left(\operatorname{ch}(\overline{K}(s))-s_{\ast}(\operatorname{Td}(\overline{s}))\right)\bullet\left(\widetilde{\operatorname{Td}}_{m}(\overline{\pi}^{\prime},\overline{\pi})\bullet\operatorname{Td}(\overline{\pi}^{\prime})\right).$
(6.12)

Observe that, by the definition of the hermitian structure of $\overline{s}$
and $\overline{s}^{\prime}$ we have

$\operatorname{Td}(\overline{s})\bullet
s^{\ast}\widetilde{\operatorname{Td}}_{m}(\overline{\pi}^{\prime},\overline{\pi})=-\widetilde{\operatorname{Td}}_{m}(\overline{s}^{\prime},\overline{s})\bullet\operatorname{Td}(\overline{s}^{\prime}).$
(6.13)

By combining equations (6.9), (6.10), (6.12) and (6.13) we obtain

$T_{S}(\overline{\pi}^{\prime},\overline{s_{\ast}\mathcal{F}},\overline{\mathcal{F}})=-\pi_{\ast}\left(T_{S}(\overline{s}^{\prime},\overline{\mathcal{F}},\overline{s_{\ast}\mathcal{F}})\bullet\operatorname{Td}(\overline{\pi}^{\prime})\right).$
(6.14)

We now prove (6.9) for a particular choice of hermitian structures. Let
$f\colon X\to{\mathbb{P}}^{n}$ denote the composition of $s$ with the
projection ${\mathbb{P}}^{n}_{X}\to{\mathbb{P}}^{n}$. Then we have a
commutative diagram with cartesian squares

$\textstyle{{\mathbb{P}}^{n}\times
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\times
f}$$\scriptstyle{\pi}$$\textstyle{{\mathbb{P}}^{n}\times{\mathbb{P}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{f}$$\textstyle{{\mathbb{P}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\scriptstyle{\operatorname{id}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{{\mathbb{P}}^{n}}$

Let $\overline{\Delta}$ and $\overline{p}_{2}$ be as in Definition 6.2. On
$\overline{\pi}$ and $\overline{s}$ we put the hermitian structures induced by
$\overline{\Delta}$. Since the Koszul complex
$\overline{K}(s)=(\operatorname{id}_{{\mathbb{P}}^{n}}\times
f)^{\ast}\overline{K}(\Delta)$, by Proposition 2.15 and functoriality,
equation (6.9) in this case follows from equation (6.3). ∎

We now study another compatibility between analytic torsion classes for closed
immersions and projective spaces. Let $\iota\colon X\to Y$ be a closed
immersion of smooth complex varieties. Consider the cartesian square

$\textstyle{{\mathbb{P}}^{n}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\iota_{1}}$$\textstyle{{\mathbb{P}}^{n}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{Y}$

Choose hermitian structures on $\pi$ and $\iota$ and put on $\pi_{1}$ and
$\iota_{i}$ the induced ones.

######  Proposition 6.15.

Let $S$ be a real additive genus. Let $T_{S}$ denote both, the theory of
analytic torsion classes for closed immersions determined by $S$, and the
theory of analytic torsion classes for projective spaces compatible with it.
Let $\overline{\mathcal{F}}$ be and object of
$\operatorname{\overline{\mathbf{D}}^{b}}({\mathbb{P}}^{n}_{X})$. Put
hermitian structures on $(\pi_{1})_{\ast}\mathcal{F}$,
$(\iota_{1})_{\ast}\mathcal{F}$ and $(\pi\circ\iota_{1})_{\ast}\mathcal{F}$.
Then

$T_{S}(\overline{\pi})+\overline{\pi}_{\flat}(T_{S}(\overline{\iota_{1}}))=T_{S}(\overline{\iota})+\overline{\iota}_{\flat}(T_{S}(\overline{\pi_{1}})).$
(6.16)

###### Proof.

By the anomaly formulas, if equation (6.16) holds for a particular choice of
metrics on $(\pi_{1})_{\ast}\mathcal{F}$, $(\iota_{1})_{\ast}\mathcal{F}$ and
$(\pi\circ\iota_{1})_{\ast}\mathcal{F}$, then it holds for any choice. Because
the sheaves $\mathcal{G}(k)$, with $\mathcal{G}$ a coherent sheaf on $X$,
constitute a generating class of
$\operatorname{\mathbf{D}^{b}}({\mathbb{P}}^{n}_{X})$ and by propositions 3.19
and 5.12, we reduce to the case when $\mathcal{F}$ is of the form
$\mathcal{G}(k)$. We choose arbitrary hermitian structures on $\mathcal{G}$
and $\iota_{\ast}\mathcal{G}$. Furthermore, we assume ${\mathcal{O}}(k)$,
$(\pi_{1})_{\ast}{\mathcal{O}}(k)$ and $\pi_{\ast}{\mathcal{O}}(k)$ endowed
with the hermitian structures of Notation 5.2. From these choices and the
projection formula, the objects $(\pi_{1})_{\ast}\mathcal{F}$,
$(\iota_{1})_{\ast}\mathcal{F}$ and $(\pi\circ\iota_{1})_{\ast}\mathcal{F}$
automatically inherit hermitian structures. Indeed, it is enough to observe
the natural isomorphisms

$\displaystyle(\pi_{1})_{\ast}\mathcal{F}\cong\mathcal{G}\otimes(\pi_{1})_{\ast}{\mathcal{O}}(k)$
(6.17)
$\displaystyle(\iota_{1})_{\ast}(\pi_{1}^{\ast}\mathcal{G}\otimes\iota_{1}^{\ast}{\mathcal{O}}(k))\cong\pi^{\ast}(\iota_{\ast}\mathcal{G})\otimes{\mathcal{O}}(k)$
(6.18)
$\displaystyle(\pi\circ\iota_{1})_{\ast}\mathcal{F}\cong\pi_{\ast}(\pi^{\ast}\iota_{\ast}\mathcal{G}\otimes{\mathcal{O}}(k))\cong\iota_{\ast}\mathcal{G}\otimes\pi_{\ast}{\mathcal{O}}(k).$
(6.19)

We now work out the left hand side of equation (6.16). Using the projection
formula for the theory $T_{S}$ for projective spaces, and equations
(6.17)–(6.19), we find

$T_{S}(\overline{\pi})=t_{n,k}\bullet\operatorname{ch}(\overline{\iota_{\ast}\mathcal{G}}).$
(6.20)

Using the functoriality of $T_{S}$ for closed immersions and the projection
formula we have

$\displaystyle T_{S}(\overline{\iota}_{1})=$
$\displaystyle\pi^{\ast}T_{S}(\iota,\overline{\mathcal{G}},\overline{\iota_{\ast}\mathcal{G}})\bullet\operatorname{ch}(\overline{{\mathcal{O}}(k)})$
$\displaystyle\overline{\pi}_{\flat}(T_{S}(\overline{\iota}_{1}))=$
$\displaystyle
T_{S}(\iota,\overline{\mathcal{G}},\overline{\iota_{\ast}\mathcal{G}})\bullet\pi_{\ast}(\operatorname{ch}(\overline{{\mathcal{O}}(k)})\bullet\operatorname{Td}(\overline{\pi})).$
(6.21)

Now for the right hand side of (6.16). The projection formula for $T_{S}$ for
closed immersions implies

$T_{S}(\overline{\iota})=T_{S}(\iota,\overline{\mathcal{G}},\overline{\iota\mathcal{G}})\bullet\operatorname{ch}(\overline{\pi_{\ast}{\mathcal{O}}(k)}).$
(6.22)

Similarly, we obtain
$T_{S}(\overline{\pi}_{1})=t_{n,k}\bullet\operatorname{ch}(\overline{\mathcal{G}}),$
and hence

$\overline{\iota}_{\flat}(T_{S}(\overline{\pi}_{1}))=t_{n,k}\bullet\iota_{\ast}(\operatorname{ch}(\overline{\mathcal{G}})\bullet\operatorname{Td}(\overline{\iota})).$
(6.23)

Using (6.20)–(6.23), the difference of the two sides of (6.16) becomes

$t_{n,k}\bullet\operatorname{d}_{\mathcal{D}}T_{S}(\iota,\overline{\mathcal{G}},\overline{\iota_{\ast}\mathcal{G}})-T_{S}(\iota,\overline{\mathcal{G}},\overline{\iota_{\ast}\mathcal{G}})\bullet\operatorname{d}_{\mathcal{D}}t_{n,k}=-\operatorname{d}_{\mathcal{D}}(t_{n,k}\bullet
T_{S}(\iota,\overline{\mathcal{G}},\overline{\iota_{\ast}\mathcal{G}}))=0$

in the group $\oplus_{p}\widetilde{\mathcal{D}}_{D}^{2p-1}(Y,N_{\iota},p)$. ∎

##  7 Generalized analytic torsion classes

In this section we will extend the definition of analytic torsion classes to
arbitrary morphisms of smooth complex varieties. Our construction is based on
the construction of analytic torsion classes by Zha in [50].

######  Definition 7.1.

A theory of generalized analytic torsion classes is an assignment that, to
each morphism $\overline{f}\colon X\to Y$ in
$\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$ and relative
metrized complex
$\overline{\xi}=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}}),$
assigns a class of currents

$T(\overline{\xi})\in\bigoplus_{p=1}^{n+1}\widetilde{\mathcal{D}}^{2p-1}_{D}(Y,N_{f},p)$

satisfying the following properties:

  1. (i)

(Differential equation) For any current $\eta\in T(\overline{\xi})$, we have

$\operatorname{d}_{\mathcal{D}}\eta=\operatorname{ch}(\overline{f_{\ast}\mathcal{F}})-\overline{f}_{\flat}[\operatorname{ch}(\overline{\mathcal{F}})].$
(7.2)

  2. (ii)

(Functoriality) If $g\colon Y^{\prime}\to Y$ is a morphism transverse to $f$,
then

$g^{\ast}T(\overline{\xi})=T(g^{\ast}\overline{\xi}).$

  3. (iii)

(Additivity and normalization) If $\overline{\xi}_{1}$, $\overline{\xi}_{2}$
are relative metrized complexes on $X$, then

$T(\overline{\xi}_{1}\oplus\overline{\xi}_{2})=T(\overline{\xi}_{1})+T(\overline{\xi}_{2}).$

  4. (iv)

(Projection formula) If $\overline{\xi}$ is a relative metrized complex, and
$\overline{\mathcal{G}}\in\operatorname{Ob}\operatorname{\overline{\mathbf{D}}^{b}}(Y)$,
then

$T(\overline{\xi}\otimes\overline{\mathcal{G}})=T(\overline{\xi})\bullet\operatorname{ch}(\overline{\mathcal{G}}).$

  5. (v)

(Transitivity) If $\overline{f}\colon X\to Y$, $\overline{g}\colon Y\to Z$ are
morphisms in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$, and
$(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$ and
$(\overline{g},\overline{f_{\ast}\mathcal{F}},\overline{(g\circ
f)_{\ast}\mathcal{F}})$ are relative metrized complexes, then

$T(\overline{g}\circ\overline{f})=T(\overline{g})+\overline{g}_{\flat}(T(\overline{f})).$
(7.3)

Propositions 7.4 and 7.6 below contain several anomaly and compatibility
formulas satisfied by an arbitrary theory of generalized analytic torsion
classes. They follow from properties (i)–(iii) and are analogous to those in
propositions 3.17 and 5.10, 3.19 and 5.12 respectively. The proofs are
omitted, as they are similar to those of the analogous statements referred to
hereinbefore.

######  Proposition 7.4.

Let $T$ be a theory of generalized analytic torsion classes. Let
$\overline{\xi}=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$
be a relative metrized complex.

  1. (i)

If $\overline{\mathcal{F}}^{\prime}$ is another choice of metric on
$\mathcal{F}$ and
$\overline{\xi}_{1}=(\overline{f},\overline{\mathcal{F}}^{\prime},\overline{f_{\ast}\mathcal{F}})$,
then

$T(\overline{\xi}_{1})=T(\overline{\xi})+\overline{f}_{\flat}[\operatorname{\widetilde{ch}}(\overline{\mathcal{F}}^{\prime},\overline{\mathcal{F}})].$

  2. (ii)

If $\overline{f}^{\prime}$ is another choice of hermitian structure on $f$ and
$\overline{\xi}_{2}=(\overline{f}^{\prime},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$,
then

$T(\overline{\xi}_{2})=T(\overline{\xi})+\overline{f}^{\prime}_{\flat}[\operatorname{ch}(\overline{\mathcal{F}})\bullet\widetilde{\operatorname{Td}}_{m}(\overline{f}^{\prime},\overline{f})].$
(7.5)

  3. (iii)

If $\overline{f_{\ast}\mathcal{F}}^{\prime}$ is a different choice of metric
on $f_{\ast}\mathcal{F}$, and
$\overline{\xi}_{3}=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}}^{\prime})$,
then

$T(\overline{\xi}_{3})=T(\overline{\xi})-\operatorname{\widetilde{ch}}(\overline{f_{\ast}\mathcal{F}}^{\prime},\overline{f_{\ast}\mathcal{F}}).$

######  Proposition 7.6.

Let $T$ be a theory of generalized analytic torsion classes. Let
$\overline{f}\colon X\to Y$ be a morphism in
$\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. Consider the
distinguished triangles in $\operatorname{\overline{\mathbf{D}}^{b}}(X)$ and
$\operatorname{\overline{\mathbf{D}}^{b}}(Y)$ respectively:

$(\overline{\tau}):\
\overline{\mathcal{F}}_{2}\to\overline{\mathcal{F}}_{1}\to\overline{\mathcal{F}}_{0}\to\overline{\mathcal{F}}_{2}[1],\
\text{ and }\ (\overline{f_{\ast}\tau}):\
\overline{f_{\ast}\mathcal{F}}_{2}\to\overline{f_{\ast}\mathcal{F}}_{1}\to\overline{f_{\ast}\mathcal{F}}_{0}\to\overline{f_{\ast}\mathcal{F}}_{2}[1],$

and the relative metrized complexes
$\overline{\xi}_{i}=(\overline{f},\overline{\mathcal{F}}_{i},\overline{f_{\ast}\mathcal{F}}_{i})$,
$i=0,1,2$. Then we have:

$\sum_{j=0,1,2}(-1)^{j}T(\overline{\xi}_{j})=\widetilde{\operatorname{ch}}(\overline{\pi_{\ast}\tau})-\overline{f}_{\flat}(\widetilde{\operatorname{ch}}(\overline{\tau})).$

The main result of this section is the following classification theorem.

######  Theorem 7.7.

Let $S$ be a real additive genus. Then there exists a unique theory of
generalized analytic torsion classes that agrees with $T_{S}$ when restricted
to the class of closed immersions. Moreover, if $T$ is a theory of generalized
analytic torsion classes, then there exists a real additive genus $S$ such
that $T=T_{S}$.

We will denote the theory associated to the additive genus $S$, whose
existence is guaranteed by the preceding theorem, by $T_{S}$. In particular,
there is a unique theory of generalized analytic torsion classes that agrees
with $T^{h}$ when restricted to the class of closed immersions. This theory
will be called homogeneous.

###### Proof.

We first prove the uniqueness. Let $T$ be a theory of analytic torsion classes
that agrees with $T_{S}$ for the class of closed immersions. Since the
restriction of $T$ to projective spaces, by the transitivity axiom, is
compatible with $T_{S}$, by Theorem 6.4, it also agrees with $T_{S}$. Finally,
the transitivity axiom implies that $T$ is determined by its values for closed
immersions and projective spaces.

