23{C}}/{\mathaccent 23{C}})$).

Using Poincaré and Serre duality, we have an isomorphism of vector bundles on
${\mathaccent 23{C}}$:

$\varphi_{n}:{\mathcal{H}}^{2(N-1)-n}(Y_{\mathaccent 23{C}}/{\mathaccent
23{C}})\longrightarrow{\mathcal{H}}^{n}(Y_{\mathaccent 23{C}}/{\mathaccent
23{C}})^{\vee},$

that sends, for every integer $p$, the subbundle
$F^{p}{\mathcal{H}}^{2(N-1)-n}$ into the orthogonal subbundle:

$(F^{N-p}{\mathcal{H}}^{n})^{\perp}\subset{\mathcal{H}}^{n}(Y_{\mathaccent
23{C}}/{\mathaccent 23{C}})^{\vee}.$

Consequently, for every integer $p$, this isomorphism induces by restriction
and quotient an isomorphism:

$\varphi_{n}^{p}:\mathrm{gr}_{F}^{p}{\mathcal{H}}^{2(N-1)-n}\longrightarrow(F^{N-p}{\mathcal{H}}^{n})^{\perp}/(F^{N-1-p}{\mathcal{H}}^{n})^{\perp}\simeq(\mathrm{gr}_{F}^{N-1-p}{\mathcal{H}}^{n})^{\vee}.$

Recall that the duality of vector bundles exchanges the upper and lower
Deligne extensions, therefore, the isomorphism $\varphi_{n}$ can be extended
into an isomorphism of vector bundles on $C$:

$\overline{\varphi_{n}}:\overline{{\mathcal{H}}^{2(N-1)-n}(Y_{\mathaccent
23{C}}/{\mathaccent
23{C}})}_{+}\longrightarrow(\overline{{\mathcal{H}}^{n}(Y_{\mathaccent
23{C}}/{\mathaccent 23{C}})}_{-})^{\vee},$

that sends for every integer $p$, the subbundle
$\overline{F^{p}{\mathcal{H}}^{2(N-1)-n}}_{+}$ into the subbundle
$(\overline{F^{N-p}{\mathcal{H}}^{n}}_{-})^{\perp}$, hence induces an
isomorphism by restriction and quotient:

$\overline{\varphi_{n}^{p}}:\mathrm{gr}_{\overline{F}_{+}}^{p}\overline{{\mathcal{H}}^{2(N-1)-n}}_{+}\longrightarrow(\mathrm{gr}_{\overline{F}_{-}}^{N-1-p}\overline{{\mathcal{H}}^{n}}_{-})^{\vee}.$

By definition, the upper Griffiths line bundle on $C$ of the VHS
$\mathbb{H}^{2(N-1)-n}(Y_{\mathaccent 23{C}}/{\mathaccent 23{C}})$ on
${\mathaccent 23{C}}$ is given by:

$\displaystyle\mathcal{GK}_{C,+}(\mathbb{H}^{2(N-1)-n}(Y_{\eta}/C_{\eta}))$
$\displaystyle=\bigotimes_{p\in\mathbb{Z}}(\det\mathrm{gr}_{\overline{F}_{+}}^{p}\overline{{\mathcal{H}}^{2(N-1)-n}}_{+})^{\otimes
p}$
$\displaystyle\simeq\bigotimes_{p\in\mathbb{Z}}(\det\mathrm{gr}_{\overline{F}_{-}}^{N-1-p}\overline{{\mathcal{H}}^{n}}_{-})^{\otimes-p}\textrm{
using the isomorphisms $(\overline{\varphi_{n}^{p}})_{p}$}$
$\displaystyle\simeq\bigotimes_{p^{\prime}\in\mathbb{Z}}(\det\mathrm{gr}_{\overline{F}_{-}}^{p^{\prime}}\overline{{\mathcal{H}}^{n}}_{-})^{\otimes
p^{\prime}-(N-1)}$
$\displaystyle\simeq\bigotimes_{p^{\prime}\in\mathbb{Z}}(\det\mathrm{gr}_{\overline{F}_{-}}^{p^{\prime}}\overline{{\mathcal{H}}^{n}}_{-})^{\otimes
p^{\prime}}\otimes\big{(}\bigotimes_{p^{\prime}\in\mathbb{Z}}(\det\mathrm{gr}_{\overline{F}_{-}}^{p^{\prime}}\overline{{\mathcal{H}}^{n}}_{-})\big{)}^{\otimes-(N-1)}$
$\displaystyle\simeq\mathcal{GK}_{C,-}(\mathbb{H}^{n}(Y_{\eta}/C_{\eta}))\otimes(\det\overline{{\mathcal{H}}^{n}(Y_{\mathaccent
23{C}}/{\mathaccent 23{C}})}_{-})^{\otimes-(N-1)}.$

Consequently, there is an isomorphism of line bundles on $C$:

$\mathcal{GK}_{C,+}(\mathbb{H}^{2(N-1)-n}(Y_{\eta}/C_{\eta}))\otimes\mathcal{GK}_{C,-}(\mathbb{H}^{n}(Y_{\eta}/C_{\eta}))^{\vee}\simeq\big{(}\det\overline{{\mathcal{H}}^{n}(Y_{\mathaccent
23{C}}/{\mathaccent 23{C}})}_{-}\big{)}^{\otimes-(N-1)}.$

Using Lemma 2.5.1, the line bundle on $C$:

$\big{(}\det\overline{{\mathcal{H}}^{n}(Y_{\mathaccent 23{C}}/{\mathaccent
23{C}})}_{-}\big{)}^{\otimes r}\otimes{\mathcal{O}}_{C}(rA^{n})$

is of $2$-torsion. Consequently, the line bundle on $C$:

$\displaystyle\mathcal{GK}_{C,+}($
$\displaystyle\mathbb{H}^{2(N-1)-n}(Y_{\eta}/C_{\eta}))^{\otimes
r}\otimes\mathcal{GK}_{C,-}(\mathbb{H}^{n}(Y_{\eta}/C_{\eta}))^{\otimes(-r)}\otimes{\mathcal{O}}_{C}(-r(N-1)A^{n})$
$\displaystyle\simeq(\det\overline{{\mathcal{H}}^{n}(Y_{\mathaccent
23{C}}/{\mathaccent
23{C}})}_{-})^{\otimes-r(N-1)}\otimes{\mathcal{O}}_{C}(-r(N-1)A^{n}),$
$\displaystyle\simeq\big{[}(\det\overline{{\mathcal{H}}^{n}(Y_{\mathaccent
23{C}}/{\mathaccent 23{C}})}_{-})^{\otimes
r}\otimes{\mathcal{O}}_{C}(rA^{n})\big{]}^{\otimes-(N-1)},$

is of $2$-torsion, as wanted.

For proving the second assertion in Proposition 2.5.2, we use that the
isomorphism $\varphi_{n}$ is also compatible with the connections of both VHS,
hence with their local monodromy. If the local monodromy at every point of
$\Delta$ of the VHS $\mathbb{H}^{n}(Y_{\mathaccent 23{C}}/{\mathaccent
23{C}})$ is unipotent, so is the local monodromy of the dual VHS
$\mathbb{H}^{n}(Y_{\mathaccent 23{C}}/{\mathaccent 23{C}})^{\vee}$, and so is
the monodromy of the VHS $\mathbb{H}^{2(N-1)-n}(Y_{\mathaccent
23{C}}/{\mathaccent 23{C}})$.

Finally, applying the first part of the proof with $A^{n}=0$ and $r=1$, we
obtain the last assertion in Proposition 2.5.2. ∎

##  3\. Characteristic classes of relative differentials and of logarithmic
relative differentials

In this section, we establish various results relating the Chern classes of
sheaves of differentials, and vector bundles of logarithmic differentials.
These results will be used in Section 4 to derive Theorems 1.3.2 and 4.2.3
from Theorems 1.3.1 and 4.2.1, and are also of independent interest.

The statements of our results are gathered in Subsection 3.1. The following
subsections, which contain their rather technical proofs and will not be
referred to in the remainder of this paper, could be skipped at first reading.

###  3.1. Comparing Chern classes and Todd classes of differentials and
logarithmic differentials

####  3.1.1. Chern classes of differentials and of logarithmic differentials:
the absolute case

Although we are ultimately interested by characteristic of sheaves of relative
differentials, we shall first prove the following results in the absolute
case.

Let $X$ be a smooth $n$-dimensional complex scheme and let $E$ be a reduced
divisor with strict normal crossings in $X$. Let us denote by $(E_{i})_{i\in
I}$ the irreducible components of $E$, and for any subset $J\subseteq I$, let
us define:191919In particular, $E_{\emptyset}=X$.

$E_{J}:=\bigcap_{i\in J}E_{i}.$

It is a smooth subscheme of codimension $|J|$ in $X$, and we shall denote by:

$i_{E_{J}}:E_{J}{\lhook\joinrel\longrightarrow}X$

the inclusion morphism.

As in [Del70, II, 3.4], for every non-negative integer $r$, we denote by
$E^{r}$ the subscheme of codimension $r$ of $X$ defined as the union of the
intersections of $r$ distinct components of $D$:

$E^{r}:=\bigcup_{J\subset I,|J|=r}E_{J}.$

Observe that for every subset $J$ in $I$, the subscheme:

$E_{J}\cap E^{|J|+1}=\bigcup_{i\in I-J}E_{J\cup\\{i\\}}$

is a divisor with normal crossings in $E_{J}$.

######  Proposition 3.1.1.

With the previous notation, for every subset $J$ of $I$, the following
relation holds between total Chern classes in ${\mathrm{CH}}^{\ast}(E_{J})$:

(3.1.1) $c\big{(}\Omega^{1}_{X}(\log
E)_{|E_{J}}\big{)}=c\big{(}\Omega^{1}_{E_{J}}(\log E_{J}\cap
E^{|J|+1})\big{)}.$

Moreover the following equality holds in ${\mathrm{CH}}^{\ast}(X)$:

(3.1.2) $c(\Omega_{X}^{1}(\log E))=\sum_{J\subseteq
I}i_{E_{J}\ast}c(\Omega^{1}_{E_{J}}),$

and the following equality holds in ${\mathrm{CH}}^{\ast}(X)_{\mathbb{Q}}$:

(3.1.3) ${\rm{Td}}(\Omega^{1}_{X}(\log
E))={\rm{Td}}(\Omega^{1}_{X})\prod_{i\in I}{\rm{td}}(-[E_{i}])^{-1},$

where ${\rm{td}}(x)$ denotes the formal series $x/(1-e^{-x})$.

The relation (3.1.2) may be reformulated as the following equalities, where
$r$ denotes an integer such that $0\leq r\leq n$:

(3.1.4) $c_{r}(\Omega^{1}_{X}(\log E))=\sum_{J\subseteq I,|J|\leq
r}i_{E_{J}\ast}c_{r-|J|}(\Omega^{1}_{E_{J}}).$

####  3.1.2. Notation

For the rest of the statements, let us adopt the following notation.

Let $Y$ be a connected smooth projective complex scheme of pure dimension $N$,
let $C$ be a connected smooth projective complex curve, and let

$g:Y\longrightarrow C$

be a surjective morphism that is smooth outside of $Y_{\Delta}$, where
$\Delta$ is a reduced divisor in $C$.

