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where $A_{\ast}$ is split acyclic and $K(F,N_{Y/X}\otimes
N^{-1}_{\infty/\mathbb{P}^{1}})$ is the Koszul resolution.

  3. (iii)

Let $f\colon X^{\prime}\longrightarrow X$ be a morphism of complex manifolds
and assume that we are in one of the following cases:

    1. (a)

The map $f$ is smooth.

    2. (b)

The map $f$ is arbitrary and $E_{\ast}$ is acyclic.

    3. (c)

$f$ is transverse to $Y$.

Then $E^{\prime}_{\ast}:=f^{\ast}(E_{\ast})$ is exact over $f^{-1}(U)$,

$W^{\prime}:=W(E^{\prime}_{\ast})=W\underset{X}{\times}X^{\prime},$

with $f_{W}\colon W^{\prime}\longrightarrow W$ the induced map, and we have
$f_{W}^{\ast}(\operatorname{tr}_{1}(E_{\ast})_{\ast})=\operatorname{tr}_{1}(f^{\ast}(E_{\ast}))_{\ast}$.

  4. (iv)

If the vector bundles $E_{i}$ are provided with hermitian metrics, then one
can choose a hermitian metric on $\operatorname{tr}_{1}(E_{\ast})_{\ast}$ such
that its restriction to $X\times\\{0\\}$ is isometric to $E_{\ast}$ and the
restriction to $U\times\\{\infty\\}$ is orthogonally split. We will denote by
$\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast}$ the complex
$\operatorname{tr}_{1}(E_{\ast})_{\ast}$ with such a choice of hermitian
metrics. Moreover, this choice of metrics can be made functorial. That is, if
$f$ is a map as in item (iii), then

$f_{W}^{\ast}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})=\operatorname{tr}_{1}(f^{\ast}(\overline{E}_{\ast}))_{\ast}$

###### Proof.

The case when $E_{\ast}$ is acyclic has already been treated. For the case
when $Y$ is non-empty and $F$ is non zero, we first recall the construction of
the Grassmannian graph of an arbitrary complex from [20], which is more
general than what we need here. If $E$ is a vector bundle over $X$ we will
denote by $E(i)$ the vector bundle over $X\times\mathbb{P}^{1}$ given by
$E(i)=p^{\ast}E\otimes q^{\ast}\mathcal{O}(i)$.

Let $\widetilde{C}_{\ast}$ be the complex of locally free sheaves given by
$\widetilde{C}_{i}=E_{i}(i)\oplus E_{i-1}(i-1)$ with differential given by
$\operatorname{d}(a,b)=(b,0)$. On
$X\times(\mathbb{P}^{1}\setminus\\{\infty\\})$ we consider, for each $i$, the
inclusion of vector bundles $\gamma_{i}\colon
E_{i}\hookrightarrow\widetilde{C}_{i}$ given by $s\longmapsto(s\otimes
y^{i},\operatorname{d}s\otimes y^{i-1})$. Let $G$ be the product of the
Grassmann bundles $Gr(n_{i},\widetilde{C}_{i})$ that parametrize rank
$n_{i}=\operatorname{rk}E_{i}$ subbundles of $\widetilde{C}_{i}$ over
$X\times\mathbb{P}^{1}$. The inclusion $\gamma_{\ast}\colon\bigoplus
E_{i}\longrightarrow\bigoplus\widetilde{C}_{i}$ induces a section $s$ of $G$
over $X\times\mathbb{A}^{1}$.

Then $W(E_{\ast})$ is defined to be the closure of $s(X\times\mathbb{A}^{1})$
in $G$. Since the projection from $G$ to $X\times\mathbb{P}^{1}$ is proper,
the same is true for the induced map $\pi\colon W\longrightarrow
X\times\mathbb{P}^{1}$. For each $i$, the induced map $W\longrightarrow
Gr(n_{i},\widetilde{C}_{i})$ defines a subbundle
$\operatorname{tr}_{1}(E_{\ast})_{i}$ of $\pi^{\ast}\widetilde{C}_{i}$. This
subbundle agrees with $E_{i}$ over $X\times\mathbb{A}^{1}$. The differential
of $\widetilde{C}_{\ast}$ induces a differential on
$\operatorname{tr}_{1}(E_{\ast})_{\ast}$.

Assume now that the bundles $E_{i}$ are provided with hermitian metrics. Using
the Fubini-Study metric of $\mathcal{O}(1)$ we obtain induced metrics on
$\widetilde{C}_{i}$. Over
$\pi^{-1}(X\times(\mathbb{P}^{1}\setminus\\{\infty\\}))$ we induce a metric on
$\operatorname{tr}_{1}(E_{\ast})_{i}$ by means of the identification with
$E_{i}$. Over $\pi^{-1}(X\times(\mathbb{P}^{1}\setminus\\{0\\}))$ we consider
on $\operatorname{tr}_{1}(E_{\ast})_{i}$ the metric induced by
$\widetilde{C}_{i}$. We glue together both metrics with the partition of unity
$\\{\sigma_{0},\sigma_{\infty}\\}$ of notation 2.2.

In the case we are interested there is a more explicit description of
$\operatorname{tr}_{1}(E_{\ast})_{\ast}$ given in [6] Section 4 (c). Namely,
$\operatorname{tr}_{1}(E_{\ast})_{i}$ is the kernel of the morphism

$\phi\colon p_{W}^{\ast}\widetilde{C}_{i}=p_{W}^{\ast}E_{i}(i)\oplus
p_{W}^{\ast}E_{i-1}(i-1)\longrightarrow p_{W}^{\ast}E_{i-1}(i)\oplus
p_{W}^{\ast}E_{i-2}(i-1)$ (5.5)

given by $\phi(s,t)=(\operatorname{d}s-t\otimes y,\operatorname{d}t)$.

The only statements that are not explicitly proved in [6] Section 4 (c) or
[20] Section 1 are the functoriality when $f$ is not smooth and the properties
of the explicit choice of metrics.

If the complex $E_{\ast}$ is acyclic, then the same is true for
$E^{\prime}_{\ast}=f^{\ast}E_{\ast}$. In this case $W=X\times\mathbb{P}^{1}$
and $W^{\prime}=X^{\prime}\times\mathbb{P}^{1}$. Then the functoriality
follows from the definition of $\operatorname{tr}_{1}(E_{\ast})_{\ast}$.

Assume now that we are in case (iii)c. We can form the Cartesian square

$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}}$$\scriptstyle{g}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{X}$

where $i^{\prime}$ is also a closed immersion of complex manifolds. Then we
have that $E^{\prime}_{\ast}$ is a resolution of $i^{\prime}_{\ast}g^{\ast}F$.
Hence $W^{\prime}=W(E^{\prime}_{\ast})$ is the deformation to the normal cone
of $Y^{\prime}$ and therefore $W^{\prime}=W\underset{X}{\times}X^{\prime}$.
Again the functoriality of $\operatorname{tr}_{1}(E_{\ast})_{\ast}$ can be
checked using the explicit construction of [20] Section 1 that we have
recalled above. ∎

######  Remark 5.6.

  1. (i)

The definition of $\operatorname{tr}_{1}(E_{\ast})$ can be extended to any
bounded chain complex over a integral scheme (see [20]).

  2. (ii)

There is a sign difference in the definition of the inclusion $\gamma$ used in
[20] and the one used in [6]. We have followed the signs of the first
reference.

##  6 Singular Bott-Chern classes

Throughout this section we will use notation 1.3. In particular we will write

$\displaystyle\widetilde{\mathcal{D}}^{n}_{D}(X,p)$
$\displaystyle=\left.\mathcal{D}^{n}_{D}(X,p)\right/\operatorname{d}_{\mathcal{D}}\mathcal{D}^{n-1}_{D}(X,p),$
$\displaystyle\widetilde{\mathcal{D}}^{n}_{D}(X,S,p)$
$\displaystyle=\left.\mathcal{D}^{n}_{D}(X,S,p)\right/\operatorname{d}_{\mathcal{D}}\mathcal{D}^{n-1}_{D}(X,S,p).$

A particularly important current is
$W_{1}\in\mathcal{D}^{1}_{D}(\mathbb{P}^{1},1)$ given by

$W_{1}=[\frac{-1}{2}\log\|t\|^{2}].$ (6.1)

With the above convention, this means that

$W_{1}(\eta)=\frac{1}{2\pi
i}\int_{\mathbb{P}^{1}}\frac{-1}{2}\log\|t\|^{2}\bullet\eta.$ (6.2)

By the Poincaré-Lelong equation

$\operatorname{d}_{\mathcal{D}}W_{1}=\delta_{\infty}-\delta_{0}.$ (6.3)

Note that the current $W_{1}$ was used in the construction of Bott-Chern
classes (definition 2.11) and will also have a role in the definition of
singular Bott-Chern classes.

Before defining singular Bott-Chern classes we need to define the objects that
give rise to them.

######  Definition 6.4.

Let $i\colon Y\longrightarrow X$ be a closed immersion of complex manifolds.
Let $N$ be the normal bundle of $Y$ and let $h_{N}$ be a hermitian metric on
$N$. We denote $\overline{N}=(N,h_{N})$. Let $r_{N}$ be the rank of $N$, that
agrees with the codimension of $Y$ in $X$. Let $\overline{F}=(F,h_{F})$ be a
hermitian vector bundle on $Y$ of rank $r_{F}$. Let $\overline{E}_{\ast}\to
i_{\ast}F$ be a metric on the coherent sheaf $i_{\ast}F$. The four-tuple

$\overline{\xi}=(i,\overline{N},\overline{F},\overline{E}_{\ast}).$ (6.5)

is called a _hermitian embedded vector bundle_. The number $r_{F}$ will be
called the _rank_ of $\overline{\xi}$ and the number $r_{N}$ will be called
the _codimension_ of $\overline{\xi}$.

By convention, any exact complex of hermitian vector bundles on $X$ will be
considered a hermitian embedded vector bundle of any rank and codimension.

Obviously, to any hermitian embedded vector bundle we can associate the
metrized coherent sheaf $(i_{\ast}F,\overline{E}_{\ast}\to i_{\ast}F)$.

######  Definition 6.6.

A _singular Bott-Chern class_ for a hermitian embedded vector bundle
$\overline{\xi}$ is a class
$\widetilde{\eta}\in\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,p)$ such
that

$\operatorname{d}_{\mathcal{D}}\eta=\sum_{i=0}^{n}(-1)^{i}[\operatorname{ch}(\overline{E}_{i})]-i_{\ast}([\operatorname{Td}^{-1}(\overline{N})\operatorname{ch}(\overline{F})])$
(6.7)

for any current $\eta\in\tilde{\eta}$.

The existence of this class is guaranteed by the Grothendieck-Riemann-Roch
theorem, which implies that the two currents in the right hand side of
equation (6.7) are cohomologous.

Even if we have defined singular Bott-Chern classes as classes of currents
with arbitrary singularities, it is an important observation that in each
singular Bott-Chern class we can find representatives with controlled
singularities. Let $N^{\ast}_{Y,0}$ be the conormal bundle of $Y$ with the
zero section deleted. It is a closed conical subset of $T^{\ast}_{0}(X)$.
Since the current

$\sum_{i=0}^{n}(-1)^{i}[\operatorname{ch}(\overline{E}_{i})]-i_{\ast}([\operatorname{Td}^{-1}(\overline{N})\operatorname{ch}(\overline{F})])\\\
=\sum_{i=0}^{n}(-1)^{i}[\operatorname{ch}(\overline{E}_{i})]-\operatorname{Td}^{-1}(\overline{N})\operatorname{ch}(\overline{F})\delta_{Y}$

belongs to $\mathcal{D}^{\ast}_{D}(X,N^{\ast}_{Y,0},p)$, by corollary 4.7, we
obtain

######  Proposition 6.8.

Let $\overline{\xi}=(i,\overline{N},\overline{F},\overline{E}_{\ast})$ be a
hermitian embedded vector bundle as before. Then any singular Bott-Chern class
for $\overline{\xi}$ belongs to the subset

$\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,N^{\ast}_{Y,0},p)\subset\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,p).$

$\square$

This result will allow us to define inverse images of singular Bott-Chern
classes for certain maps.

Let $f\colon X^{\prime}\longrightarrow X$ be a morphism of complex manifolds
that is transverse to $Y$. We form the Cartesian square

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0.0pt\hbox{\hbox{\kern 3.0pt\raise
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Observe that, by the transversality hypothesis, the normal bundle to
$Y^{\prime}$ on $X^{\prime}$ is the inverse image of the normal bundle to $Y$
on $X$ and $f^{\ast}E_{\ast}$ is a resolution of $i^{\prime}_{\ast}g^{\ast}F$.
Thus we write
$f^{\ast}\overline{\xi}=(i^{\prime},f^{\ast}\overline{N},g^{\ast}\overline{F},f^{\ast}\overline{E}_{\ast})$,
which is a hermitian embedded vector bundle.

By proposition 6.8, given any singular Bott-Chern class $\widetilde{\eta}$ for
$\xi$, we can find a representative
$\eta\in\bigoplus_{p}\mathcal{D}^{2p-1}_{D}(X,N^{\ast}_{Y,0},p)$. By theorem
4.3, there is a well defined current $f^{\ast}\eta$ and it is a singular Bott-
Chern current for $f^{\ast}\xi$. Therefore we can define
$f^{\ast}(\widetilde{\eta})=\widetilde{f^{\ast}(\eta)}$. Again by theorem 4.3,
this class does not depend on the choice of the representative $\eta$.

Our next objective is to study the possible definitions of functorial singular
Bott-Chern classes.

######  Definition 6.9.

Let $r_{F}$ and $r_{N}$ be two integers. A _theory of singular Bott-Chern
classes of rank $r_{F}$ and codimension $r_{N}$_ is an assignment which, to
each hermitian embedded vector bundle $\overline{\xi}=(i\colon
Y\longrightarrow X,\overline{N},\overline{F},\overline{E}_{\ast})$ of rank
$r_{F}$ and codimension $r_{N}$, assigns a class of currents

$T(\overline{\xi})\in\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,p)$

satisfying the following properties

  1. (i)

(Differential equation) The following equality holds

$\operatorname{d}_{\mathcal{D}}T(\overline{\xi})=\sum_{i}(-1)^{i}[\operatorname{ch}(\overline{E}_{i})]-i_{\ast}([\operatorname{Td}^{-1}(\overline{N})\operatorname{ch}(\overline{F})]).$
(6.10)

  2. (ii)

(Functoriality) For every morphism $f\colon X^{\prime}\longrightarrow X$ of
complex manifolds that is transverse to $Y$, then

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

  3. (iii)

(Normalization) Let $\overline{A}=(A_{\ast},g_{\ast})$ be a non-negatively
graded orthogonally split complex of vector bundles. Write
$\overline{\xi}\oplus\overline{A}=(i\colon Y\longrightarrow
X,\overline{N},\overline{F},\overline{E}_{\ast}\oplus\overline{A}_{\ast})$.
Then $T(\overline{\xi})=T(\overline{\xi}\oplus\overline{A})$. Moreover, if
$X=\operatorname{Spec}\mathbb{C}$ is one point, $Y=\emptyset$ and
$\overline{E}_{\ast}=0$, then $T(\overline{\xi})=0$.

A _theory of singular Bott-Chern classes_ is an assignment as before, for all
positive integers $r_{F}$ and $r_{M}$. When the inclusion $i$ and the bundles
$F$ and $N$ are clear from the context, we will denote $T(\overline{\xi})$ by
$T(\overline{E}_{\ast})$. Sometimes we will have to restrict ourselves to
complex algebraic manifolds and algebraic vector bundles. In this case we will
talk of _theory of singular Bott-Chern classes for algebraic vector bundles_.

######  Remark 6.11.

  1. (i)

Recall that the case when $Y=\emptyset$ and $\overline{E}_{\ast}$ is any
bounded exact sequence of hermitian vector bundles is considered a hermitian
embedded vector bundle of arbitrary rank. In this case, the properties above
imply that

$T(\overline{\xi})=[\widetilde{\operatorname{ch}}(\overline{E}_{\ast})],$

where $\widetilde{\operatorname{ch}}$ is the Bott-Chern class associated to
the Chern character. That is, for acyclic complexes, any theory of singular
Bott-Chern classes agrees with the Bott-Chern classes associated to the Chern
character.