We now prove the existence. For the moment, let $T_{S}$ be the theory of
analytic torsion classes for closed immersions and projective spaces
determined by $S$. Let $\overline{f}\colon X\to Y$ be a morphism in
$\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$, and let
$\overline{\xi}=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$
be a relative metrized complex. Since $f$ is assumed to be projective, there
is a factorization $f=\pi\circ\iota$, where $\iota\colon
X\to{\mathbb{P}}^{n}_{Y}$ is a closed immersion and
$\pi\colon{\mathbb{P}}_{Y}^{n}\to Y$ is the projection. Choose auxiliary
hermitian structures on $\iota$, $\pi$ and $\iota_{\ast}\mathcal{F}$. Then we
define

$T_{S}(\overline{\xi})=T_{S}(\overline{\pi})+\overline{\pi}_{\flat}(T_{S}(\overline{\iota}))+\overline{f}_{\flat}\left[\operatorname{ch}(\overline{\mathcal{F}})\bullet\widetilde{\operatorname{Td}}_{m}(\overline{f},\overline{\pi}\circ\overline{\iota})\right]$
(7.8)

To simplify the notations, in the sequel we will also refer to it simply by
$T(\overline{\xi})$. The anomaly formulas easily imply that this definition
does not depend on the choice of hermitian structures on $\iota$, $\pi$ and
$\iota_{\ast}\mathcal{F}$. We next show that this definition is independent of
the factorization of $f$. Let $f=\pi_{1}\circ\iota_{1}=\pi_{2}\circ\iota_{2}$
be two different factorizations, being ${\mathbb{P}}^{n_{i}}$, the target of
$\iota_{i}$, $i=1,2$. Since equation (7.8) is independent of the choice of
auxiliary hermitian structures, by [17, Lem. 5.12], we may assume that
$\overline{f}=\overline{\pi}_{1}\circ\overline{\iota}_{1}=\overline{\pi}_{2}\circ\overline{\iota}_{2}$.

We consider the commutative diagram with cartesian square

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{1}}$$\scriptstyle{\operatorname{id}_{X}}$$\textstyle{X\underset{Y}{\times}{\mathbb{P}}_{Y}^{n_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k_{1}}$$\scriptstyle{q_{1}}$$\textstyle{{\mathbb{P}}_{Y}^{n_{1}}\underset{Y}{\times}{\mathbb{P}}_{Y}^{n_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{1}}$$\scriptstyle{f}$$\textstyle{{\mathbb{P}}_{Y}^{n_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\textstyle{Y}$

where $j_{1}(x)=(x,\iota_{2}(x))$, $p_{1}$ is the first projection and $q_{1}$
and $k_{1}$ are defined by the cartesian square. The hermitian structure of
$\overline{\pi}_{2}$ induces a hermitian structure on $p_{1}$ that, in turn,
induces a hermitian structure on $q_{1}$. The hermitian structure of
$\iota_{1}$ induces a hermitian structure on $k_{1}$ and the hermitian
structure of $\iota_{2}$ induces one on $j_{1}$. We will denote the
corresponding morphisms of
$\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$ by
$\overline{p}_{1}$, $\overline{q}_{1}$, $\overline{k}_{1}$ and
$\overline{j}_{1}$. We consider also the analogous diagram obtained swapping 1
and 2. Finally, we write
$\overline{p}=\overline{\pi}_{1}\circ\overline{p}_{1}=\overline{\pi}_{2}\circ\overline{p}_{2}$
and
$\overline{j}=\overline{k}_{1}\circ\overline{j}_{1}=\overline{k}_{2}\circ\overline{j}_{2}$.
Then we have

$\displaystyle
T(\overline{\pi}_{1})+(\overline{\pi}_{1})_{\flat}(T(\overline{\iota}_{1}))$
$\displaystyle=T(\overline{\pi}_{1})+(\overline{\pi}_{1})_{\flat}(T(\overline{\iota}_{1}))+\overline{f}_{\flat}\left(T(\overline{q}_{1})+(\overline{q}_{1})_{\flat}(T(\overline{j}_{1}))\right)$
$\displaystyle=T(\overline{\pi}_{1})+(\overline{\pi}_{1})_{\flat}\big{(}T(\overline{\iota}_{1})+(\overline{\iota}_{1})_{\flat}(T(\overline{q}_{1}))\big{)}+\overline{p}_{\flat}(\overline{k}_{1})_{\flat}(T(\overline{j}_{1}))$
$\displaystyle=T(\overline{\pi}_{1})+(\overline{\pi}_{1})_{\flat}\big{(}T(\overline{p}_{1})+(\overline{p}_{1})_{\flat}(T(\overline{k}_{1}))\big{)}+\overline{p}_{\flat}(\overline{k}_{1})_{\flat}(T(\overline{j}_{1}))$
$\displaystyle=T(\overline{p})+\overline{p}_{\flat}(T(\overline{j})).$

Analogously, we obtain

$T(\overline{\pi}_{2})+(\overline{\pi}_{2})_{\flat}(T(\overline{\iota}_{2}))=T(\overline{p})+\overline{p}_{\flat}(T(\overline{j})).$

Hence $T_{S}$ is well defined for all relative metrized complexes.

It remains to prove that it satisfies the properties of a theory of analytic
torsion classes. The properties (i) to (iv) are clear. We thus focus on
property (v). Let $\overline{f}\colon X\to Y$ and $\overline{g}\colon Y\to Z$
be morphisms in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$.
We choose factorizations of $\overline{g}\circ\overline{f}$ and
$\overline{g}$:

where the hermitian structures on $\overline{p}$ and $\overline{r}$ come from
fixed hermitan structures on the tangent bundles
$T_{{\mathbb{P}}^{m}_{{\mathbb{C}}}}$ and
$T_{{\mathbb{P}}^{n}_{{\mathbb{C}}}}$, and the hermitian structures
$\overline{i}$ and $\overline{\ell}$ are obtained by using [17, Lem. 5.12]. We
define $\varphi:X\to{\mathbb{P}}^{m}_{{\mathbb{C}}}$ to be the morphism
obtained from $i$ by composing with the projection to
${\mathbb{P}}^{m}_{{\mathbb{C}}}$. Then we see that the morphism
$j:=(\varphi,f)\colon X\to{\mathbb{P}}^{m}_{Y}$ is a closed immersion. Indeed,
it is enough to realize that the composition

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\varphi,f)}$$\textstyle{{\mathbb{P}}^{m}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\operatorname{id},g)}$$\textstyle{{\mathbb{P}}^{m}_{Z}}$

agrees with the closed immersion $i$ and that $G:=(\operatorname{id},g)$ is
separated (since proper). We can thus decompose $\overline{f}$ as

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{j}}$$\scriptstyle{\overline{f}}$$\textstyle{{\mathbb{P}}^{m}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{q}}$$\textstyle{Y.}$

Again, in this factorization the hermitian structure $\overline{q}$ comes from
the previously fixed hermitian structure on
$T_{{\mathbb{P}}^{m}_{{\mathbb{C}}}}$ and the hermitian structure
$\overline{j}$ is obtained by using [17, Lem. 5.12]. Because
$\overline{g}\circ\overline{f}=\overline{p}\circ\overline{i}$ and by the very
construction of $T$ for arbitrary projective morphisms (7.8), we have

$T(\overline{g}\circ\overline{f})=T(\overline{p})+\overline{p}_{\flat}(T(\overline{i})).$
(7.9)

We proceed to work on $T(\overline{i})$. For this we write the commutative
diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\scriptstyle{i}$$\textstyle{{\mathbb{P}}^{m}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k}$$\scriptstyle{G}$$\textstyle{{\mathbb{P}}^{m}_{{\mathbb{P}}^{n}_{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{P}}^{n}_{{\mathbb{P}}^{m}_{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{{\mathbb{P}}^{m}_{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}}$$\textstyle{{\mathbb{P}}^{m}_{Z}.}$

We recall that $G=(\operatorname{id},g)$ and $k=(\operatorname{id},\ell)$.
Below, $G$, $k$ and $\pi$ will be endowed with the obvious hermitian
structures. With these choices, we observe that
$\overline{i}=\overline{G}\circ\overline{j}$ and
$\overline{G}=\overline{\pi}\circ\overline{k}$. Taking also into account the
construction of $T$ and the fact that $T=T_{S}$ is transitive for compositions
of closed immersions, we find

$T(\overline{i})=T(\overline{\pi}\circ\overline{k}\circ\overline{j})=T(\overline{\pi})+\overline{\pi}_{\flat}(T(\overline{k}))+\overline{G}_{\flat}(T(\overline{j}))=T(\overline{G})+\overline{G}_{\flat}(T(\overline{j})).$
(7.10)

Therefore, from equations (7.9), (7.10) and the identity
$\overline{p}_{\flat}\overline{G}_{\flat}=\overline{g}_{\flat}\overline{q}_{\flat}$
we derive

$T(\overline{g}\circ\overline{f})=T(\overline{p})+\overline{p}_{\flat}(T(\overline{G}))+\overline{g}_{\flat}\overline{q}_{\flat}(T(\overline{j})).$
(7.11)

We claim that

$T(\overline{p})+\overline{p}_{\flat}(T(\overline{G}))=T(\overline{g})+\overline{g}_{\flat}(T(\overline{q})).$
(7.12)

Assuming this for a while, we combine (7.11) and (7.12) into

$T(\overline{g}\circ\overline{f})=T(\overline{g})+\overline{g}_{\flat}(T(\overline{q})+\overline{q}_{\flat}(T(\overline{j})))\\\
=T(\overline{g})+\overline{g}_{\flat}(T(\overline{f})).$ (7.13)

Hence we are lead to prove (7.12). For this we construct the commutative
diagram with cartesian squares

$\textstyle{{\mathbb{P}}^{m}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\ell}}$$\scriptstyle{\overline{q}}$$\textstyle{{\mathbb{P}}^{m}_{Z}\times_{Z}{\mathbb{P}}^{n}_{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
14.22636pt\tilde{r}}$$\scriptstyle{\tilde{p}}$$\textstyle{{\mathbb{P}}^{m}_{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{p}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\ell}}$$\textstyle{{\mathbb{P}}^{n}_{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{r}}$$\textstyle{Z.}$

Observe that $\overline{G}=\tilde{r}\circ\tilde{\ell}$. Recall now Proposition
5.41 and Proposition 6.15. We then have the chain of equalities

$T(\overline{p})+\overline{p}_{\flat}(T(\overline{G}))=T(\overline{p})+\overline{p}_{\flat}(T(\tilde{r})+\tilde{r}_{\flat}(T(\tilde{\ell})))=T(\overline{r})+\overline{r}_{\flat}(T(\tilde{p})+\tilde{p}_{\flat}(T(\tilde{\ell}))\\\
=T(\overline{r})+\overline{r}_{\flat}(T(\overline{\ell})+\overline{\ell}_{\flat}(T(\overline{q})))=T(\overline{g})+\overline{g}_{\flat}(T(\overline{q})).$

This proves the claim.

The last assertion of the statement of the theorem follows from the
uniqueness. ∎

######  Theorem 7.14.

  1. (i)

Let $T$ be a theory of generalized analytic torsion classes. Then there is a
unique real additive genus $S$ such that, for any relative metrized complex
$\overline{\xi}:=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$,
we have

$T(\overline{\xi})-T^{h}(\overline{\xi})=-f_{\ast}[\operatorname{ch}(\mathcal{F})\bullet\operatorname{Td}(T_{f})\bullet
S(T_{f})\bullet\mathbf{1}_{1}].$ (7.15)

  2. (ii)

Conversely, any real additive genus $S$ defines, by means of equation (7.15),
a unique theory of generalized analytic torsion classes $T_{S}$.

###### Proof.

We prove the first item, the second being immediate. Let $S$ be the real
additive genus corresponding to $T$, provided by Theorem 7.7. Then (7.15)
holds for embedded metrized complexes. Because $T$ and $T^{h}$ are both
transitive, it suffices to show that (7.15) holds whenever
$f\colon{\mathbb{P}}^{n}_{X}\to X$ is a trivial projective bundle. Observe $T$
and $T^{h}$ satisfy the same anomaly formulas. Then, since the sheaves
$\mathcal{G}(k)$, $k=-n,\dots,0$ form a generating class for
$\operatorname{\mathbf{D}^{b}}({\mathbb{P}}^{n}_{X})$, and by the projection
formula for $T$ and $T^{h}$, we easily reduce to the case
$\overline{\xi}=\overline{\xi}(k)$. Let $t_{n,k}$, $t_{n,k}^{h}$ be the
characteristic numbers of $T$, $T^{h}$ respectively. We have to establish the
equality

$t_{n,-i}-t^{h}_{n,-i}=-\pi_{\ast}(\operatorname{ch}(\overline{{\mathcal{O}}}(-i))\operatorname{Td}(\overline{\pi})S(T_{\overline{\pi}})),\quad
i=-n,\dots,0.$ (7.16)

This is an equation of real numbers. By functoriality, this equation is
equivalent to the analogous equation in
$\oplus_{p}H^{2p-1}_{\mathcal{D}}({\mathbb{P}}^{n}_{{\mathbb{C}}},{\mathbb{R}}(p))$,
for the second projection
$p_{2}:{\mathbb{P}}^{n}_{{\mathbb{C}}}\times{\mathbb{P}}^{n}_{{\mathbb{C}}}\to{\mathbb{P}}^{n}_{{\mathbb{C}}}$
instead of $\pi$. Because the classes $\operatorname{ch}(\Lambda^{i}Q^{\vee})$
constitute a basis for
$\oplus_{p}H^{2p-1}_{\mathcal{D}}({\mathbb{P}}^{n}_{{\mathbb{C}}},{\mathbb{R}}(p))$,
(7.16) is equivalent to the equation in cohomology

$\sum_{i}(-1)^{i}(t_{n,-i}-t^{h}_{n,-i})\operatorname{ch}(\Lambda^{i}\overline{Q}^{\vee})=\\\
-p_{2\ast}(\sum_{i}(-1)^{i}\operatorname{ch}(p_{1}^{\ast}\overline{{\mathcal{O}}}(-i)\otimes\Lambda^{i}p_{2}^{\ast}\overline{Q}^{\vee})\operatorname{Td}(\overline{p}_{2})S(T_{\overline{p}_{2}})\bullet\mathbf{1}_{1}).$
(7.17)

Recalling the exact sequence (6.1), minus the right hand side of (7.17)
becomes

$p_{2\ast}(\operatorname{ch}(\overline{\Delta_{\ast}{\mathcal{O}}_{{\mathbb{P}}^{n}}})\operatorname{Td}(\overline{p}_{2})S(T_{\overline{p}_{2}})\bullet\mathbf{1}_{1})=\\\
p_{2\ast}(\Delta_{\ast}(\operatorname{ch}(\overline{{\mathcal{O}}}_{{\mathbb{P}}^{n}})\operatorname{Td}(\overline{\Delta}))\operatorname{Td}(\overline{p}_{2})S(T_{\overline{p}_{2}})\bullet\mathbf{1}_{1})=S(T_{{\mathbb{P}}^{n}})\bullet\mathbf{1}_{1}.$

On the other hand, using the compatibility condition (Definition 6.2), the
left hand side of (7.17) can be equivalently written as

$T(\overline{p}_{2},\overline{\Delta_{\ast}{\mathcal{O}}_{{\mathbb{P}}^{n}}},\overline{{\mathcal{O}}_{{\mathbb{P}}^{n}}})-T^{h}(\overline{p}_{2},\overline{\Delta_{\ast}{\mathcal{O}}_{{\mathbb{P}}^{n}}},\overline{{\mathcal{O}}_{{\mathbb{P}}^{n}}})=\\\
-p_{2\flat}(T(\Delta,\overline{{\mathcal{O}}}_{{\mathbb{P}}^{n}},\overline{\Delta_{\ast}{\mathcal{O}}_{{\mathbb{P}}^{n}}})-T^{h}(\Delta,\overline{{\mathcal{O}}}_{{\mathbb{P}}^{n}},\overline{\Delta_{\ast}{\mathcal{O}}_{{\mathbb{P}}^{n}}})).$
(7.18)

The genus $S$ is additive, so in Deligne cohomology we have the relation

$S(T_{\overline{\Delta}})=S(T_{{\mathbb{P}}^{n}})-\Delta^{\ast}S(T_{{\mathbb{P}}^{n}\times{\mathbb{P}}^{n}})=S(T_{{\mathbb{P}}^{n}})-\Delta^{\ast}p_{1}^{\ast}S(T_{{\mathbb{P}}^{n}})-\Delta^{\ast}p_{2}^{\ast}S(T_{{\mathbb{P}}^{n}})=-S(T_{{\mathbb{P}}^{n}}).$

Hence, since the statement is known for closed immersions, the right hand side
of (7.18) becomes

$p_{2\ast}(\Delta_{\ast}(\operatorname{ch}(\overline{{\mathcal{O}}_{{\mathbb{P}}^{n}}})\operatorname{Td}(T_{\overline{\Delta}})S(T_{\overline{\Delta}})\bullet\mathbf{1}_{1})\operatorname{Td}(\overline{p}_{2}))=-S(T_{{\mathbb{P}}^{n}})\bullet\mathbf{1}_{1}.$

This concludes the proof. ∎

##  8 Higher analytic torsion forms of Bismut and Köhler

We now explain the relationship between the theory of analytic torsion forms
of Bismut-Köhler [13] and the theory of generalized analytic torsion classes
developed so far.