Let us assume that the divisor $Y_{\Delta}$ is a divisor with strict normal
crossings, and let us write it as:

$D=\sum_{i\in I}m_{i}D_{i},$

where $I$ is a finite set and for every $i$ in $I$, $m_{i}$ is a positive
integer, and where $D_{i}$ is a smooth divisor such that the $(D_{i})_{i\in
I}$ intersect each other transversally.

As before, for every subset $J$ of $I$, we denote:

$D_{J}:=\bigcap_{i\in J}D_{i},$

and for every integer $r\geq 1$, we define the subscheme:

$D^{r}:=\bigcup_{J\subset I,|J|=r}D_{J}.$

The subscheme:

$D_{J}\cap D^{|J|+1}=\bigcup_{i\in I-J}D_{J\cup\\{i\\}}$

is a divisor with normal crossings in $D_{J}$.

For every element $i$ in $I$, we denote by

${\mathcal{N}}_{i}:={\mathcal{N}}_{D_{i}}Y\simeq{\mathcal{O}}_{Y}(D_{i})_{|D_{i}}$

the normal vector bundle of the smooth divisor $D_{i}$ in $Y$.

Let $\preceq$ be a total order on $I$.

####  3.1.3. Normal bundles to the strata $D_{J}$ and Chern classes

######  Proposition 3.1.2.

For every non-empty subset $J$ in $I$, we have the equality of total Chern
classes in ${\mathrm{CH}}^{\ast}(D_{J})$:

(3.1.5) $c\big{(}\omega^{1}_{Y/C|D_{J}}\big{)}=c\big{(}\Omega^{1}_{D_{J}}(\log
D_{J}\cap D^{|J|+1})\big{)}.$

######  Proposition 3.1.3.

Let $i$ be an element in $I$, we have the equality in
${\mathrm{CH}}^{1}(D_{i})$:

$m_{i}c_{1}({\mathcal{N}}_{i})=-\sum_{j\in I-\\{i\\}}m_{j}[D_{ij}],$

where $D_{ij}:=D_{\\{i,j\\}}.$

######  Corollary 3.1.4.

Let $i$ be an element in $I$ and let $\beta$ be a class in
${\mathrm{CH}}^{\ast}(D_{i})$, we have the equality in
${\mathrm{CH}}^{\ast}(Y)$:

$m_{i}i_{D_{i}\ast}(c_{1}({\mathcal{N}}_{i})\cap\beta)=-\sum_{j\in
I-\\{i\\}}m_{j}i_{D_{ij}\ast}\beta_{|D_{ij}}.$

######  Corollary 3.1.5.

Let $(i,j)$ be a pair in $I^{2}$ such that $i\prec j$.

We have the equality in ${\mathrm{CH}}^{1}(D_{ij})$:

(3.1.6)
$m_{i}c_{1}({\mathcal{N}}_{i|D_{ij}})+m_{j}c_{1}({\mathcal{N}}_{j|D_{ij}})=-\sum_{k\in
I-\\{i,j\\}}m_{k}[D_{ijk}],$

where $D_{ijk}:=D_{\\{i,j,k\\}}.$

####  3.1.4. Comparing the characteristic classes of $[T_{g}]$ and
${\omega^{1\vee}_{Y/C}}$

Finally, we shall show the following results of comparison of the Chern
classes of $\omega^{1}_{Y/C}$ and of the relative tangent class in $K$-theory:

$[T_{g}]:=[T_{Y}]-g^{\ast}[T_{C}].$

The duality operation on vector bundles on $Y$ extends to an involution
$.^{\vee}$ of the abelian group $K^{0}(Y)$, which sends $[T_{g}]$ to the class
$[\Omega^{1}_{Y}]-g^{\ast}[\Omega^{1}_{C}]$. Observe that we have a sequence
of coherent sheaves on $Y$:

$0\longrightarrow
g^{\ast}\Omega^{1}_{C}\overset{Dg}{\longrightarrow}\Omega^{1}_{Y}\longrightarrow\Omega^{1}_{Y/C}\longrightarrow
0.$

This sequence is exact: everything but the injectivity of the differential
morphism $Dg$ is a standard property of the sheaf of Kähler differentials, and
since the morphism $g$ is smooth on a dense open subset of $Y$, the morphism
$Dg$ does not vanish on a dense open subset, hence is injective.

Consequently, the following equality holds in $K^{0}(Y)$:

$[T_{g}]=[\Omega^{1}_{Y/C}]^{\vee}.$

######  Proposition 3.1.6.

We have the equality in ${\mathrm{CH}}^{\ast}(Y)_{\mathbb{Q}}$:

(3.1.7)
$\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}=\prod_{i\in
I}{\rm{td}}([D_{i}])-1/2\sum_{i\in I}m_{i}[D_{i}],$

where ${\rm{td}}(x)$ denotes the formal series with rational coefficients
$x/(1-e^{-x})$.

In particular, we have the equality in ${\mathrm{CH}}^{1}(Y)_{\mathbb{Q}}$:

(3.1.8)
$\bigg{[}\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}\bigg{]}^{(1)}=-1/2\sum_{i\in
I}(m_{i}-1)[D_{i}],$

and the equalities in ${\mathrm{CH}}^{2}(Y)_{\mathbb{Q}}$:

(3.1.9)
$\displaystyle\bigg{[}\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}\bigg{]}^{(2)}$
$\displaystyle=1/12\sum_{i\in
I}i_{D_{i}\ast}c_{1}({\mathcal{N}}_{i})+1/4\sum_{(i,j)\in I^{2},i\prec
j}[D_{ij}],$ (3.1.10) $\displaystyle=1/12\sum_{(i,j)\in I^{2},i\prec
j}(3-m_{i}/m_{j}-m_{j}/m_{i})[D_{ij}].$

######  Proposition 3.1.7.

For every non-negative integer $r$, we have the equality in
${\mathrm{CH}}^{r}(Y)$:

(3.1.11) $c_{r}(\omega^{1\vee}_{Y/C})=c_{r}([T_{g}])+\sum_{i\in
I}m_{i}i_{D_{i}\ast}[c_{r-1}(T_{D_{i}})+c_{1}({\mathcal{N}}_{i})c_{r-2}(T_{D_{i}})]+\sum_{\emptyset\neq
J\subset I}(-1)^{|J|}i_{D_{J}\ast}c_{r-|J|}(T_{D_{J}}).$

Furthermore, if the subscheme $D^{3}$ is empty, then for every non-negative
integer $r$, we have the equality in ${\mathrm{CH}}^{r}(Y)$:

(3.1.12) $c_{r}(\omega^{1\vee}_{Y/C})=c_{r}([T_{g}])+\sum_{i\in
I}(m_{i}-1)i_{D_{i}\ast}c_{r-1}(T_{D_{i}})-\sum_{(i,j)\in I^{2},i\prec
j}(m_{i}+m_{j}-1)i_{D_{ij}\ast}c_{r-2}(T_{D_{ij}}).$

The remainder of this section shall be devoted to the proofs of these results.

More specifically, Subsection 3.2 shall be devoted to the proof of Proposition
3.1.1; Subsection 3.3 shall be devoted to the proofs of Propositions 3.1.2,
3.1.3 and Corollaries 3.1.4 and 3.1.5; and Subsection 3.4 shall be devoted to
the proofs of Propositions 3.1.6 and 3.1.7.

###  3.2. Characteristic classes of differentials and logarithmic
differentials: the absolute case

Let us adopt the notation of Proposition 3.1.1.

####  3.2.1. Proof of (3.1.1)

Let $J$ be a subset of $I$. Let

${\mathcal{N}}_{J}:={\mathcal{N}}_{E_{J}}X$

be the normal vector bundle of the subscheme $E_{J}$ in $X$.

One checks in local coordinates that the composition of morphisms of vector
bundles on $E_{J}$:

${\mathcal{N}}_{J}^{\vee}\longrightarrow\Omega^{1}_{X|E_{J}}\longrightarrow\Omega^{1}_{X}(\log
E)_{|E_{J}}$

vanishes, so using the commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{N}}_{J}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\textstyle{\Omega^{1}_{X|E_{J}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{E_{J}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\Omega^{1}_{X}(\log
E)_{|E_{J}}}$

and the exactness of the horizontal exact sequence of vector bundles, we
obtain a map of vector bundles on $E_{J}$:

$\alpha_{J}:\Omega^{1}_{E_{J}}\longrightarrow\Omega^{1}_{X}(\log E)_{|E_{J}}.$

One checks in local coordinates that the map $\alpha_{J}$ factors through a
map of vector bundles:

$\tilde{\alpha_{J}}:\Omega^{1}_{E_{J}}(\log E_{J}\cap
E^{|J|+1})\longrightarrow\Omega^{1}_{X}(\log E)_{|E_{J}}.$

On the other hand, let us define a map of vector bundles on $E_{J}$ by:

$\beta_{J}:=(\mathrm{Res}_{E_{i}})_{i\in J}:\quad\Omega^{1}_{X}(\log
E)_{|E_{J}}\longrightarrow{\mathcal{O}}_{E_{J}}^{J}\simeq{\mathcal{O}}_{E_{J}}^{\oplus|J|},$

where, for every $i$ in $J$, $\mathrm{Res}_{E_{i}}$ denotes the residue
morphism relative to the divisor $E_{i}$.

Equality (3.1.1) follows immediately from the following proposition.

######  Proposition 3.2.1.

The morphisms $\tilde{\alpha_{J}}$ and $\beta_{J}$ fit together in an exact
sequence of vector bundles on $E_{J}$:

$0\longrightarrow\Omega^{1}_{E_{J}}(\log E_{J}\cap
E^{|J|+1})\overset{\tilde{\alpha_{J}}}{\longrightarrow}\Omega^{1}_{X}(\log
E)_{|E_{J}}\overset{\beta_{J}}{\longrightarrow}{\mathcal{O}}_{E_{J}}^{J}\longrightarrow
0.$

###### Proof.

We can check this on a neighborhood of a point $P$ of $E_{J}$. Let
$i_{1},...,i_{s}$ be the elements of $J$, with $s:=|J|$.

Let $r$ be the largest integer such that $P$ is in $E^{r}$, in particular,
$s\leq r\leq n$. Let $E_{i_{1}}$,…, $E_{i_{s}}$, $E_{i_{s+1}}$,…,$E_{i_{r}}$
be the components of $E$ that meet in $P$, with $\\{i_{s+1},...,i_{r}\\}$ a
subset of $I-J$.

Let us consider local coordinates $(x_{1},...,x_{n})$ of $X$ centered in $P$,
such that for every integer $1\leq l\leq r$, $E_{i_{l}}$ is locally defined by
$(x_{l}=0)$.

The vector bundle $\Omega^{1}_{E_{J}}$ admits a local frame
$(\mathrm{d}x_{s+1},...,\mathrm{d}x_{n})$, the vector bundle
$\Omega^{1}_{X|E_{J}}$ admits a local frame
($\mathrm{d}x_{1},$,…,$\mathrm{d}x_{n}$), the vector bundle
$\Omega^{1}_{X}(\log E)_{|E_{J}}$ admits a local frame
($\mathrm{d}x_{1}/x_{1}$,…,$\mathrm{d}x_{r}/x_{r}$,
$\mathrm{d}x_{r+1}$,…,$\mathrm{d}x_{n}$), and the canonical map from
$\Omega^{1}_{X|E_{J}}$ to $\Omega^{1}_{X}(\log E)_{|E_{J}}$ sends, for every
integer $l$, $\mathrm{d}x_{l}$ to $x_{l}(\mathrm{d}x_{l}/x_{l})$ if $l\leq r$,
else to $\mathrm{d}x_{l}$.