  2. (ii)

If the map $f$ is transverse to $Y$, then either $f^{-1}(Y)$ is empty or it
has the same codimension as $Y$. Moreover, it is clear that $f^{\ast}F$ has
the same rank as $F$. Therefore, the properties of singular Bott-Chern classes
do not mix rank or codimension. This is why we have defined singular Bott-
Chern classes for a particular rank and codimension.

  3. (iii)

By contrast with the case of Bott-Chern classes, the properties above are not
enough to characterize singular Bott-Chern classes.

For the rest of this section we will assume the existence of a theory of
singular Bott-Chern classes and we will obtain some consequences of the
definition.

We start with the compatibility of singular Bott-Chern classes with exact
sequences and Bott-Chern classes.

Let

$\overline{\chi}\colon
0\longrightarrow\overline{F}_{n}\longrightarrow\dots\longrightarrow\overline{F}_{1}\longrightarrow\overline{F}_{0}\longrightarrow
0$ (6.12)

be a bounded exact sequence of hermitian vector bundles on $Y$. For
$j=0,\dots,n$, let $\overline{E}_{j,\ast}\longrightarrow i_{\ast}F_{j}$ be a
resolution, and assume that they fit in a commutative diagram

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with exact rows. We write $\overline{\xi_{j}}=(i\colon Y\longrightarrow
X,\overline{N},\overline{F}_{j},\overline{E}_{j,\ast})$. For each $k$, we
denote by $\overline{\eta}_{k}$ the exact sequence

$0\longrightarrow\overline{E}_{n,k}\longrightarrow\dots\longrightarrow\overline{E}_{1,k}\longrightarrow\overline{E}_{0,k}\longrightarrow
0.$

######  Proposition 6.13.

With the above notations, the following equation holds:

$T(\bigoplus_{j\text{ even}}\overline{\xi}_{j})-T(\bigoplus_{j\text{
odd}}\overline{\xi}_{j})=\sum_{k}(-1)^{k}[\widetilde{\operatorname{ch}}(\overline{\eta}_{k})]-i_{\ast}([\operatorname{Td}^{-1}(\overline{N})\widetilde{\operatorname{ch}}(\overline{\chi})]).$

Here the direct sum of hermitian embedded vector bundles, involving the same
embedding and the same hermitian normal bundle, is defined in the obvious
manner.

###### Proof.

We consider the construction of theorem 5.4 for each of the exact sequences
$\overline{\eta}_{k}$ and the exact sequence $\overline{\chi}$. For each $k$,
we have $W_{X}:=W(\overline{\eta}_{k})=X\times\mathbb{P}^{1}$ and we denote
$W_{Y}:=W(\overline{\chi})=Y\times\mathbb{P}^{1}$. On $W_{Y}$ we consider the
transgression exact sequence $\operatorname{tr}_{1}(\overline{\chi})_{\ast}$
and on $W_{X}$ we consider the transgression exact sequences
$\operatorname{tr}_{1}(\overline{\eta_{k}})_{\ast}$. We denote by $j\colon
W_{Y}\longrightarrow W_{X}$ the induced morphism. Then there is an exact
sequence (of exact sequences)

$\dots\longrightarrow\operatorname{tr}_{1}(\overline{\eta}_{1})_{\ast}\longrightarrow\operatorname{tr}_{1}(\overline{\eta}_{0})_{\ast}\longrightarrow
j_{\ast}\operatorname{tr}_{1}(\overline{\chi})_{\ast}\longrightarrow 0.$

We denote

$\displaystyle\operatorname{tr}_{1}(\overline{\chi})_{+}$
$\displaystyle=\bigoplus_{j\text{
even}}\operatorname{tr}_{1}(\overline{\chi})_{j},\quad$
$\displaystyle\operatorname{tr}_{1}(\overline{\chi})_{-}$
$\displaystyle=\bigoplus_{j\text{
odd}}\operatorname{tr}_{1}(\overline{\chi})_{j},$
$\displaystyle\operatorname{tr}_{1}(\overline{\eta}_{k})_{+}$
$\displaystyle=\bigoplus_{j\text{
even}}\operatorname{tr}_{1}(\overline{\eta}_{k})_{j},\quad$
$\displaystyle\operatorname{tr}_{1}(\overline{\eta}_{k})_{-}$
$\displaystyle=\bigoplus_{j\text{
odd}}\operatorname{tr}_{1}(\overline{\eta}_{k})_{j},$

and

$\displaystyle\operatorname{tr}_{1}(\overline{\xi})_{+}$
$\displaystyle=(j\colon W_{Y}\longrightarrow
W_{X},p_{Y}^{\ast}\overline{N},\operatorname{tr}_{1}(\overline{\chi})_{+},\operatorname{tr}_{1}(\overline{\eta}_{\ast})_{+}),$
$\displaystyle\operatorname{tr}_{1}(\overline{\xi})_{-}$
$\displaystyle=(j\colon W_{Y}\longrightarrow
W_{X},p_{Y}^{\ast}\overline{N},\operatorname{tr}_{1}(\overline{\chi})_{-},\operatorname{tr}_{1}(\overline{\eta}_{\ast})_{-}),$

where here $p_{Y}\colon W_{Y}\longrightarrow Y$ denotes the projection.

We consider the current on $X\times\mathbb{P}^{1}$ given by
$W_{1}\bullet\left(T(\operatorname{tr}_{1}(\overline{\xi})_{+})-T(\operatorname{tr}_{1}(\overline{\xi})_{-})\right)$.
This current is well defined because the wave front set of $W_{1}$ is the
conormal bundle of $(X\times\\{0\\})\cup(X\times\\{\infty\\})$, whereas the
wave front set of $T(\operatorname{tr}_{1}(\overline{\xi})_{\pm})$ is the
conormal bundle of $Y\times\mathbb{P}^{1}$.

By the functoriality of the transgression exact sequences, we obtain that

$\operatorname{tr}_{1}(\overline{\xi})_{+}\mid_{X\times\\{0\\}}=\bigoplus_{j\text{
even}}\overline{\xi}_{j},\quad\operatorname{tr}_{1}(\overline{\xi})_{-}\mid_{X\times\\{0\\}}=\bigoplus_{j\text{
odd}}\overline{\xi}_{j}.$

Moreover, using the fact that, for any bounded acyclic complex of hermitian
vector bundles $\overline{E}_{\ast}$, the exact sequence
$\operatorname{tr}_{1}(\overline{E}_{\ast})\mid_{X\times\\{\infty\\}}$ is
orthogonally split, we have an isometry

$\operatorname{tr}_{1}(\overline{\xi})_{+}\mid_{X\times\\{\infty\\}}\cong\operatorname{tr}_{1}(\overline{\xi})_{-}\mid_{X\times\\{\infty\\}}.$

We now denote by $p_{X}\colon W_{X}\longrightarrow X$ the projection. Using
the properties that define a theory of singular Bott-Chern classes, in the
group $\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,N^{\ast}_{Y,0},p)$,
the following holds

$\displaystyle 0$
$\displaystyle=\operatorname{d}_{\mathcal{D}}(p_{X})_{\ast}\left(W_{1}\bullet
T(\operatorname{tr}_{1}(\overline{\xi})_{+})-W_{1}\bullet
T(\operatorname{tr}_{1}(\overline{\xi})_{-})\right)$
$\displaystyle=\left(T(\operatorname{tr}_{1}(\overline{\xi})_{+})-T(\operatorname{tr}_{1}(\overline{\xi})_{-})\right)\mid_{X\times\\{\infty\\}}-\left(T(\operatorname{tr}_{1}(\overline{\xi})_{+})-T(\operatorname{tr}_{1}(\overline{\xi})_{-})\right)\mid_{X\times\\{0\\}}$
$\displaystyle-(p_{X})_{\ast}\sum_{k}(-1)^{k}W_{1}\bullet\left(\operatorname{ch}(\operatorname{tr}_{1}(\overline{\eta}_{k})_{+})-\operatorname{ch}(\operatorname{tr}_{1}(\overline{\eta}_{k})_{-})\right)$
$\displaystyle+(p_{X})_{\ast}\left(W_{1}\bullet
j_{\ast}\left[\operatorname{Td}^{-1}(p_{Y}^{\ast}\overline{N})\operatorname{ch}(\operatorname{tr}_{1}(\overline{\chi})_{+})-\operatorname{Td}^{-1}(p_{Y}^{\ast}\overline{N})\operatorname{ch}(\operatorname{tr}_{1}(\overline{\chi})_{-})\right]\right)$
$\displaystyle=-T(\bigoplus_{j\text{
even}}\overline{\xi}_{j})+T(\bigoplus_{j\text{
odd}}\overline{\xi}_{j})+\sum(-1)^{k}[\widetilde{\operatorname{ch}}(\overline{\eta}_{k})]-i_{\ast}[\operatorname{Td}^{-1}(\overline{N})\bullet\widetilde{\operatorname{ch}}(\overline{\chi})],$

which implies the proposition. ∎

The following result is a consequence of proposition 6.13 and theorem 2.24.

######  Corollary 6.14.

Let $Y\longrightarrow X$ be a closed immersion of complex manifolds. Let
$\overline{\chi}$ be an exact sequence of hermitian vector bundles on $Y$ as
(6.12). For each $j$, let $\xi_{j}=(i\colon Y\longrightarrow
X,\overline{N},\overline{F}_{j},\overline{E}_{j,\ast})$ be a hermitian
embedded vector bundle. We denote by $\overline{\varepsilon}$ the induced
exact sequence of metrized coherent sheaves. Then

$T(\bigoplus_{j\text{ even}}\overline{\xi}_{j})-T(\bigoplus_{j\text{
odd}}\overline{\xi}_{j})=[\widetilde{\operatorname{ch}}(\overline{\varepsilon})]-i_{\ast}([\operatorname{Td}^{-1}(\overline{N})\widetilde{\operatorname{ch}}(\overline{\chi})]).$

$\square$

We now study the effect of changing the metric of the normal bundle $N$.

######  Proposition 6.15.

Let $\overline{\xi}_{0}=(i,\overline{N}_{0},\overline{F},\overline{E}_{\ast})$
be a hermitian embedded vector bundle, where $\overline{N}_{0}=(N,h_{0})$. Let
$h_{1}$ be another metric in the vector bundle $N$ and write
$\overline{N}_{1}=(N,h_{1})$,
$\overline{\xi}_{1}=(i,\overline{N}_{1},\overline{F},\overline{E}_{\ast})$.
Then

$T(\overline{\xi}_{0})-T(\overline{\xi}_{1})=-i_{\ast}[\widetilde{\operatorname{Td}^{-1}}(N,h_{0},h_{1})\operatorname{ch}(\overline{F})].$

###### Proof.

The proof is completely analogous to the proof of proposition 6.13. ∎

We now study the case when $Y$ is the zero section of a completed vector
bundle. Let $\overline{F}$ and $\overline{N}$ be hermitian vector bundles over
$Y$. We denote $P=\mathbb{P}(N\oplus\mathbb{C})$, the projective bundle of
lines in $N\oplus\mathcal{O}_{Y}$. Let $s\colon Y\longrightarrow P$ denote the
zero section and let $\pi_{P}\colon P\longrightarrow Y$ denote the projection.
Let $K(\overline{F},\overline{N})$ be the Koszul resolution of definition 5.3.
We will use the notations before this definition.

The following result is due to Bismut, Gillet and Soulé for the particular
choice of singular Bott-Chern classes defined in [6].

######  Theorem 6.16.

Let $T$ be a theory of singular Bott-Chern classes of rank $r_{F}$ and
codimension $r_{N}$. Let $Y$ be a complex manifold and let $\overline{F}$ and
$\overline{N}$ be hermitian vector bundles of rank $r_{F}$ and $r_{N}$
respectively. Then the current
$(\pi_{P})_{\ast}(T(K(\overline{F},\overline{N})))$ is closed. Moreover the
cohomology class that it represents does not depend on the metric of $N$ and
$F$ and determines a characteristic class for pairs of vector bundles of rank
$r_{F}$ and $r_{N}$. We denote this class by $C_{T}(F,N)$.

###### Proof.

We have that

$\displaystyle\operatorname{d}_{\mathcal{D}}(\pi_{P})_{\ast}$
$\displaystyle(T(K(\overline{F},\overline{N})))$
$\displaystyle=(\pi_{P})_{\ast}(\operatorname{d}_{\mathcal{D}}T(K(\overline{F},\overline{N})))$
$\displaystyle=(\pi_{P})_{\ast}\left(\sum_{k=0}^{r}(-1)^{k}[\operatorname{ch}(\bigwedge^{k}\overline{Q}^{\vee})\pi_{P}^{\ast}\operatorname{ch}(\overline{F})]-s_{\ast}[\operatorname{Td}^{-1}(\overline{N})\operatorname{ch}(\overline{F})]\right)$
$\displaystyle=\left((\pi_{P})_{\ast}[c_{r}(\overline{Q})\operatorname{Td}^{-1}(\overline{Q})]-[\operatorname{Td}^{-1}(\overline{N})]\right)\operatorname{ch}(\overline{F})).$

Therefore, the fact that the current
$(\pi_{P})_{\ast}(T(K(\overline{F},\overline{N})))$ is closed follows from
corollary 3.8. The fact that this class is functorial on
$(Y,\overline{N},\overline{F})$ is clear from the construction Thus, the fact
that it does not depend on the hermitian metrics of $N$ and $F$ follows from
proposition 1.7. ∎

######  Remark 6.17.

By theorem 1.8 we know that, if we restrict ourselves to the algebraic
category, $C_{T}(F,N)$ is given by a power series on the Chern classes with
coefficients in $\mathbb{D}$. By degree reasons

$C_{T}(F,N)\in\bigoplus_{p}H_{\mathcal{D}^{\text{{\rm
an}}}}^{2p-1}(Y,\mathbb{R}(p)).$

Let ${\bf 1}_{1}\in H^{1}_{\mathcal{D}}(\ast,\mathbb{R}(1))$ be the element
determined by the constant function with value 1 in $\mathcal{D}^{1}(\ast,1)$.
Then $C_{T}(F,N)/{\bf 1}_{1}$ is a power series in the Chern classes of $N$
and $F$ with real coefficients.

##  7 Classification of theories of singular Bott-Chern classes

The aim of this section is to give a complete classification of the possible
theories of singular Bott-Chern classes. This classification is given in terms
of the characteristic class $C_{T}$ introduced in the previous section.

######  Theorem 7.1.

Let $r_{F}$ and $r_{N}$ be two positive integers. Let $C$ be a characteristic
class for pairs of vector bundles of rank $r_{F}$ and $r_{N}$. Then there
exists a unique theory $T_{C}$ of singular Bott-Chern classes of rank $r_{F}$
and codimension $r_{N}$ such that $C_{T_{C}}=C$.

###### Proof.

We first prove the uniqueness. Assume that $T$ is a theory of singular Bott-
Chern classes such that $C_{T}=C$. Let $\overline{\xi}=(i\colon
Y\longrightarrow X,\overline{N},\overline{F},\overline{E}_{\ast})$ be a
hermitian embedded vector bundle as in section 6. Let $W$ be the deformation
to the normal cone of $Y$. We will use all the notations of section 5. In
particular, we will denote by
$p_{\widetilde{X}}\colon\widetilde{X}\longrightarrow X$ and $p_{P}\colon
P\longrightarrow X$ the morphisms induced by restricting $p_{W}$. Recall that
$p_{P}$ can be factored as

$P\overset{\pi_{P}}{\longrightarrow}Y\overset{i}{\longrightarrow}X.$

The normal vector bundle to the inclusion $j\colon
Y\times\mathbb{P}^{1}\longrightarrow W$ is isomorphic to $p_{Y}^{\ast}N\otimes
q_{Y}^{\ast}\mathcal{O}(-1)$. We provide it with the hermitian metric induced
by the metric of $N$ and the Fubini-Study metric of $\mathcal{O}(-1)$ and we
denote it by $\overline{N}^{\prime}$.