Let $\pi\colon X\to Y$ be a smooth projective morphism (a projective
submersion) of smooth complex varieties. Let $\omega$ be a closed $(1,1)$-form
on $X$ that induces a Kähler metric on the fibers of $\pi$. Then
$(\pi,\omega)$ is called a Kähler fibration. The form $\omega$ defines a
hermitian structure on $T_{\pi}$, and we will abusively write
$\overline{\pi}=(\pi,\omega)$ for the corresponding morphism in
$\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$.

Let $\overline{F}$ be a hermitian vector bundle on $X$ such that for every
$i\geq 0$, $R^{i}\pi_{\ast}F$ is locally free. We consider on
$R^{i}\pi_{\ast}F$ the $L^{2}$ metric obtained using Hodge theory on the
fibers of $\pi$. Using [17, Def. 3.47] we obtain a hermitian structure on
$\pi_{\ast}F$, denoted by $\overline{\pi_{\ast}F}_{L^{2}}$. Then
$\overline{\xi}=(\overline{\pi},\overline{F},\overline{\pi_{\ast}F}_{L^{2}})$
is a relative metrized complex. The relative metrized complexes that arise in
this way will be said to be _Kähler_.

In the paper [13], Bismut and Köhler associate to every Kähler relative
metrized complex $\overline{\xi}$ a differential form, that we temporarily
denote by $\tau(\overline{\xi})$. Since in [13] the authors use real valued
characteristic classes, while we use characteristic classes in the Deligne
complex, we have to change the normalization of this form. To this end, if
$\tau(\overline{\xi})^{(p-1,p-1)}$ is the component of degree $(p-1,p-1)$ of
$\tau(\overline{\xi})$, then we put

$T^{BK}(\overline{\xi})^{(2p-1,p)}=\frac{1}{2}(2\pi
i)^{p-1}[\tau(\overline{\xi})^{(p-1,p-1)}]\in\widetilde{\mathcal{D}}_{D}^{2p-1}(Y,\emptyset,p).$

We recall that $[\cdot]$ converts differential forms into currents according
with the conventions in [19, §1] (compare with equation (3.20)). We define

$T^{BK}(\overline{\xi})=\sum_{p\geq 1}T^{BK}(\overline{\xi})^{(2p-1,p)}.$

The first main result of [13] is that this class satisfies the differential
equation

$d_{\mathcal{D}}T^{BK}(\overline{\xi})=\operatorname{ch}(\overline{\pi_{\ast}F}_{L^{2}})-\overline{\pi}_{\flat}[\operatorname{ch}(\overline{F})].$

Thus, $T^{BK}(\overline{\xi})$ is an example of analytic torsion class.

Let now $\omega^{\prime}$ be another closed $(1,1)$-form on $X$ that induces a
Kähler metric on the fibers of $\pi$. We denote
$\overline{\pi}^{\prime}=(\pi,\omega^{\prime})$. Let $\overline{F}^{\prime}$
be the vector bundle $F$ with another choice of metric and define
$\overline{\pi_{\ast}F}^{\prime}_{L^{2}}$ to be the object $\pi_{\ast}F$ with
the $L^{2}$ metric induced by $\omega^{\prime}$ and $\overline{F}^{\prime}$.
We write $\overline{\xi}^{\prime}$ for the Kähler relative metrized complex
$(\overline{\pi}^{\prime},\overline{F}^{\prime},\overline{\pi_{\ast}F}^{\prime}_{L^{2}})$.

The second main result of [13] is the following anomaly formula.

######  Theorem 8.1 ([13] Theorem 3.10).

The following formula holds:

$T^{BK}(\overline{\xi}^{\prime})-T^{BK}(\overline{\xi})=\operatorname{\widetilde{ch}}(\overline{\pi_{\ast}F}_{L^{2}},\overline{\pi_{\ast}F}^{\prime}_{L^{2}})+\overline{\pi}^{\prime}_{\flat}\left[\operatorname{ch}(\overline{F})\bullet\widetilde{\operatorname{Td}}_{m}(\overline{\pi}^{\prime},\overline{\pi})-\operatorname{\widetilde{ch}}(\overline{F},\overline{F}^{\prime})\right].$

In the book [3], Bismut studies the compatibility of higher analytic torsion
forms with complex immersions. Before stating his result we have to recall the
definition of the $R$-genus of Gillet and Soulé [29]. It is the additive genus
attached to the power series

$R(x)=\sum_{\begin{subarray}{c}m\text{ odd}\\\ m\geq
1\end{subarray}}\left(2\zeta^{\prime}(-m)+\left(1+\frac{1}{2}+\dots+\frac{1}{m}\right)\zeta(-m)\right)\frac{x^{m}}{m!}.$
(8.2)

Let $T_{-R/2}$ be the theory of analytic torsion classes for closed immersions
associated to $\frac{-1}{2}R$.

######  Remark 8.3.

The fact that we obtain the additive genus $-R/2$ instead of $R$ is due to two
facts. The signs comes from the minus sign in equation (3.12), while the
factor $1/2$ comes from the difference of the normalization of Green forms
used in this paper and the one used in [27]. Note however that the arithmetic
intersection numbers computed using both normalizations agree, because the
definition of arithmetic degree in [27, §3.4.3] has a factor $1/2$ while the
definition of arithmetic degree in [18, (6.24)] does not.

Consider a commutative diagram of smooth complex varieties

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\iota}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Z}$

where $f$ and $g$ are projective submersions and $\iota$ is a closed
immersion. Let $\overline{F}$ be a hermitian vector bundle on $X$ such that
the sheaves $R^{i}f_{\ast}F$ are locally free and let

$0\to\overline{E}_{n}\to\dots\to\overline{E}_{0}\to\iota_{\ast}F\to 0$

be a resolution of $\iota_{\ast}F$ by hermitian vector bundles. We assume that
for all $i,j$, $R^{i}g_{\ast}E_{j}$ is locally free. We will denote by $E$ the
complex $E_{n}\to\dots\to E_{0}$. Let $\omega^{X}$ and $\omega^{Y}$ be closed
$(1,1)$ forms that define a structure of Kähler fibration on $f$ and $g$
respectively. As before we write $\overline{f}=(f,\omega^{X})$ and
$\overline{g}=(g,\omega^{Y})$. The exact sequence

$0\longrightarrow T_{f}\longrightarrow f^{\ast}T_{g}\longrightarrow
N_{X/Y}\longrightarrow 0$

induces a hermitian structure on $N_{X/Y}$. We will denote $\overline{\iota}$
the inclusion $\iota$ with this hermitian structure. Finally we denote by
$\overline{f_{\ast}F}_{E}$ the hermitian structure on $f_{\ast}F$ induced by
the hermitian structures $\overline{g_{\ast}E_{j}}_{L^{2}}$, $j=0,\dots,n$.

Then, adapted to our language, the main result of [3] can be stated as
follows.

######  Theorem 8.4 ([3] Theorems 0.1 and 0.2).

The following equation holds in the group
$\bigoplus_{p}\widetilde{\mathcal{D}}_{D}^{2p-1}(Z,\emptyset,p)$:

$T^{BK}(\overline{f},\overline{F},\overline{f_{\ast}F}_{L_{2}})=\sum_{j=0}^{n}(-1)^{j}T^{BK}(\overline{g},\overline{E}_{j},\overline{f_{\ast}E_{j}}_{L_{2}})\\\
+\overline{g}_{\flat}(T_{-R/2}(\overline{\iota},\overline{F},\overline{E}))+\operatorname{\widetilde{ch}}(\overline{f_{\ast}F}_{E},\overline{f_{\ast}F}_{L^{2}}).$

We can particularize the previous result to the case when $F=0$. Then $E$ and
$g_{\ast}E$ are acyclic objects. The hermitian structures of
$\overline{E}_{j}$ and $\overline{g_{\ast}E_{j}}_{L^{2}}$ induce hermitian
structures on them. We denote these hermitian structures as $\overline{E}$ and
$\overline{g_{\ast}E}_{L^{2}}$.

######  Corollary 8.5.

Let $\overline{E}$ be a bounded acyclic complex of hermitian vector bundles on
$Y$ such that the direct images $R^{i}g_{\ast}E_{j}$ are locally free on $Z$.
Then

$\sum_{j=0}^{n}(-1)^{j}T^{BK}(\overline{g},\overline{E_{j}},\overline{g_{\ast}E_{j}}_{L^{2}})=\operatorname{\widetilde{ch}}(\overline{g_{\ast}E}_{L^{2}})-\overline{g}_{\flat}(\operatorname{\widetilde{ch}}(\overline{E}))$

in $\bigoplus_{p}\widetilde{\mathcal{D}}_{D}^{2p-1}(Z,\emptyset,p)$.

We will also need a particular case of functoriality and projection formula
for the higher analytic torsion forms of Bismut-Köhler proved by Rössler [43].

The relative metrized complexes $\overline{\xi}_{n}(k)$ of Notation 5.2 are
Kähler. Therefore we can apply the construction of Bismut-Köhler to them. We
denote

$t^{BK}_{n,k}=T^{BK}(\overline{\xi}_{n}(k)).$ (8.6)

By Corollary 8.5, the numbers $t^{BK}_{n,k}$ satisfy the relation (5.19).
Hence they are determined by the main characteristic numbers $t^{BK}_{n,k}$
for $-n\leq k\leq 0$.

######  Theorem 8.7 ([43] Lemma 7.15).

Let $\pi\colon{\mathbb{P}}^{n}_{X}\to X$ be a trivial projective bundle. Let
$\overline{G}$ be a hermitian vector bundle on $X$. Then

$T^{BK}(\overline{\xi}_{n}(k)\otimes\overline{G})=t^{BK}_{n,k}\bullet\operatorname{ch}(\overline{G}).$

###### Proof.

In [43] this result is proved for $k\gg 0$. Using Corollary 8.5 and the Koszul
resolution (5.18) one can extend the result to all $k\in{\mathbb{Z}}$. ∎

We have all the ingredients we need to prove the main result of this section.

######  Theorem 8.8.

Let $T_{-R/2}$ be the theory of generalized analytic torsion classes
associated to the additive genus $\frac{-1}{2}R$. Then, for every Kähler
relative metrized complex $\overline{\xi}$, we have

$T^{BK}(\overline{\xi})=T_{-R/2}(\overline{\xi}).$

In particular $T_{-R/2}$ extends the construction of Bismut-Köhler to
arbitrary projective morphisms of smooth complex varieties and arbitrary
smooth metrics.

###### Proof.

Let $\mathfrak{t}^{BK}=\\{t^{BK}_{n,k}\mid n\geq 0,-n\leq k\leq 0\\}$ and let
$T_{\mathfrak{t}^{BK}}$ be the theory of analytic torsion classes for
projective spaces associated to it. Let $\pi\colon{\mathbb{P}}^{n}_{X}\to X$
be a relative projective space and let
$\overline{\xi}=(\overline{\pi},\overline{E},\overline{\pi_{\ast}E}_{L^{2}})$
be a Kähler relative metrized complex. By choosing $d\gg 0$ we may assume that
all the coherent sheaves of the resolution $\gamma_{d}(F)$ of Corollary 4.3
are locally free. Using theorems 8.7 and 8.1, Proposition 5.10 and corollaries
5.15 and 8.5 we obtain that

$T^{BK}(\overline{\xi})=T_{\mathfrak{t}^{BK}}(\overline{\xi}).$

By Theorem 8.4, the theories $T_{\mathfrak{t}^{BK}}$ and $T_{-R/2}$ are
compatible in the sense of Definition 6.2. Therefore, $T^{BK}=T_{-R/2}$ when
restricted to projective spaces.

Finally, by factoring a smooth projective morphism as a closed immersion
followed by the projection of a relative projective space, Theorem 8.4 implies
that $T^{BK}=T_{-R/2}$ for all smooth projective morphisms. ∎

######  Remark 8.9.

  1. (i)

The construction of Bismut-Köhler applies to a wider class of varieties and
morphisms: complex analytic manifolds and proper Kähler submersions. However
for the comparison we have to restrict to smooth algebraic varieties and
smooth projective morphisms.

  2. (ii)

The results of Bismut and his coworkers are more precise. Here the class
$T^{BK}(\overline{\xi})$ is well defined up to the image of
$\operatorname{d}_{\mathcal{D}}$. In contrast, the higher analytic torsion
form of Bismut and Köhler is a well defined differential form, local on the
base and whose class modulo $\operatorname{d}_{\mathcal{D}}$ agrees with
$T^{BK}(\overline{\xi})$.

As a consequence of Theorem 8.8, we obtain the following results that,
although they should follow from the definition of higher analytic torsion
classes, we have not been able to find them explicitly in the literature.

######  Corollary 8.10.

Let $f\colon X\to Y$ be a smooth projective morphism of smooth complex
varieties, and let
$\overline{\xi}=(\overline{f},\overline{E},\overline{f_{\ast}E}_{L^{2}})$ be a
Kähler relative metrized complex.

  1. (i)

Let $g\colon Y^{\prime}\to Y$ be a morphism of smooth complex varieties. Then

$T^{BK}(g^{\ast}\overline{\xi})=g^{\ast}T^{BK}(\overline{\xi}).$

  2. (ii)

Let $\overline{G}$ be a hermitian vector bundle on $Y$. Then

$T^{BK}(\overline{\xi}\otimes\overline{G})=T^{BK}(\overline{\xi})\bullet\operatorname{ch}(\overline{G}).$

The last consequence we want to discuss generalizes results already proved by
Berthomieu-Bismut [1, Thm 3.1] and Ma [35, Thm. 0.1], [36, Thm. 0.1]. However
we note that while we stay within the algebraic category and work with
projective morphisms, these authors deal with proper Kähler holomorphic
submersions of complex manifolds. Let $\overline{g}\colon X\to Y$ and
$\overline{h}\colon Y\to Z$ be morphisms in the category
$\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$, such that the
composition $\overline{f}=\overline{h}\circ\overline{g}$ is a smooth morphism.
We choose a structure of Kähler fibration on $f$, that we denote
$\overline{f}^{\prime}$. Let $\overline{E}$ be a hermitian vector bundle on
$X$ and assume that the higher direct images $R^{i}f_{\ast}E$ are locally
free. Then we may consider the analytic torsion
$T^{BK}(\overline{f}^{\prime})$ attached to the Kähler relative metrized
complex $(\overline{f}^{\prime},\overline{E},\overline{f_{\ast}E}_{L^{2}})$.
Also, we choose an auxiliary hermitian structure on $g_{\ast}E$. We can
consider the torsion classes $T_{R/2}(\overline{g})$ and
$T_{R/2}(\overline{h})$ of the relative metrized complexes
$(\overline{g},\overline{E},\overline{g_{\ast}E})$ and
$(\overline{h},\overline{g_{\ast}E},\overline{f_{\ast}E}_{L^{2}})$.

We make the following additional assumption in some particular situations:

(*)

The morphisms $g$ and $h$ are Kähler fibrations, the higher direct images
$R^{i}g_{\ast}E$ and $R^{j}h_{\ast}R^{i}g_{\ast}E$ are locally free and the
auxiliary hermitian structure on $g_{\ast}E$ is the $L^{2}$ hermitian
structure.