Consequently, the map

$\alpha_{J}:\Omega^{1}_{E_{J}}\longrightarrow\Omega^{1}_{X}(\log E)_{|E_{J}}$

sends, for every integer $l\geq s+1$, $\mathrm{d}x_{l}$ to
$x_{l}(\mathrm{d}x_{l}/x_{l})$ if $l\leq r$, else to $\mathrm{d}x_{l}$.

The divisor $E_{J}\cap E^{|J|+1}$ in $E_{J}$ is locally defined by the
equation $(x_{s+1}...x_{r}=0)$, so the vector bundle $\Omega^{1}_{E_{J}}(\log
E_{J}\cap E^{|J|+1})$ admits a local frame
$(\mathrm{d}x_{s+1}/x_{s+1},...,\mathrm{d}x_{r}/x_{r},\mathrm{d}x_{r+1},...,\mathrm{d}x_{n})$.

Consequently, the map

$\tilde{\alpha_{J}}:\Omega^{1}_{E_{J}}(\log E_{J}\cap
E^{|J|+1})\longrightarrow\Omega^{1}_{X}(\log E)_{|E_{J}}$

sends, for every integer $s+1\leq l\leq r$, $\mathrm{d}x_{l}/x_{l}$ to itself,
and for every integer $l>r$, $\mathrm{d}x_{l}$ to itself. It is an injective
morphism of vector bundles, whose image is the vector subbundle of
$\Omega^{1}_{X}(\log E)_{|E_{J}}$ generated by
$(\mathrm{d}x_{l}/x_{l})_{s+1\leq l\leq r}$ and by $(\mathrm{d}x_{l})_{l>r}$.

On the other hand, the morphism of vector bundles:

$\beta_{J}:\Omega^{1}_{X}(\log
E)_{|E_{J}}\longrightarrow{\mathcal{O}}_{E_{J}}^{J}$

sends, for every integer $l$ such that $1\leq l\leq s$,
$\mathrm{d}x_{l}/x_{l}$ to $(0,...,1,...,0)$ with the $1$ in $i_{l}$-th
position; for every integer $l$ such that $s+1\leq l\leq r$,
$\mathrm{d}x_{l}/x_{l}$ to $0$, and for every integer $l$ such that $l>r$,
$\mathrm{d}x_{l}$ to $0$.

Consequently, it is surjective, and its kernel is the vector subbundle of
$\Omega^{1}_{X}(\log E)_{|E_{J}}$ generated by
$(\mathrm{d}x_{l}/x_{l})_{s+1\leq l\leq r}$ and by $(\mathrm{d}x_{l})_{l>r}$,
which is exactly the image of $\tilde{\alpha_{J}}$.

So the sequence is exact, as wanted. ∎

####  3.2.2. Proof of (3.1.2)

We shall reason by induction over the dimension $n$ of $X$.

If $n=0$, then $X$ is a finite scheme and the divisor $E$ is empty, so that
equality (3.1.2) holds.

Let $n\geq 1$ be an integer, let us assume that equality (3.1.2) holds for
schemes of dimension at most $n-1$, and let us prove it for a scheme $X$ of
dimension $n$.

One easily checks in local coordinates that we have an exact sequence of
coherent sheaves on $X$:

(3.2.1)
$0\longrightarrow\Omega^{1}_{X}\overset{u}{\longrightarrow}\Omega^{1}_{X}(\log
E)\overset{(\mathrm{Res}_{E_{i}})_{i\in I}}{\longrightarrow}\bigoplus_{i\in
I}i_{E_{i}\ast}{\mathcal{O}}_{E_{i}}\longrightarrow 0,$

where $u$ is the canonical inclusion morphism, and for every $i$ in $I$,
$\mathrm{Res}_{E_{i}}$ is the residue morphism.

Consequently, we have the equality in ${\mathrm{CH}}^{\ast}(X)$:

(3.2.2) $c(\Omega^{1}_{X})=c(\Omega^{1}_{X}(\log E))\prod_{i\in
I}c([i_{E_{i}\ast}{\mathcal{O}}_{E_{i}}])^{-1}.$

For every element $i$ in $I$, we have an exact sequence of coherent sheaves on
$X$:

(3.2.3)
$0\longrightarrow{\mathcal{O}}_{X}(-E_{i})\longrightarrow{\mathcal{O}}_{X}\longrightarrow
i_{E_{i}\ast}{\mathcal{O}}_{E_{i}}\longrightarrow 0,$

hence the equality in ${\mathrm{CH}}^{\ast}(X)$:

$c([i_{E_{i}\ast}{\mathcal{O}}_{E_{i}}])^{-1}=c({\mathcal{O}}_{X}(-E_{i}))=1-[E_{i}].$

Hence replacing in (3.2.2):

$\displaystyle c(\Omega^{1}_{X})$ $\displaystyle=c(\Omega^{1}_{X}(\log
E))\prod_{i\in I}(1-[E_{i}]),$ $\displaystyle=c(\Omega^{1}_{X}(\log
E))\sum_{K\subset I}\prod_{i\in K}(-[E_{i}]),$
$\displaystyle=c(\Omega^{1}_{X}(\log E))\sum_{K\subset
I}(-1)^{|K|}[E_{K}]\textrm{ because the $(E_{i})_{i\in I}$ intersect each
other transversally,}$ $\displaystyle=\sum_{K\subset
I}(-1)^{|K|}i_{E_{K}\ast}c(\Omega^{1}_{X}(\log E)_{|E_{K}}),$
$\displaystyle=\sum_{K\subset
I}(-1)^{|K|}i_{E_{K}\ast}c(\Omega^{1}_{E_{K}}(\log E_{K}\cap
E^{|K|+1}))\textrm{ using equality \eqref{eq Omega (log) _E_J},}$ (3.2.4)
$\displaystyle=c(\Omega^{1}_{X}(\log E))+\sum_{\emptyset\neq K\subset
I}(-1)^{|K|}i_{E_{K}\ast}c(\Omega^{1}_{E_{K}}(\log E_{K}\cap E^{|K|+1})).$

Let $K$ be a non-empty subset of $I$. Applying the induction hypothesis to the
scheme $E_{K}$ of dimension at most $n-1$ and to the divisor with strict
normal crossings $E_{K}\cap E^{|K|+1}$, whose components are the
$(E_{K\cup\\{i\\}})_{i\in I-K}$, yields the equality in
${\mathrm{CH}}^{\ast}(E_{K})$:

$c(\Omega^{1}_{E_{K}}(\log E_{K}\cap E^{|K|+1}))=\sum_{K\subset J\subset
I}i_{E_{J},E_{K}\ast}c(\Omega^{1}_{E_{J}}).$

Hence replacing in (3.2.4):

$\displaystyle c(\Omega^{1}_{X})$ $\displaystyle=c(\Omega^{1}_{X}(\log
E))+\sum_{\emptyset\neq K\subset J\subset
I}(-1)^{|K|}i_{E_{J}\ast}c(\Omega^{1}_{E_{J}}),$
$\displaystyle=c(\Omega^{1}_{X}(\log E))+\sum_{\emptyset\neq J\subset
I}\Big{[}\sum_{\emptyset\neq K\subset
J}(-1)^{|K|}\Big{]}i_{E_{J}\ast}c(\Omega^{1}_{E_{J}}),$
$\displaystyle=c(\Omega^{1}_{X}(\log E))+\sum_{\emptyset\neq J\subset
I}\Big{[}\prod_{i\in J}(1-1)-1\Big{]}i_{E_{J}\ast}c(\Omega^{1}_{E_{J}}),$
$\displaystyle=c(\Omega^{1}_{X}(\log E))-\sum_{\emptyset\neq J\subset
I}i_{E_{J}\ast}c(\Omega^{1}_{E_{J}}),$

hence we have the equality in ${\mathrm{CH}}^{\ast}(X)$:

$\displaystyle c(\Omega^{1}_{X}(\log E))$
$\displaystyle=c(\Omega^{1}_{X})+\sum_{\emptyset\neq J\subset
I}i_{E_{J}\ast}c(\Omega^{1}_{E_{J}}),$ $\displaystyle=\sum_{J\subset
I}i_{E_{J}\ast}c(\Omega^{1}_{E_{J}}),$

which concludes the induction and the proof of equality (3.1.2).

####  3.2.3. Proof of (3.1.3)

Similarly to the proof of (3.1.2), using exact sequences (3.2.1), (3.2.3) and
the multiplicativity of the Todd class ${\rm{Td}}$, we obtain the equality in
${\mathrm{CH}}^{\ast}(X)_{\mathbb{Q}}$:

${\rm{Td}}(\Omega^{1}_{X}(\log E))={\rm{Td}}(\Omega^{1}_{X})\prod_{i\in
I}{\rm{Td}}([i_{E_{i}\ast}{\mathcal{O}}_{E_{i}}])={\rm{Td}}(\Omega^{1}_{X})\prod_{i\in
I}{\rm{td}}(-[E_{i}])^{-1},$

as wanted.

###  3.3. Normal bundles to the strata of the singular fibers and Chern
classes

####  3.3.1. Proof of Proposition 3.1.2

Let $J$ be a non-empty subset of $I$. Since the line bundle
$\Omega^{1}_{C}(\log\Delta)_{|\Delta}$ can be trivialised on $\Delta$, the
line bundle $(g^{\ast}\Omega^{1}_{C}(\log\Delta))_{|D_{J}}$ can be trivialised
on $D_{J}$, and we obtain the equality in ${\mathrm{CH}}^{\ast}(D_{J})$:

$c(g^{\ast}\Omega^{1}_{C}(\log\Delta))_{|D_{J}}=1,$

hence by definition of the relative logarithmic vector bundle
$\omega^{1}_{Y/C}$ on $Y$, the equality in ${\mathrm{CH}}^{\ast}(D_{J})$:

$c(\omega^{1}_{Y/C})_{|D_{J}}=c(\Omega^{1}_{Y}(\log Y_{\Delta}))_{|D_{J}},$

hence applying equality (3.1.1) from Proposition 3.1.1 to the scheme $X:=Y$
and the reduced divisor with strict normal crossings $E:=|D|=\sum_{i\in
I}D_{i}$, and to the subset $J$ in $I$:

$c(\omega^{1}_{Y/C})_{|D_{J}}=c(\Omega^{1}_{D_{J}}(\log D_{J}\cap
D^{|J|+1})),$

as wanted.