By theorem 5.4 we have a complex of hermitian vector bundles,
$\operatorname{tr}_{1}(E_{\ast})_{\ast}$ such that the restriction
$\operatorname{tr}_{1}(E_{\ast})_{\ast}|_{X\times\\{0\\}}$ is isometric to
$E_{\ast}$, the restriction
$\operatorname{tr}_{1}(E_{\ast})_{\ast}|_{\widetilde{X}}$ is orthogonally
split and there is an exact sequence on $P$

$0\longrightarrow
A_{\ast}\longrightarrow\operatorname{tr}_{1}(E_{\ast})_{\ast}|_{P}\longrightarrow
K(F,N)\longrightarrow 0,$

where $A_{\ast}$ is split acyclic and $K(F,N)$ is the Koszul resolution.
Recall that we have trivialized $N^{-1}_{\infty/\mathbb{P}^{1}}$ by means of
the section $y$ of $\mathcal{O}_{\mathbb{P}^{1}}(1)$. We choose a hermitian
metric in every bundle of $A_{\ast}$ such that it becomes orthogonally split.
For each $k$ we will denote by $\overline{\eta}_{k}$ the exact sequence of
hermitian vector bundles

$0\longrightarrow\overline{A}_{k}\longrightarrow\operatorname{tr}_{1}(\overline{E}_{\ast})_{k}|_{P}\longrightarrow
K(\overline{F},\overline{N})_{k}\longrightarrow 0.$ (7.2)

Observe that the current $W_{1}$ is defined as the current associated to a
locally integrable differential form. The pull-back of this form to $W$ is
also locally integrable. Therefore it defines a current on $W$ that we also
denote by $W_{1}$. Moreover, since the wave front sets of $W_{1}$ and of
$T(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})$ are disjoint, there is
a well defined current $W_{1}\bullet
T(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})$. Then, using the
properties of singular Bott-Chern classes in definition 6.9, the equality

$\displaystyle 0$
$\displaystyle=\operatorname{d}_{\mathcal{D}}(p_{W})_{\ast}\left(W_{1}\bullet
T(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})\right)$
$\displaystyle=(p_{\widetilde{X}})_{\ast}(T(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})|_{\widetilde{X}})+(p_{P})_{\ast}(T(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})|_{P})-T(\overline{\xi})$
$\displaystyle-(p_{W})_{\ast}\left(W_{1}\bullet\left(\sum_{k}(-1)^{k}\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})-(j_{\ast}(\operatorname{ch}(p_{Y}^{\ast}\overline{F})\operatorname{Td}^{-1}(\overline{N}^{\prime}))\right)\right)$

holds in the group $\bigoplus_{k}\widetilde{\mathcal{D}}^{2k-1}(X,k)$. By
properties 6.9(ii) and 6.9(iii),
$T(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})|_{\widetilde{X}}=T(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast}|_{\widetilde{X}})=0$.

By proposition 6.13 we have

$T(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast}|_{P})=T(K(\overline{F},\overline{N}))-\sum_{k}(-1)^{k}[\widetilde{\operatorname{ch}}(\overline{\eta}_{k})].$

Moreover, we have

$(p_{P})_{\ast}(T(K(\overline{F},\overline{N})))=i_{\ast}(\pi_{P})_{\ast}(T(K(\overline{F},\overline{N})))=i_{\ast}C_{T}(F,N).$

By the definition of $N^{\prime}$ and the choice of its metric, there are two
differential forms $a,b$ on $Y$, such that

$\operatorname{ch}(p_{Y}^{\ast}\overline{F})\operatorname{Td}^{-1}(\overline{N}^{\prime})=p_{Y}^{\ast}(a)+p_{Y}^{\ast}(b)\land
q_{Y}^{\ast}(c_{1}(\mathcal{O}(-1))).$

We denote $\omega=-c_{1}(\mathcal{O}(-1))$. By the properties of the Fubini-
Study metric, $\omega$ is invariant under the involution of $\mathbb{P}^{1}$
that sends $t$ to $1/t$. Then

$(p_{W})_{\ast}\left(W_{1}\bullet(j_{\ast}(\operatorname{ch}(p_{Y}^{\ast}\overline{F})\operatorname{Td}^{-1}(\overline{N}^{\prime}))\right)=i_{\ast}(p_{Y})_{\ast}(W_{1}\bullet(p_{Y}^{\ast}a+p_{Y}^{\ast}b\omega))=0$

because the current $W_{1}$ changes sign under the involution $t\longmapsto
1/t$.

Summing up, we have obtained the equation

$T(\overline{\xi})=-(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{k})\right)\\\
-\sum_{k}(-1)^{k}(p_{P})_{\ast}[\widetilde{\operatorname{ch}}(\overline{\eta}_{k})]+i_{\ast}C_{T}(F,N).$
(7.3)

Hence the singular Bott-Chern class is characterized by the properties of
definition 6.9 and the characteristic class $C_{T}$.

In order to prove the existence of a theory of singular Bott-Chern classes, we
use equation (7.3) to define a class $T_{C}(\xi)$ as follows.

######  Definition 7.4.

Let $C$ be a characteristic class for pairs of vector bundles of rank $r_{F}$
and $r_{N}$ as in theorem 7.1. Let $\overline{\xi}=(i\colon Y\longrightarrow
X,\overline{N},\overline{F},\overline{E}_{\ast})$ be as in definition 6.9. Let
$\overline{A}_{\ast}$, $\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast}$ and
$\overline{\eta}_{\ast}$ be as in (7.2). Then we define

$T_{C}(\overline{\xi})=-(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{k})\right)\\\
-\sum_{k}(-1)^{k}(p_{P})_{\ast}[\widetilde{\operatorname{ch}}(\overline{\eta}_{k})]+i_{\ast}C(F,N).$
(7.5)

We have to prove that this definition does not depend on the choice of the
metric of $\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast}$ or the metric of
$\overline{A}_{\ast}$, that $T_{C}$ satisfies the properties of definition 6.9
and that the characteristic class $C_{T_{C}}$ agrees with $C$.

First we prove the independence from the metrics. We denote by $h_{k}$ the
hermitian metric on $\operatorname{tr}_{1}(\overline{E}_{\ast})_{k}$ and by
$g_{k}$ the hermitian metric on $A_{k}$. Let $h^{\prime}_{k}$ and
$g^{\prime}_{k}$ be another choice of metrics satisfying also that
$(A_{\ast},g^{\prime}_{\ast})$ is orthogonally split, that
$(\operatorname{tr}_{1}(E_{\ast})_{k},h^{\prime}_{k})|_{X\times\\{0\\}}$ is
isometric to $\overline{E}_{k}$ and that
$(\operatorname{tr}_{1}(E_{\ast})_{k},h^{\prime}_{k})|_{\widetilde{X}}$ is
orthogonally split. We denote by $\overline{\eta}^{\prime}_{k}$ the exact
sequence $\eta_{k}$ provided with the metrics $g^{\prime}$ and $h^{\prime}$.
Then, in the group $\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}(X,p)$, we have

$\sum_{k}(-1)^{k}(p_{P})_{\ast}[\widetilde{\operatorname{ch}}(\overline{\eta}_{k})]-\sum_{k}(-1)^{k}(p_{P})_{\ast}[\widetilde{\operatorname{ch}}(\overline{\eta}^{\prime}_{k})]=\\\
\sum_{k}(-1)^{k}(p_{P})_{\ast}\left[\widetilde{\operatorname{ch}}(A_{k},g_{k},g^{\prime}_{k})\right]-\sum_{k}(-1)^{k}(p_{P})_{\ast}\left[\widetilde{\operatorname{ch}}(\operatorname{tr}_{1}(E_{\ast})_{k}|_{P},h_{k},h^{\prime}_{k})\right].$
(7.6)

Observe that the first term of the right hand side vanishes due to the
hypothesis of $A_{\ast}$ being orthogonally split for both metrics.

Moreover, we also have,

$(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\operatorname{tr}_{1}(E_{\ast})_{k},h_{k})\right)-\\\
(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\operatorname{tr}_{1}(E_{\ast})_{k},h^{\prime}_{k})\right)=\\\
(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{d}_{\mathcal{D}}\widetilde{\operatorname{ch}}(\operatorname{tr}_{1}(E_{\ast})_{k},h_{k},h^{\prime}_{k})\right).$
(7.7)

But, in the group $\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}(X,p)$,

$(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{d}_{\mathcal{D}}\widetilde{\operatorname{ch}}(\operatorname{tr}_{1}(E_{\ast})_{k},h_{k},h^{\prime}_{k})\right)=\\\
\sum_{k}(-1)^{k}(p_{\widetilde{X}})_{\ast}[\widetilde{\operatorname{ch}}(\operatorname{tr}_{1}(E_{\ast})_{k},h_{k},h^{\prime}_{k})]|_{\widetilde{X}}\\\
+\sum_{k}(-1)^{k}(p_{P})_{\ast}[\widetilde{\operatorname{ch}}(\operatorname{tr}_{1}(E_{\ast})_{k},h_{k},h^{\prime}_{k})]|_{P})\\\
-\sum_{k}(-1)^{k}[\widetilde{\operatorname{ch}}(\operatorname{tr}_{1}(E_{\ast})_{k},h_{k},h^{\prime}_{k})]|_{X\times\\{0\\}}.$
(7.8)

The last term of the right hand side vanishes because the metrics $h_{k}$ and
$h^{\prime}_{k}$ agree when restricted to $X\times\\{0\\}$ and the first term
vanishes by the hypothesis that
$\operatorname{tr}_{1}(E_{\ast})_{\ast}|_{\widetilde{X}}$ is orthogonally
split with both metrics. Combining equations (7.6), (7.7) and (7.8) we obtain
that the right hand side of equation (7.5) does not depend on the choice of
metrics.

We next prove the property (i) of definition 6.9. We compute

$\operatorname{d}_{\mathcal{D}}T_{C}(\overline{\xi})=-\sum_{k}(-1)^{k}\left((p_{\widetilde{X}})_{\ast}\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{k}|_{\widetilde{X}})+(p_{P})_{\ast}\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{k}|_{P})\right)\\\
+\sum_{k}(-1)^{k}\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{k}|_{X\times\\{0\\}})\\\
-\sum_{k}(-1)^{k}(p_{P})_{\ast}\left(\operatorname{ch}(\overline{A}_{k})+\operatorname{ch}(K(\overline{F},\overline{N})_{k})-\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{k}|_{P})\right).$

Using that $\overline{A}_{\ast}$ and that
$\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast}|_{\widetilde{X}}$ are
orthogonally split and corollary 3.8 we obtain

$\displaystyle\operatorname{d}_{\mathcal{D}}T_{C}(\overline{\xi})$
$\displaystyle=\sum_{k}(-1)^{k}\operatorname{ch}(\overline{E}_{k})-\sum_{k}(-1)^{k}(p_{P})_{\ast}\operatorname{ch}(K(\overline{F},\overline{N})_{k})$
$\displaystyle=\sum_{k}(-1)^{k}[\operatorname{ch}(\overline{E}_{k})]-(p_{P})_{\ast}[c_{r}(\overline{Q})\operatorname{Td}^{-1}(\overline{Q})]$
$\displaystyle=\sum_{k}(-1)^{k}[\operatorname{ch}(\overline{E}_{k})]-i_{\ast}[\operatorname{ch}(\overline{F})\operatorname{Td}^{-1}(\overline{N})].$

We now prove the normalization property. We consider first the case when
$Y=\emptyset$ and $\overline{E}_{\ast}$ is a non-negatively graded
orthogonally split complex. We denote by

$\overline{K}_{i}=\operatorname{Ker}(\operatorname{d}_{i}\colon
E_{i}\longrightarrow E_{i-1})$

with the induced metric. By hypothesis there are isometries

$\overline{E}_{i}=\overline{K}_{i}\oplus\overline{K}_{i-1}.$

Under these isometries, the differential is $\operatorname{d}(s,t)=(t,0)$.
Following the explicit construction of $\operatorname{tr}_{1}(E_{\ast})$ given
in [20], recalled in definition 2.5, we see that

$\operatorname{tr}_{1}(E_{\ast})_{i}=p^{\ast}K_{i}\otimes
q^{\ast}\mathcal{O}(i)\oplus p^{\ast}K_{i-1}\otimes
q^{\ast}\mathcal{O}(i-1)=K_{i}(i)\oplus K_{i-1}(i-1).$

Moreover, we can induce a metric on $\operatorname{tr}_{1}(E_{\ast})_{\ast}$
satisfying the hypothesis of definition 2.9 by means of the metric of the
bundles $K_{i}$ and the Fubini-Study metric on the bundles $\mathcal{O}(i)$.
It is clear that the second and third terms of the right hand side of equation
(7.3) are zero. For the first term we have

$\displaystyle\sum_{k}(-1)^{k}$
$\displaystyle(p_{W})_{\ast}W_{1}\bullet\left(\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{k})\right)$
$\displaystyle=(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\overline{K}_{k}(k)\overset{\perp}{\oplus}\overline{K}_{k-1}(k-1))\right)$
$\displaystyle=(p_{W})_{\ast}\left(W_{1}\bullet(a+b\land\omega)\right),$

where $\omega$ is the Fubini-Study $(1,1)$-form on $\mathbb{P}^{1}$ and $a,b$
are inverse images of differential forms on $X$. Therefore we obtain that
$T_{C}(\overline{E}_{\ast})=0$.

Now let $\overline{\xi}=(i\colon Y\longrightarrow
X,\overline{N},\overline{F},\overline{E}_{\ast})$ and let
$\overline{B}_{\ast}$ be a non-negatively graded orthogonally split complex of
vector bundles. By [20] section 1.1, we have that $W(E_{\ast}\oplus
B_{\ast})=W(E_{\ast})$ and that

$\operatorname{tr}_{1}(E_{\ast}\oplus
B_{\ast})=\operatorname{tr}_{1}(E_{\ast})\oplus\pi^{\ast}\operatorname{tr}_{1}(B_{\ast}).$

In order to compute $T_{C}(\overline{\xi})$, we have to consider the exact
sequences of hermitian vector bundles over $P$

$\overline{\eta}_{k}\colon
0\longrightarrow\overline{A}_{k}\longrightarrow\operatorname{tr}_{1}(\overline{E}_{\ast})_{k}|_{P}\longrightarrow
K(\overline{F},\overline{N})_{k}\longrightarrow 0,$

whereas, in order to compute $T_{C}(\overline{\xi}\oplus\overline{B}_{\ast})$,
we consider the sequences

$\overline{\eta}^{\prime}_{k}\colon\\\
0\longrightarrow\overline{A}_{k}\oplus\pi^{\ast}(\operatorname{tr}_{1}(\overline{B})_{k})|_{P}\longrightarrow\operatorname{tr}_{1}(\overline{E}_{\ast})_{k}\oplus\pi^{\ast}(\operatorname{tr}_{1}(\overline{B})_{k})|_{P}\longrightarrow
K(\overline{F},\overline{N})_{k}\longrightarrow 0.$

By the additivity of Bott-Chern classes, we have that
$\widetilde{\operatorname{ch}}(\overline{\eta}_{k})=\widetilde{\operatorname{ch}}(\overline{\eta}^{\prime}_{k})$.
Therefore

$\displaystyle
T_{C}(\overline{\xi}\oplus\bar{B}_{\ast})-T_{C}(\overline{\xi})$
$\displaystyle=-(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast}\oplus\overline{B}_{\ast})_{k})\right)$
$\displaystyle\phantom{AAA}+(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{k})\right)$
$\displaystyle=-(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\operatorname{tr}_{1}(\overline{B}_{\ast})_{k})\right)$
$\displaystyle=0.$

The proof of the functoriality is left to the reader.

Finally we prove that $C_{T_{C}}=C$. Let $Y$ be a complex manifold and let
$\overline{F}$ and $\overline{N}$ be two hermitian vector bundles. We write
$X=\mathbb{P}(N\oplus\mathbb{C})$. Let $i\colon Y\longrightarrow X$ be the
inclusion given by the zero section and let $\pi_{X}\colon X\longrightarrow Y$
be the projection. On $X$ we have the tautological exact sequence

$0\longrightarrow\mathcal{O}(-1)\longrightarrow\pi_{X}^{\ast}(N\oplus\mathbb{C})\longrightarrow
Q\longrightarrow 0$

and the Koszul resolution, denoted $K(\overline{F},\overline{N})$. We denote

$\overline{\xi}=(i\colon Y\longrightarrow
X,\overline{N},\overline{F},K(\overline{F},\overline{N})).$

Using the definition of $T_{C}$, that is, equation (7.5), and the fact that
$T_{C}$ satisfies the properties of definition 6.9, hence equation (7.3) is
satisfied, we obtain that

$i_{\ast}C(F,N)=i_{\ast}C_{T_{C}}(F,N)$

Applying $(\pi_{X})_{\ast}$ we obtain that $C(F,N)=C_{T_{C}}(F,N)$ which
finishes the proof of theorem 7.1. ∎

##  8 Transitivity and projection formula

We now investigate how different properties of the characteristic class
$C_{T}$ are reflected in the corresponding theory of singular Bott-Chern
classes.