When the hypothesis (*) is satisfied we denote by
$\overline{h_{\ast}g_{\ast}E}_{L^{2}}$ the $L^{2}$ hermitian structure
attached to the Kähler structure on $\overline{h}$ and the $L^{2}$ metric on
$\overline{g_{\ast}E}_{L^{2}}$. Observe that this last structure may differ
from the $L^{2}$ structure on $\overline{f_{\ast}E}_{L^{2}}$. In this
situation we can consider the torsion classes $T^{BK}(\overline{g})$ and
$T^{BK}(\overline{h}^{\prime})$ attached to
$(\overline{g},\overline{E},\overline{g_{\ast}E}_{L^{2}})$ and
$(\overline{h},\overline{g_{\ast}E}_{L^{2}},\overline{h_{\ast}g_{\ast}E}_{L^{2}})$.
By Proposition 5.10, we have the relation

$T^{BK}(\overline{h}^{\prime})=T_{R/2}(\overline{h})-\widetilde{\operatorname{ch}}(\overline{h_{\ast}g_{\ast}E}_{L^{2}},\overline{f_{\ast}E}_{L^{2}}).$

The properties of the generalized analytic torsion classes imply immediately:

######  Corollary 8.11.

Under these assumptions, we have the equality

$T^{BK}(\overline{f}^{\prime})=T_{-R/2}(\overline{h})+\overline{h}_{\flat}(T_{-R/2}(\overline{g}))+\overline{f}^{\prime}_{\flat}(\operatorname{ch}(\overline{E})\bullet\widetilde{\operatorname{Td}}_{m}(\overline{f}^{\prime},\overline{f})).$

If in addition the hypothesis (*) is satisfied, then we have

$T^{BK}(\overline{f}^{\prime})=T^{BK}(\overline{h}^{\prime})+\overline{h}_{\flat}(T^{BK}(\overline{g}))+\overline{f}^{\prime}_{\flat}(\operatorname{ch}(\overline{E})\bullet\widetilde{\operatorname{Td}}_{m}(\overline{f}^{\prime},\overline{f}))+\widetilde{\operatorname{ch}}(\overline{h_{\ast}g_{\ast}E}_{L^{2}},\overline{f_{\ast}E}_{L^{2}}).$

Since $T_{-R/2}$ extends the theory of analytic torsion classes $T^{BK}$, we
will denote $T_{-R/2}$ by $T^{BK}$ for arbitrary relative metrized complexes.

##  9 Grothendieck duality and analytic torsion

We will study now the compatibility of the analytic torsion with Grothendieck
duality.

######  Definition 9.1.

Let
$\overline{\mathcal{F}}=(\mathcal{F},\overline{E}\dashrightarrow\mathcal{F})$
be an object of $\operatorname{\overline{\mathbf{D}}^{b}}(X)$. Then the _rank_
of $\overline{\mathcal{F}}$ is

$\operatorname{rk}(\overline{\mathcal{F}})=\sum_{i}(-1)^{i}\dim(E_{i}).$

This is just the Euler characteristic of the complex. _The determinant_ of
$\overline{\mathcal{F}}$ is the complex

$\det(\overline{\mathcal{F}})=\bigotimes_{i}\left(\Lambda^{\dim
E^{i}}\overline{E}^{i}\right)^{(-1)^{i}}[-\operatorname{rk}(\overline{\mathcal{F}})].$

It consists of a single line bundle concentrated in degree
$\operatorname{rk}(\overline{\mathcal{F}})$.

######  Definition 9.2.

Let $\overline{f}\colon X\to Y$ be a morphism in
$\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$ of relative
dimension $e$. The _metrized dualizing complex_ , is the complex given by

$\boldsymbol{\omega}_{\overline{f}}=(\det T_{\overline{f}})^{\vee}.$

This complex is concentrated in degree $-e$. The underlying object of
$\operatorname{\mathbf{D}^{b}}(X)$ will be denoted by
$\boldsymbol{\omega}_{f}$. If we are interested in the dualizing sheaf as a
sheaf and not as an element of $\operatorname{\mathbf{D}^{b}}(X)$ we will
denote it by $\omega_{f}$ or $\omega_{X/Y}$. Finally, if
$Y=\operatorname{Spec}{\mathbb{C}}$, we will denote $\boldsymbol{\omega}_{f}$
(respectively $\omega_{f}$) by $\boldsymbol{\omega}_{X}$ (respectively
$\omega_{X}$).

######  Definition 9.3.

Let $\mathcal{D}^{\ast}(\ast)$ be the Deligne complex associated to a
Dolbeault complex. The _sign operator_ is

$\sigma\colon\mathcal{D}^{\ast}(\ast)\longrightarrow\mathcal{D}^{\ast}(\ast),\quad\omega\in\mathcal{D}^{n}(p)\mapsto\sigma(\omega)=(-1)^{p}\omega.$

The sign operator satisfies the following compatibilities.

######  Proposition 9.4.

  1. (i)

Let $(\mathcal{D}^{\ast}(\ast),\operatorname{d}_{\mathcal{D}})$ be a Deligne
algebra. Then the sign operator is a morphism of differential algebras. That
is

$\operatorname{d}_{\mathcal{D}}\circ\,\sigma=\sigma\circ\operatorname{d}_{\mathcal{D}},\quad\sigma(\omega\bullet\eta)=\sigma(\omega)\bullet\sigma(\eta).$

  2. (ii)

Let $\overline{\mathcal{F}}$ be an object of
$\operatorname{\overline{\mathbf{D}}^{b}}(X)$. Then the following equalities
are satisfied

$\displaystyle\sigma\operatorname{ch}(\overline{\mathcal{F}})$
$\displaystyle=\operatorname{ch}(\overline{\mathcal{F}}^{\vee}),$ (9.5)
$\displaystyle\sigma\operatorname{ch}(\det(\overline{\mathcal{F}}))$
$\displaystyle=\operatorname{ch}(\det(\overline{\mathcal{F}})^{\vee})=\operatorname{ch}(\det(\overline{\mathcal{F}}))^{-1},$
(9.6) $\displaystyle\sigma\operatorname{Td}(\overline{\mathcal{F}})$
$\displaystyle=(-1)^{\operatorname{rk}(\overline{\mathcal{F}})}\operatorname{Td}(\overline{\mathcal{F}})\bullet\operatorname{ch}(\det(\overline{\mathcal{F}})^{\vee}.$
(9.7)

###### Proof.

The first statement is clear because if $\omega\in\mathcal{D}^{n}(p)$ and
$\eta\in\mathcal{D}^{m}(q)$ then
$\operatorname{d}_{\mathcal{D}}\omega\in\mathcal{D}^{n+1}(p)$ and
$\omega\bullet\eta\in\mathcal{D}^{n+m}(p+q)$.

For the second statement, let $\overline{E}\dashrightarrow\mathcal{F}$ be the
hermitian structure of $\mathcal{F}$. Write

$\overline{E}^{+}=\bigoplus_{i\text{
even}}\overline{E}^{i},\qquad\overline{E}^{-}=\bigoplus_{i\text{
odd}}\overline{E}^{i}.$

Since this statement is local on $X$, we can chose trivializations of
$\overline{E}^{+}$ and $\overline{E}^{-}$ over an open subset $U$. Let $H^{+}$
and $H^{-}$ be the matrices of the hermitian metrics on $\overline{E}^{+}$ and
$\overline{E}^{-}$. The curvature matrices of $\overline{E}^{+}$ and
$\overline{E}^{-}$, whose entries are elements of $\mathcal{D}^{2}(U,1)$, are

$K^{\pm}=K^{\pm}(\overline{\mathcal{F}})=-\overline{\partial}(H^{\pm})^{-1}\partial
H^{\pm}.$

The characteristic forms can be computed from the curvature matrix:

$\displaystyle\operatorname{ch}(\overline{\mathcal{F}})$
$\displaystyle=\operatorname{tr}(\exp(K^{+}))-\operatorname{tr}(\exp(K^{-})),$
$\displaystyle\operatorname{ch}(\det(\overline{\mathcal{F}}))$
$\displaystyle=(-1)^{\operatorname{rk}(\overline{\mathcal{F}})}\det(\exp(K^{+}))\bullet\det(\exp(K^{-}))^{-1},$
$\displaystyle\operatorname{Td}(\overline{\mathcal{F}})$
$\displaystyle=\det\left(\frac{K^{+}}{1-\exp(-K^{+})}\right)\bullet\det\left(\frac{K^{-}}{1-\exp(-K^{-})}\right)^{-1}.$

The sign in the second equation comes from the fact that
$\det(\overline{\mathcal{F}})$ is concentrated in degree
$\operatorname{rk}(\overline{\mathcal{F}})$. Therefore, since
$\sigma(K^{\pm})=-K^{\pm}=K^{\pm}(\overline{\mathcal{F}}^{\vee})$, we have

$\displaystyle\sigma\operatorname{ch}(\overline{\mathcal{F}})$
$\displaystyle=\sigma\operatorname{tr}(\exp(K^{+}))-\sigma\operatorname{tr}(\exp(K^{-}))$
$\displaystyle=\operatorname{tr}(\exp(K^{+}(\overline{\mathcal{F}}^{\vee})))-\operatorname{tr}(\exp(K^{-}(\overline{\mathcal{F}}^{\vee})))=\operatorname{ch}(\overline{\mathcal{F}}^{\vee}),$
$\displaystyle\sigma\operatorname{ch}(\det(\overline{\mathcal{F}}))$
$\displaystyle=\det(\exp(-K^{+}))\bullet\det(\exp(-K^{-}))^{-1}=\operatorname{ch}(\det(\overline{\mathcal{F}}))^{-1},$
$\displaystyle\sigma\operatorname{Td}(\overline{\mathcal{F}})$
$\displaystyle=\det\left(\frac{-K^{+}}{1-\exp(K^{+})}\right)\bullet\det\left(\frac{-K^{-}}{1-\exp(K^{-})}\right)^{-1}$
$\displaystyle=\det\left(\frac{K^{+}}{1-\exp(-K^{+})}\right)\bullet\det(\exp(-K^{+}))$
$\displaystyle\phantom{AAA}\bullet\det\left(\frac{K^{-}}{1-\exp(-K^{-})}\right)^{-1}\bullet\det(\exp(-K^{-}))^{-1}$
$\displaystyle=\operatorname{Td}(\overline{\mathcal{F}})\bullet\operatorname{ch}(\det(\overline{\mathcal{F}}))^{-1}.$

∎

######  Corollary 9.8.

Let $[\overline{E}]\in\operatorname{\mathbf{KA}}(X)$. Then
$\widetilde{\operatorname{ch}}(\overline{E}^{\vee})=\sigma\widetilde{\operatorname{ch}}(\overline{E}).$

###### Proof.

Due to Proposition 9.4, the assignment sending $[\overline{E}]$ to
$\sigma\widetilde{\operatorname{ch}}(\overline{E})$ satisfies the
characterizing properties of $\widetilde{\operatorname{ch}}$. ∎

In the particular case of a projective morphism between smooth complex
varieties or, more generally, smooth varieties over a field, Grothendieck
duality takes a very simple form (see for instance [32, §3.4] and the
references therein). If $\mathcal{F}$ is an object of
$\operatorname{\mathbf{D}^{b}}(X)$ and $f\colon X\to Y$ is a projective
morphism of smooth complex varieties, then there is a natural functorial
isomorphism

$f_{\ast}(\mathcal{F}^{\vee}\otimes\boldsymbol{\omega}_{f})\cong(f_{\ast}\mathcal{F})^{\vee}.$
(9.9)

The compatibility between analytic torsion and Grothendieck duality is given
by the following result.

######  Theorem Definition 9.10.

Let $T$ be a theory of generalized analytic torsion classes. Then the
assignment that, to a relative metrized complex
$\overline{\xi}=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$,
associates the class

$T^{\vee}(\overline{\xi})=\sigma
T(\overline{f},\overline{\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}},\overline{f_{\ast}\mathcal{F}}^{\vee})$

is a theory of generalized analytic torsion classes that we call the _theory
dual to_ $T$.

###### Proof.

We have to show that, if $T$ satisfies the conditions of Definition 7.1, then
the same is true for $T^{\vee}$. We start with the differential equation. Let
$e$ be the relative dimension of $f$.

$\displaystyle\operatorname{d}_{\mathcal{D}}T^{\vee}(\overline{\xi})$
$\displaystyle=\operatorname{d}_{\mathcal{D}}\sigma
T(\overline{f},\overline{\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}},\overline{f_{\ast}\mathcal{F}}^{\vee})$
$\displaystyle=\sigma\operatorname{d}_{\mathcal{D}}T(\overline{f},\overline{\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}},\overline{f_{\ast}\mathcal{F}}^{\vee})$
$\displaystyle=\sigma\operatorname{ch}(\overline{f_{\ast}\mathcal{F}}^{\vee})-\sigma
f_{\ast}\left[\operatorname{ch}(\overline{\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}})\bullet\operatorname{Td}(\overline{f})\right]$
$\displaystyle=\operatorname{ch}(\overline{f_{\ast}\mathcal{F}})-(-1)^{e}f_{\ast}\left[\sigma\operatorname{ch}(\overline{\mathcal{F}}^{\vee})\bullet\sigma(\operatorname{ch}(\det(T_{\overline{f}})^{\vee})\bullet\operatorname{Td}(T_{\overline{f}}))\right]$
$\displaystyle=\operatorname{ch}(\overline{f_{\ast}\mathcal{F}})-f_{\ast}\left[\operatorname{ch}(\overline{\mathcal{F}})\bullet\operatorname{Td}(\overline{f})\right]$

The functoriality and the additivity are clear. We next check the projection
formula. Let $\overline{\mathcal{G}}$ be an object of
$\operatorname{\overline{\mathbf{D}}^{b}}(Y)$. Then

$T^{\vee}(\overline{\xi}\otimes\overline{\mathcal{G}})=\sigma
T(\overline{f},\overline{\mathcal{F}}^{\vee}\otimes
f^{\ast}\overline{\mathcal{G}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}},\overline{f_{\ast}\mathcal{F}}^{\vee}\otimes\overline{\mathcal{G}}^{\vee})\\\
=\sigma\left(T(\overline{f},\overline{\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}},\overline{f_{\ast}\mathcal{F}}^{\vee})\bullet\operatorname{ch}(\overline{\mathcal{G}}^{\vee})\right)=T^{\vee}(\overline{\xi})\bullet\operatorname{ch}(\overline{\mathcal{G}}).$

Finally we check the transitivity. Let $\overline{g}\colon Y\to Z$ be another
morphism in $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. By
the definition of $\overline{g}\circ\overline{f}$ we have
$\boldsymbol{\omega}_{\overline{g}\circ\overline{f}}=f^{\ast}\boldsymbol{\omega}_{\overline{g}}\otimes\boldsymbol{\omega}_{\overline{f}}.$
Therefore,

$f_{\ast}\left(\mathcal{F}^{\vee}\otimes\boldsymbol{\omega}_{g\circ
f}\right)=f_{\ast}\left(\mathcal{F}^{\vee}\otimes
f^{\ast}\boldsymbol{\omega}_{g}\otimes\boldsymbol{\omega}_{f})\right)\\\
=f_{\ast}\left(\mathcal{F}^{\vee}\otimes\boldsymbol{\omega}_{f})\right)\otimes\boldsymbol{\omega}_{g}=(f_{\ast}\mathcal{F})^{\vee}\otimes\boldsymbol{\omega}_{g}.$

On $f_{\ast}\left(\mathcal{F}^{\vee}\otimes\boldsymbol{\omega}_{g\circ
f}\right)$ we put the hermitian structure of
$\overline{f_{\ast}\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{g}}$.
Then we have

$\displaystyle T^{\vee}(\overline{g}\circ\overline{f})$ $\displaystyle=\sigma
T(\overline{g}\circ\overline{f},\overline{\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{g}\circ\overline{f}},\overline{(g\circ
f)_{\ast}\mathcal{F}}^{\vee})$ $\displaystyle=\sigma
T(\overline{g},\overline{f_{\ast}\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{g}},\overline{(g\circ
f)_{\ast}\mathcal{F}}^{\vee})$
$\displaystyle\phantom{AAA}+\sigma\overline{g}_{\flat}T(\overline{f},\overline{\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}}\otimes
f^{\ast}\boldsymbol{\omega}_{\overline{g}},\overline{f_{\ast}\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{g}})$
$\displaystyle=T^{\vee}(\overline{g},\overline{f_{\ast}\mathcal{F}},\overline{(g\circ
f)_{\ast}\mathcal{F}})$ $\displaystyle\phantom{AAA}+\sigma
g_{\ast}(T(\overline{f},\overline{\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}},\overline{f_{\ast}\mathcal{F}}^{\vee})\bullet\operatorname{ch}(\boldsymbol{\omega}_{\overline{g}})\bullet\operatorname{Td}(\overline{g}))$
$\displaystyle=T^{\vee}(\overline{g})+\overline{g}_{\flat}T^{\vee}(\overline{f}).$

Therefore, $T^{\vee}$ satisfies also the transitivity property. Hence it is a
generalized theory of analytic torsion classes. ∎

######  Definition 9.11.

A theory of generalized analytic torsion classes $T$ is called _self-dual_
when $T^{\vee}=T$.