####  3.3.2. Proofs of Proposition 3.1.3 and Corollaries 3.1.4 and 3.1.5

Let $i$ be an element of $I$. Since $\Delta$ is a scheme of dimension $0$, the
line bundle ${\mathcal{N}}_{\Delta}C$ on $\Delta$ can be trivialized.
Consequently, we can trivialize the following line bundle on $D_{i}$:

$\displaystyle(g^{\ast}{\mathcal{N}}_{\Delta}C)_{|D_{i}}$
$\displaystyle\simeq{\mathcal{O}}_{Y}(Y_{\Delta})_{|D_{i}}$
$\displaystyle\simeq{\mathcal{O}}_{Y}(m_{i}D_{i})_{|D_{i}}\otimes\bigotimes_{j\in
I-\\{i\\}}{\mathcal{O}}_{Y}(m_{j}D_{j})_{|D_{i}}$
$\displaystyle\simeq{\mathcal{N}}_{i}^{\otimes m_{i}}\otimes\bigotimes_{j\in
I-\\{i\\}}{\mathcal{O}}_{D_{i}}(m_{j}D_{ij}).$

Taking Chern classes, we obtain the following equality in
${\mathrm{CH}}^{1}(D_{i})$:

(3.3.1) $m_{i}c_{1}({\mathcal{N}}_{i})=-\sum_{j\in I-\\{i\\}}m_{j}[D_{ij}],$

which shows Proposition 3.1.3.

If $\beta$ is a class in ${\mathrm{CH}}^{\ast}(D_{i})$, intersecting (3.3.1)
with $\beta$ then pushing forward by the inclusion $i_{D_{i}}$ yields
Corollary 3.1.4.

Now, let $(i,j)$ be a pair in $I^{2}$ such that $i\prec j$. Applying equality
(3.3.1) with the element $i$ then restricting to $D_{ij}$ yields the equality
in ${\mathrm{CH}}^{1}(D_{ij})$:

$m_{i}c_{1}({\mathcal{N}}_{i|D_{ij}})=-\sum_{k\in
I-\\{i\\}}m_{k}[D_{ik}]_{|D_{ij}},$

where $[D_{ik}]$ is seen as a divisor in $D_{i}$.

If $k\neq j$, this divisor intersects transversally the divisor $D_{ij}$, so
that we have the equality in ${\mathrm{CH}}^{1}(D_{ij})$:

$[D_{ik}]_{|D_{ij}}=[D_{ik}\cap D_{ij}]=[D_{ijk}].$

If $k=j$, the self-intersection of the divisor $D_{ik}$ is given by:

$[D_{ik}]_{|D_{ij}}=c_{1}({\mathcal{N}}_{D_{ij}}D_{i})=c_{1}({\mathcal{N}}_{j|D_{ij}}).$

We have used the canonical isomorphism of line bundles on $D_{ij}$:

${\mathcal{N}}_{D_{ij}}D_{i}\simeq{\mathcal{N}}_{j|D_{ij}}.$

Hence replacing, we have the equality in ${\mathrm{CH}}^{1}(D_{ij})$:

$m_{i}c_{1}({\mathcal{N}}_{i|D_{ij}})=-m_{j}c_{1}({\mathcal{N}}_{j|D_{ij}})-\sum_{k\in
I-\\{i,j\\}}m_{k}[D_{ijk}],$

which shows Corollary 3.1.5.

###  3.4. Characteristic classes of $[T_{g}]$ and $\omega^{1\vee}_{Y/C}$

####  3.4.1. Proof of Proposition 3.1.6

Using equality (3.1.3) from Proposition 3.1.1 applied to the scheme $X:=Y$ and
the reduced divisor with strict normal crossings $E:=|D|=\sum_{i\in I}D_{i}$,
we have the equality in ${\mathrm{CH}}^{\ast}(Y)_{\mathbb{Q}}$:

${\rm{Td}}(\Omega^{1}_{Y}(\log D))={\rm{Td}}(\Omega^{1}_{Y})\prod_{i\in
I}{\rm{td}}(-[D_{i}])^{-1},$

hence the equality:

${\rm{Td}}(\Omega^{1}_{Y})={\rm{Td}}(\Omega^{1}_{Y}(\log D))\prod_{i\in
I}{\rm{td}}(-[D_{i}]),$

hence, using the definition of the relative logarithmic bundle
$\omega^{1}_{Y/C}$ and the multiplicativity of ${\rm{Td}}$:

${\rm{Td}}(\Omega^{1}_{Y})={\rm{Td}}(\omega^{1}_{Y/C})\,g^{\ast}{\rm{Td}}(\Omega^{1}_{C}(\log\Delta))\,\prod_{i\in
I}{\rm{td}}(-[D_{i}]).$

Since $C$ is a curve, we have a canonical isomorphism of line bundles on $C$:

$\Omega^{1}_{C}(\log\Delta)\simeq\Omega^{1}_{C}(\Delta),$

hence replacing:

$\displaystyle{\rm{Td}}(\Omega^{1}_{Y})$
$\displaystyle={\rm{Td}}(\omega^{1}_{Y/C})\,g^{\ast}{\rm{td}}(c_{1}(\Omega^{1}_{C})+[\Delta])\,\prod_{i\in
I}{\rm{td}}(-[D_{i}]).$

Taking duals, we obtain the equality in
${\mathrm{CH}}^{\ast}(Y)_{\mathbb{Q}}$:

(3.4.1)
${\rm{Td}}(T_{Y})={\rm{Td}}(\omega^{1\vee}_{Y/C})\,g^{\ast}{\rm{td}}(c_{1}(T_{C})-[\Delta])\,\prod_{i\in
I}{\rm{td}}([D_{i}]).$

By definition of the relative tangent class $[T_{g}]$ in $K^{0}(Y)$, we have
the identity in ${\mathrm{CH}}^{\ast}(Y)_{\mathbb{Q}}$:

$\displaystyle{\rm{Td}}([T_{g}])$
$\displaystyle={\rm{Td}}(T_{Y})g^{\ast}{\rm{Td}}(T_{C})^{-1},$
$\displaystyle={\rm{Td}}(\omega^{1\vee}_{Y/C})\,g^{\ast}[{\rm{td}}(c_{1}(T_{C})-[\Delta]){\rm{td}}(c_{1}(T_{C}))^{-1}]\prod_{i\in
I}{\rm{td}}([D_{i}])\textrm{ using \eqref{first equality td omega},}$

hence the equality:

(3.4.2)
$\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}=g^{\ast}[{\rm{td}}(c_{1}(T_{C})-[\Delta]){\rm{td}}(c_{1}(T_{C}))^{-1}]\prod_{i\in
I}{\rm{td}}([D_{i}]).$

Since $C$ is a curve, all the classes in ${\mathrm{CH}}^{2}(C)_{\mathbb{Q}}$
vanish. Consequently, we have the equality in
${\mathrm{CH}}^{\ast}(C)_{\mathbb{Q}}$:

$\displaystyle{\rm{td}}(c_{1}(T_{C})-[\Delta]){\rm{td}}(c_{1}(T_{C}))^{-1}$
$\displaystyle=[1+1/2\,(c_{1}(T_{C})-[\Delta])][1+1/2\,c_{1}(T_{C})]^{-1},$
$\displaystyle=[1+1/2(c_{1}(T_{C})-[\Delta])][1-1/2\ c_{1}(T_{C})],$
$\displaystyle=1-1/2\,[\Delta].$

Hence replacing in (3.4.2), we obtain the equality in
${\mathrm{CH}}^{\ast}(Y)_{\mathbb{Q}}$:

$\displaystyle\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}$
$\displaystyle=g^{\ast}[1-1/2\,[\Delta]]\prod_{i\in I}{\rm{td}}([D_{i}]),$
(3.4.3) $\displaystyle=\prod_{i\in
I}{\rm{td}}([D_{i}])-1/2\,(g^{\ast}[\Delta])\prod_{i\in I}{\rm{td}}([D_{i}]).$

The line bundle ${\mathcal{O}}_{C}(\Delta)_{|\Delta}$ on $\Delta$ can be
trivialized, so for every element $i$ in $I$, the line bundle
$(g^{\ast}{\mathcal{O}}_{C}(\Delta))_{|D_{i}}$ on $D_{i}$ can be trivialized.
Consequently, for every element $i$ in $I$, we have the equality in
${\mathrm{CH}}^{2}(Y)$:

$(g^{\ast}[\Delta])[D_{i}]=0,$

hence we have the equality in ${\mathrm{CH}}^{\ast}(Y)_{\mathbb{Q}}$:

$(g^{\ast}[\Delta])\prod_{i\in
I}{\rm{td}}([D_{i}])=g^{\ast}[\Delta]=\sum_{i\in I}m_{i}[D_{i}].$

Hence replacing in (3.4.3), we have the equality in
${\mathrm{CH}}^{\ast}(Y)_{\mathbb{Q}}$:

$\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}=\prod_{i\in
I}{\rm{td}}([D_{i}])-1/2\sum_{i\in I}m_{i}[D_{i}],$

which shows equality (3.1.7).

Taking the terms of codimension 1, we obtain the equality in
${\mathrm{CH}}^{1}(Y)_{\mathbb{Q}}$:

$\displaystyle\Big{[}\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}\Big{]}^{(1)}$
$\displaystyle=\sum_{i\in I}{\rm{td}}([D_{i}])^{(1)}-1/2\sum_{i\in
I}m_{i}[D_{i}],$ $\displaystyle=\sum_{i\in I}(1/2[D_{i}])-1/2\sum_{i\in
I}m_{i}[D_{i}],$ $\displaystyle=-1/2\sum_{i\in I}(m_{i}-1)[D_{i}],$

which shows equality (3.1.8).

Taking instead the terms of codimension 2, we obtain the equality in
${\mathrm{CH}}^{2}(Y)_{\mathbb{Q}}$:

$\displaystyle\Big{[}\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}\Big{]}^{(2)}$
$\displaystyle=\big{[}\prod_{i\in I}{\rm{td}}([D_{i}])\big{]}^{(2)},$
$\displaystyle=\sum_{i\in I}{\rm{td}}([D_{i}])^{(2)}+\sum_{(i,j)\in
I^{2},i\prec j}{\rm{td}}([D_{i}])^{(1)}.{\rm{td}}([D_{j}])^{(1)},$
$\displaystyle=\sum_{i\in I}(1/12\,[D_{i}]^{2})+\sum_{(i,j)\in I^{2},i\prec
j}(1/2\,[D_{i}]).(1/2[D_{j}]),$ (3.4.4) $\displaystyle=1/12\,\sum_{i\in
I}[D_{i}]^{2}+1/4\sum_{(i,j)\in I^{2},i\prec j}[D_{i}].[D_{j}].$

For every element $i$ in $I$, the self-intersection of the divisor $D_{i}$ is
given in ${\mathrm{CH}}^{2}(Y)$ by:

$[D_{i}]^{2}=i_{D_{i}\ast}c_{1}({\mathcal{N}}_{i}).$

On the other hand, for every pair $(i,j)$ in $I^{2}$ such that $i\prec j$, the
divisors $D_{i}$ and $D_{j}$ intersect transversally, so that we have the
equality in ${\mathrm{CH}}^{2}(Y)$:

$[D_{i}].[D_{j}]=[D_{ij}].$

Hence replacing in (3.4.4):

$\Big{[}\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}\Big{]}^{(2)}=1/12\sum_{i\in
I}i_{D_{i}\ast}c_{1}({\mathcal{N}}_{i})+1/4\sum_{(i,j)\in I^{2},i\prec
j}[D_{ij}],$

which shows equality (3.1.9).