######  Proposition 8.1.

Let $i\colon Y\hookrightarrow X$ be a closed immersion of complex manifolds.
Let $\overline{F}$ be a hermitian vector bundle on $Y$ and $\overline{G}$ a
hermitian vector bundle on $X$. Let $\overline{N}$ denote the normal bundle to
$Y$ provided with a hermitian metric. Let $\overline{E}_{\ast}$ be a finite
resolution of $i_{\ast}F$ by hermitian vector bundles. We denote
$\overline{\xi}=(i\colon Y\longrightarrow
X,\overline{N},\overline{F},\overline{E}_{\ast})$ and
$\overline{\xi}\otimes\overline{G}=(i\colon Y\longrightarrow
X,\overline{N},\overline{F}\otimes
i^{\ast}\overline{G},\overline{E}_{\ast}\otimes\overline{G})$. Then

$T(\overline{\xi}\otimes\overline{G})-T(\overline{\xi})\bullet\operatorname{ch}(\overline{G})=i_{\ast}(C_{T}(F\otimes
i^{\ast}G,N))-i_{\ast}(C_{T}(F,N))\bullet\operatorname{ch}(\overline{G}).$

###### Proof.

Since the construction of $\operatorname{tr}_{1}(E_{\ast})_{\ast}$ is local on
$X$ and $Y$ and compatible with finite sums, we have that

$W(E_{\ast})=W(E_{\ast}\otimes
G),\qquad\operatorname{tr}_{1}(\overline{E}_{\ast}\otimes\overline{G})_{\ast}=\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast}\otimes
p_{W}^{\ast}\overline{G}.$

We first compute

$(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast}\otimes\overline{G})_{\ast})\right)\\\
=(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})p^{\ast}_{W}\operatorname{ch}(\overline{G})\right)\\\
=(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})\right)\operatorname{ch}(\overline{G}).$
(8.2)

The Koszul resolution of $i_{\ast}(F\otimes i^{\ast}G)$ is given by

$K(F\otimes i^{\ast}G,N)=K(F,N)\otimes p_{P}^{\ast}G.$

For each $k\geq 0$, we will denote by $\overline{\eta}_{k}\otimes
p_{P}^{\ast}\overline{G}$ the exact sequence

$0\longrightarrow\overline{A}_{k}\otimes
p_{P}^{\ast}\overline{G}\longrightarrow\operatorname{tr}_{1}(\overline{E}_{\ast}\otimes\overline{G})_{k}|_{P}\longrightarrow
K(\overline{F},\overline{N})_{k}\otimes
p_{P}^{\ast}\overline{G}\longrightarrow 0.$

Then, we have

$(p_{P})_{\ast}[\widetilde{\operatorname{ch}}(\overline{\eta}_{k}\otimes
p_{P}^{\ast}\overline{G})]=(p_{P})_{\ast}[\widetilde{\operatorname{ch}}(\overline{\eta}_{k})\bullet
p_{P}^{\ast}\operatorname{ch}(\overline{G})]=(p_{P})_{\ast}[\widetilde{\operatorname{ch}}(\overline{\eta}_{k})]\bullet\operatorname{ch}(\overline{G})$
(8.3)

Thus the proposition follows from equation (8.2), equation (8.3) and formula
(7.3). ∎

######  Definition 8.4.

We will say that a theory of singular Bott-Chern classes is _compatible with
the projection formula_ if, whenever we are in the situation of proposition
8.1, the following equality holds:

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

We will say that a characteristic class $C$ (of pairs of vector bundles) is
_compatible with the projection formula_ if it satisfies

$C(F,N)=C(\mathcal{O}_{Y},N)\bullet\operatorname{ch}(F).$

######  Corollary 8.5.

A theory of singular Bott-Chern classes $T$ is compatible with the projection
formula if and only if it is the case for the associated characteristic class
$C_{T}$.

###### Proof.

Assume that $C_{T}$ is compatible with the projection formula and that we are
in the situation of proposition 8.1. Then

$\displaystyle i_{\ast}C_{T}(F\otimes i^{\ast}G,N))$
$\displaystyle=i_{\ast}(C_{T}(\mathcal{O}_{Y},N)\bullet\operatorname{ch}(F\otimes
i^{\ast}G))$
$\displaystyle=i_{\ast}(C_{T}(\mathcal{O}_{Y},N)\bullet\operatorname{ch}(F)i^{\ast}\operatorname{ch}(G))$
$\displaystyle=i_{\ast}(C_{T}(\mathcal{O}_{Y},N)\bullet\operatorname{ch}(F))\operatorname{ch}(G)$
$\displaystyle=i_{\ast}(C_{T}(F,N))\bullet\operatorname{ch}(G).$

Thus, by proposition 8.1, $T$ is compatible with the projection formula.

Assume that $T$ is compatible with the projection formula. Let $\overline{F}$
and $\overline{N}$ be hermitian vector bundles over a complex manifold $Y$.
Let $s\colon Y\hookrightarrow P:=\mathbb{P}(N\oplus\mathbb{C})$ be the zero
section and let $\pi\colon P\longrightarrow Y$ be the projection. Then

$\displaystyle C_{T}(F,N)$
$\displaystyle=\pi_{\ast}(T(K(\overline{F},\overline{N})))$
$\displaystyle=\pi_{\ast}(T(K(\overline{\mathcal{O}}_{Y},\overline{N})\otimes\pi^{\ast}\overline{F}))$
$\displaystyle=\pi_{\ast}(T(K(\overline{\mathcal{O}}_{Y},\overline{N}))\bullet\pi^{\ast}\operatorname{ch}(F))$
$\displaystyle=\pi_{\ast}(T(K(\overline{\mathcal{O}}_{Y},\overline{N})))\bullet\operatorname{ch}(F)$
$\displaystyle=C_{T}(\mathcal{O}_{Y},N)\bullet\operatorname{ch}(F).$

∎

We will next investigate the relationship between singular Bott-Chern classes
and compositions of closed immersions. Thus, let

$\textstyle{Y\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{Y/X}}$$\scriptstyle{i_{Y/M}}$$\textstyle{\,X\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{X/M}}$$\textstyle{\,M}$

be a composition of closed immersions. Assume that the normal bundles
$N_{Y/X}$, $N_{X/M}$ and $N_{Y/M}$ are provided with hermitian metrics. We
will denote by $\overline{\varepsilon}$ the exact sequence

$\overline{\varepsilon}\colon
0\rightarrow\overline{N}_{Y/X}\rightarrow\overline{N}_{Y/M}\rightarrow
i_{Y/X}^{\ast}\overline{N}_{X/M}\rightarrow 0.$ (8.6)

Let $P_{X/M}=\mathbb{P}(N_{X/M}\oplus\mathbb{C})$ be the projective completion
of the normal cone to $X$ in $M$. Then there is an isomorphism

$N_{Y/P_{X/M}}\cong N_{Y/X}\oplus i^{\ast}_{Y/X}N_{X/M}.$ (8.7)

We denote by $\overline{N}_{Y/P_{X/M}}$ the vector bundle on the left hand
side with the hermitian metric induced by the isomorphism (8.7).

Let $\overline{F}$ be a hermitian vector bundle over $Y$, let
$\overline{E}_{\ast}\longrightarrow(i_{Y/X})_{\ast}F$ be a resolution by
hermitian vector bundles. Let $\overline{E}^{\prime}_{\ast,\ast}$ be a complex
of complexes of vector bundles over $M$, such that, for each $k\geq 0$,
$\overline{E}^{\prime}_{k,\ast}\longrightarrow(i_{X/M})_{\ast}E_{k}$ is a
resolution, and there is a commutative diagram of resolutions

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247.22421pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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30.75pt\raise-41.21109pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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It follows that we have a resolution
$\operatorname{Tot}(\overline{E}^{\prime}_{\ast,\ast})\longrightarrow(i_{Y/M})_{\ast}F$
of $(i_{Y/M})_{\ast}F$ by hermitian vector bundles.

######  Notation 8.8.

We will denote

$\displaystyle\overline{\xi}_{Y\hookrightarrow X}$
$\displaystyle=(i_{Y/X},\overline{N}_{Y/X},\overline{F},\overline{E}_{\ast}),$
$\displaystyle\overline{\xi}_{Y\hookrightarrow M}$
$\displaystyle=(i_{Y/M},\overline{N}_{Y/M},\overline{F},\operatorname{Tot}(\overline{E}^{\prime}_{\ast,\ast})),$
$\displaystyle\overline{\xi}_{X\hookrightarrow M,k}$
$\displaystyle=(i_{X/M},\overline{N}_{X/M},\overline{E}_{k},\overline{E}^{\prime}_{k,\ast}).$

We will also denote by $\overline{\xi}_{Y\hookrightarrow P_{X/M}}$ the
hermitian embedded vector bundle

$\left(Y\hookrightarrow
P_{X/M},\overline{N}_{Y/P_{X/M}},\overline{F},\operatorname{Tot}(\pi_{P_{X/M}}^{\ast}\overline{E}_{\ast}\otimes
K(\mathcal{O}_{X},\overline{N}_{X/M}))\right).$

Let $T$ be a theory of singular Bott-Chern classes, and let $C_{T}$ be its
associated characteristic class. Our aim now is to relate
$T(\overline{\xi}_{Y\hookrightarrow X})$, $T(\overline{\xi}_{Y\hookrightarrow
M})$ and $T(\overline{\xi}_{X\hookrightarrow M,k})$.

Let $W_{X}$ be the deformation to the normal cone of $X$ in $M$. As before we
denote by $j_{X}\colon X\times\mathbb{P}^{1}\longrightarrow W_{X}$ the
inclusion.

We denote by $W$ the deformation to the normal cone of
$j_{X}(Y\times\mathbb{P}^{1})$ in $W_{X}$.

$M$

$W_{Y}$

$P_{Y/M}$$\widetilde{W}_{X}$$P_{Y\times\mathbb{P}^{1}}$

$\widetilde{M}_{X}\times\mathbb{P}^{1}$

$W_{Y/P}$

$W_{X}$$\widetilde{M}_{X}\times\\{\infty\\}$$\widetilde{P}_{X/M}$$P_{Y/P_{X/M}}$$W$$\mathbb{P}^{1}\times\mathbb{P}^{1}$$(0,\infty)$$(0,0)$$(\infty,0)$$(\infty,\infty)$$P_{X/M}$$\widetilde{M}_{X}\times\\{0\\}$$\widetilde{M}_{Y}$

Figure 1: Double deformation

This double deformation is represented in figure 1. There is a proper map
$q_{W}\colon W\longrightarrow\mathbb{P}^{1}\times\mathbb{P}^{1}$. The fibers
of $q_{W}$ over the corners of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ are as
follows:

$\displaystyle q_{W}^{-1}(0,0)$ $\displaystyle=M,$ $\displaystyle
q_{W}^{-1}(\infty,0)$ $\displaystyle=\widetilde{M}_{X}\times\\{0\\}\cup
P_{X/M},$ $\displaystyle q_{W}^{-1}(0,\infty)$
$\displaystyle=\widetilde{M}_{Y}\cup P_{Y/M},$ $\displaystyle
q_{W}^{-1}(\infty,\infty)$
$\displaystyle=\widetilde{M}_{X}\times\\{\infty\\}\cup\widetilde{P}_{X/M}\cup
P_{Y/P_{X/M}},$

where $\widetilde{M}_{X}$ and $\widetilde{M}_{Y}$ are the blow-up of $M$ along
$X$ and $Y$ respectively, $P_{Y/M}=\mathbb{P}(N_{Y/M}\oplus\mathbb{C})$ is the
projective completion of the normal cone to $Y$ in $M$, $P_{Y/P_{X/M}}$ of the
normal cone to $Y$ in $P_{X/M}$ and $\widetilde{P}_{X/M}$ is the blow-up of
$P_{X/M}$ along $Y$. The preimages by $\pi$ of the different faces of
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ are as follows:

$\displaystyle q_{W}^{-1}(\mathbb{P}^{1}\times\\{0\\})$ $\displaystyle=W_{X},$
$\displaystyle q_{W}^{-1}(\\{0\\}\times\mathbb{P}^{1})$ $\displaystyle=W_{Y},$
$\displaystyle q_{W}^{-1}(\mathbb{P}^{1}\times\\{\infty\\})$
$\displaystyle=\widetilde{W}_{X}\cup P_{Y\times\mathbb{P}^{1}},$
$\displaystyle q_{W}^{-1}(\\{\infty\\}\times\mathbb{P}^{1})$
$\displaystyle=\widetilde{M}_{X}\times\mathbb{P}^{1}\cup W_{Y/P},$

where $W_{Y}$ is the deformation to the normal cone of $Y$ in $M$, the
component $\widetilde{W}_{X}$ is the blow-up of $W_{X}$ along
$j_{X}(Y\times\mathbb{P}^{1})$, while
$P_{Y\times\mathbb{P}^{1}}=\mathbb{P}(N_{Y\times\mathbb{P}^{1}/W_{X}}\oplus\mathbb{C})$
is the projective completion of the normal cone to
$j_{X}(Y\times\mathbb{P}^{1})$ in $W_{X}$ and $W_{Y/P}$ is the deformation to
the normal cone of $Y$ inside $P_{X/M}$. All the above subvarieties will be
called boundary components of $W$.

We will use the following notations for the different maps.

$\displaystyle p_{X}\colon X\times\mathbb{P}^{1}\longrightarrow X$
$\displaystyle p_{Y}\colon Y\times\mathbb{P}^{1}\longrightarrow Y$
$\displaystyle p_{Y\times\mathbb{P}^{1}}\colon
Y\times\mathbb{P}^{1}\times\mathbb{P}^{1}\longrightarrow
Y\times\mathbb{P}^{1}$ $\displaystyle
p_{\widetilde{M}_{X}\times\mathbb{P}^{1}}\colon\widetilde{M}_{X}\times\mathbb{P}^{1}\longrightarrow
M$ $\displaystyle p_{W_{Y/P}}\colon W_{Y/P}\longrightarrow M$ $\displaystyle
p_{W_{Y}}\colon W_{Y}\longrightarrow M$ $\displaystyle p_{W_{X}}\colon
W_{X}\longrightarrow M$ $\displaystyle p_{P_{Y\times\mathbb{P}^{1}}}\colon
P_{Y\times\mathbb{P}^{1}}\longrightarrow M$ $\displaystyle
p_{\widetilde{W}_{X}}\colon\widetilde{W}_{X}\longrightarrow M$ $\displaystyle
p_{P_{Y/P_{X/M}}}\colon P_{Y/P_{X/M}}\longrightarrow M$ $\displaystyle
p_{P_{X/M}}\colon P_{X/M}\longrightarrow M$ $\displaystyle
p_{\widetilde{P}_{X/M}}\colon\widetilde{P}_{X/M}\longrightarrow M$
$\displaystyle p_{P_{Y/M}}\colon P_{Y/M}\longrightarrow M$ $\displaystyle
p_{W}\colon W\longrightarrow M$ $\displaystyle j_{Y}\colon
Y\times\mathbb{P}^{1}\longrightarrow W_{Y}$ $\displaystyle
j^{\prime}_{Y}\colon Y\times\mathbb{P}^{1}\longrightarrow W_{X}$
$\displaystyle j_{Y\times\mathbb{P}^{1}}\colon
Y\times\mathbb{P}^{1}\times\mathbb{P}^{1}\longrightarrow W$ $\displaystyle
i_{Y/P_{X/M}}\colon Y\longrightarrow P_{X/M}$
$\displaystyle\pi_{P_{X/M}}\colon P_{X/M}\longrightarrow X$
$\displaystyle\pi_{P_{Y/M}}\colon P_{Y/M}\longrightarrow Y$
$\displaystyle\pi_{P_{Y/P}}\colon P_{Y/P_{X/M}}\longrightarrow Y$
$\displaystyle\pi_{P_{Y\times\mathbb{P}^{1}}}\colon
P_{Y\times\mathbb{P}^{1}}\longrightarrow Y\times\mathbb{P}^{1}$
$\displaystyle\pi_{\widetilde{M}_{X}}\colon\widetilde{M}_{X}\longrightarrow M$
$\displaystyle\pi_{\widetilde{M}_{Y}}\colon\widetilde{M}_{Y}\longrightarrow M$

Note that the map $p_{\widetilde{M}_{X}\times\mathbb{P}^{1}}$ factors through
the blow-up $\widetilde{M}_{X}\longrightarrow M$ and the map
$p_{\widetilde{W}_{X}}$ factors through the blow-up
$\widetilde{M}_{Y}\longrightarrow M$, whereas the maps $p_{W_{Y/P}}$,
$p_{P_{X/M}}$ and $p_{\widetilde{P}_{X/M}}$ factor through the inclusion
$X\hookrightarrow M$ and the maps $p_{P_{Y\times\mathbb{P}^{1}}}$,
$p_{P_{Y/M}}$ and $p_{P_{Y/P_{X/M}}}$ factor through the inclusion
$Y\hookrightarrow M$.