We want to characterize the self-dual theories of generalized analytic torsion
classes.

######  Theorem 9.12.

The homogeneous theory of generalized analytic torsion classes is self-dual.

###### Proof.

By the uniqueness of the homogeneous theory, it is enough to prove that, if
$T$ is homogeneous then $T^{\vee}$ is homogeneous. Let $X$ be a smooth complex
variety and let $\overline{N}$ be a hermitian vector bundle of rank $r$ on
$X$. Put $P=\mathbb{P}(N\oplus 1)$ and let $s\colon X\to P$ be the zero
section and $\pi\colon P\to X$ the projection. Let $\overline{Q}$ be the
tautological quotient bundle with the induced metric and $\overline{K}(s)$ the
Koszul resolution associated to the section $s$. Since the normal bundle
$N_{X/P}$ can be identified with $N$, on the map $s$ we can consider the
hermitian structure given by the hermitian metric on $N$. Then
$\det\overline{Q}$ is a complex concentrated in degree $r$. Moreover

$s^{\ast}\det\overline{Q}=\det\overline{N}=\boldsymbol{\omega}_{\overline{s}}.$

The Koszul resolution satisfies the duality property
$\overline{K}(s)^{\vee}=\overline{K}(s)\otimes\det\overline{Q}.$ The theory
$T$ is homogeneous if and only if the class

$T(\overline{s},\overline{\mathcal{O}}_{X},\overline{K}(s))\bullet\operatorname{Td}(\overline{Q})$

is homogeneous of bidegree $(2r-1,r)$ in the Deligne complex. Then

$\displaystyle
T^{\vee}(\overline{s},\overline{\mathcal{O}}_{X},\overline{K}(s))\bullet\operatorname{Td}(\overline{Q})$
$\displaystyle=\sigma
T(\overline{s},\boldsymbol{\omega}_{\overline{s}},\overline{K}(s)^{\vee})\bullet\operatorname{Td}(\overline{Q})$
$\displaystyle=\sigma
T(\overline{s},s^{\ast}\det\overline{Q},\overline{K}(s)\otimes\det\overline{Q})\bullet\operatorname{Td}(\overline{Q})$
$\displaystyle=\sigma(T(\overline{s},\overline{\mathcal{O}}_{X},\overline{K}(s))\bullet\operatorname{ch}(\det\overline{Q}))\bullet\operatorname{Td}(\overline{Q})$
$\displaystyle=\sigma
T(\overline{s},\overline{\mathcal{O}}_{X},\overline{K}(s))\bullet\operatorname{ch}(\det\overline{Q}^{\vee})\bullet\operatorname{Td}(\overline{Q})$
$\displaystyle=(-1)^{r}\sigma(T(\overline{s},\overline{\mathcal{O}}_{X},\overline{K}(s))\bullet\operatorname{Td}(\overline{Q}))$

is homogeneous of bidegree $(2r-1,r)$ in the Deligne complex. ∎

######  Proposition 9.13.

Let

$S(x)=\sum_{n=0}^{\infty}a_{n}x^{n}\in{\mathbb{R}}[[x]]$

be a power series in one variable with real coefficients. Denote by $S$ the
corresponding real additive genus and by $T_{S}$ the associated theory of
analytic torsion classes. Then the dual theory $T^{\vee}_{S}$ has
corresponding real additive genus $S^{\sigma}(x):=-S(-x)$.

###### Proof.

Let
$\overline{\xi}=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})$
be a relative metrized complex. If $e$ is the relative dimension of $f$, then
we have $\sigma f_{\ast}=(-1)^{e}f_{\ast}\sigma$. Then the proposition readily
follows from the definition of $T^{\vee}_{S}$, the self-duality of $T^{h}$ and
Proposition 9.4. ∎

We can now characterize the self-dual theories of analytic torsion classes.

######  Corollary 9.14.

The theory of analytic torsion classes $T_{S}$ attached to the real additive
genus $S(x)=\sum_{n\geq 0}a_{n}x^{n}$ is self-dual if and only if $a_{n}=0$
for $n$ even.

###### Proof.

By the proposition, $T_{S}^{\vee}=T_{S^{\sigma}}$, hence $T$ is self-dual if,
and only if, $S^{\sigma}=S$. The corollary follows. ∎

In particular we recover the following fact, which is well known if we
restrict to Kähler relative metrized complexes.

######  Corollary 9.15.

The theory of analytic torsion classes of Bismut-Köhler $T^{BK}$ is self-dual.

###### Proof.

We just remark that the even coefficients of the $R$-genus vanish (8.2). ∎

We now elaborate on an intimate relation between self-duality phenomena and
the analytic torsion of de Rham complexes. Let $f\colon X\to Y$ be a smooth
projective morphism of smooth algebraic varieties, of relative dimension $e$.
Let $\overline{T}_{X/Y}$ denote the vertical tangent bundle, endowed with a
hermitian metric. Write $\overline{f}$ for the corresponding morphism in
$\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. On the locally
free sheaves $\Omega_{X/Y}^{p}=\Lambda^{p}\Omega_{X/Y}$ we put the induced
hermitian structures. The metrized de Rham complex is

$0\to\overline{{\mathcal{O}}}_{X}\overset{0}{\to}\overline{\Omega}_{X/Y}\overset{0}{\to}\overline{\Omega}_{X/Y}^{2}\overset{0}{\to}\dots\overset{0}{\to}\overline{\Omega}_{X/Y}^{e}\to
0$

with 0 differentials. In fact, we are really considering the de Rham graded
sheaf and converting it into a complex in a trivial way. We refer to the
corresponding object of $\operatorname{\overline{\mathbf{D}}^{b}}(X)$ by
$\overline{\Omega}_{X/Y}^{\bullet}$ ([17, Def. 3.37]). The individual terms
$\overline{\Omega}_{X/Y}^{p}$ will be considered as complexes concentrated in
degree $p$. We then obviously have:

######  Lemma 9.16.

The objects
$(\overline{\Omega}_{X/Y}^{\bullet})^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}}$
and $\overline{\Omega}_{X/Y}^{\bullet}[2e]$ are tightly isomorphic.

For every $p$, $q$, the cohomology sheaf $R^{q}f_{\ast}\Omega_{X/Y}^{p}$ is
locally free, because the Hodge numbers $h^{p,q}$ of the fibers of $f$ (which
are projective, hence Kähler) are known to be locally constant. Every stalk of
this sheaf is endowed with the usual $L^{2}$ metric of Hodge theory. This
family of $L^{2}$ metrics on $R^{q}f_{\ast}\Omega_{X/Y}^{p}$ glue into a
smooth metric. Because the Hodge star operators $\ast$ act by isometries, it
is easily shown that Serre duality becomes an isometry for the $L^{2}$
structures: the isomorphism

$(R^{q}f_{\ast}\Omega_{X/Y}^{p})^{\vee}\overset{\sim}{\longrightarrow}R^{e-q}f_{\ast}((\Omega_{X/Y}^{p})^{\vee}\otimes\boldsymbol{\omega}_{f})=R^{e-q}f_{\ast}\Omega_{X/Y}^{e-p}$

preserves the $L^{2}$ hermitian structures. For every $p$, let
$\overline{f_{\ast}\Omega_{X/Y}^{p}}$ denote the object of
$\operatorname{\overline{\mathbf{D}}^{b}}(Y)$ with the metric induced by the
$L^{2}$ metrics on its cohomology pieces ([17, Def. 3.47]). Here $f_{\ast}$
stands for the derived direct image. By [17, Prop. 3.48], Grothendieck duality

$(f_{\ast}\Omega_{X/Y}^{p})^{\vee}\overset{\sim}{\longrightarrow}f_{\ast}\Omega_{X/Y}^{e-p}[2e]$

is a tight isomorphism. Finally, let
$[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}]$ be the object of
$\operatorname{\overline{\mathbf{D}}^{b}}(Y)$ provided by [17, Def. 3.39]. The
next lemma follows easily from the construction of [17, Def. 3.39].

######  Lemma 9.17.

Grothendieck duality defines a tight isomorphism in
$\operatorname{\overline{\mathbf{D}}^{b}}(Y)$

$[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}]^{\vee}\cong[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}][2e].$

######  Theorem 9.18.

Let $T$ be a theory of analytic torsion classes. The following assertions are
equivalent:

  1. (i)

the theory $T$ is self-dual;

  2. (ii)

for every $f$, $\overline{T}_{f}$, $\overline{\Omega}_{X/Y}^{\bullet}$ and
$[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}]$ as above and for every odd
integer $p\geq 1$, the part of bidegree $(2p-1,p)$ (in the Deligne complex) of
$T(\overline{f},\overline{\Omega}_{X/Y}^{\bullet},[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}])$
vanishes.

###### Proof.

Assume first of all that $T$ is self-dual. We apply the definition of
$T^{\vee}$, the self-duality assumption and lemmas 9.16 and 9.17. We find the
equality

$\begin{split}T(\overline{f},\overline{\Omega}_{X/Y}^{\bullet},[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}])&=\sigma
T(\overline{f},\overline{\Omega}_{X/Y}^{\bullet}[2e],[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}][2e])\\\
&=(-1)^{2e}\sigma
T(\overline{f},\overline{\Omega}_{X/Y}^{\bullet},[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}])\\\
&=\sigma
T(\overline{f},\overline{\Omega}_{X/Y}^{\bullet},[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}]).\end{split}$

The sign operator $\sigma$ changes the sign of the components of bidegree
$(2p-1,p)$ for odd $p$. Hence
$T(\overline{f},\overline{\Omega}_{X/Y}^{\bullet},[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}])^{(2p-1,p)}$
vanishes for $p\geq 1$ odd.

For the converse implication, let $S(x)=\sum_{n\geq 0}a_{n}x^{n}$ be the real
additive genus attached to $T$ via Theorem 7.14. By Corollary 9.14, we have to
show that the coefficients $a_{n}$ with $n$ even vanish. Let us look at a
smooth morphism $f\colon X\to Y$ of relative dimension $1$, with an arbitrary
metric on $T_{f}$. Then, developing the power series of $\operatorname{ch}$
and $\operatorname{Td}$ and taking into account that
$\Omega^{1}_{X/Y}=T_{f}^{\vee}=\omega_{X/Y}$, we compute

$f_{\ast}[\operatorname{ch}(\Omega_{X/Y}^{\bullet})\operatorname{Td}(T_{f})S(T_{f})\bullet\mathbf{1}_{1}]=\sum_{n\geq
0}(-1)^{n+1}a_{n}f_{\ast}[c_{1}(\omega_{X/Y})^{n+1}\bullet\mathbf{1}_{1}].$

Therefore, for $p\geq 1$ odd, we have

$(-1)^{p}a_{p-1}f_{\ast}[c_{1}(\omega_{X/Y})^{p}\bullet\mathbf{1}_{1}]=\\\
(T(\overline{f},\overline{\Omega_{X/Y}^{\bullet}},[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}])-T^{h}(\overline{f},\overline{\Omega_{X/Y}^{\bullet}},[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}]))^{(2p-1,p,p)}=0.$
(9.19)

Hence it is enough that for every odd integer $p\geq 1$, we find a relative
curve $f\colon X\to Y$ such that $f_{\ast}(c_{1}(\omega_{X/Y})^{p})\neq 0$ in
the cohomology group $H^{2p}(Y,{\mathbb{C}})$. Let $d=p-1$ and choose $Y$ to
be a smooth projective variety of dimension $d$. Let $L$ be an ample line
bundle on $Y$ and take $X={\mathbb{P}}(L\oplus{\mathcal{O}}_{Y})$. Consider
the tautological exact sequence

$0\longrightarrow{\mathcal{O}}(-1)\longrightarrow
f^{\ast}(L\oplus{\mathcal{O}}_{Y})\longrightarrow Q\longrightarrow 0.$

We easily derive the relations

$\displaystyle\pi^{\ast}c_{1}(L)=c_{1}(Q)-c_{1}({\mathcal{O}}(1))$ (9.20)
$\displaystyle c_{1}({\mathcal{O}}(-1))c_{1}(Q)=0.$ (9.21)

Moreover we have

$c_{1}(\omega_{X/Y})=-c_{1}(Q)-c_{1}({\mathcal{O}}(1)).$ (9.22)

From (9.20)–(9.22) and because $d=p-1$ is even, we compute

$c_{1}(\omega_{X/Y})^{d}=c_{1}(Q)^{d}+c_{1}({\mathcal{O}}(1))^{d}=\pi^{\ast}c_{1}(L)^{d}.$

Therefore we find

$c_{1}(\omega_{X/Y})^{p}=\pi^{\ast}c_{1}(L)^{d}c_{1}(\omega_{X/Y}).$ (9.23)

Finally, $f$ is a fibration in curves of genus $0$, hence
$f_{\ast}(c_{1}(\omega_{X/Y}))=-2$. We infer that (9.23) leads to

$f_{\ast}(c_{1}(\omega_{X/Y})^{p})=-2c_{1}(L)^{d}.$

This class does not vanish, since $Y$ is projective of dimension $d$ and $L$
is ample. ∎

We end with a characterization of the theory of analytic torsion classes of
Bismut-Köhler.

######  Theorem 9.24.

The theory of analytic torsion classes of Bismut-Köhler $T^{BK}$ is the unique
theory of generalized analytic torsion classes such that, for every
$\overline{f}\colon X\to Y$, a Kähler fibration in
$\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$, we have the
vanishing

$T^{BK}(\overline{f},\overline{\Omega}_{X/Y}^{\bullet},[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}])=0.$

###### Proof.

That the theory $T^{BK}$ vanishes for de Rham complexes of Kähler fibrations
is a theorem of Bismut [5]. For the uniqueness, let $T$ be a theory of
generalized analytic torsion classes vanishing on de Rham complexes of Kähler
fibrations. Denote by $S(x)=\sum_{k\geq 0}a_{k}x^{k}$ its corresponding genus.
If $\overline{f}$ is a relative curve with a structure of Kähler fibration,
then by Theorem 7.14

$T^{h}(\overline{f},\overline{\Omega}_{X/Y}^{\bullet},[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}])=\sum_{k\geq
0}(-1)^{k}a_{k}f_{\ast}[c_{1}(\omega_{X/Y})^{k+1}\bullet\mathbf{1}_{1}].$
(9.25)

It is enough to find, for every $k\geq 0$, a relative curve $f$ such that
$f_{\ast}(c_{1}(\omega_{X/Y})^{k+1})$ does not vanish. The elementary
construction in the proof of Theorem 9.18 works whenever $k$ is even, but one
easily sees it fails for $k$ odd. Fortunately, there is an alternative
argument. Let $g\geq 2$ and $n\geq 3$ be integers. Consider the fine moduli
scheme of smooth curves of genus $g$ with a Jacobi structure of level $n$ [21,
Def. 5.4], to be denoted $\mathcal{M}_{g}^{n}$. Let
$\pi\colon\mathcal{C}_{g}^{n}\to\mathcal{M}_{g}^{n}$ be the universal curve.
An example of Kähler fibration structure on $\pi$ is provided by Teichmüller
theory (see for instance [48, Sec. 5]). By [22, Thm. 1], the tautological
class $\kappa_{g-2}:=\pi_{\ast}(c_{1}(\omega_{\pi})^{g-1})\in
H^{2(g-2)}(\mathcal{M}_{g}^{n},{\mathbb{C}})$ does not vanish. Taking $g=k+2$
and $f=\pi$, we conclude the proof of the theorem. ∎

We note that in the previous theorem, the existence is provided by Bismut’s
theorem. It would be interesting to have a proof of the existence of a theory
satisfying the condition of Theorem 9.24 from the axiomatic point of view.