Finally, for every element $i$ in $I$, applying Proposition 3.1.3 and dividing
by $m_{i}$, we have the equality in ${\mathrm{CH}}^{1}(D_{i})_{\mathbb{Q}}$:

$c_{1}({\mathcal{N}}_{i})=-\sum_{j\in I-\\{i\\}}m_{j}/m_{i}[D_{ij}],$

hence pushing forward by the inclusion of $D_{i}$ in $Y$, then summing over
$i$, we have the equality in ${\mathrm{CH}}^{2}(Y)_{\mathbb{Q}}$:

$\sum_{i\in I}i_{D_{i}\ast}c_{1}({\mathcal{N}}_{i})=-\sum_{(i,j)\in
I^{2},i\prec j}(m_{i}/m_{j}+m_{j}/m_{i})[D_{ij}],$

hence replacing, we obtain the equality in
${\mathrm{CH}}^{2}(Y)_{\mathbb{Q}}$:

$\displaystyle\Big{[}\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}\Big{]}^{(2)}$
$\displaystyle=-1/12\sum_{(i,j)\in I^{2},i\prec
j}(m_{i}/m_{j}+m_{j}/m_{i})[D_{ij}]+1/4\sum_{(i,j)\in I^{2},i\prec
j}[D_{ij}],$ $\displaystyle=1/12\sum_{(i,j)\in I^{2},i\prec
j}(3-m_{i}/m_{j}-m_{j}/m_{i})[D_{ij}],$

which shows equality (3.1.10).

This concludes the proof of Proposition 3.1.6.

####  3.4.2. Proof of Proposition 3.1.7

Using equality (3.1.2) from Proposition 3.1.1 applied to the scheme $X:=Y$ and
the reduced divisor with strict normal crossings $E:=|D|=\sum_{i\in I}D_{i}$,
we have the equality in ${\mathrm{CH}}^{\ast}(Y)$:

$c(\Omega^{1}_{Y}(\log D))=c(\Omega^{1}_{Y})+\sum_{\emptyset\neq J\subset
I}i_{D_{J}\ast}c(\Omega^{1}_{D_{J}}),$

hence, by definition of the relative logarithmic bundle $\omega^{1}_{Y/C}$,
the equality in ${\mathrm{CH}}^{\ast}(Y)$:

$\displaystyle c(\omega^{1}_{Y/C})$
$\displaystyle=\big{[}c(\Omega^{1}_{Y})+\sum_{\emptyset\neq J\subset
I}i_{D_{J}\ast}c(\Omega^{1}_{D_{J}})\big{]}\,g^{\ast}c(\Omega^{1}_{C}(\log\Delta))^{-1}.$

Since $C$ is a curve, we have a canonical isomorphism of line bundles on $C$:

$\Omega^{1}_{C}(\log\Delta)\simeq\Omega^{1}_{C}(\Delta),$

hence replacing:

$c(\omega^{1}_{Y/C})=\big{[}c(\Omega^{1}_{Y})+\sum_{\emptyset\neq J\subset
I}i_{D_{J}\ast}c(\Omega^{1}_{D_{J}})\big{]}\,g^{\ast}(1+c_{1}(\Omega^{1}_{C})+[\Delta])^{-1},$

hence taking duals:

(3.4.5) $c(\omega^{1\vee}_{Y/C})=\big{[}c(T_{Y})+\sum_{\emptyset\neq J\subset
I}(-1)^{|J|}i_{D_{J}\ast}c(T_{D_{J}})\big{]}\,g^{\ast}(1+c_{1}(T_{C})-[\Delta])^{-1}.$

The line bundles $T_{C|\Delta}$ and ${\mathcal{O}}_{C}(\Delta)_{|\Delta}$ on
$\Delta$ can be trivialized, so for every non-empty subset $J$ in $I$, the
line bundles $(g^{\ast}T_{C})_{|D_{J}}$ and
$(g^{\ast}{\mathcal{O}}_{C}(\Delta))_{|D_{J}}$ on $D_{J}$ can be trivialized.
Consequently, for every non-empty subset $J$ in $I$, we have the equalities in
${\mathrm{CH}}^{\ast}(Y)$:

$[i_{D_{J}\ast}c(T_{D_{J}})](g^{\ast}c_{1}(T_{C}))=[i_{D_{J}\ast}c(T_{D_{J}})](g^{\ast}[\Delta])=0,$

hence we have the equality in ${\mathrm{CH}}^{\ast}(Y)$:

$\bigg{[}\sum_{\emptyset\neq J\subset
I}(-1)^{|J|}i_{D_{J}\ast}c(T_{D_{J}})\bigg{]}\,g^{\ast}(1+c_{1}(T_{C})-[\Delta])^{-1}=\sum_{\emptyset\neq
J\subset I}(-1)^{|J|}i_{D_{J}\ast}c(T_{D_{J}}).$

Hence replacing in (3.4.5), we obtain the equality:

(3.4.6)
$c(\omega^{1\vee}_{Y/C})=c(T_{Y})g^{\ast}(1+c_{1}(T_{C})-[\Delta])^{-1}+\sum_{\emptyset\neq
J\subset I}(-1)^{|J|}i_{D_{J}\ast}c(T_{D_{J}}).$

Since $C$ is a curve, all the classes in ${\mathrm{CH}}^{2}(C)$ vanish.
Consequently, we have the equality in ${\mathrm{CH}}^{\ast}(C)$:

$(1+c_{1}(T_{C})-[\Delta])^{-1}=(1+c_{1}(T_{C}))^{-1}+[\Delta].$

Hence replacing in (3.4.6):

$\displaystyle c(\omega^{1\vee}_{Y/C})$
$\displaystyle=c(T_{Y})g^{\ast}[(1+c_{1}(T_{C}))^{-1}+[\Delta]]+\sum_{\emptyset\neq
J\subset I}(-1)^{|J|}i_{D_{J}\ast}c(T_{D_{J}}),$
$\displaystyle=c(T_{Y})g^{\ast}c(T_{C})^{-1}+c(T_{Y})\big{(}\sum_{i\in
I}m_{i}[D_{i}]\big{)}+\sum_{\emptyset\neq J\subset
I}(-1)^{|J|}i_{D_{J}\ast}c(T_{D_{J}}),$ $\displaystyle=c([T_{g}])+\sum_{i\in
I}m_{i}i_{D_{i}\ast}c(T_{Y|D_{i}})+\sum_{\emptyset\neq J\subset
I}(-1)^{|J|}i_{D_{J}\ast}c(T_{D_{J}})\textrm{ by definition of the class
$[T_{g}]$.}$

Let $r$ be a non-negative integer, taking the terms of codimension $r$, we
obtain the equality in ${\mathrm{CH}}^{r}(Y)$:

(3.4.7) $c_{r}(\omega^{1\vee}_{Y/C})=c_{r}([T_{g}])+\sum_{i\in
I}m_{i}i_{D_{i}\ast}c_{r-1}(T_{Y|D_{i}})+\sum_{\emptyset\neq J\subset
I}(-1)^{|J|}i_{D_{J}\ast}c_{r-|J|}(T_{D_{J}}).$

For every element $i$ in $I$, we have an exact sequence of vector bundles on
$D_{i}$:

$0\longrightarrow T_{D_{i}}\longrightarrow
T_{Y|D_{i}}\longrightarrow{\mathcal{N}}_{i}\longrightarrow 0,$

hence the equality in ${\mathrm{CH}}^{\ast}(D_{i})$:

$c(T_{Y|D_{i}})=c(T_{D_{i}})(1+c_{1}({\mathcal{N}}_{i})),$

hence the equality in ${\mathrm{CH}}^{r-1}(D_{i})$,

$c_{r-1}(T_{Y|D_{i}})=c_{r-1}(T_{D_{i}})+c_{1}({\mathcal{N}}_{i})c_{r-2}(T_{D_{i}}).$

Hence replacing in (3.4.7), we obtain the equality in ${\mathrm{CH}}^{r}(Y)$:

$c_{r}(\omega^{1\vee}_{Y/C})=c_{r}([T_{g}])+\sum_{i\in
I}m_{i}i_{D_{i}\ast}[c_{r-1}(T_{D_{i}})+c_{1}({\mathcal{N}}_{i})c_{r-2}(T_{D_{i}})]+\sum_{\emptyset\neq
J\subset I}(-1)^{|J|}i_{D_{J}\ast}c_{r-|J|}(T_{D_{J}}),$

which shows equality (3.1.11).

Now, let us assume that the subscheme $D^{3}$ is empty. In this case, equality
(3.1.11) can be rewritten:

$\displaystyle c_{r}(\omega^{1\vee}_{Y/C})$
$\displaystyle=c_{r}([T_{g}])+\sum_{i\in
I}m_{i}i_{D_{i}\ast}[c_{r-1}(T_{D_{i}})+c_{1}({\mathcal{N}}_{i})c_{r-2}(T_{D_{i}})]$
$\displaystyle-\sum_{i\in I}i_{D_{i}\ast}c_{r-1}(T_{D_{i}})+\sum_{(i,j)\in
I^{2},i\prec j}i_{D_{ij}\ast}c_{r-2}(T_{D_{ij}}),$
$\displaystyle=c_{r}([T_{g}])+\sum_{i\in
I}(m_{i}-1)i_{D_{i}\ast}c_{r-1}(T_{D_{i}})$ (3.4.8) $\displaystyle+\sum_{i\in
I}m_{i}i_{D_{i}\ast}(c_{1}({\mathcal{N}}_{i})c_{r-2}(T_{D_{i}}))+\sum_{(i,j)\in
I^{2},i\prec j}i_{D_{ij}\ast}c_{r-2}(T_{D_{ij}}).$

For every element $i$ in $I$, applying Corollary 3.1.4 to the class
$\beta:=c_{r-2}(T_{D_{i}})$ in ${\mathrm{CH}}^{r-2}(D_{i})$ yields the
equality in ${\mathrm{CH}}^{r}(Y)$:

(3.4.9)
$m_{i}i_{D_{i}\ast}(c_{1}({\mathcal{N}}_{i})c_{r-2}(T_{D_{i}}))=-\sum_{j\in
I-\\{i\\}}m_{j}i_{D_{ij}\ast}c_{r-2}(T_{D_{i}|D_{ij}}),$

hence summing over $i$, we obtain the equality in ${\mathrm{CH}}^{r}(Y)$:

$\sum_{i\in
I}m_{i}i_{D_{i}\ast}(c_{1}({\mathcal{N}}_{i})c_{r-2}(T_{D_{i}}))=-\sum_{(i,j)\in
I^{2},i\prec
j}i_{D_{ij}\ast}(m_{j}c_{r-2}(T_{D_{i}|D_{ij}})+m_{i}c_{r-2}(T_{D_{j}|D_{ij}})).$

For every pair $(i,j)$ in $I^{2}$ such that $i\prec j$, we have the following
normal exact sequences of vector bundles on $D_{ij}$:

$0\longrightarrow T_{D_{ij}}\longrightarrow
T_{D_{i}|D_{ij}}\longrightarrow{\mathcal{N}}_{D_{ij}}D_{i}\longrightarrow 0,$
$0\longrightarrow T_{D_{ij}}\longrightarrow
T_{D_{j}|D_{ij}}\longrightarrow{\mathcal{N}}_{D_{ij}}D_{j}\longrightarrow 0,$

hence the following equalities in ${\mathrm{CH}}^{r-2}(D_{ij})$:

$c_{r-2}(T_{D_{i}|D_{ij}})=c_{r-2}(T_{D_{ij}})+c_{1}({\mathcal{N}}_{D_{ij}}D_{i})c_{r-3}(T_{D_{ij}}),$
$c_{r-2}(T_{D_{j}|D_{ij}})=c_{r-2}(T_{D_{ij}})+c_{1}({\mathcal{N}}_{D_{ij}}D_{j})c_{r-3}(T_{D_{ij}}),$

hence using the canonical isomorphisms of line bundles on $D_{ij}$ between
${\mathcal{N}}_{D_{ij}}D_{j}$ and ${\mathcal{N}}_{i|D_{ij}}$, as well as
between ${\mathcal{N}}_{D_{ij}}D_{i}$ and ${\mathcal{N}}_{j|D_{ij}}$:

$c_{r-2}(T_{D_{i}|D_{ij}})=c_{r-2}(T_{D_{ij}})+c_{1}({\mathcal{N}}_{j|D_{ij}})c_{r-3}(T_{D_{ij}}),$
$c_{r-2}(T_{D_{j}|D_{ij}})=c_{r-2}(T_{D_{ij}})+c_{1}({\mathcal{N}}_{i|D_{ij}})c_{r-3}(T_{D_{ij}}),$

hence replacing in (3.4.9), we obtain the equality in ${\mathrm{CH}}^{r}(Y)$:

$\sum_{i\in
I}m_{i}i_{D_{i}\ast}(c_{1}({\mathcal{N}}_{i})c_{r-2}(T_{D_{i}}))=-\sum_{(i,j)\in
I^{2},i\prec j}(m_{j}+m_{i})i_{D_{ij}\ast}c_{r-2}(T_{D_{ij}})\\\
-\sum_{(i,j)\in I^{2},i\prec
j}i_{D_{ij}\ast}[\\{m_{i}c_{1}({\mathcal{N}}_{i|D_{ij}})+m_{j}c_{1}({\mathcal{N}}_{j|D_{ij}})\\}c_{r-3}(T_{D_{ij}}];$

using Corollary 3.1.5, and using that $D^{3}$ is empty, we get:

$\sum_{i\in
I}m_{i}i_{D_{i}\ast}(c_{1}({\mathcal{N}}_{i})c_{r-2}(T_{D_{i}}))=-\sum_{(i,j)\in
I^{2},i\prec j}(m_{i}+m_{j})i_{D_{ij}\ast}c_{r-2}(T_{D_{ij}}).$

Hence replacing in (3.4.8), we obtain the equality in
${\mathrm{CH}}^{r}(Y)_{\mathbb{Q}}$:

$\displaystyle c_{r}(\omega^{1\vee}_{Y/C})$
$\displaystyle=c_{r}([T_{g}])+\sum_{i\in
I}(m_{i}-1)i_{D_{i}\ast}c_{r-1}(T_{D_{i}})$ $\displaystyle-\sum_{(i,j)\in
I^{2},i\prec j}(m_{i}+m_{j})i_{D_{ij}\ast}c_{r-2}(T_{D_{ij}})+\sum_{(i,j)\in
I^{2},i\prec j}i_{D_{ij}\ast}c_{r-2}(T_{D_{ij}}),$
$\displaystyle=c_{r}([T_{g}])+\sum_{i\in
I}(m_{i}-1)i_{D_{i}\ast}c_{r-1}(T_{D_{i}})-\sum_{(i,j)\in I^{2},i\prec
j}(m_{i}+m_{j}-1)i_{D_{ij}\ast}c_{r-2}(T_{D_{ij}}).$

This proves equality (3.1.12).

##  4\. The alternating product of Griffiths line bundles associated to a
pencil of complete varieties with DNC degenerations

In this section, we work over a base field $k$ of characteristic zero, and by
the word “scheme”, we mean a separated algebraic scheme of finite type over
$k$.

###  4.1. The characteristic classes $\rho$ and $\rho_{r}$ and the Riemann-
Roch-Grothendieck formula

In this paragraph, we will work on a fixed smooth $k$-scheme $X$.

####  4.1.1. The classes $\lambda_{y}$, $\varphi_{y}$, and $\rho$

Recall that the _Grothendieck $\lambda$-map_:

$\lambda_{y}:K^{0}(X)\longrightarrow 1+y\,K^{0}(X)[[y]],$

where $y$ denotes an indeterminate, is defined as the only map that is
multiplicative for exact sequences, and that is given on (the $K$-theory class
of) a vector bundle $V$ by:

$\lambda_{y}([V]):=\sum\limits_{i=0}^{{\rm
rk}(V)}y^{i}\left[\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{i}V\right]\in
1+y\,K^{0}(X)[y];$

see for instance [SGA71, app. to Exposé 0].

######  Definition 4.1.1.

For every vector bundle $V$ on $X$, we define $\varphi_{y}(V)$ as the element
of ${\mathrm{CH}}^{\ast}(X)_{\mathbb{Q}}[y]$ given by:

(4.1.1)
$\varphi_{y}(V):={\rm{ch}}\big{(}\lambda_{y}(V^{\vee})\big{)}\,\,{\rm{Td}}(V)=\sum\limits_{i=0}^{{\rm
rk}(V)}y^{i}{\rm{ch}}\left(\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{i}V^{\vee}\right)\,{\rm{Td}}(V).$

The ring ${\mathrm{CH}}^{\ast}(X)_{\mathbb{Q}}[y]$ has a graduation given by
the sum of the cohomological degree in ${\mathrm{CH}}^{\ast}$ and the degree
in the indeterminate $y$:

$\left({\mathrm{CH}}^{\ast}(X)_{\mathbb{Q}}[y]\right)^{(d)}:=\overset{d}{\underset{e=0}{\bigoplus}}{\mathrm{CH}}^{e}(X)_{\mathbb{Q}}\otimes_{\mathbb{Q}}\mathbb{Q}[y]_{d-e}.$

For every vector bundle $V$ over $X$, the term of total degree $0$ of the
element $\varphi_{y}(V)$ is $1$, so this element is in
$1+({\mathrm{CH}}^{\ast}(X)_{\mathbb{Q}}[y])^{(\geq 1)}.$

######  Proposition 4.1.2.

The map $\varphi_{y}$ is multiplicative for short exact sequences.

###### Proof.

The $\lambda$-map $\lambda_{y}$ and the Todd class ${\rm{Td}}$ are both
multiplicative for short exact sequences, and the Chern character ${\rm{ch}}$
is a morphism of rings. ∎

######  Remark 4.1.3.

We could use this to prolong $\varphi_{y}$ to a morphism from $K^{0}(X)$ to
$1+\left({\mathrm{CH}}^{\ast}(X)_{\mathbb{Q}}[[y]]\right)^{(\geq 1)}$, but we
would then lose the property that $\varphi_{y}(V)$ is a polynomial, which is
necessary for our purpose.

######  Proposition 4.1.4.

Let $L$ be a line bundle on $X$, we have the identity in
${\mathrm{CH}}^{\ast}(X)_{\mathbb{Q}}[y]$:

(4.1.2) $\varphi_{y}(L)={\rm{td}}(c_{1}(L))\,\left[1+y\,e^{-c_{1}(L)}\right].$

In particular, replacing $y$ with $-1$:

(4.1.3) $\varphi_{-1}(L)=c_{1}(L).$

###### Proof.

The first equality (4.1.2) comes from a direct computation of
$\varphi_{y}(L)$, using the equality in
${\mathrm{CH}}^{\ast}(X)_{\mathbb{Q}}$:

${\rm{ch}}(L^{\vee})=e^{-c_{1}(L)}.$

The second equality (4.1.3) follows by definition of the series ${\rm{td}}$. ∎

The identity (4.1.3) is also a particular case of [Ful98, Ex. 3.2.5]. Indeed,
the upcoming Proposition 4.1.8 can be seen as a variant of this example.

######  Definition 4.1.5.

If $V$ is a vector bundle on $X$, the class $\rho(V)$ is defined by:

$\rho(V):=\dot{\varphi}_{-1}(V)=\Big{(}\frac{\mathrm{d}}{\mathrm{d}y}{\rm{ch}}(\lambda_{y}(V^{\vee}))\Big{)}_{|y=-1}{\rm{Td}}(V)\in{\mathrm{CH}}^{\ast}(X)_{\mathbb{Q}}.$

the derivative of $\varphi$ applied to $y=-1$, it is well-defined since
$\varphi_{y}(V)$ is a polynomial.

If $r\geq 1$ is an integer, we also define the characteristic class:

$\rho_{r}:=c_{r-1}-\frac{r}{2}c_{r}+\frac{1}{12}c_{1}c_{r}:K^{0}(X)\longrightarrow{\mathrm{CH}}^{\ast}(X)_{\mathbb{Q}}.$

We are interested in comparing the classes $\rho(V)$ and $\rho_{r}([V])$ when
$V$ is a vector bundle of rank $r$.

######  Proposition 4.1.6.

Let $L$ be a line bundle on $X$, we have the identity in
${\mathrm{CH}}^{\ast}(X)_{\mathbb{Q}}$:

(4.1.4) $\rho(L)={\rm{td}}(-c_{1}(L)).$

###### Proof.

We differentiate the equality (4.1.2). Since $\varphi_{y}(L)$ is a polynomial
of degree at most 1, its derivative is its coefficient of degree $1$, which
gives the equality:

$\rho(L)={\rm{td}}(c_{1}(L))\,e^{-c_{1}(L)}.$

Now, we have an equality of formal series in an indeterminate $x$:

${\rm{td}}(x)\,e^{-x}={\rm{td}}(-x),$

that we can show using the definition of ${\rm{td}}$ in terms of exponentials,
as follows:

$\displaystyle{\rm{td}}(x)\,e^{-x}$
$\displaystyle=\bigg{(}\frac{x}{1-e^{-x}}\bigg{)}\,e^{-x},$
$\displaystyle=\frac{x}{e^{x}-1},$ $\displaystyle=-\frac{x}{1-e^{-(-x)}},$
$\displaystyle={\rm{td}}(-x).$

Replacing $x$ with $c_{1}(L)$, this becomes the result. ∎

######  Corollary 4.1.7.

Let $V$ be a vector bundle of rank $r$ on $X$, and let

$V=V_{r}\supset...\supset V_{0}=0$

be a filtration such that $L_{i}:=V_{i}/V_{i-1}$ is a line bundle for every
integer $i\in\\{1,\dots,r\\}.$

We have the equality in ${\mathrm{CH}}^{\ast}(X)_{\mathbb{Q}}$:

$\rho(V)=\sum\limits_{i=1}^{r}{\rm{td}}(-c_{1}(L_{i}))\,\prod\limits_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{r}c_{1}(L_{j}).$

###### Proof.

Using Proposition 4.1.2, the map $\varphi_{y}$ is multiplicative for exact
sequences, so we have the equality:

$\varphi_{y}(V)=\prod\limits_{i=1}^{r}\varphi_{y}(L_{i}).$

Differentiating and applying Leibniz’s rule:

$\dot{\varphi}_{-1}(V)=\sum\limits_{i=1}^{r}\dot{\varphi}_{-1}(L_{i})\,\prod\limits_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{r}\varphi_{-1}(L_{j}).$

Using (4.1.3) and (4.1.4), we obtain the result. ∎

This allows us to compare $\rho$ and $\rho_{r}$:

######  Proposition 4.1.8.

Let $V$ be a vector bundle of rank $r$ on $X$, we have the identity in
${\mathrm{CH}}^{\ast}(X)_{\mathbb{Q}}$:

(4.1.5) $\rho(V)=\rho_{r}([V])+(\textrm{terms of codimension}\geq r+2).$

###### Proof.

First, suppose that $V$ has a filtration by subbundles
$V=V_{r}\supset...\supset V_{0}=0$, such that for $1\leq i\leq r$,
$L_{i}:=V_{i}/V_{i-1}$ is a line bundle. Using Corollary 4.1.7, we have the
equality:

$\rho(V)=\sum\limits_{i=1}^{r}{\rm{td}}(-c_{1}(L_{i}))\,\prod\limits_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{r}c_{1}(L_{j}).$

Recall the first terms of the development of ${\rm{td}}$:

${\rm{td}}(x)=1+\frac{x}{2}+\frac{x^{2}}{12}+...\quad.$

Accordingly we have:

$\displaystyle\rho(V)$
$\displaystyle=\sum\limits_{i=1}^{r}\Bigg{[}\Big{(}1-\frac{c_{1}(L_{i})}{2}+\frac{c_{1}(L_{i})^{2}}{12}+(\textrm{terms
of codimension}\geq 3)\Big{)}\prod\limits_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{r}c_{1}(L_{j})\Bigg{]},$
$\displaystyle=\sum\limits_{i=1}^{r}\prod\limits_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{r}c_{1}(L_{j})-\frac{1}{2}\sum\limits_{i=1}^{r}\prod\limits_{j=1}^{r}c_{1}(L_{j})+\frac{1}{12}\sum\limits_{i=1}^{r}\Big{(}c_{1}(L_{i})\,\prod\limits_{j=1}^{r}c_{1}(L_{j})\Big{)}+(\textrm{codim}\geq
r+2)),$
$\displaystyle=\sum\limits_{i=1}^{r}\prod\limits_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{r}c_{1}(L_{j})-\frac{r}{2}\prod\limits_{j=1}^{r}c_{1}(L_{j})+\frac{1}{12}\Big{(}\sum\limits_{i=1}^{r}c_{1}(L_{i})\Big{)}\prod\limits_{j=1}^{r}c_{1}(L_{j})+(\textrm{codim}\geq
r+2),$

where by $(\textrm{codim}\geq r+2)$ we mean an element in ${\mathrm{CH}}^{\geq
r+2}(X)_{\mathbb{Q}}$.

By multiplicativity of the total Chern class, the elementary symmetric
polynomials of $(c_{1}(L_{i}))_{i}$ are simply the Chern classes of $V$, which
gives the equality:

$\displaystyle\rho(V)$
$\displaystyle=c_{r-1}(V)-\frac{r}{2}c_{r}(V)+\frac{1}{12}c_{1}(V)c_{r}(V)+(\textrm{codim}\geq
r+2),$ $\displaystyle=\rho_{r}([V])+(\textrm{codim}\geq r+2)).$

This shows the desired formula when $V$ has such a filtration.

In the general case, we are reduced to this situation by the “splitting
principle” ([Ful98, Remark 3.2.3]). ∎

####  4.1.2. The class $\rho$ and the Riemann-Roch-Grothendieck formula

The Riemann-Roch formula implies the following result.

######  Proposition 4.1.9.

Let $X,S$ be two smooth $k$-schemes of pure dimension, let:

$f:X\longrightarrow S$

be a proper morphism of relative dimension $m\geq 0$, and let $V$ be a vector
bundle on $X$. Let us denote by

$[T_{f}]\in K^{0}(X)$

the relative tangent class associated with the l.c.i. morphism $f$.202020see
for instance [Ful98, B.7.6].

We have the equality in ${\mathrm{CH}}^{\ast}(S)_{\mathbb{Q}}$:

$f_{\ast}\left(\rho(V)\frac{{\rm{Td}}([T_{f}])}{{\rm{Td}}(V)}\right)=\sum\limits_{\begin{subarray}{c}0\leq
p\leq{\rm rk}(V),\\\ 0\leq q\leq
m\end{subarray}}p(-1)^{p+q-1}{\rm{ch}}(R^{q}f_{\ast}\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{p}V^{\vee}).$

In particular, taking the terms of codimension $1$, we have the equality in
${\mathrm{CH}}^{1}(S)_{\mathbb{Q}}$:

$f_{\ast}\left[\left(\rho(V)\frac{{\rm{Td}}([T_{f}])}{{\rm{Td}}(V)}\right)^{(m+1)}\right]=\sum\limits_{\begin{subarray}{c}0\leq
p\leq{\rm rk}(V),\\\ 0\leq q\leq
m\end{subarray}}p(-1)^{p+q-1}c_{1}(R^{q}f_{\ast}\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{p}V^{\vee}).$

###### Proof.

We have the equality:

(4.1.6) $\displaystyle
f_{\ast}\left(\rho(V)\frac{{\rm{Td}}([T_{f}])}{{\rm{Td}}(V)}\right)$
$\displaystyle=f_{\ast}\left(\Big{(}\frac{\mathrm{d}}{\mathrm{d}y}{\rm{ch}}(\lambda_{y}(V^{\vee}))\Big{)}_{|y=-1}{\rm{Td}}(V)\frac{{\rm{Td}}[T_{f}]}{{\rm{Td}}(V)}\right),$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}y}f_{\ast}\left({\rm{ch}}(\lambda_{y}(V^{\vee})){\rm{Td}}([T_{f}])\right)_{|y=-1},$
(4.1.7) $\displaystyle=\frac{\mathrm{d}}{\mathrm{d}y}f_{\ast}\left(\sum_{0\leq
p\leq{\rm
rk}(V)}y^{p}{\rm{ch}}(\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{p}V^{\vee}){\rm{Td}}([T_{f}])\right)_{|y=-1},$
(4.1.8) $\displaystyle=\frac{\mathrm{d}}{\mathrm{d}y}\left(\sum_{0\leq
p\leq{\rm
rk}(V)}y^{p}{\rm{ch}}(R^{\bullet}f_{\ast}\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{p}V^{\vee})\right)_{|y=-1},$
$\displaystyle=\sum_{0\leq p\leq{\rm
rk}(V)}p(-1)^{p-1}{\rm{ch}}(R^{\bullet}f_{\ast}\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{p}V^{\vee}),$
(4.1.9) $\displaystyle=\sum\limits_{\begin{subarray}{c}0\leq p\leq{\rm
rk}(V)\\\ 0\leq q\leq
m\end{subarray}}p(-1)^{p-1}(-1)^{q}{\rm{ch}}(R^{q}f_{\ast}\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{p}V^{\vee}),$

where in (4.1.6), we have used the definition of the class $\rho$; in (4.1.7),
we have used the definition of the class $\lambda_{y}$; in (4.1.8), we have
used the Riemann-Roch-Grothendieck formula; and in (4.1.9), we have used the
definition of the morphism of abelian groups $R^{\bullet}f_{\ast}$. ∎

###  4.2. Application to Griffiths line bundles of fibrations over curves with
DNC fibers

####  4.2.1. An application of Steenbrink’s theory

We are now able to compute a certain combination of the Griffiths line bundles
in terms of the function $\rho_{r}$.

######  Theorem 4.2.1.

Let $C$ be a connected smooth projective complex curve with generic point
$\eta$, $Y$ be a connected smooth projective $N$-dimensional complex scheme,
and let

$g:Y\longrightarrow C$

be a surjective morphism of complex schemes. Let us assume that there exists a
finite subset $\Delta$ in $C$ such that $g$ is smooth over $C-\Delta$, and
such that the divisor $Y_{\Delta}$ is a divisor with strict normal crossings
in $Y$.

We have the equality in ${\mathrm{CH}}^{1}(C)_{\mathbb{Q}}$:

$\sum_{n=0}^{2(N-1)}(-1)^{n-1}c_{1}(\mathcal{GK}_{C,-}(\mathbb{H}^{n}(Y_{\eta}/C_{\eta})))=g_{\ast}\left[\left(\rho_{N-1}(\omega^{1\vee}_{Y/C})\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}\right)^{(N)}\right].$

###### Proof.

Using Proposition 2.1.4, for every integer $n$, we have an isomorphism of line
bundles over $C$:

$\mathcal{GK}_{C,-}(\mathbb{H}^{n}(Y_{\eta}/C_{\eta}))\simeq\bigotimes_{0\leq
p\leq n}\big{(}\det R^{n-p}g_{\ast}\omega^{p}_{Y/C}\big{)}^{\otimes p}.$

Consequently, taking Chern classes and taking the alternate sum, we obtain the
equality in ${\mathrm{CH}}^{1}(C)_{\mathbb{Q}}$:

$\sum_{n=0}^{2(N-1)}(-1)^{n-1}c_{1}(\mathcal{GK}_{C,-}(\mathbb{H}^{n}(Y_{\eta}/C_{\eta})))=\sum_{n=0}^{2(N-1)}(-1)^{n-1}\sum_{0\leq
p\leq n}pc_{1}(R^{n-p}g_{\ast}\omega^{p}_{Y/C}).$

We reindex this sum on $p$ and $q:=n-p$, and we get:

$\sum_{n=0}^{2(N-1)}(-1)^{n-1}c_{1}(\mathcal{GK}_{C,-}(\mathbb{H}^{n}(Y_{\eta}/C_{\eta})))=\sum\limits_{\begin{subarray}{c}p,q\geq
0,\\\ p+q\leq
2(N-1)\end{subarray}}p(-1)^{p+q-1}c_{1}(R^{q}g_{\ast}\omega^{p}_{Y/C}).$

Since the morphism $g$ is of relative dimension $N-1$, the term in the sum
vanishes unless $p,q\leq N-1$, so we can rewrite:

$\displaystyle\sum_{n=0}^{2(N-1)}(-1)^{n-1}c_{1}(\mathcal{GK}_{C,-}(\mathbb{H}^{n}(Y_{\eta}/C_{\eta})))$
$\displaystyle=\sum_{0\leq p,q\leq
N-1}p(-1)^{p+q-1}c_{1}(R^{q}g_{\ast}\omega^{p}_{Y/C}),$
$\displaystyle=\sum_{0\leq p,q\leq
N-1}p(-1)^{p+q-1}c_{1}(R^{q}g_{\ast}\mathchoice{{\textstyle\bigwedge}}{{\bigwedge}}{{\textstyle\wedge}}{{\scriptstyle\wedge}}^{p}(\omega^{1\vee}_{Y/C})^{\vee}).$

Now, we apply Proposition 4.1.9 to the morphism $g$ of relative dimension
$N-1$ and the locally free sheaf

$V:=\omega^{1\vee}_{Y/C}$

of rank $N-1$. This yields the equality:

$\sum_{n=0}^{2(N-1)}(-1)^{n-1}c_{1}(\mathcal{GK}_{C,-}(\mathbb{H}^{n}(Y_{\eta}/C_{\eta})))=g_{\ast}\left[\left(\rho(\omega^{1\vee}_{Y/C})\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}\right)^{(N)}\right],$

hence applying Proposition 4.1.8 to the vector bundle $V$ of rank $N-1$:

$\displaystyle\sum_{n=0}^{2(N-1)}(-1)^{n-1}c_{1}(\mathcal{GK}_{C,-}(\mathbb{H}^{n}(Y_{\eta}/C_{\eta})))$
$\displaystyle=g_{\ast}\left[\left(\\{\rho_{N-1}(\omega^{1\vee}_{Y/C})+(\textrm{terms
of codimension}\geq
N+1)\\}\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}\right)^{(N)}\right],$
$\displaystyle=g_{\ast}\left[\left(\rho_{N-1}(\omega^{1\vee}_{Y/C})\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}\right)^{(N)}\right].\qed$

####  4.2.2. The alternating product of Griffiths line bundles and the
geometry of singular fibers

We adopt the notation of the previous subsection, and assume, without loss of
generality, that the divisor $\Delta$ in $C$ is reduced.