The normal bundle to $X\times\mathbb{P}^{1}$ in $W_{X}$ is isomorphic to
$p_{X}^{\ast}N_{X/M}\otimes q_{X}^{\ast}\mathcal{O}(-1)$ and we consider on it
the metric induced by the metric on $\overline{N}_{X/M}$ and the Fubini-Study
metric on $\mathcal{O}(-1)$. We denote it by
$\overline{N}_{X\times\mathbb{P}^{1}/W_{X}}$. The normal bundle to
$Y\times\mathbb{P}^{1}$ in $W_{X}$ satisfies

$\displaystyle N_{Y\times\mathbb{P}^{1}/W_{X}}|_{Y\times\\{0\\}}$
$\displaystyle\cong N_{Y/M}$ $\displaystyle
N_{Y\times\mathbb{P}^{1}/W_{X}}|_{Y\times\\{\infty\\}}$ $\displaystyle\cong
N_{Y/X}\oplus i^{\ast}_{Y/X}N_{X/M}.$

On $N_{Y\times\mathbb{P}^{1}/W_{X}}$ we choose a hermitian metric such that
the above isomorphisms are isometries. Finally, on the normal bundle to
$Y\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ in $W$, we define a metric using
the same procedure as the definition of the metric of
$\overline{N}_{X\times\mathbb{P}^{1}/W_{X}}$.

On $W_{X}$ we obtain a sequence of resolutions
$\operatorname{tr}_{1}(\overline{E}^{\prime})_{n,\ast}\longrightarrow(j_{X})_{\ast}p_{X}^{\ast}E_{n}$.
They form a complex of complexes
$\operatorname{tr}_{1}(\overline{E}^{\prime})_{\ast,\ast}$ and the associated
total complex
$\operatorname{Tot}(\operatorname{tr}_{1}(\overline{E}^{\prime})_{\ast,\ast})$
provides us with a resolution

$\operatorname{Tot}(\operatorname{tr}_{1}(\overline{E}^{\prime})_{\ast,\ast})_{\ast}\longrightarrow(j^{\prime}_{Y})_{\ast}p_{Y}^{\ast}F.$
(8.9)

The restriction of
$\operatorname{Tot}(\operatorname{tr}_{1}(\overline{E}^{\prime})_{\ast,\ast})$
to $M$ is $\operatorname{Tot}(\overline{E}^{\prime}_{\ast,\ast})$. The
restriction of each complex
$\operatorname{tr}_{1}(\overline{E}^{\prime})_{n,\ast}$ to
$\widetilde{M}_{X}\times\\{0\\}$ is orthogonally split. Therefore the
restriction of
$\operatorname{Tot}(\operatorname{tr}_{1}(\overline{E}^{\prime}))$ to
$\widetilde{M}_{X}\times\\{0\\}$ is the total complex of a complex of
orthogonally split complexes. So it is acyclic although not necessarily
orthogonally split. The restriction of each complex
$\operatorname{tr}_{1}(\overline{E}^{\prime})_{n,\ast}$ to $P_{X/M}$ fits in
an exact sequence

$0\longrightarrow\overline{A}_{n,\ast}\longrightarrow\operatorname{tr}_{1}(\overline{E}^{\prime})_{n,\ast}|_{P_{X/M}}\longrightarrow\pi_{P_{X/M}}^{\ast}\overline{E}_{n}\otimes
K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})_{\ast}\longrightarrow 0.$

These exact sequences glue together giving a commutative diagram

$\textstyle{\operatorname{Tot}(\overline{A}_{\ast,\ast})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Tot}(\operatorname{tr}_{1}(\overline{E}^{\prime})_{\ast,\ast}|_{P_{X/M}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Tot}(\pi_{P_{X/M}}^{\ast}\overline{E}_{\ast}\otimes
K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})_{\ast})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(i_{Y/P_{X/M}})_{\ast}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(i_{Y/P_{X/M}})_{\ast}F}$

where the rows are short exact sequences. Even if the complexes
$(\overline{A}_{n})_{\ast}$ are orthogonally split, this is not necessarily
the case for $\operatorname{Tot}(\overline{A}_{\ast,\ast})$. To ease the
notation we will denote
$\overline{A}_{\ast}=\operatorname{Tot}(\overline{A}_{\ast,\ast})$.

Applying theorem 5.4 to the resolution (8.9), we obtain a complex of hermitian
vector bundles
$\widetilde{E}^{\prime}_{\ast}=\operatorname{tr}_{1}(\operatorname{Tot}(\operatorname{tr}_{1}(\overline{E}^{\prime})_{\ast,\ast}))$
which is a resolution of the coherent sheaf
$(j_{Y\times\mathbb{P}^{1}})_{\ast}p_{Y\times\mathbb{P}^{1}}^{\ast}p_{Y}^{\ast}F$.

We now study the restriction of $\widetilde{E}^{\prime}_{\ast}$ to each of the
boundary components of $W$.

  * •

The restriction of $\widetilde{E}^{\prime}_{\ast}$ to $W_{X}$ is just
$\operatorname{Tot}(\operatorname{tr}_{1}(\overline{E}^{\prime}))$ which has
already been described. For each $k\geq 0$, we will denote by $\eta^{1}_{k}$
the short exact sequence of hermitian vector bundles on $P_{X/M}$

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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133.93396pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{\operatorname{Tot}(\pi_{P_{X/M}}^{\ast}\overline{E}\otimes
K(\mathcal{O}_{X},\overline{N}_{X/M}))_{k}}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

whereas, for each $n,k\geq 0$ we will denote by $\eta^{1}_{n,k}$ the short
exact sequence

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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33.50427pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{tr}_{1}(\overline{E}^{\prime})_{n,k}|_{P_{X/M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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111.09811pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\pi_{P_{X/M}}^{\ast}\overline{E}_{n}\otimes
K(\mathcal{O}_{X},\overline{N}_{X/M})_{k}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

  * •

Its restriction to $W_{Y}$ is
$\operatorname{tr}_{1}(\operatorname{Tot}(\overline{E}^{\prime}))$. It is a
resolution of $(j_{Y})_{\ast}p_{Y}^{\ast}F$. Its restriction to
$\widetilde{M}_{Y}$ is orthogonally split, whereas its restriction to
$P_{Y/M}$ fits in an exact sequence

$0\longrightarrow\overline{B}_{\ast}\longrightarrow\operatorname{tr}_{1}(\operatorname{Tot}(\overline{E}^{\prime}))_{\ast}|_{P_{Y/M}}\longrightarrow\pi_{P_{Y/M}}^{\ast}\overline{F}\otimes
K(\overline{\mathcal{O}}_{Y},\overline{N}_{Y/M})\longrightarrow 0.$

For each $k\geq 0$ we will denote by $\eta^{2}_{k}$ the degree $k$ piece of
the above exact sequence.

  * •

Its restriction to $\widetilde{M}_{X}\times\mathbb{P}^{1}$ is an acyclic
complex, such that its further restriction to $\widetilde{M}_{X}\times\\{0\\}$
is acyclic and its restriction to $\widetilde{M}_{X}\times\\{\infty\\}$ is
orthogonally split.

  * •

Its restriction to $W_{Y/P}$ fits in a short exact sequence

$0\rightarrow\operatorname{tr}_{1}(\overline{A}_{\ast})\rightarrow\widetilde{E}^{\prime}_{\ast}|_{W_{Y/P}}\rightarrow\operatorname{tr}_{1}(\operatorname{Tot}(\pi_{P_{X/M}}^{\ast}\overline{E}\otimes
K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})))\rightarrow 0.$

For each $k\geq 0$, we will denote by $\mu^{1}_{k}$ the exact sequence of
hermitian vector bundles over $W_{Y/P}$ given by the piece of degree $k$ of
this exact sequence. The three terms of the above exact sequence become
orthogonally split when restricted to $\widetilde{P}_{X/M}$. By contrast, when
restricted to $P_{Y/P_{X/M}}$ they fit in a commutative diagram

$\textstyle{\overline{C}^{1}_{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{C}^{2}_{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{C}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{tr}_{1}(\overline{A})_{\ast}|_{P_{Y/P_{X/M}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{E}^{\prime}_{\ast}|_{P_{Y/P_{X/M}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{D}^{2}_{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{D}^{1}_{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{D}^{1}_{\ast}}$

where the complexes $\overline{C}^{i}_{\ast}$ are orthogonally split, and

$\displaystyle\overline{D}^{1}_{\ast}$
$\displaystyle=\pi^{\ast}_{P_{Y/P}}\overline{F}\otimes
K(\overline{\mathcal{O}}_{Y},\overline{N}_{Y/P_{X/M}}),$
$\displaystyle\overline{D}^{2}_{\ast}$
$\displaystyle=\operatorname{tr}_{1}(\operatorname{Tot}(\pi_{P_{X/M}}^{\ast}\overline{E}\otimes
K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})))|_{P_{Y/P_{X/M}}}.$

For each $k\geq 0$, we will denote by $\eta^{3}_{k}$ the exact sequence
corresponding to the piece of degree $k$ of the second row of the above
diagram, by $\eta^{4}_{k}$ that of the second column and by $\eta^{5}_{k}$
that of the third column. Notice that the map in the third row is an isometry.
We assume that the metric on $C^{1}_{\ast}$ is chosen in such a way that the
first column is an isometry. Since the complexes $\overline{C}^{i}_{\ast}$ are
orthogonally split, by lemma 2.17 we obtain

$\sum_{k}(-1)^{k}\left(\widetilde{\operatorname{ch}}(\eta^{3}_{k})-\widetilde{\operatorname{ch}}(\eta^{4}_{k})+\widetilde{\operatorname{ch}}(\eta^{5}_{k})\right)=0.$
(8.10)

Note that the restriction of $\mu^{1}_{k}$ to $P_{X/M}$ agrees with
$\eta^{1}_{k}$, whereas its restriction to $P_{Y/P_{X/M}}$ agrees with
$\eta^{3}_{k}$.

  * •

Its restriction to $\widetilde{W}_{X}$ is orthogonally split.

  * •

Finally its restriction to $P_{Y\times\mathbb{P}^{1}}$ fits in an exact
sequence

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30.90001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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83.66197pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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K(\mathcal{O}_{Y\times\mathbb{P}^{1}},\overline{N}_{Y\times\mathbb{P}^{1}/W_{X}})}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

where $\overline{D}_{\ast}$ is orthogonally split. For each $k\geq 0$ we will
denote by $\mu^{2}_{k}$ the piece of degree $k$ of this exact sequence. Note
that the restriction of $\mu^{2}_{k}$ to $P_{Y/M}$ agrees with $\eta^{2}_{k}$
and the restriction of $\mu^{2}_{k}$ to $P_{Y/P_{X/M}}$ agrees with
$\eta^{4}_{k}$.

On $\mathbb{P}^{1}\times\mathbb{P}^{1}$ we denote the two projections by
$p_{1}$ and $p_{2}$. Since the currents $p_{1}^{\ast}W_{1}$ and
$p_{2}^{\ast}W_{1}$ have disjoint wave front sets we can define the current
$W_{2}=p_{1}^{\ast}W_{1}\bullet
p_{2}^{\ast}W_{1}\in\mathcal{D}^{2}_{D}(\mathbb{P}^{1}\times\mathbb{P}^{1},2)$
which satisfies

$\operatorname{d}_{\mathcal{D}}W_{2}=(\delta_{\\{\infty\\}\times\mathbb{P}^{1}}-\delta_{\\{0\\}\times\mathbb{P}^{1}})\bullet
p_{2}^{\ast}W_{1}-p_{1}^{\ast}W_{1}\bullet(\delta_{\mathbb{P}^{1}\times\\{\infty\\}}-\delta_{\mathbb{P}^{1}\times\\{0\\}}).$
(8.11)

The key point in order to study the compatibility of singular Bott-Chern
classes and composition of closed immersions is that, in the group
$\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}(M,p)$, we have

$\operatorname{d}_{\mathcal{D}}(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{2}\bullet\operatorname{ch}(\widetilde{E}^{\prime}_{k})\right)=0.$

We compute this class using the equation (8.11). It can be decomposed as
follows.

$\displaystyle\operatorname{d}_{\mathcal{D}}(p_{W})_{\ast}$
$\displaystyle\left(\sum_{k}(-1)^{k}W_{2}\bullet\operatorname{ch}(\widetilde{E}^{\prime}_{k})\right)=$
$\displaystyle(p_{\widetilde{M}_{X}\times\mathbb{P}^{1}})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\widetilde{E}^{\prime}_{k}|_{\widetilde{M}_{X}\times\mathbb{P}^{1}})\right)$
(a)
$\displaystyle+(p_{W_{Y/P}})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\widetilde{E}^{\prime}_{k}|_{W_{Y/P}})\right)$
(b)
$\displaystyle-(p_{W_{Y}})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\widetilde{E}^{\prime}_{k}|_{W_{Y}})\right)$
(c)
$\displaystyle-(p_{\widetilde{W}_{X}})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\widetilde{E}^{\prime}_{k}|_{\widetilde{W}_{X}})\right)$
(d)
$\displaystyle-(p_{P_{Y\times\mathbb{P}^{1}}})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\widetilde{E}^{\prime}_{k}|_{P_{Y\times\mathbb{P}^{1}}})\right)$
(e)
$\displaystyle+(p_{W_{X}})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\widetilde{E}^{\prime}_{k}|_{W_{X}})\right)$
(f) $\displaystyle\phantom{AAAA}=\colon I_{a}+I_{b}-I_{c}-I_{d}-I_{e}+I_{f}$

We compute each of the above terms.

(a) Since the restriction
$\widetilde{E}^{\prime}|_{\widetilde{M}_{X}\times\\{\infty\\}}$ is
orthogonally split, we have

$I_{a}=-(\pi_{\widetilde{M}_{X}})_{\ast}\widetilde{\operatorname{ch}}(\widetilde{E}^{\prime}|_{\widetilde{M}_{X}\times\\{0\\}}).$

But, using lemma 2.17 and the fact, for each $k$, the complexes
$\operatorname{tr}_{1}(\overline{E}^{\prime})_{k,\ast}|_{\widetilde{M}_{X}}$
are orthogonally split, we obtain that $I_{a}=0$.

(b) We compute

$\displaystyle I_{b}=$
$\displaystyle(p_{W_{Y/P}})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\widetilde{E}^{\prime}_{k}|_{W_{Y/P}})\right)$
$\displaystyle=$
$\displaystyle(p_{W_{Y/P}})_{\ast}\left(W_{1}\bullet\sum_{k}(-1)^{k}(-\operatorname{d}_{\mathcal{D}}\widetilde{\operatorname{ch}}(\mu^{1}_{k})+\operatorname{ch}(\operatorname{tr}_{1}(\overline{A}_{\ast})_{k})\right.$
$\displaystyle\left.\phantom{(p_{W_{Y/P}})_{\ast}\sum_{k}(-1)^{k}}+\operatorname{ch}(\operatorname{tr}_{1}(\operatorname{Tot}(\pi_{P_{X/M}}^{\ast}\overline{E}\otimes
K(\mathcal{O}_{X},\overline{N}_{X/M})))_{k}))\right)$ $\displaystyle=$
$\displaystyle\sum_{k}(-1)^{k}(-(p_{P_{Y/P_{X/M}}})_{\ast}\widetilde{\operatorname{ch}}(\eta^{3}_{k})-(p_{\widetilde{P}_{X/M}})_{\ast}\widetilde{\operatorname{ch}}(\mu^{1}_{k}|_{\widetilde{P}_{X/M}})+(p_{P_{X/M}})_{\ast}\widetilde{\operatorname{ch}}(\eta^{1}_{k}))$
$\displaystyle-\widetilde{\operatorname{ch}}(\overline{A})$
$\displaystyle-(i_{X/M})_{\ast}(\pi_{P_{X/M}})_{\ast}T(\overline{\xi}_{Y\hookrightarrow
P_{X/M}})+(i_{Y/M})_{\ast}C_{T}(F,N_{Y/P_{X/M}})$
$\displaystyle-\sum_{k}(-1)^{k}(p_{P_{Y/P_{X/M}}})\widetilde{\operatorname{ch}}(\eta^{5}_{k}),$

where $\xi_{Y\hookrightarrow P_{X/M}}$ is as in notation 8.8.