##  10 Direct images of hermitian structures

As another application of a theory of generalized analytic torsion classes, we
construct direct images of metrized complexes. It turns out that the natural
place to define direct images is not the category
$\operatorname{\overline{\mathbf{D}}^{b}}(\cdot)$ but a new category
$\operatorname{\mathbf{\widehat{D}}^{b}}(\cdot)$ that is the analogue to the
arithmetic $K$-theory of Gillet and Soulé [28].

Let $X$ be a smooth complex variety. The fibers of the forgetful functor
$\operatorname{\overline{\mathbf{D}}^{b}}(X)\to\operatorname{\mathbf{D}^{b}}(X)$
have a structure of $\operatorname{\mathbf{KA}}(X)$-torsor, for the action of
$\operatorname{\mathbf{KA}}(X)$ by translation of the hermitian structures
([17, Thm. 3.13]). At the same time, the group $\operatorname{\mathbf{KA}}(X)$
acts on the group $\oplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,p)$ by
translation, via the Bott-Chern character $\widetilde{\operatorname{ch}}$
([17, Prop. 4.6]). Observe that all Bott-Chern classes live in these groups,
as for the analytic torsion classes. It is thus natural to build a product
category over $\operatorname{\mathbf{KA}}(X)$.

######  Definition 10.1.

Let $S\subset T^{\ast}X_{0}$ be a closed conical subset. We define

$\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)=\operatorname{\overline{\mathbf{D}}^{b}}(X)\times_{\operatorname{\mathbf{KA}}(X)}\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,S,p)$

to be the category whose objects are equivalence classes
$[\overline{\mathcal{F}},\eta]$ of pairs $(\overline{\mathcal{F}},\eta)$
belonging to
$\operatorname{Ob}\operatorname{\overline{\mathbf{D}}^{b}}(X)\times\oplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,S,p)$,
under the equivalence relation

$(\overline{\mathcal{F}},\eta)\sim(\overline{\mathcal{F}}+[\overline{E}],\eta-\widetilde{\operatorname{ch}}(\overline{E}))$

for $[\overline{E}]\in\operatorname{\mathbf{KA}}(X)$, and with morphisms

$\operatorname{Hom}_{\operatorname{\mathbf{\widehat{D}}^{b}}(X)}([\overline{\mathcal{F}},\eta],[\overline{\mathcal{G}},\nu])=\operatorname{Hom}_{\operatorname{\mathbf{D}^{b}}(X)}(\mathcal{F},\mathcal{G}).$

If $S\subset T$ are closed conical subsets of $T^{\ast}X_{0}$, then
$\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$ is naturally a full subcategory
of $\operatorname{\mathbf{\widehat{D}}^{b}}(X,T)$.

In the sequel, we extend the main operations in
$\operatorname{\mathbf{D}^{b}}(X)$ to the categories
$\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$. In particular, we use the
theory of generalized analytic torsion classes to construct push-forward
morphisms attached to morphisms in
$\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$.

The category $\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$ has a natural
additive structure. More generally, if $S,T$ are closed conical subsets of
$T^{\ast}X_{0}$, then there is an obvious addition functor

$\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)\times\operatorname{\mathbf{\widehat{D}}^{b}}(X,T)\overset{\oplus}{\longrightarrow}\operatorname{\mathbf{\widehat{D}}^{b}}(X,S\cup
T).$

The object $[\overline{0},0]$ is a neutral element for this operation. If
$S+T$ is disjoint with the zero section in $T^{\ast}X$, then there is a
product defined by the functor

$\begin{split}\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)\times\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,T)&\overset{\otimes}{\longrightarrow}\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,(S+T)\cup
S\cup T)\\\
([\overline{\mathcal{F}},\eta],[\overline{\mathcal{G}},\nu])&\longmapsto[\overline{\mathcal{F}}\otimes\overline{\mathcal{G}},\operatorname{ch}(\overline{\mathcal{F}})\bullet\nu+\eta\bullet\operatorname{ch}(\overline{\mathcal{G}})+\operatorname{d}_{D}\eta\bullet\nu]\end{split}$
(10.2)

and the obvious assignment for morphisms. This product is commutative up to
natural isomorphism. It induces on
$\operatorname{\mathbf{\widehat{D}}^{b}}(X,\emptyset)$ a structure of
commutative and associative ring category. Also,
$[\overline{\mathcal{O}}_{X},0]$ is a unity object for the product structure.
More generally, the category $\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$
becomes a left and right
$\operatorname{\mathbf{\widehat{D}}^{b}}(X,\emptyset)$ module. Under the same
assumptions on $S$, $T$ we may define an internal $\operatorname{Hom}$. For
this, let
$[\overline{\mathcal{F}},\eta]\in\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$
and
$[\overline{\mathcal{G}},\nu]\in\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,T)$.
Then we put

$\underline{\operatorname{Hom}}([\overline{\mathcal{F}},\eta],[\overline{\mathcal{G}},\nu])=[\underline{\operatorname{Hom}}(\overline{\mathcal{F}},\overline{\mathcal{G}}),(\sigma\operatorname{ch}(\overline{\mathcal{F}}))\bullet\nu+(\sigma\eta)\bullet\operatorname{ch}(\overline{\mathcal{G}})+(\operatorname{d}_{D}\sigma\eta)\bullet\nu],$

where we recall that $\sigma$ is the sign operator (Definition 9.3). Using
Corollary 9.8, it is easily seen this is well defined. In particular, we put

$[\overline{\mathcal{F}},\eta]^{\vee}:=\underline{\operatorname{Hom}}([\overline{\mathcal{F}},\eta],[\overline{{\mathcal{O}}}_{X},0])=[\overline{\mathcal{F}}^{\vee},\sigma\eta].$

The shift $[1]$ on $\operatorname{\overline{\mathbf{D}}^{b}}(X)$ induces a
well defined shift functor on $\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$,
whose action on objects is

$[\overline{\mathcal{F}},\eta][1]=[\overline{\mathcal{F}}[1],-\eta].$

There is a Chern character

$\operatorname{ch}:\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)\longrightarrow\bigoplus_{p}\widetilde{\mathcal{D}}_{D}^{2p}(X,S,p),\quad[\overline{\mathcal{F}},\eta]\longmapsto\operatorname{ch}(\overline{\mathcal{F}})+\operatorname{d}_{D}\eta,$

which is well defined because
$\operatorname{d}_{D}\widetilde{\operatorname{ch}}(\overline{E})=\operatorname{ch}(\overline{E})$
for $[\overline{E}]\in\operatorname{\mathbf{KA}}(X)$. The Chern character is
additive and compatible with the product structure:

$\operatorname{ch}([\overline{\mathcal{F}},\eta]\otimes[\overline{\mathcal{G}},\nu])=\operatorname{ch}([\overline{\mathcal{F}},\eta])\bullet\operatorname{ch}([\overline{\mathcal{G}},\nu]).$

Notice the relations

$\displaystyle\operatorname{ch}([\overline{\mathcal{F}},\eta]^{\vee})=\sigma\operatorname{ch}([\overline{\mathcal{F}},\eta]),$
$\displaystyle\operatorname{ch}([\overline{\mathcal{F}},\eta][1])=-\operatorname{ch}([\overline{\mathcal{F}},\eta]).$

We may also define Bott-Chern classes for isomorphisms and distinguished
triangles. Let
$\widehat{\varphi}\colon[\overline{\mathcal{F}},\eta]\dashrightarrow[\overline{\mathcal{G}},\nu]$
be an isomorphism in $\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$, whose
underlying morphism in $\operatorname{\mathbf{D}^{b}}(X)$ we denote $\varphi$.
While the class
$\widetilde{\operatorname{ch}}(\varphi\colon\overline{\mathcal{F}}\dashrightarrow\overline{\mathcal{G}})$
depends on the representatives $(\overline{\mathcal{F}},\eta)$,
$(\overline{\mathcal{G}},\nu)$, the class

$\widetilde{\operatorname{ch}}(\widehat{\varphi}):=\widetilde{\operatorname{ch}}(\varphi\colon\overline{\mathcal{F}}\dashrightarrow\overline{\mathcal{G}})+\nu-\eta$

is well defined.

######  Lemma 10.3.

Let
$\widehat{\varphi}:[\overline{\mathcal{F}},\eta]\dashrightarrow[\overline{\mathcal{G}},\nu]$
be an isomorphism in $\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$, with
underlying morphism $\varphi$ in $\operatorname{\mathbf{D}^{b}}(X)$. Then, the
following conditions are equivalent:

  1. (i)

there exists $[\overline{E}]\in\operatorname{\mathbf{KA}}(X)$ such that
$\varphi$ induces a tight isomorphism between
$\overline{\mathcal{F}}+[\overline{E}]$ and $\overline{\mathcal{G}}$, and
$\nu=\eta-\widetilde{\operatorname{ch}}(\overline{E})$;

  2. (ii)

$\widetilde{\operatorname{ch}}(\widehat{\varphi})=0$.

###### Proof.

This is actually a tautology. Because $\operatorname{\mathbf{KA}}(X)$ acts
freely and transitively on the possible hermitian structures on $\mathcal{F}$,
there exists a unique $[\overline{E}]\in\operatorname{\mathbf{KA}}(X)$ such
that $\overline{\mathcal{F}}+[\overline{E}]$ is tightly isomorphic to
$\overline{\mathcal{G}}$ via the morphism $\varphi$. Then we have

$\widetilde{\operatorname{ch}}(\widehat{\varphi})=\widetilde{\operatorname{ch}}(\overline{E})+\nu-\eta.$

The lemma follows. ∎

######  Definition 10.4.

Let $\widehat{\varphi}$ be an isomorphism in
$\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$. We say that
$\widehat{\varphi}$ is tight if the equivalent conditions of Lemma 10.3 are
satisfied.

In particular, if
$\varphi\colon\overline{\mathcal{F}}\dashrightarrow\overline{\mathcal{G}}$ is
a tight isomorphism in $\operatorname{\overline{\mathbf{D}}^{b}}(X)$, then
$\varphi$ induces a tight isomorphism
$[\overline{\mathcal{F}},\eta]\dashrightarrow[\overline{\mathcal{G}},\nu]$ if
and only if $\eta=\nu$.

The following lemma provides an example involving the notion of tight
isomorphism.

######  Lemma 10.5.

Let
$[\overline{\mathcal{F}},\eta]\in\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$
and
$[\overline{\mathcal{G}},\nu]\in\operatorname{\mathbf{\widehat{D}}^{b}}(X,T)$.
Assume that $S+T$ does not cross the zero section. Then there is a functorial
tight isomorphism

$[\overline{\mathcal{F}},\eta]^{\vee}\otimes[\overline{\mathcal{G}},\nu]\cong\underline{\operatorname{Hom}}([\overline{\mathcal{F}},\eta],[\overline{\mathcal{G}},\nu]).$

Assume now given a distinguished triangle

$\widehat{\tau}\colon\quad[\overline{\mathcal{F}},\eta]\dashrightarrow[\overline{\mathcal{G}},\nu]\dashrightarrow[\overline{\mathcal{H}},\mu]\dashrightarrow[\overline{\mathcal{F}},\eta][1].$

Let $\overline{\tau}$ denote the distinguished triangle
$\overline{\mathcal{F}}\dashrightarrow\overline{\mathcal{G}}\dashrightarrow\overline{\mathcal{H}}\dashrightarrow$
in $\operatorname{\overline{\mathbf{D}}^{b}}(X)$. Then we put

$\widetilde{\operatorname{ch}}(\widehat{\tau})=\widetilde{\operatorname{ch}}(\overline{\mathcal{\tau}})+\eta-\nu+\mu.$

By [17, Thm. 3.33 (vii)], this class does not depend on the representatives
and is thus well defined.

We study now the functoriality of
$\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$ with respect to inverse and
direct images. Let $f\colon X\to Y$ be a morphism of smooth complex varieties.
Let $T\subset T^{\ast}Y_{0}$ be a closed conical subset disjoint with $N_{f}$.
The action of the left inverse image functor on objects is

$f^{\ast}\colon\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(Y,T)\longrightarrow\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,f^{\ast}T),\qquad[\overline{\mathcal{F}},\eta]\longmapsto[f^{\ast}\overline{\mathcal{F}},f^{\ast}\eta].$

That this assignment is well defined amounts to the functoriality of
$\widetilde{\operatorname{ch}}$.

Let $\overline{f}$ be a morphism in the category
$\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. The definition of
a direct image functor attached to $\overline{f}$ depends upon the choice of a
theory of generalized analytic torsion classes. Let $T$ be such a theory. Then
we define a functor $\overline{f}_{\ast}$ whose action on objects is

$\begin{split}\overline{f}_{\ast}\colon\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)&\longrightarrow\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(Y,f_{\ast}S)\\\
[\overline{\mathcal{F}},\eta]&\longmapsto[\overline{f_{\ast}\mathcal{F}},\overline{f}_{\flat}(\eta)-T(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})],\end{split}$
(10.6)

where $\overline{f_{\ast}\mathcal{F}}$ is an arbitrary choice of hermitian
structure on $f_{\ast}\mathcal{F}$. By the anomaly formulas, this definition
does not depend on the representative $(\overline{\mathcal{F}},\eta)$ nor on
the choice of hermitian structure on $\overline{f_{\ast}\mathcal{F}}$.

######  Theorem 10.7.

Let $\overline{f}:X\to Y$ and $\overline{g}:Y\to Z$ be morphisms in
$\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. Let $S\subset
T^{\ast}X_{0}$ and $T\subset T^{\ast}Y_{0}$ be closed conical subsets.

  1. (i)

Let
$[\overline{\mathcal{F}},\eta]\in\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$.
Then there is a functorial tight isomorphism

$(\overline{g}\circ\overline{f})_{\ast}([\overline{\mathcal{F}},\eta])\cong\overline{g}_{\ast}\overline{f}_{\ast}([\overline{\mathcal{F}},\eta]).$

  2. (ii)

(Projection formula) Assume that $T\cap N_{f}=\emptyset$ and that
$T+f_{\ast}S$ does not cross the zero section of $T^{\ast}Y$. Let
$[\overline{\mathcal{F}},\eta]\in\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$
and
$[\overline{\mathcal{G}},\nu]\in\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(Y,T)$.
Then there is a functorial tight isomorphism

$\overline{f}_{\ast}([\overline{\mathcal{F}},\eta]\otimes
f^{\ast}[\overline{\mathcal{G}},\nu])\cong\overline{f}_{\ast}[\overline{\mathcal{F}},\eta]\otimes[\overline{\mathcal{G}},\nu]$

in $\operatorname{\mathbf{\widehat{D}}^{b}}(Y,W)$, where
$W=f_{\ast}(S+f^{\ast}T)\cup f_{\ast}S\cup f_{\ast}f^{\ast}T.$

  3. (iii)

(Base change) Consider a cartesian diagram

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{\prime}}$$\scriptstyle{f^{\prime}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{Y.}$

Suppose that $f$ and $h$ are transverse and that $N_{h^{\prime}}$ is disjoint
with $S$. Equip $f^{\prime}$ with the hermitian structure induced by the
natural isomorphism $h^{\ast}T_{f}\dashrightarrow T_{f^{\prime}}$. Let
$[\overline{\mathcal{F}},\eta]\in\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$.
Then there is a functorial tight isomorphism

$h^{\ast}\overline{f}_{\ast}[\overline{\mathcal{F}},\eta]\cong\overline{f}^{\prime}_{\ast}{h^{\prime}}^{\ast}[\overline{\mathcal{F}},\eta]$

in
$\operatorname{\mathbf{\widehat{D}}^{b}}(Y^{\prime},f^{\prime}_{\ast}{h^{\prime}}^{\ast}S)$.