By corollary 2.19 and the fact that the exact sequences
$\overline{A}_{k,\ast}$ are orthogonally split, the term
$\widetilde{\operatorname{ch}}(\overline{A})$ vanishes.

Also by corollary 2.19 we can see that

$\sum_{k}(-1)^{k}(p_{\widetilde{P}_{X/M}})_{\ast}\widetilde{\operatorname{ch}}(\mu^{1}_{k}|_{\widetilde{P}_{X/M}})$

vanishes.

Therefore we conclude

$\displaystyle I_{b}=$
$\displaystyle\sum_{k}(-1)^{k}(-(p_{P_{Y/P_{X/M}}})_{\ast}\widetilde{\operatorname{ch}}(\eta^{3}_{k})+(p_{P_{X/M}})_{\ast}\widetilde{\operatorname{ch}}(\eta^{1}_{k}))-(p_{P_{Y/P_{X/M}}})\widetilde{\operatorname{ch}}(\eta^{5}_{k})$
$\displaystyle-(i_{X/M})_{\ast}(\pi_{P_{X/M}})_{\ast}T(\overline{\xi}_{Y\hookrightarrow
P_{X/M}})+(i_{Y/M})_{\ast}C_{T}(F,N_{Y/P_{X/M}}).$

(c) By the definition of singular Bott-Chern forms we have

$I_{c}=-T(\overline{\xi}_{Y\hookrightarrow
M})+(i_{Y/M})_{\ast}C_{T}(F,N_{Y/M})-\sum_{k}(-1)^{k}(p_{P_{Y/M}})_{\ast}\widetilde{\operatorname{ch}}(\eta^{2}_{k}),$

(d) Since the restriction of $\widetilde{E}^{\prime}_{\ast}$ to
$\widetilde{W}_{X}$ is orthogonally split, we have $I_{d}=0$.

(e) We compute

$\displaystyle I_{e}=$
$\displaystyle(p_{P_{Y\times\mathbb{P}^{1}}})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\widetilde{E}^{\prime}_{k}|_{P_{Y\times\mathbb{P}^{1}}})\right)$
$\displaystyle=$
$\displaystyle(p_{P_{Y\times\mathbb{P}^{1}}})_{\ast}\left(W_{1}\bullet\sum_{k}(-1)^{k}\big{(}-\operatorname{d}_{\mathcal{D}}\widetilde{ch}(\mu^{2}_{k})+\operatorname{ch}(\overline{D}_{k})\right.$
$\displaystyle\left.\phantom{(p_{P_{Y\times\mathbb{P}^{1}}})_{\ast}\sum_{k}}+\operatorname{ch}(\pi^{\ast}_{P_{Y\times\mathbb{P}^{1}}}p^{\ast}_{Y}\overline{F}\otimes
K(\mathcal{O}_{Y\times\mathbb{P}^{1}},\overline{N}_{Y\times\mathbb{P}^{1}/W_{X}})_{k})\big{)}\right).$

The term $\sum(-1)^{k}\operatorname{ch}(\overline{D}_{k})$ vanishes because
the complex $D_{\ast}$ is orthogonally split. We have

$\sum_{k}(-1)^{k}(p_{P_{Y\times\mathbb{P}^{1}}})_{\ast}(W_{1}\bullet\operatorname{ch}(\pi^{\ast}_{P_{Y\times\mathbb{P}^{1}}}p^{\ast}_{Y}\overline{F}\otimes
K(\overline{\mathcal{O}}_{Y\times\mathbb{P}^{1}},\overline{N}_{Y\times\mathbb{P}^{1}/W_{X}})_{k}))\\\
=(i_{Y/M})_{\ast}\operatorname{ch}(\overline{F})\bullet(p_{Y})_{\ast}\left(W_{1}\bullet\pi^{\ast}_{P_{Y\times\mathbb{P}^{1}}}\sum_{k}(-1)^{k}\operatorname{ch}(K(\overline{\mathcal{O}}_{Y\times\mathbb{P}^{1}},\overline{N}_{Y\times\mathbb{P}^{1}/W_{X}})_{k})\right)\\\
=(i_{Y/M})_{\ast}\operatorname{ch}(\overline{F})\bullet(p_{Y})_{\ast}\left(W_{1}\bullet\operatorname{Td}^{-1}(\overline{N}_{Y\times\mathbb{P}^{1}/W_{X}})\right)\\\
=(i_{Y/M})_{\ast}\operatorname{ch}(\overline{F})\bullet\widetilde{\operatorname{Td}^{-1}}(\overline{\varepsilon}_{N}),$
(8.12)

where $\overline{\varepsilon}_{N}$ is the exact sequence (8.6).

Therefore we obtain

$I_{e}=-\sum_{k}(-1)^{k}(p_{P_{Y/P_{X/M}}})_{\ast}\widetilde{\operatorname{ch}}(\eta_{k}^{4})+\sum_{k}(-1)^{k}(p_{P_{Y/M}})_{\ast}\widetilde{\operatorname{ch}}(\eta_{k}^{2})\\\
+(i_{Y/M})_{\ast}\operatorname{ch}(\overline{F})\bullet\widetilde{\operatorname{Td}^{-1}}(\overline{\varepsilon}_{N}).$

(f) Finally we have

$\displaystyle I_{f}=$
$\displaystyle-\sum_{k}(-1)^{k}T(\overline{\xi}_{X\hookrightarrow
M,k})+\sum_{k}(-1)^{k}(i_{X/M})_{\ast}C_{T}(E_{k},N_{X/M})$
$\displaystyle-\sum_{k,l}(-1)^{k+l}(p_{P_{X/M}})_{\ast}\widetilde{\operatorname{ch}}(\eta^{1}_{k,l}).$

By corollary 2.19 we have that

$\sum_{m,l}(-1)^{m+l}(p_{P_{X/M}})_{\ast}\widetilde{\operatorname{ch}}(\eta^{1}_{m,l})=\sum_{k}(-1)^{k}(p_{P_{X/M}})_{\ast}\widetilde{\operatorname{ch}}(\eta^{1}_{k}).$

Thus

$\displaystyle I_{f}=$
$\displaystyle-\sum_{k}(-1)^{k}T(\overline{\xi}_{X\hookrightarrow
M,k})+\sum_{k}(-1)^{k}(i_{X/M})_{\ast}C_{T}(E_{k},N_{X/M})$
$\displaystyle-\sum_{k}(-1)^{k}(p_{P_{X/M}})_{\ast}\widetilde{\operatorname{ch}}(\eta^{1}_{k}).$

Summing up all the terms we have computed, and taking into account equation
(8.10) and the fact that

$C_{T}(F,N_{Y/M})=C_{T}(F,N_{Y/P_{X/M}})$

we have obtained the following partial result.

######  Lemma 8.13.

Let $i_{Y/M}=i_{X/M}\circ i_{Y/X}$ be a composition of closed immersions of
complex manifolds. Let $T$ be a theory of singular Bott-Chern classes with
$C_{T}$ its associated characteristic class. Let
$\overline{\xi}_{Y\hookrightarrow M}$, $\overline{\xi}_{X\hookrightarrow M,k}$
and $\overline{\xi}_{Y\hookrightarrow P_{X/M}}$ be as in notation 8.8, and let
$\overline{\varepsilon}$ be as in (8.6). Then, in the group
$\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}(M,p)$, the equation

$T(\overline{\xi}_{Y\hookrightarrow
M})=\sum_{k}(-1)^{k}T(\overline{\xi}_{X\hookrightarrow
M,k})-\sum_{k}(-1)^{k}(i_{X/M})_{\ast}C_{T}(E_{k},N_{X/M})\\\
+(i_{X/M})_{\ast}(\pi_{P_{X/M}})_{\ast}T(\overline{\xi}_{Y\hookrightarrow
P_{X/M}})+(i_{Y/M})_{\ast}\operatorname{ch}(\overline{F})\bullet\widetilde{\operatorname{Td}^{-1}}(\overline{\varepsilon}_{N})$
(8.14)

holds.

In order to compute the third term of the right hand side of equation (8.14)
we consider the following situation

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To ease the notation, we denote $P_{X/M}$ by $P$,
$Y\underset{X}{\times}P_{X/M}$ by $X^{\prime}$ and we denote by $P^{\prime}$
the projective completion of the normal cone to $X^{\prime}$ in $P$ and by
$\pi_{P^{\prime}}\colon P^{\prime}\longrightarrow X^{\prime}$,
$\pi_{X^{\prime}/Y}\colon X^{\prime}\longrightarrow Y$ and
$\pi_{P^{\prime}/Y}\colon P^{\prime}\longrightarrow Y$ the projections.
Observe that $X$ and $X^{\prime}$ intersect transversely along $Y$. Moreover,
$N_{Y/X^{\prime}}=i^{\ast}_{Y/X}N_{X/M}$,
$N_{X^{\prime}/P}=\pi^{\ast}_{X^{\prime}/Y}N_{Y/X}$ and $N_{Y/P}=N_{Y/X}\oplus
N_{Y/X^{\prime}}$. We use these identifications to define metrics on
$N_{Y/X^{\prime}}$, $N_{X^{\prime}/P}$ and $N_{Y/P}$. Therefore the exact
sequence

$0\longrightarrow\overline{N}_{Y/X^{\prime}}\longrightarrow\overline{N}_{Y/P}\longrightarrow
i^{\ast}_{Y/X^{\prime}}\overline{N}_{X^{\prime}/P}\longrightarrow 0$

is orthogonally split.

We apply the previous lemma to the composition of closed inclusions

$Y\hookrightarrow X^{\prime}\hookrightarrow P,$

the vector bundle $\overline{F}$ over $Y$ and the resolutions

$\displaystyle\pi^{\ast}\overline{F}\otimes
j^{\ast}K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})_{\ast}\longrightarrow
s_{\ast}F$ $\displaystyle\pi^{\ast}\overline{E}_{\ast}\otimes
K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})_{k}\longrightarrow
j_{\ast}(\pi^{\ast}F\otimes j^{\ast}K(\mathcal{O}_{X},N_{X/M})_{k}).$

We denote by $\overline{\xi}_{Y\hookrightarrow P}$ and
$\overline{\xi}_{X^{\prime}\hookrightarrow P,k}$ the hermitian embedded vector
bundles corresponding to the above resolutions. If $i_{Y/P^{\prime}}\colon
Y\hookrightarrow P^{\prime}$ is the induced inclusion, we denote by
$\overline{\xi}_{Y\hookrightarrow P^{\prime}}$ the hermitian embedded vector
bundle

$\left(i_{Y/P^{\prime}},\overline{N}_{Y/P^{\prime}},\overline{F},\operatorname{Tot}(\pi^{\ast}_{P^{\prime}}j^{\ast}K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})\otimes
K(\overline{\mathcal{O}}_{X^{\prime}},\overline{N}_{X^{\prime}/P})\otimes(\pi_{P^{\prime}/Y})^{\ast}\overline{F})\right).$

Note that the hermitian embedded vector bundle
$\overline{\xi}_{Y\hookrightarrow P}$ agrees with the hermitian embedded
vector bundle denoted $\overline{\xi}_{Y\hookrightarrow P_{X/M}}$ in lemma
8.13. Moreover, we have that

$\overline{\xi}_{X^{\prime}\hookrightarrow
P,k}=\pi^{\ast}\overline{\xi}_{Y\hookrightarrow X}\otimes
K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})_{k}.$

Applying lemma 8.13, we obtain

$T(\overline{\xi}_{Y\hookrightarrow
P_{X/M}})=\sum_{k}(-1)^{k}T(\overline{\xi}_{X^{\prime}\hookrightarrow
P_{X/M},k})\\\ -\sum_{k}(-1)^{k}j_{\ast}C_{T}(\pi^{\ast}F\otimes
j^{\ast}K(\mathcal{O}_{X},N_{X/M})_{k},N_{X^{\prime}/P})\\\
+j_{\ast}(\pi_{P^{\prime}})_{\ast}T(\overline{\xi}_{Y\hookrightarrow
P^{\prime}})$ (8.15)

By proposition 8.1,

$\sum_{k}(-1)^{k}T(\overline{\xi}_{X^{\prime}\hookrightarrow
P_{X/M},k})=\sum_{k}(-1)^{k}T(\pi^{\ast}\overline{\xi}_{Y\hookrightarrow
X}\otimes K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})_{k})\\\
=T(\pi^{\ast}\overline{\xi}_{Y\hookrightarrow
X})\bullet\sum_{k}(-1)^{k}\operatorname{ch}(K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})_{k})\\\
+\sum_{k}(-1)^{k}j_{\ast}C_{T}(\pi^{\ast}F\otimes
j^{\ast}K(\mathcal{O}_{X},N_{X/M})_{k},N_{X^{\prime}/P})\\\
-\sum_{k}(-1)^{k}j_{\ast}C_{T}(\pi^{\ast}F,N_{X^{\prime}/P})\bullet\operatorname{ch}(K(\mathcal{O}_{X},N_{X/M})_{k})$
(8.16)

We now want to compute the term
$(i_{X/M})_{\ast}(\pi_{P_{X/M}})_{\ast}j_{\ast}(\pi_{P^{\prime}})_{\ast}T(\overline{\xi}_{Y\hookrightarrow
P^{\prime}})$.

Observe that we can identify

$P^{\prime}=\mathbb{P}(i_{Y/X}^{\ast}N_{X/M}\oplus\mathbb{C})\underset{Y}{\times}\mathbb{P}(s^{\ast}N_{X^{\prime}/P}\oplus\mathbb{C}),$

where $s^{\ast}N_{X^{\prime}/P}$ is canonically isomorphic to $N_{Y/X}$.

Moreover

$(i_{X/M})_{\ast}(\pi_{P_{X/M}})_{\ast}j_{\ast}(\pi_{P^{\prime}})_{\ast}T(\overline{\xi}_{Y\hookrightarrow
P^{\prime}})=(i_{Y/M})_{\ast}(\pi_{P^{\prime}/Y})_{\ast}T(\overline{\xi}_{Y\hookrightarrow
P^{\prime}}).$

######  Definition 8.17.

We denote

$C_{T}^{\operatorname{ad}}(F,N_{Y/X},i_{Y/X}^{\ast}N_{X/M})=(\pi_{P^{\prime}/Y})_{\ast}T(\overline{\xi}_{Y\hookrightarrow
P^{\prime}})$

and we define

$\rho(F,N_{Y/X},i_{Y/X}^{\ast}N_{X/M})=C_{T}(F,N_{Y/M})-C_{T}^{\operatorname{ad}}(F,N_{Y/X},i_{Y/X}^{\ast}N_{X/M}).$
(8.18)

######  Lemma 8.19.

The current $C_{T}^{\operatorname{ad}}(F,N_{Y/X},i_{Y/X}^{\ast}N_{X/M})$ is
closed and defines a characteristic class of triples of vector bundles.
Therefore $\rho$ is also a characteristic class. Moreover the class $\rho$
does not depend on the theory of singular Bott-Chern classes $T$.

###### Proof.

The fact that $C_{T}^{\operatorname{ad}}(F,N_{Y/X},i_{Y/X}^{\ast}N_{X/M})$ is
closed and determines a characteristic class is proved as in 6.16. The
independence of $\rho$ from to $T$ is seen as follows. We denote by
$\overline{K}^{\prime}_{\ast}$ the complex

$\operatorname{Tot}(\pi^{\ast}_{P^{\prime}}j^{\ast}K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})\otimes
K(\overline{\mathcal{O}}_{X^{\prime}},\overline{N}_{X^{\prime}/P}))\otimes(\pi_{P^{\prime}/Y})^{\ast}\overline{F}.$

This complex is a resolution of $(i_{Y/P^{\prime}})_{\ast}\overline{F}$

Let $W$ be the blow-up of $P^{\prime}\times\mathbb{P}^{1}$ along
$Y\times\infty$, and let $\operatorname{tr}_{1}(\overline{K}^{\prime})_{\ast}$
be the deformation of complexes on $W$ given by theorem 5.4. Just by looking
at the rank of the different vector bundles we see that the restriction of
$\operatorname{tr}_{1}(\overline{K}^{\prime})_{\ast}$ to $P_{Y/P^{\prime}}$,
the exceptional divisor of this blow-up, is isomorphic (although not
necessarily isometric) to the Koszul complex
$K(\overline{F},\overline{N}_{X/M})_{\ast}$. Then, by equation (7.3)

$T(\overline{\xi}_{Y\hookrightarrow
P^{\prime}})-(i_{Y/P^{\prime}})_{\ast}C_{T}(F,N_{Y/M})=\\\
-(p_{W})_{\ast}\left(W_{1}\bullet\sum_{k}(-1)^{k}\operatorname{ch}(\operatorname{tr}_{1}(\overline{K}^{\prime})_{k})\right)\\\
-\sum_{k}(-1)^{k}(p_{P})_{\ast}\widetilde{\operatorname{ch}}(\operatorname{tr}_{1}(\overline{K}^{\prime})_{k}|_{P_{Y/P^{\prime}}},K(\overline{F},\overline{N}_{X/M})_{k}).$

Since the right hand side of this equation does not depend on the theory $T$,
the result is proved. ∎

Using equations (8.15), (8.16), lemma 8.19 and the projection formula, we
obtain

$\displaystyle(\pi_{P_{X/M}})_{\ast}T(\overline{\xi}_{Y\hookrightarrow
P_{X/M}})=$ $\displaystyle\left(T(\overline{\xi}_{Y\hookrightarrow
X})-(i_{Y/X})_{\ast}C_{T}(F,N_{Y/X})\right)$
$\displaystyle\phantom{AA}\bullet(\pi_{P_{X/M}})_{\ast}\sum_{k}(-1)^{k}\operatorname{ch}(K(\mathcal{O}_{X},\overline{N}_{X/M})_{k})$
$\displaystyle+(\pi_{P_{X/M}})_{\ast}j_{\ast}(\pi_{P^{\prime}})_{\ast}T(\overline{\xi}_{Y\hookrightarrow
P^{\prime}})$ $\displaystyle=$
$\displaystyle\left(T(\overline{\xi}_{Y\hookrightarrow
X})-(i_{Y/X})_{\ast}C_{T}(F,N_{Y/X})\right)\bullet\operatorname{Td}^{-1}(\overline{N}_{X/M})$
$\displaystyle+(i_{Y/X})_{\ast}C^{\operatorname{ad}}_{T}(F,N_{Y/X},i_{Y/X}^{\ast}N_{X/M})$
$\displaystyle=$ $\displaystyle\left(T(\overline{\xi}_{Y\hookrightarrow
X})-(i_{Y/X})_{\ast}C_{T}(F,N_{Y/X})\right)\bullet\operatorname{Td}^{-1}(\overline{N}_{X/M})$
$\displaystyle+(i_{Y/X})_{\ast}C_{T}(F,N_{Y/M})-\rho(F,N_{Y/X},i_{Y/X}^{\ast}N_{X/M}).$
(8.20)

Joining this equation and lemma 8.13 we obtain the main relationship between
singular Bott-Chern classes and composition of closed immersions.

######  Proposition 8.21.

Let $i_{Y/M}=i_{X/M}\circ i_{Y/X}$ be a composition of closed immersions of
complex manifolds. Let $T$ be a theory of singular Bott-Chern classes with
$C_{T}$ its associated characteristic class. Let
$\overline{\xi}_{Y\hookrightarrow M}$, $\overline{\xi}_{X\hookrightarrow M,k}$
and $\overline{\xi}_{Y\hookrightarrow P_{X/M}}$ be as in notation 8.8 and let
$\overline{\varepsilon}$ be as in (8.6). Then, in the group
$\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}(M,p)$, we have the equation

$T(\overline{\xi}_{Y\hookrightarrow
M})=\sum_{k}(-1)^{k}T(\overline{\xi}_{X\hookrightarrow
M,k})+(i_{X/M})_{\ast}(T(\overline{\xi}_{Y\hookrightarrow
X})\bullet\operatorname{Td}^{-1}(\overline{N}_{X/M}))\\\
+(i_{Y/M})_{\ast}\operatorname{ch}(\overline{F})\bullet\widetilde{\operatorname{Td}^{-1}}(\overline{\varepsilon}_{N})\\\
+(i_{Y/M})_{\ast}C^{\operatorname{ad}}_{T}(F,N_{Y/X},i_{Y/X}^{\ast}N_{X/M})\\\
-(i_{X/M})_{\ast}((i_{Y/X})_{\ast}C_{T}(F,N_{Y/X})\bullet\operatorname{Td}^{-1}(N_{X/M}))\\\
-(i_{X/M})_{\ast}\sum_{k}(-1)^{k}C_{T}(E_{k},N_{X/M})$

We can simplify the formula of proposition 8.21 if we assume that our theory
of singular Bott-Chern classes is compatible with the projection formula.

######  Corollary 8.22.

With the hypothesis of proposition 8.21, assume furthermore that $T$ is
compatible with the projection formula. Then

$T(\overline{\xi}_{Y\hookrightarrow
M})=\sum_{k}(-1)^{k}T(\overline{\xi}_{X\hookrightarrow
M,k})+(i_{X/M})_{\ast}(T(\overline{\xi}_{Y\hookrightarrow
X})\bullet\operatorname{Td}^{-1}(\overline{N}_{X/M}))\\\
+(i_{Y/M})_{\ast}\operatorname{ch}(\overline{F})\bullet\widetilde{\operatorname{Td}^{-1}}(\overline{\varepsilon}_{N})\\\
+(i_{Y/M})_{\ast}\left[C^{\operatorname{ad}}_{T}(F,N_{Y/X},i_{Y/X}^{\ast}N_{X/M})-C_{T}(F,N_{Y/X})\bullet\operatorname{Td}^{-1}(i_{Y/X}^{\ast}N_{X/M}))\right.\\\
\left.-C_{T}(F,i_{Y/X}^{\ast}N_{X/M})\bullet\operatorname{Td}^{-1}(N_{Y/X})\right]$

###### Proof.

Since $T$ is compatible with the projection formula, then $C_{T}$ is also.
Therefore, using the Grothendieck-Riemann-Roch theorem for closed immersions
at the level of analytic Deligne cohomology classes, we have

$\displaystyle\sum_{k}(-1)^{k}C_{T}(E_{k},$ $\displaystyle
N_{X/M})=C_{T}(\mathcal{O}_{X},N_{X/M})\bullet\sum_{k}(-1)^{k}\operatorname{ch}(E_{k})$
$\displaystyle=C_{T}(\mathcal{O}_{X},N_{X/M})\bullet(i_{Y/X})_{\ast}(\operatorname{ch}(F)\bullet\operatorname{Td}^{-1}(N_{Y/X}))$
$\displaystyle=(i_{Y/X})_{\ast}(i_{Y/X}^{\ast}C_{T}(\mathcal{O}_{X},N_{X/M})\bullet\operatorname{ch}(F)\bullet\operatorname{Td}^{-1}(N_{Y/X}))$
$\displaystyle=(i_{Y/X})_{\ast}(C_{T}(F,i_{Y/X}^{\ast}N_{X/M})\bullet\operatorname{Td}^{-1}(N_{Y/X})),$

which implies the result. ∎

######  Definition 8.23.

Let $T$ be a theory of singular Bott-Chern classes. We will say that $T$ is
_transitive_ if the equation

$T(\overline{\xi}_{Y\hookrightarrow
M})=\sum_{k}(-1)^{k}T(\overline{\xi}_{X\hookrightarrow
M,k})+(i_{X/M})_{\ast}(T(\overline{\xi}_{Y\hookrightarrow
X})\bullet\operatorname{Td}^{-1}(\overline{N}_{X/M}))\\\
+(i_{Y/M})_{\ast}\operatorname{ch}(\overline{F})\bullet\widetilde{\operatorname{Td}^{-1}}(\overline{\varepsilon}_{N})$
(8.24)

holds. When equation (8.24) is satisfied for a particular choice of complex
immersions and resolutions, we say that the theory $T$ is _transitive with
respect to this particular choice_.

We now introduce an abstract version of definition 8.17.

######  Definition 8.25.

Given any characteristic class $C$ of pairs of vector bundles, we will denote

$C^{\rho}(F,N_{1},N_{2}):=C(F,N_{1}\oplus N_{2})-\rho(F,N_{1},N_{2}),$

where $\rho$ is the characteristic class of definition 8.17.

Note that, when $T$ is a theory of singular Bott-Chern classes we have

$C^{\rho}_{T}(F,N_{1},N_{2})=C^{\operatorname{ad}}_{T}(F,N_{1},N_{2}).$

######  Definition 8.26.

We will say that a characteristic class $C$ (of pairs of vector bundles) is
_$\rho$ -Todd additive_ (in the second variable) if it satisfies

$C(F,N_{1}\oplus
N_{2})=C(F,N_{1})\bullet\operatorname{Td}^{-1}(N_{2})+C(F,N_{2})\bullet\operatorname{Td}^{-1}(N_{1})+\rho(F,N_{1},N_{2})$

or, equivalently,

$C^{\rho}(F,N_{1},N_{2})=C(F,N_{1})\bullet\operatorname{Td}^{-1}(N_{2})+C(F,N_{2})\bullet\operatorname{Td}^{-1}(N_{1}).$

A direct consequence of corollary 8.22 is

######  Corollary 8.27.

Let $T$ be a theory of singular Bott-Chern classes that is compatible with the
projection formula. Then it is transitive if and only if the associated
characteristic class $C_{T}$ is $\rho$-Todd additive.

Since we are mainly interested in singular Bott-Chern classes that are
transitive and compatible with the projection formula, we will study
characteristic classes that are compatible with the projection formula and
$\rho$-Todd-additive in the second variable. Since we want to express any
characteristic class in terms of a power series we will restrict ourselves to
the algebraic category.

######  Proposition 8.28.

Let $C$ be a class that is compatible with the projection formula and
$\rho$-Todd additive in the second variable. Then $C$ determines a power
series $\phi_{C}(x)$ given by

$C(\mathcal{O}_{Y},L)=\phi_{C}(c_{1}(L)),$ (8.29)

for every complex algebraic manifold $Y$ and algebraic line bundle $Y$.
Conversely, given any power series in one variable $\phi(x)$, there exists a
unique characteristic class for algebraic vector bundles that is compatible
with the projection formula and $\rho$-Todd additive in the second variable
such that equation (8.29) holds.

###### Proof.

This result follows directly from the splitting principle and theorem 1.8. ∎

######  Remark 8.30.

The utility of corollary 8.27 and proposition 8.28 is limited by the fact that
we do not know an explicit formula for the class
$\rho(\mathcal{O}_{Y},N_{1},N_{2})$. This class is related with the arithmetic
difference between $\mathbb{P}_{Y}(N_{1}\oplus N_{2}\oplus\mathbb{C})$ and
$\mathbb{P}_{Y}(N_{1}\oplus\mathbb{C})\underset{Y}{\times}\mathbb{P}_{Y}(N_{2}\oplus\mathbb{C})$,
the second space being simpler than the first. The main ingredients needed to
compute this class are the Bott-Chern classes of the tautological exact
sequence. Therefore the work of Mourougane [29] might be useful for computing
this class.

Recall that an additive genus is a characteristic class for algebraic vector
bundles $S$ such that

$S(N_{1}\oplus N_{2})=S(N_{1})+S(N_{2}).$

Let $\phi(x)=\sum_{i=0}^{\infty}a_{i}x^{i}$ be a power series in one variable.
There is a one to one correspondence between additive genus and power series
characterized by the condition that $S(L)=\phi(c_{1}(L))$, for each line
bundle $L$.

Since the class $\rho$ does not depend on the theory $T$ it cancels out when
considering the difference between two different theories of singular Bott-
Chern classes.

######  Proposition 8.31.

Let $C_{1}$ and $C_{2}$ be two characteristic classes for pairs of algebraic
vector bundles that are compatible with the projection formula and
$\rho$-Todd-additive in the second variable. Then there is a unique additive
genus $S_{12}$ such that

$C_{1}(F,N)-C_{2}(F,N)=\operatorname{ch}(F)\bullet\operatorname{Td}(N)^{-1}\bullet
S_{12}(N).$ (8.32)

We can summarize the results of this section in the following theorem.

######  Theorem 8.33.

There is a one to one correspondence between theories of singular Bott-Chern
classes for complex algebraic manifolds that are transitive and compatible
with the projection formula, and formal power series
$\phi(x)\in\mathbb{R}[[x]]$. To each theory of singular Bott-Chern classes
corresponds the power series $\phi$ such that

$C_{T}(\mathcal{O}_{Y},L)={\bf 1}_{1}\bullet\phi(c_{1}(L)),$ (8.34)

for every complex algebraic manifold $Y$ and every algebraic line bundle $L$.
To each power series $\phi$ it corresponds a unique class $C$, compatible with
the projection formula and $\rho$-Todd-additive in the second variable,
characterized by equation (8.34) and a theory of singular Bott-Chern given by
definition 7.4.

Even if we do not know the exact value of the class $\rho$ another consequence
of corollary 8.27 is that, in order to prove the transitivity of a theory of
singular Bott-Chern classes it is enough to check it for a particular class of
compositions.

######  Corollary 8.35.

Let $T$ be a theory of singular Bott-Chern classes compatible with the
projection formula. Then $T$ is transitive if and only if for any compact
complex manifold $Y$ and vector bundles $N_{1}$, $N_{2}$, the theory $T$ is
transitive with respect to the composition of inclusions

$Y\hookrightarrow\mathbb{P}_{Y}(N_{1}\oplus\mathbb{C})\hookrightarrow\mathbb{P}_{Y}(N_{1}\oplus\mathbb{C})\times_{Y}\mathbb{P}_{Y}(N_{2}\oplus\mathbb{C})$

and the Koszul resolutions. $\square$

We can make the previous corollary a little more explicit. Let $\pi_{1}$ and
$\pi_{2}$ be the projections from
$P:=\mathbb{P}_{Y}(N_{1}\oplus\mathbb{C})\times_{Y}\mathbb{P}_{Y}(N_{2}\oplus\mathbb{C})$
to $P_{1}:=\mathbb{P}_{Y}(N_{1}\oplus\mathbb{C})$ and
$P_{2}:=\mathbb{P}_{Y}(N_{2}\oplus\mathbb{C})$ respectively. Let
$\overline{K}_{1}=K(\overline{\mathcal{O}}_{Y},\overline{N}_{1})$ and
$\overline{K}_{2}=K(\overline{\mathcal{O}}_{Y},\overline{N}_{2})$ be the
Koszul resolutions in $P_{1}$ and $P_{2}$ respectively. Then,

$\overline{K}=\pi_{1}^{\ast}K_{1}\otimes\pi_{2}^{\ast}K_{2}$

is a resolution of $\mathcal{O}_{Y}$ in $P$. Then the theory $T$ is transitive
in this case if

$T(\overline{K})=\pi_{2}^{\ast}T(\overline{K}_{2})\bullet\pi_{1}^{\ast}(c_{r_{1}}(\overline{Q}_{1})\bullet\operatorname{Td}^{-1}(\overline{Q}_{1}))+(i_{1})_{\ast}(T(\overline{K}_{1})\bullet
p_{1}^{\ast}\operatorname{Td}^{-1}(\overline{N}_{2})),$

where $r_{1}$ is the rank of $N_{1}$, $\overline{Q}_{1}$ is the tautological
quotient bundle in $P_{1}$ with the induced metric, $i_{1}\colon
P_{1}\longrightarrow P$ is the inclusion and $p_{1}\colon P_{1}\longrightarrow
Y$ is the projection.