###### Proof.

The first and the second assertions follow from Proposition 2.14, the
transitivity and the projection formula for $T$. For the third item, one uses
the functoriality of the analytic torsion classes and Proposition 2.15. ∎

We close this section with an extension of Grothendieck duality to
$\operatorname{\mathbf{\widehat{D}}^{b}}$. Let $\overline{f}:X\to Y$ be a
morphism is $\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. To
enlighten notations, we denote by $\boldsymbol{\omega}_{\overline{f}}$ the
object $[\boldsymbol{\omega}_{\overline{f}},0]$ in
$\operatorname{\mathbf{\widehat{D}}^{b}}(X,\emptyset)$ (Definition 9.2).
Suppose given a closed conical subset $T\subset T^{\ast}Y_{0}$ such that
$T\cap N_{f}=\emptyset$. Then we define the functor $\overline{f}^{!}$ whose
action on objects is

$\overline{f}^{!}:\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(Y,T)\longrightarrow\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,f^{\ast}T),\qquad[\overline{\mathcal{F}},\eta]\longmapsto
f^{\ast}[\overline{\mathcal{F}},\eta]\otimes\boldsymbol{\omega}_{\overline{f}}.$

Observe the equality

$[\overline{\mathcal{G}},\nu]\otimes\boldsymbol{\omega}_{\overline{f}}=[\overline{\mathcal{G}}\otimes\boldsymbol{\omega}_{\overline{f}},\nu\bullet\operatorname{ch}(\boldsymbol{\omega}_{\overline{f}})].$
(10.8)

Now fix a theory of generalized analytic torsion classes. To the morphism
$\overline{f}$ we have attached the direct image functor
$\overline{f}_{\ast}$. We denote by $\overline{f}^{\vee}_{\ast}$ the direct
image functor associated to $\overline{f}$ and the dual theory (Theorem
Definition 9.10).

######  Theorem 10.9 (Grothendieck duality for
$\operatorname{\mathbf{\widehat{D}}^{b}}$).

Let $\overline{f}:X\to Y$ be a morphism in
$\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. Let $S\subset
T^{\ast}X_{0}$ and $T\subset T^{\ast}Y_{0}$ be closed conical subsets such
that $T\cap N_{f}=\emptyset$ and $T+f_{\ast}S$ is disjoint with the zero
section. Let
$[\overline{\mathcal{F}},\eta]\in\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)$
and
$[\overline{\mathcal{G}},\sigma]\in\operatorname{Ob}\operatorname{\mathbf{\widehat{D}}^{b}}(Y,T)$.
Then there is a functorial tight isomorphism

$\underline{\operatorname{Hom}}(\overline{f}_{\ast}[\overline{\mathcal{F}},\eta],[\overline{\mathcal{G}},\nu])\cong\overline{f}^{\vee}_{\ast}\underline{\operatorname{Hom}}([\overline{\mathcal{F}},\eta],\overline{f}^{!}[\overline{\mathcal{G}},\nu])$

in $\operatorname{\mathbf{\widehat{D}}^{b}}(Y,W)$, where
$W=f_{\ast}(S+f^{\ast}T)\cup f_{\ast}S\cup f_{\ast}f^{\ast}T.$

In particular, we have

$(\overline{f}_{\ast}[\overline{\mathcal{F}},\eta])^{\vee}\cong\overline{f}^{\vee}_{\ast}([\overline{\mathcal{F}},\eta]^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}}).$
(10.10)

###### Proof.

By Lemma 10.5 and Proposition 10.7, we are reduced to establish the functorial
tight isomorphism (10.10). The proof follows readily from the definitions,
Grothendieck duality and the following two observations. First of all, if $T$
is the theory of analytic torsion classes, then by the very definition of
$T^{\vee}$ we find

$\sigma
T(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}})=T^{\vee}(\overline{f},\overline{\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}},\overline{f_{\ast}(\mathcal{F}^{\vee}\otimes\boldsymbol{\omega}_{f})}),$

where the metric on
$\overline{f_{\ast}(\mathcal{F}^{\vee}\otimes\boldsymbol{\omega}_{f})}$ is
chosen so that Grothendieck duality provides a tight isomorphism

$\overline{f_{\ast}\mathcal{F}}^{\vee}\cong\overline{f_{\ast}(\mathcal{F}^{\vee}\otimes\boldsymbol{\omega}_{f})}.$

Secondly, for direct images of currents, we compute

$\sigma\overline{f}_{\flat}(\eta)=\sigma
f_{\ast}(\eta\bullet\operatorname{Td}(T_{\overline{f}}))=(-1)^{e}f_{\ast}(\sigma\eta\bullet\sigma\operatorname{Td}(T_{\overline{f}}))=f_{\ast}(\sigma\eta\bullet\operatorname{ch}(\boldsymbol{\omega}_{\overline{f}})\bullet\operatorname{Td}(T_{\overline{f}})).$

Here $e$ is the relative dimension of $f$, and to derive the last equality we
appeal to Proposition 9.4. To conclude, we recall equation (10.8). ∎

######  Corollary 10.11.

Let $T$ be a self-dual theory of generalized analytic torsion classes.

  1. (i)

Then there is a functorial isomorphism
$(\overline{f}_{\ast}[\overline{\mathcal{F}},\eta])^{\vee}\cong\overline{f}_{\ast}([\overline{\mathcal{F}},\eta]^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}}).$

  2. (ii)

If the hermitian structure of $\overline{f}$ comes from chosen metrics on
$T_{X}$, $T_{Y}$ and $\boldsymbol{\omega}_{X}$, $\boldsymbol{\omega}_{Y}$ are
equipped with the induced metrics, then we have a commutative diagram

$\textstyle{\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{f}_{\ast}}$$\scriptstyle{(\cdot)^{\vee}\otimes\overline{\boldsymbol{\omega}}_{X}}$$\textstyle{\operatorname{\mathbf{\widehat{D}}^{b}}(X,S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{f}_{\ast}}$$\textstyle{\operatorname{\mathbf{\widehat{D}}^{b}}(Y,f_{\ast}S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\cdot)^{\vee}\otimes\overline{\boldsymbol{\omega}}_{Y}}$$\textstyle{\operatorname{\mathbf{\widehat{D}}^{b}}(Y,f_{\ast}S).}$

###### Proof.

The first claim is immediate from Theorem 10.9. The second item follows from
the first one and the projection formula (Proposition 10.7). ∎

##  11 Analytic torsion for degenerating families of curves

As a second example of application of the theory developed in this article, we
describe the singularities of the analytic torsion for degenerating families
of curves. The results we prove are particular instances of those obtained by
Bismut-Bost [6], Bismut [4] and Yoshikawa [49]. Although the methods of this
section can be extended to recover the results of Yoshikawa in [49], for
simplicity, we will restrict ourselves to fibrations in curves over a curve.

In fact, our proof is not very different from the one in [4] and [49]. For
instance, one of the main ingredients of the proof of the results in [4] and
[49] is the Bismut-Lebeau immersion formula. Our approach implicitly uses
Bismut’s generalization of the immersion formula, encoded in the existence of
analytic torsion theories for arbitrary projective morphisms. We expect that
the techniques of this section can be used to generalize the above results to
situations more general than the ones considered by Yoshikawa.

Let $S$ be a smooth complex curve and $f\colon X\to S$ a projective morphism
of smooth complex varieties, whose fibers are reduced curves with at most
ordinary double singular points. We assume that $f$ is generically smooth.
Following Bismut-Bost [6, Sec. 2(b)], we call such a family an f.s.o.
(_famille à singularités ordinaires_). The singular locus of $f$, to be
denoted $\Sigma$, is a zero dimensional reduced closed subset of $X$. Its
direct image $\Delta=f_{\ast}(\Sigma)$ is the Weil divisor

$\Delta=\sum_{p\in S}n_{p}p,$

where $n_{p}$ is the number of singular points of the fiber $f^{-1}(p)$. We
will abusively identify $\Delta$ with its support. With these notations, we
put $V=S\setminus\Delta$. Locally for the analytic topology, the morphism $f$
can be written in complex coordinates either as $f(z_{0},z_{1})=z_{0}$ or
$f(z_{0},z_{1})=z_{0}z_{1}$ [6, Sec. 3(a)]. In the second case, the point of
coordinates $(z_{0},z_{1})=(0,0)$ belongs to the singular locus $\Sigma$.

For a vector bundle $F$ over $X$, let ${\mathbb{P}}(F)$ be the projective
space of lines in $F$. The differential $df\colon T_{X}\to f^{\ast}T_{S}$
induces a section ${\mathcal{O}}_{X}\to\Omega_{X}\otimes f^{\ast}T_{S}$.
Because $f$ is smooth over $X\setminus\Sigma$, this section does not vanish on
$X\setminus\Sigma$. Therefore there is an induced map

$\mu\colon X\setminus\Sigma\longrightarrow{\mathbb{P}}(\Omega_{X}\otimes
f^{\ast}T_{S})\cong{\mathbb{P}}(\Omega_{X}),$

called the _Gauss map_. Notice that this map was already used in [4] and [49].

We next study the blow-up $\widetilde{X}=\text{Bl}_{\Sigma}(X)$ of $X$ at
$\Sigma$ and relate it to the Gauss map. Let $\pi\colon\widetilde{X}\to X$ be
the natural projection. Let $E$ be the exceptional divisor of $\pi$,

$E=\bigsqcup_{p\in\Sigma}E_{p},\quad E_{p}\cong{\mathbb{P}}(T_{p}X),$

with the reduced scheme structure. For every $p\in\Sigma$, there is an
identification $T_{p}X\cong\Omega_{X,p}$ provided by the hessian of $f$, which
is a non-degenerate bilinear form on $T_{p}X$. The local description of the
blow-up at a point implies:

######  Lemma 11.1.

There is a commutative diagram

$\textstyle{E_{p}={\mathbb{P}}(T_{p}X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
8.5359pt\sim}$$\textstyle{{\mathbb{P}}(\Omega_{X,p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\mu}}$$\scriptstyle{\pi}$$\textstyle{{\mathbb{P}}(\Omega_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{X}$$\textstyle{X\setminus\Sigma.\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$

Denote by ${\mathcal{O}}(-1)$ the tautological divisor either on
${\mathbb{P}}(\Omega_{X})$ or on $E_{p}$. Then there is a natural isomorphism
$\widetilde{\mu}^{\ast}{\mathcal{O}}(-1)\mid_{E_{p}}\cong{\mathcal{O}}(-1).$

Consider now the short exact sequence of vector bundles on
${\mathbb{P}}(\Omega_{X})$

$0\to{\mathcal{O}}(-1)\to p^{\ast}\Omega_{X}\to Q\to 0,$

where $Q$ is the universal quotient bundle. Observe that $Q$ is of rank 1. The
dual exact sequence is

$0\to U\to p^{\ast}T_{X}\to{\mathcal{O}}(1)\to 0,$

$U$ being the universal vector subsheaf. We denote by $\eta$ the induced exact
sequence on $\widetilde{X}$

$\eta\colon
0\to\widetilde{\mu}^{\ast}U\to\pi^{\ast}T_{X}\to\widetilde{\mu}^{\ast}{\mathcal{O}}(1)\to
0,$ (11.2)

From (11.2) and the definition $\omega_{X/S}=\omega_{X}\otimes f^{\ast}T_{S}$,
we derive a natural isomorphism

$\widetilde{\mu}^{\ast}U\otimes\pi^{\ast}\omega_{X/S}\cong\widetilde{\mu}^{\ast}{\mathcal{O}}(-1)\otimes\widetilde{f}^{\ast}T_{S}.$
(11.3)

######  Lemma 11.4.

We have

$\widetilde{\mu}^{\ast}{\mathcal{O}}(-1)\otimes\widetilde{f}^{\ast}T_{S}={\mathcal{O}}(E).$
(11.5)

###### Proof.

First of all we observe that
$\widetilde{\mu}^{\ast}U\otimes\pi^{\ast}\omega_{X/S}$ is trivial on the open
$W=\widetilde{X}\setminus E$. Indeed, by construction of the Gauss map we have

$\widetilde{\mu}^{\ast}U\mid_{W}=\ker(df\colon T_{X}\to
f^{\ast}T_{S})\mid_{W}=\omega_{X/S}^{\vee}\mid W.$

Hence by equation (11.3) we can write

$\widetilde{\mu}^{\ast}{\mathcal{O}}(-1)\otimes\widetilde{f}^{\ast}T_{S}={\mathcal{O}}(\sum_{p\in\Sigma}m_{p}E_{p}).$

To compute the multiplicities $m_{p}$ we use that
$\widetilde{\mu}^{\ast}{\mathcal{O}}(-1)\mid_{E_{p}}={\mathcal{O}}(-1)$,
$(E_{p}\cdot\widetilde{f}^{\ast}T_{S})=0$ and $(E_{p}\cdot E_{p})=-1$:

$-m_{p}=\deg(\widetilde{\mu}^{\ast}{\mathcal{O}}(-1)\otimes\widetilde{f}^{\ast}T_{S})\mid_{E_{p}}=-1+0=-1.$

The lemma follows. ∎

Later we will need the commutative diagram of exact sequences

$\textstyle{\eta\mid_{W}\colon}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\mu}^{\ast}U\mid_{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{T_{X}\mid_{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{\widetilde{\mu}^{\ast}{\mathcal{O}}(1)\mid_{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{0}$$\textstyle{\varepsilon\colon}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\omega_{X/S}^{\vee}\mid_{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{X}\mid_{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{f^{\ast}T_{S}\mid_{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$
(11.6)

After the identification
$\widetilde{\mu}^{\ast}{\mathcal{O}}(-1)\otimes\widetilde{f}^{\ast}T_{S}={\mathcal{O}}(E)$
provided by the lemma, the morphism $\gamma$ is the restriction to $W$ of the
natural inclusion
$\widetilde{\mu}^{\ast}{\mathcal{O}}(1)\to\widetilde{\mu}^{\ast}{\mathcal{O}}(1)\otimes{\mathcal{O}}(E)$.
This fact will be used below.

We now proceed to introduce the hermitian vector bundles and the analytic
torsion classes we aim to study. We fix a theory of generalized analytic
torsion classes $T$.

Let $f\colon X\to S$, $\widetilde{f}\colon\widetilde{X}\to S$ be f.s.o. as
above. Recall that we write $W=X\setminus\Sigma=\widetilde{X}\setminus E$ and
$V=S\setminus\Delta$, so that $f^{-1}(V)\subset W$. We endow the tangent
spaces $T_{X}$ and $T_{S}$ with smooth hermitian metrics. We will denote by
$\overline{f}$ the corresponding morphism in the category
$\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. On the open
subset $W$, there is a quasi-isomorphism

$\omega_{X/S}^{\vee}\mid_{W}=\boldsymbol{\omega}_{X/S}^{\vee}[1]\mid_{W}\to
T_{f}$

induced by the identification
$\omega_{X/S}^{\vee}\mid_{W}=\ker(T_{X}\mid_{W}\to f^{\ast}T_{S})$. On
$\omega_{X/S}^{\vee}\mid_{W}$, and in particular on
$\omega_{f^{-1}(V)/V}^{\vee}$, we will put the metric induced by
$\overline{T_{X}}\mid_{W}$. We will write $\overline{f}^{\prime}\colon
f^{-1}(V)\to V$ for the corresponding morphism in
$\overline{\operatorname{\mathbf{Sm}}}_{\ast/{\mathbb{C}}}$. Observe that the
restriction of $f$ to $W$, and hence to $f^{-1}(V)$, may be identified with
the restriction of $\widetilde{f}$. Let $\overline{\mathcal{F}}$ be an object
in $\operatorname{\overline{\mathbf{D}}^{b}}(X)$ and fix a hermitian structure
on $f_{\ast}\mathcal{F}$. Then we consider the relative metrized complexes

$\overline{\xi}=(\overline{f},\overline{\mathcal{F}},\overline{f_{\ast}\mathcal{F}}),\qquad\overline{\xi}^{\prime}=(\overline{f}^{\prime},\overline{\mathcal{F}}\mid_{f^{-1}(V)},\overline{f_{\ast}\mathcal{F}}\mid_{V}),$

and the corresponding analytic torsion classes

$T(\overline{\xi})\in\bigoplus_{p}\widetilde{\mathcal{D}}_{D}^{2p-1}(S,N_{f},p),\qquad
T(\overline{\xi}^{\prime})\in\bigoplus_{p}\widetilde{\mathcal{D}}_{D}^{2p-1}(V,\emptyset,p).$

By the functoriality of analytic torsion classes and the anomaly formulas, we
have

$T(\overline{\xi}^{\prime})=T(\overline{\xi})\mid_{V}-\overline{f}_{\flat}[\operatorname{ch}(\overline{\mathcal{F}}\mid_{f^{-1}(V)})\widetilde{\operatorname{Td}}_{m}(\overline{\varepsilon}\mid_{f^{-1}(V)})].$
(11.7)

Here $\overline{\varepsilon}$ is the exact sequence in (11.6), with the
hermitian metrics we have just defined. From now on we will omit the reference
to $f^{-1}(V)$ and $V$ in the formulas.

We consider the hermitian structures on the sheaves $U$ and ${\mathcal{O}}(1)$
on ${\mathbb{P}}(\Omega_{X})$ induced by $p^{\ast}\overline{T_{X}}$. We will
write $\overline{\eta}$ for the exact sequence in (11.2) and
$\overline{\alpha}$, $\overline{\beta}$ and $\overline{\gamma}$ for the
vertical isomorphisms in diagram (11.6), all provided with the corresponding
metrics. Notice that $\overline{\alpha}$ and $\overline{\beta}$ are
isometries. By the properties of the Bott-Chern class
$\widetilde{\operatorname{Td}}_{m}$, we have

$\widetilde{\operatorname{Td}}_{m}(\overline{\varepsilon})=\widetilde{\operatorname{Td}}_{m}(\overline{\eta})+\operatorname{Td}(\overline{\eta})\widetilde{\operatorname{Td}}_{m}(\overline{\gamma}).$
(11.8)

Hence, from (11.7)–(11.8) and identifying $f$ with $\widetilde{f}$ over $V$,
we have

$T(\overline{\xi}^{\prime})=T(\overline{\xi})-\widetilde{f}_{\ast}[\pi^{\ast}\operatorname{ch}(\overline{\mathcal{F}})\pi^{\ast}\operatorname{Td}(\overline{f})\widetilde{\operatorname{Td}}_{m}(\overline{\eta})]\\\
-\widetilde{f}_{\ast}[\pi^{\ast}\operatorname{ch}(\overline{\mathcal{F}})\pi^{\ast}\operatorname{Td}(\overline{f})\operatorname{Td}(\overline{\eta})\widetilde{\operatorname{Td}}_{m}(\overline{\gamma})].$
(11.9)

It will be convenient to have a precise description of
$\widetilde{\operatorname{Td}}_{m}(\overline{\gamma})$ at our disposal.

For shorthand, we write $L:=\widetilde{\mu}^{\ast}{\mathcal{O}}(1)$ and
$\|\cdot\|_{0}$ for its hermitian structure constructed before. We denote by
$\|\cdot\|_{1}$ the metric on ${\mathcal{O}}(E)$ such that the isomorphism
$\overline{{\mathcal{O}}(E)}_{1}=\overline{L}_{0}^{-1}\otimes\widetilde{f}^{\ast}\overline{T_{S}}$
(Lemma 11.4) is an isometry. Recall that $\gamma$ gets identified with the
restriction to $W$ of the natural inclusion $L\to L\otimes{\mathcal{O}}(E)$.
We let $\|\cdot\|_{\infty}$ be the hermitian metric on $L\mid_{W}$ such that
$\gamma$ is an isometry. Hence, if $\mathbf{1}$ denotes the canonical section
of ${\mathcal{O}}(E)$ and $\ell$ is any section of $L\mid_{W}$, then we have

$\|\ell\|_{\infty}=\|\ell\|_{0}\|\mathbf{1}\|_{1}.$

To simplify the notations, we will skip the reference to $W$. We then have on
$W$

$\widetilde{\operatorname{Td}}_{m}(\overline{\gamma})=\widetilde{\operatorname{Td}}_{m}(\overline{L}_{0}\overset{\operatorname{id}}{\to}\overline{L}_{\infty}).$

To compute a representative of this class, we fix a smooth function
$h\colon{\mathbb{P}}^{1}_{{\mathbb{C}}}\to{\mathbb{R}}$ such that $h(0)=0$ and
$h(\infty)=1$. Then we proceed by a deformation argument. Let $q\colon
W\times{\mathbb{P}}^{1}_{{\mathbb{C}}}\to W$ be the projection to the first
factor. On the line bundle $q^{\ast}L$ we put the metric that, on the fiber at
the point $(w,t)\in W\times{\mathbb{P}}^{1}_{{\mathbb{C}}}$, is determined by
the formula

$\|\ell\|_{(w,t)}=\|\ell\|_{0,w}\|\mathbf{1}\|_{1,w}^{h(t)}.$

We will write $\|\cdot\|_{t}$ for this family of metrics parametrized by
${\mathbb{P}}^{1}_{{\mathbb{C}}}$. Define

$\overline{\operatorname{Td}}(\overline{L}_{0}\to\overline{L}_{\infty})=\frac{1}{2\pi
i}\int_{{\mathbb{P}}^{1}_{{\mathbb{C}}}}\frac{-1}{2}\log(t\overline{t})(\operatorname{Td}(\overline{q^{\ast}L}_{t})-\operatorname{Td}(\overline{q^{\ast}L}_{0})).$

Then

$\overline{\operatorname{Td}}_{m}(\overline{\gamma})=\operatorname{Td}^{-1}(\overline{L}_{0})\overline{\operatorname{Td}}(\overline{L}_{0}\to\overline{L}_{\infty})$
(11.10)

represents the class $\widetilde{\operatorname{Td}}_{m}(\gamma)$. Let us
develop $\overline{\operatorname{Td}}_{m}(\overline{\gamma})$. If
$\overline{{\mathcal{O}}}_{t}$ denotes the trivial line bundle on
$W\times{\mathbb{P}}^{1}_{{\mathbb{C}}}$ with the norm
$\|\mathbf{1}\|_{t}=\|\mathbf{1}\|_{1}^{h(t)}$, then we compute

$\operatorname{Td}(\overline{q^{\ast}L}_{t})-\operatorname{Td}(\overline{q^{\ast}L}_{0})=\frac{1}{2}c_{1}(\overline{{\mathcal{O}}}_{t})+\frac{1}{6}c_{1}(\overline{{\mathcal{O}}}_{t})q^{\ast}c_{1}(\overline{L}_{0})+\frac{1}{12}c_{1}(\overline{{\mathcal{O}}}_{t})^{2}.$

By the very definition of $c_{1}$, we find

$c_{1}(\overline{{\mathcal{O}}}_{t})=\operatorname{\partial}\operatorname{\overline{\partial}}\log\|\mathbf{1}\|_{t}^{2}=\operatorname{\partial}\operatorname{\overline{\partial}}(h(t)\log\|\mathbf{1}\|_{1}^{2})\\\
=h(t)c_{1}(\overline{{\mathcal{O}}(E)}_{1})+\log\|\mathbf{1}\|_{1}^{2}\operatorname{\partial}\operatorname{\overline{\partial}}h(t)+\operatorname{\partial}h(t)\wedge\operatorname{\overline{\partial}}\log\|\mathbf{1}\|_{1}^{2}+\operatorname{\partial}\log\|\mathbf{1}\|_{1}^{2}\wedge\operatorname{\overline{\partial}}h(t).$

We easily obtain

$\displaystyle\frac{1}{2\pi
i}\int_{{\mathbb{P}}^{1}_{{\mathbb{C}}}}\frac{-1}{2}\log(t\overline{t})\frac{1}{2}c_{1}(\overline{{\mathcal{O}}}_{t})=-\frac{1}{2}\log\|\mathbf{1}\|_{1},$
(11.11) $\displaystyle\frac{1}{2\pi
i}\int_{{\mathbb{P}}^{1}_{{\mathbb{C}}}}\frac{-1}{2}\log(t\overline{t})\frac{1}{6}q^{\ast}c_{1}(\overline{L}_{0})c_{1}(\overline{{\mathcal{O}}}_{t})=-\frac{1}{6}\log\|\mathbf{1}\|_{1}c_{1}(\overline{L}_{0}).$
(11.12)

With some more work, we have

$\begin{split}\frac{1}{2\pi
i}\int_{{\mathbb{P}}^{1}_{{\mathbb{C}}}}\frac{-1}{2}\log(t\overline{t})\frac{1}{12}c_{1}(\overline{{\mathcal{O}}}_{t})^{2}=&-\frac{a}{6}\log\|\mathbf{1}\|_{1}c_{1}(\overline{{\mathcal{O}}(E)}_{1})\\\
&+\frac{b}{3}\operatorname{\partial}(\log\|\mathbf{1}\|_{1}\
\operatorname{\overline{\partial}}\log\|\mathbf{1}\|_{1}),\end{split}$ (11.13)

where

$a=\frac{1}{2\pi
i}\int_{{\mathbb{P}}^{1}_{{\mathbb{C}}}}\log(t\overline{t})\frac{1}{2}\operatorname{\partial}\operatorname{\overline{\partial}}(h(t)^{2}),\qquad
b=\frac{1}{2\pi
i}\int_{{\mathbb{P}}^{1}_{{\mathbb{C}}}}\log(t\overline{t})\operatorname{\partial}h(t)\wedge\operatorname{\overline{\partial}}h(t).$
(11.14)

We observe that

$a=\frac{1}{2\pi
i}\int_{{\mathbb{P}}^{1}_{{\mathbb{C}}}}\log(t\overline{t})\frac{1}{2}\operatorname{\partial}\operatorname{\overline{\partial}}(h(t)^{2})=\frac{1}{2},$

which is independent of $h$. All in all, equations (11.10)–(11.14) provide the
following expression for the representative
$\overline{\operatorname{Td}}_{m}(\overline{\gamma})$ of
$\widetilde{\operatorname{Td}}_{m}(\overline{\gamma})$:

$\begin{split}\overline{\operatorname{Td}}_{m}(\overline{\gamma})=\operatorname{Td}^{-1}(\overline{L}_{0})\Big{(}&-\frac{1}{2}\log\|\mathbf{1}\|_{1}-\frac{1}{6}\log\|\mathbf{1}\|_{1}c_{1}(\overline{L}_{0})\\\
&-\frac{1}{12}\log\|\mathbf{1}\|_{1}c_{1}(\overline{{\mathcal{O}}(E)}_{1})+\frac{b}{3}\operatorname{\partial}(\log\|\mathbf{1}\|_{1}\operatorname{\overline{\partial}}\log\|\mathbf{1}\|_{1})\Big{)}.\end{split}$
(11.15)

Given a current $\eta\in\mathcal{D}_{D}^{n}(X,p)$, we will call $(n,p)$ its
_Deligne bidegree_ , while we will call the _Dolbeault bidegree_ to the
bidegree in the Dolbeault complex. When it is clear from the context to which
bidegree we are referring, we call it bidegree.

We now study the singularities of the component of Deligne bidegree $(1,1)$ of
$T(\overline{\xi}^{\prime})$ near the divisor $\Delta$. For this we first
recall the decomposition of equation (11.9). Observe that
$\widetilde{\mathcal{D}}_{D}^{1}(V,\emptyset,1)$ gets identified with the
space of smooth real functions on $V$. In the sequel, for an element
$\vartheta\in\oplus_{p}\widetilde{D}_{D}^{2p-1}(\ast,p)$, we write
$\vartheta^{(2r-1,r)}$ to refer to its component of bidegree $(2r-1,r)$. By
construction of the Deligne complex, an element of Deligne bidegree $(2r-1,r)$
is just a current of Dolbeault bidegree $(r-1,r-1)$.

The following assertion is well-known. See for instance [47, Lemma 2.1, Cor.
2.2].

######  Lemma 11.16.

Let $\Omega\subset{\mathbb{C}}$ be an open subset and $\vartheta$ a current of
Dolbeault bidegree $(0,0)$ on $\Omega$. Let $\Delta$ be the standard
laplacian. If the current $\Delta\vartheta$ is represented by a locally
bounded measurable function, then $\vartheta$ is represented by a continuous
function.

######  Proposition 11.17.

The current
$T(\overline{\xi})^{(1,1)}\in\widetilde{\mathcal{D}}_{D}^{1}(S,N_{f},1)$ is
represented by a continuous function on $S$.

###### Proof.

The differential equation satisfied by $T(\overline{\xi})^{(1,1)}$ is

$\operatorname{d}_{\mathcal{D}}T(\overline{\xi})^{(1,1)}=\operatorname{ch}(\overline{f_{\ast}\mathcal{F}})^{(2,1)}-f_{\ast}[\operatorname{ch}(\overline{\mathcal{F}})\operatorname{Td}(\overline{f})]^{(2,1)}.$
(11.18)

In local coordinates, the operator
$\operatorname{d}_{\mathcal{D}}=-2\operatorname{\partial}\operatorname{\overline{\partial}}$
is a rescaling of the laplacian $\Delta$. By the lemma, it is enough we prove
that the current at the right hand side of (11.18) is represented by a locally
bounded measurable differential form. Because
$\operatorname{ch}(\overline{f_{\ast}\mathcal{F}})^{(2,1)}$ and
$\operatorname{ch}(\overline{\mathcal{F}})\operatorname{Td}(\overline{f})$ are
smooth differential forms, we are reduced to study currents of the form
$f_{\ast}[\theta]^{(2,1)}$, where $\theta$ is a smooth differential form. By a
partition of unity argument, we reduce to the case where
$f\colon{\mathbb{C}}^{2}\to{\mathbb{C}}$ is the morphism
$f(z_{0},z_{1})=z_{0}z_{1}$ and $\theta$ is a differential form of Dolbeault
bidegree (2,2) with compact support. Then we need to prove that the fiber
integral

$G(w)=\int_{z_{0}z_{1}=w}\theta$

is a bounded form in a neighborhood of $w=0$. Write
$\theta=h(z_{0},z_{1})dz_{0}\wedge d\overline{z}_{0}\wedge dz_{1}\wedge
d\overline{z}_{1}.$ We reduce to study integrals of the form

$G(w)=\left(\int_{|w|<|z_{0}|<1}h(z_{0},z_{0}/w)\frac{|w|^{2}}{|z_{0}|^{4}}dz_{0}\wedge\overline{z}_{0}\right)dw\wedge
d\overline{w}.$

The property follows from an easy computation in polar coordinates. ∎

######  Proposition 11.19.

Let $\theta$ be a differential form of Dolbeault bidegree (1,1) on
$\widetilde{X}$. Then the current $\widetilde{f}_{\ast}[\theta]$ is
represented by a bounded function on $S$.

###### Proof.

The proof is the same as in [6, Prop. 5.2]. One only has to show that the
argument in _loc. cit._ carries over to the case of the non-reduced fibres
that have appeared when blowing up the nodes. ∎

######  Corollary 11.20.

The current
$\widetilde{f}_{\ast}[\pi^{\ast}\operatorname{ch}(\overline{\mathcal{F}})\pi^{\ast}\operatorname{Td}(\overline{f})\widetilde{\operatorname{Td}}_{m}(\overline{\eta})]$
is represented by a bounded function on $S$.

###### Proof.

It suffices to observe that the differential form
$\pi^{\ast}\operatorname{ch}(\overline{\mathcal{F}})\pi^{\ast}\operatorname{Td}(\overline{f})\widetilde{\operatorname{Td}}_{m}(\overline{\eta})$
is actually smooth on the whole $\widetilde{X}$. ∎

According to (11.9), it remains to study the current

$\widetilde{f}_{\ast}[\pi^{\ast}\operatorname{ch}(\overline{\mathcal{F}})\pi^{\ast}\operatorname{Td}(\overline{f})\operatorname{Td}(\overline{\eta})\widetilde{\operatorname{Td}}_{m}(\overline{\gamma})]\mid_{V}.$

The main difference with the situation in Corollary 11.20 is that the class
$\widetilde{\operatorname{Td}}_{m}(\overline{\gamma})$ is not defined on the
whole $\widetilde{X}$, but only on $W=\widetilde{X}\setminus E$. In the
following discussion we will use the representative
$\overline{\operatorname{Td}}_{m}(\overline{\gamma})$ defined in (11.10) at
the place of $\widetilde{\operatorname{Td}}_{m}(\overline{\gamma})$. In view
of equations (11.11)–(11.13), the first result we need is the following
statement.

######  Proposition 11.21.

Let $\theta$ be a smooth and
$\operatorname{\partial},\operatorname{\overline{\partial}}$ closed
differential form on $\widetilde{X}$, of Dolbeault bidegree $(1,1)$. Let $w$
be an analytic coordinate in a neighborhood of $p\in\Delta$ with $w(p)=0$.
Write $D_{p}=E\cap\widetilde{f}^{-1}(p)$. Then, the current