The singular Bott-Chern classes that we have defined depend on the choice of a
hermitian metric on the normal bundle and behave well with respect inverse
images. Nevertheless, when one is interested in covariant functorial
properties and, in particular, in a composition of closed immersions, it might
be interesting to consider a variant of singular Bott-Chern classes that
depend on the choice of metrics on the tangent bundles to $Y$ and $X$.

######  Notation 8.36.

Let $\overline{\xi}=(i\colon Y\longrightarrow
X,\overline{N},\overline{F},\overline{E}_{\ast}\to i_{\ast}F)$ be a hermitian
embedded vector bundle. Let $\overline{T}_{X}$ and $\overline{T}_{Y}$ be the
tangent bundles to $X$ and $Y$ provided with hermitian metrics. As usual we
write $\operatorname{Td}(Y)=\operatorname{Td}(\overline{T}_{Y})$ and
$\operatorname{Td}(X)=\operatorname{Td}(\overline{T}_{X})$. We put

$\overline{\xi}_{c}=(i\colon Y\longrightarrow
X,\overline{T}_{X},\overline{T}_{Y},\overline{F},\overline{E}_{\ast}\to
i_{\ast}F).$

By abuse of notation we will also say that $\overline{\xi}_{c}$ is a hermitian
embedded vector bundle. In this situation we denote by
$\overline{\xi}_{N_{Y/X}}$ the exact sequence of hermitian vector bundles

$\overline{\xi}_{N_{Y/X}}\colon
0\longrightarrow\overline{T}_{Y}\longrightarrow
i^{\ast}\overline{T}_{X}\longrightarrow\overline{N}_{Y/X}\longrightarrow 0.$

If there is no danger of confusion we will denote
$\overline{N}=\overline{N}_{Y/X}$ and therefore
$\overline{\xi}_{N}=\overline{\xi}_{N_{Y/X}}$.

######  Definition 8.37.

Let $T$ be a theory of singular Bott-Chern classes. Then the _covariant
singular Bott-Chern class_ associated to $T$ is given by

$T_{c}(\overline{\xi}_{c})=T(\overline{\xi})+i_{\ast}(\operatorname{ch}(\overline{F})\bullet\widetilde{\operatorname{Td}^{-1}}(\overline{\xi}_{N_{Y/X}})\operatorname{Td}(Y))$
(8.38)

######  Proposition 8.39.

The covariant singular Bott-Chern classes satisfy the following properties

  1. (i)

The class $T_{c}(\overline{\xi}_{c})$ does not depend on the choice of the
metric on $N_{Y/X}$.

  2. (ii)

The differential equation

$\operatorname{d}_{\mathcal{D}}T_{c}(\overline{\xi}_{c})=\sum_{k}(-1)^{k}\operatorname{ch}(\overline{E}_{k})-i_{\ast}(\operatorname{ch}(\overline{F})\bullet\operatorname{Td}(Y))\bullet\operatorname{Td}^{-1}(X)$
(8.40)

holds.

  3. (iii)

If the theory $T$ is compatible with the projection formula, then

$T_{c}(\overline{\xi}_{c}\otimes\overline{G})=T_{c}(\overline{\xi}_{c})\bullet\operatorname{ch}(\overline{G}).$

  4. (iv)

If, moreover, the theory $T$ is transitive, then, using notation 8.8 adapted
to the current setting, we have

$T_{c}(\overline{\xi}_{Y\hookrightarrow
M,c})=\sum_{k}(-1)^{k}T_{c}(\overline{\xi}_{X\hookrightarrow M,k,c})\\\
+(i_{X/M})_{\ast}(T_{c}(\overline{\xi}_{Y\hookrightarrow
X,c})\bullet\operatorname{Td}(X))\bullet\operatorname{Td}^{-1}(M).$ (8.41)

  5. (v)

With the hypothesis of corollary 6.14, we have

$T_{c}(\bigoplus_{j\text{ even}}\overline{\xi}_{j,c})-T_{c}(\bigoplus_{j\text{
odd}}\overline{\xi}_{j,c})=[\widetilde{\operatorname{ch}}(\overline{\varepsilon})]-i_{\ast}([\widetilde{\operatorname{ch}}(\overline{\chi})\bullet\operatorname{Td}(Y)])\bullet\operatorname{Td}^{-1}(X).$
(8.42)

###### Proof.

All the statements follow from straightforward computations. ∎

##  9 Homogeneous singular Bott-Chern classes

In this section we will show that, by adding a natural fourth axiom to
definition 6.9, we obtain a unique theory of singular Bott-Chern classes that
we call homogeneous singular Bott-Chern classes, and we will compare it with
the classes previously defined by Bismut, Gillet and Soulé and by Zha.

In the paper [6], Bismut, Gillet and Soulé introduced a theory of singular
Bott-Chern classes that is the main ingredient in their construction of direct
images for closed immersions.

Strictly speaking, the construction of [6] only produces a theory of singular
Bott-Chern classes in the sense of this paper when the metrics involved
satisfy a technical condition, called Condition (A) of Bismut. Nevertheless,
there is a unique way to extend the definition of [6] from metrics satisfying
Bismut’s condition (A) to general metrics in such a way that one obtains a
theory of singular Bott-Chern classes in the sense of this paper.

In his thesis [32], Zha gave another definition of singular Bott-Chern
classes, and he also used them to define direct images for closed immersions
in Arakelov theory.

We will recall the construction of both theories of singular Bott-Chern
classes and we will show that they agree with the theory of homogeneous
singular Bott-Chern classes.

We warn the reader that the normalizations we use differ from the
normalizations in [6] and [32]. The main difference is that we insist on using
the algebro-geometric twist in cohomology, whereas in the other two papers the
authors use cohomology with real coefficients.

Let $r_{F}$ and $r_{N}$ be two positive integers. Let $Y$ be a complex
manifold and let $\overline{F}$ and $\overline{N}$ be two hermitian vector
bundles of rank $r_{F}$ and $r_{N}$ respectively. Let
$P=\mathbb{P}(N\oplus\mathbb{C})$ and let $s$ be the zero section. We will
follow the notations of definition 5.3. Then $T(K(\overline{F},\overline{N}))$
satisfies the differential equation

$\operatorname{d}_{\mathcal{D}}T(K(\overline{F},\overline{N}))=c_{r_{N}}(\overline{Q})\operatorname{Td}^{-1}(\overline{Q})\operatorname{ch}(\pi_{P}^{\ast}\overline{F})-s_{\ast}(\operatorname{ch}(\overline{F})\operatorname{Td}^{-1}(\overline{N})).$

Therefore, the class

$\widetilde{e}_{T}(\overline{F},\overline{N}):=T(K(\overline{F},\overline{N}))\bullet\operatorname{Td}(\overline{Q})\bullet\operatorname{ch}^{-1}(\pi_{P}^{\ast}\overline{F})$

satisfies the simpler equation

$\operatorname{d}_{\mathcal{D}}\widetilde{e}_{T}(\overline{F},\overline{N})=[c_{r_{N}}(\overline{Q})]-\delta_{Y}.$
(9.1)

Observe that the right hand side of this equation belongs to
$\mathcal{D}^{2r_{N}}_{D}(P,r_{N})$. Thus it seems natural to introduce the
following definition.

######  Definition 9.2.

Let $T$ be a theory of singular Bott-Chern classes of rank $r_{F}>0$ and
codimension $r_{N}$. Then the class

$\widetilde{e}_{T}(\overline{F},\overline{N})=T(K(\overline{F},\overline{N}))\bullet\operatorname{Td}(\overline{Q})\bullet\operatorname{ch}^{-1}(\pi_{P}^{\ast}\overline{F})$

is called the _Euler-Green class associated to_ $T$. The class
$T(K(\overline{F},\overline{N}))$ is said to be _homogeneous_ if

$\widetilde{e}_{T}(\overline{F},\overline{N})\in\widetilde{\mathcal{D}}^{2r_{N}-1}_{D}(P,r_{N}).$

A theory of singular Bott-Chern classes of rank $0$ is said to be
_homogeneous_ if it agrees with the theory of Bott-Chern classes associated to
the Chern character. Finally, a theory of singular Bott-Chern classes is said
to be _homogeneous_ if its restrictions to all ranks and codimensions are
homogeneous.

The main interest of the above definition is the following result.

######  Theorem 9.3.

Given two positive integers $r_{F}$ and $r_{N}$ there exists a unique theory
of homogeneous singular Bott-Chern classes of rank $r_{F}$ and codimension
$r_{N}$.

###### Proof.

The proof of this result is based on the theory of Euler-Green classes.

Let $P=\mathbb{P}(N\oplus\mathbb{C})$ be as before, and let $s$ denote the
zero section of $P$. Let $D_{\infty}$ be the subvariety of $P$ that
parametrizes the lines contained in $N$. Then $D_{\infty}=\mathbb{P}(N)$.

######  Lemma 9.4.

There exists a unique class
$\widetilde{e}(P,\overline{Q},s)\in\mathcal{D}^{2r_{N}-1}_{D}(P,r_{N})$ such
that

  1. (i)

It satisfies

$\operatorname{d}_{\mathcal{D}}\widetilde{e}(P,\overline{Q},s)=[c_{r_{N}}(\overline{Q})]-\delta_{Y}.$
(9.5)

  2. (ii)

The restriction $\widetilde{e}(P,\overline{Q},s)|_{D_{\infty}}=0.$

###### Proof.

We first show the uniqueness. Assume that $\widetilde{e}$ and
$\widetilde{e}^{\prime}$ are two classes that satisfy the hypothesis of the
theorem. Then $\widetilde{e}^{\prime}-\widetilde{e}$ is closed. Hence it
determines a cohomology class in $H^{2r_{N}-1}_{\mathcal{D}^{\text{{\rm
an}}}}(P,r_{N})$. Since, by theorem 1.2, the restriction

$H^{2r_{N}-1}_{\mathcal{D}^{\text{{\rm an}}}}(P,r_{N})\longrightarrow
H^{2r_{N}-1}_{\mathcal{D}^{\text{{\rm an}}}}(D_{\infty},r_{N})$ (9.6)

is an isomorphism, condition (ii) implies that
$\widetilde{e}^{\prime}=\widetilde{e}$. Now we prove the existence. Since $Y$
is the zero locus of the section $s$, that is transversal to the zero section
of $Q$, we know that the currents $[c_{r_{N}}]$ and $\delta_{Y}$ are
cohomologous. Therefore there exists an element
$\widetilde{a}\in\widetilde{\mathcal{D}}^{2r_{N}-1}_{D}(P,r_{N})$ such that
$\operatorname{d}_{\mathcal{D}}\widetilde{a}=[c_{r_{N}}(\overline{Q})]-\delta_{Y}$.
Since $\overline{Q}$ restricted to $D_{\infty}$ splits as an orthogonal direct
sum

$\overline{Q}|_{D_{\infty}}=\overline{S}\oplus\overline{\mathbb{C}}$ (9.7)

where the metric on the factor $\mathbb{C}$ is trivial, and the section $s$
restricts to the constant section $1$, we obtain that
$([c_{r_{N}}(\overline{Q})]-\delta_{Y})|_{D_{\infty}}=0$. Therefore
$\widetilde{a}$ determines a class in $H^{2r_{N}-1}_{\mathcal{D}^{\text{{\rm
an}}}}(P,r_{N})$. Using again that (9.6) is an isomorphism, we find an element
$\widetilde{b}\in H^{2r_{N}-1}_{\mathcal{D}^{\text{{\rm an}}}}(P,r_{N})$, such
that $\widetilde{e}=\widetilde{a}-\widetilde{b}$ satisfies the conditions of
the lemma. ∎

We continue with the proof of theorem 9.3. We first prove the uniqueness. Let
$T$ be a theory of homogeneous singular Bott-Chern classes. The splitting
(9.7) implies easily that the restriction of the Koszul resolution
$K(\overline{F},\overline{N})$ to $D_{\infty}$ is orthogonally split. By the
functoriality of singular Bott-Chern classes,
$T(K(\overline{F},\overline{N}))|_{D_{\infty}}=0$. Thus the class

$\widetilde{e}_{T}(\overline{F},\overline{N}):=T(K(\overline{F},\overline{N}))\bullet\operatorname{Td}(\overline{Q})\bullet\operatorname{ch}^{-1}(\pi_{P}^{\ast}\overline{F})\in\widetilde{\mathcal{D}}^{2r_{N}-1}_{D}(P,r_{N})$

satisfies the two conditions of lemma 9.4. Therefore
$\widetilde{e}_{T}(\overline{F},\overline{N})=\widetilde{e}(P,\overline{Q},s)$
and

$T(K(\overline{F},\overline{N}))=\widetilde{e}(P,\overline{Q},s)\bullet\operatorname{Td}^{-1}(\overline{Q})\bullet\operatorname{ch}(\pi_{P}^{\ast}\overline{F}),$
(9.8)

where the right hand side does not depend on the theory $T$. In consequence we
have that

$C_{T}(F,N)=(\pi_{P})_{\ast}T(K(\overline{F},\overline{N}))$ (9.9)

does not depend on the theory $T$. Thus by the uniqueness in theorem 7.1 we
obtain the uniqueness here.

For the existence we observe

######  Lemma 9.10.

The current

$C(F,N)=(\pi_{P})_{\ast}(\widetilde{e}(P,\overline{Q},s)\bullet\operatorname{Td}^{-1}(\overline{Q}))\bullet\operatorname{ch}(\overline{F})$

is a characteristic class for pairs of vector bundles of rank $r_{F}$ and
$r_{N}$.

###### Proof.

We first compute, using equation (9.5) and corollary 3.8,

$\displaystyle\operatorname{d}_{\mathcal{D}}C(F,N)$
$\displaystyle=(\pi_{P})_{\ast}\left(\operatorname{d}_{\mathcal{D}}\widetilde{e}(P,\overline{Q},s)\bullet\operatorname{Td}^{-1}(\overline{Q})\right)\bullet\operatorname{ch}(\overline{F})$
$\displaystyle=(\pi_{P})_{\ast}\left(([c_{r_{N}}(\overline{Q})]-\delta_{Y})\bullet\operatorname{Td}^{-1}(\overline{Q})\right)\bullet\operatorname{ch}(\overline{F})$
$\displaystyle=(\pi_{P})_{\ast}\left(c_{r_{N}}(\overline{Q})\bullet\operatorname{Td}^{-1}(\overline{Q})\right)\bullet\operatorname{ch}(\overline{F})-\operatorname{Td}^{-1}(\overline{N})\bullet\operatorname{ch}(\overline{F})$
$\displaystyle=0.$

Thus $C(F,N)$ determines a cohomology class. This class is functorial by
construction. By proposition 1.7 this class does not depend on the metric and
defines a characteristic class. ∎

By the existence in theorem 7.1 we obtain a theory of singular Bott-Chern
classes $T_{C}$ that is easily seen to be homogeneous. ∎

A reformulation of theorem 9.3 is

######  Theorem 9.11.

There exists a unique way to associate to each hermitian embedded vector
bundle $\overline{\xi}=(i\colon Y\longrightarrow
X,\overline{N},\overline{F},\overline{E}_{\ast})$ a class of currents

$T^{h}(\overline{\xi})\in\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,N^{\ast}_{Y,0},p)$

that we call homogeneous singular Bott-Chern class, satisfying the following
properties

  1. (i)

(Differential equation) The equality

$\operatorname{d}_{\mathcal{D}}T^{h}(\overline{\xi})=\sum_{i}(-1)^{i}[\operatorname{ch}(\overline{E}_{i})]-i_{\ast}([\operatorname{Td}^{-1}(\overline{N})\operatorname{ch}(\overline{F})])$
(9.12)

holds.

  2. (ii)

(Functoriality) For every morphism $f\colon X^{\prime}\longrightarrow X$ of
complex manifolds that is transverse to $Y$,

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

  3. (iii)

(Normalization) Let $\overline{A}=(A_{\ast},g_{\ast})$ be a non-negatively
graded orthogonally split complex of vector bundles. Write
$\overline{\xi}\oplus\overline{A}=(i\colon Y\longrightarrow
X,\overline{N},\overline{F},\overline{E}_{\ast}\oplus\overline{A}_{\ast})$.
Then $T^{h}(\overline{\xi})=T^{h}(\overline{\xi}\oplus\overline{A})$.
Moreover, if $X=\operatorname{Spec}\mathbb{C}$ is one point, $Y=\emptyset$ and
$\overline{E}_{\ast}=0$, then $T^{h}(\overline{\xi})=0$.

  4. (iv)