Let us adopt the notation of Paragraph 3.1.2. Namely, we write the divisor
with strict normal crossings $Y_{\Delta}$ as:

$D=\sum_{i\in I}m_{i}D_{i},$

where $I$ is a finite set, for every $i$ in $I$, $m_{i}\geq 1$ is an integer,
and $D_{i}$ is a smooth divisor, such that the $(D_{i})_{i\in I}$ intersect
each other transversally. We denote by ${\mathcal{N}}_{i}$ the normal bundle
$N_{D_{i}}Y$ of $D_{i}$ in $Y$.

The set $I$ may be written as the disjoint union:

$I=\bigcup_{x\in\Delta}I_{x},$

where, for every $x\in\Delta,$ $I_{x}$ denotes the non-empty subset of $I$
defined by:

$I_{x}:=\big{\\{}i\in I\mid g(D_{i})=\\{x\\}\big{\\}}.$

For every subset $J$ of $I$, let us denote:

$D_{J}:=\bigcap_{i\in J}D_{i},$

it is a smooth subscheme of codimension $|J|$.

We adopt the notation of [Del70, II, 3.4]. Namely, we denote, for every
integer $r\geq 1$, $D^{r}$ the subscheme of codimension $r$ of $Y$ defined by
the union of all the intersections of $r$ different components $(D_{i})$:

$D^{r}:=\bigcup_{J\subset I,|J|=r}D_{J}.$

Let us choose a total order $\preceq$ on $I$.

For every element $i$ in $I$, let us define an open subset ${\mathaccent
23{D}}_{i}$ of the divisor $D_{i}$ by:

${\mathaccent 23{D}}_{i}:=D_{i}-D_{i}\cap D^{2}.$

Similarly, for every pair $(i,j)$ in $I^{2}$ such that $i\prec j$, let us
define an open subset ${\mathaccent 23{D}}_{ij}$ of the subscheme

$D_{ij}:=D_{\\{i,j\\}}$

by:

${\mathaccent 23{D}}_{ij}:=D_{ij}-D_{ij}\cap D^{3}.$

Let $\chi_{top}$ denote the topological Euler characteristic.

First recall the following classical theorem:

######  Theorem 4.2.2.

Let $X$ be a proper smooth $n$-dimensional complex scheme and $E$ be a reduced
divisor with strict normal crossings in $X$. We have the equality of integers:

$\chi_{top}(X-E):=\sum_{i=0}^{n}\dim_{\mathbb{C}}H^{i}(X-E,\mathbb{C})=\int_{X}c_{n}(\Omega^{1\vee}_{X}(\log
E)).$

###### Proof.

We can give a proof of this theorem using the results of this paper. Let us
adopt the notation of Proposition 3.1.1, we have the equality of integers:

$\displaystyle\int_{X}c_{n}(\Omega^{1\vee}_{X}(\log E))$
$\displaystyle=(-1)^{n}\int_{X}c_{n}(\Omega^{1}_{X}(\log E)),$
$\displaystyle=(-1)^{n}\sum_{J\subset
I}\int_{E_{J}}c_{n-|J|}(\Omega^{1}_{E_{J}})\textrm{ using equality
\eqref{Chern class Omega1log} from Proposition \ref{comparison Omega log and
Omega easy cases general},}$ $\displaystyle=\sum_{J\subset
I}(-1)^{|J|}\int_{E_{J}}c_{n-|J|}(T_{E_{J}}),$ $\displaystyle=\sum_{J\subset
I}(-1)^{|J|}\chi_{top}(E_{J}),$
$\displaystyle=\chi_{top}(X)-\chi_{top}(E)\textrm{ using the inclusion-
exclusion formula,}$ $\displaystyle=\chi_{top}(X-E).\qed$

######  Theorem 4.2.3.

With the notation of this subsection, we have the equality in
${\mathrm{CH}}_{0}(C)_{\mathbb{Q}}$:

(4.2.1)
$\sum_{n=0}^{2(N-1)}(-1)^{n-1}c_{1}(\mathcal{GK}_{C,-}(\mathbb{H}^{n}(Y_{\eta}/C_{\eta})))=\frac{1}{12}g_{\ast}(c_{1}c_{N-1})(\omega^{1\vee}_{Y/C})+\sum_{x\in\Delta}\alpha_{x}[x],$

where for every $x$ in $\Delta$, $\alpha_{x}$ is the rational number given by:

(4.2.2) $\displaystyle\alpha_{x}=\frac{N-1}{4}\sum_{i\in
I_{x}}(m_{i}-1)\chi_{top}({\mathaccent
23{D}}_{i})+\frac{1}{12}\sum_{\begin{subarray}{c}(i,j)\in I_{x}^{2},\\\ i\prec
j\end{subarray}}(3-m_{i}/m_{j}-m_{j}/m_{i})\chi_{top}({\mathaccent
23{D}}_{ij}),$ (4.2.3) $\displaystyle=\frac{1}{12}\sum_{i\in
I_{x}}\Big{[}3(N-1)(m_{i}-1)\chi_{top}({\mathaccent
23{D}}_{i})+\int_{D_{i}}c_{1}({\mathcal{N}}_{i})c_{N-2}(\Omega^{1\vee}_{D_{i}}(\log
D_{i}\cap D^{2}))\Big{]}+\frac{1}{4}\sum_{\begin{subarray}{c}(i,j)\in
I_{x}^{2},\\\ i\prec j\end{subarray}}\chi_{top}({\mathaccent 23{D}}_{ij}).$

The expression (4.2.2) for $\alpha_{x}$ involves only the topology of the open
strata ${\mathaccent 23{D}}_{i}$ and ${\mathaccent 23{D}}_{ij}$ of the
singular fibers of $g$ and the multiplicities $m_{i}$ of their components
$D_{i}$. The expression (4.2.3) makes clear that $\alpha_{x}$ belongs to
$(1/12)\mathbb{Z}$.

###### Proof.

By Theorem 4.2.1 and by definition of the class $\rho_{N-1}$, the following
equalities hold in ${\mathrm{CH}}_{0}(C)_{\mathbb{Q}}$:

$\displaystyle\sum_{n=0}^{2(N-1)}$
$\displaystyle(-1)^{n-1}c_{1}(\mathcal{GK}_{C,-}(\mathbb{H}^{n}(Y_{\eta}/C_{\eta})))$
$\displaystyle=g_{\ast}\Big{[}\Big{(}\rho_{N-1}(\omega^{1\vee}_{Y/C}).\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}\Big{)}^{(N)}\Big{]},$
$\displaystyle=g_{\ast}\Big{[}\Big{(}(c_{N-2}(\omega^{1\vee}_{Y/C})-\frac{N-1}{2}c_{N-1}(\omega^{1\vee}_{Y/C})+\frac{1}{12}(c_{1}c_{N-1})(\omega^{1\vee}_{Y/C}))\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}\Big{)}^{(N)}\Big{]},$
$\displaystyle=\frac{1}{12}g_{\ast}(c_{1}c_{N-1})(\omega^{1\vee}_{Y/C})-\frac{N-1}{2}g_{\ast}\Big{[}c_{N-1}(\omega^{1\vee}_{Y/C})\,\Big{(}\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}\Big{)}^{(1)}\Big{]}$
(4.2.4)
$\displaystyle+g_{\ast}\Big{[}c_{N-2}(\omega^{1\vee}_{Y/C})\Big{(}\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}\Big{)}^{(2)}\Big{]}.$

Using successively equality (3.1.8) from Proposition 3.1.6 and Proposition
3.1.2, we get the following equality in ${\mathrm{CH}}^{N}(Y)_{\mathbb{Q}}$:

$\displaystyle
c_{N-1}(\omega^{1\vee}_{Y/C})\,\Big{(}\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}\Big{)}^{(1)}$
$\displaystyle=-\frac{1}{2}\sum_{i\in
I}(m_{i}-1)i_{D_{i}\ast}c_{N-1}(\omega^{1\vee}_{Y/C|D_{i}}),$
$\displaystyle=-\frac{1}{2}\sum_{i\in
I}(m_{i}-1)i_{D_{i}\ast}c_{N-1}(\Omega^{1\vee}_{D_{i}}(\log D_{i}\cap D^{2}))$

hence pushing forward by $g$, the equality in
${\mathrm{CH}}_{0}(C)_{\mathbb{Q}}$:

$\displaystyle
g_{\ast}\Big{[}c_{N-1}(\omega^{1\vee}_{Y/C})\,\Big{(}\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}\Big{)}^{(1)}\Big{]}$
$\displaystyle=-\frac{1}{2}\sum_{i\in
I}(m_{i}-1)g_{\ast}i_{D_{i}\ast}c_{N-1}(\Omega^{1\vee}_{D_{i}}(\log D_{i}\cap
D^{2})),$ $\displaystyle=\sum_{x\in\Delta}\alpha_{1,x}[x],$

where for every point $x$ in $\Delta$, $\alpha_{1,x}$ is the rational number
given by:

$\displaystyle\alpha_{1,x}$ $\displaystyle=-\frac{1}{2}\sum_{i\in
I_{x}}(m_{i}-1)\int_{D_{i}}c_{N-1}(\Omega^{1\vee}_{D_{i}}(\log D_{i}\cap
D^{2})),$ $\displaystyle=-\frac{1}{2}\sum_{i\in
I_{x}}(m_{i}-1)\,\chi_{top}({\mathaccent 23{D}}_{i})\textrm{ \quad\quad using
Theorem \ref{chi_top complement of DNC 2}.}$

Similarly, using equality (3.1.10) from Proposition 3.1.6 and Proposition
3.1.2, the following equalities hold in ${\mathrm{CH}}^{N}(Y)_{\mathbb{Q}}$:

$\displaystyle
c_{N-2}(\omega^{1\vee}_{Y/C})\Big{(}\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}\Big{)}^{(2)}$
$\displaystyle=\frac{1}{12}\sum_{\begin{subarray}{c}(i,j)\in I^{2},\\\ i\prec
j\end{subarray}}(3-m_{i}/m_{j}-m_{j}/m_{i})i_{D_{ij}\ast}c_{N-2}(\omega^{1\vee}_{Y/C|D_{ij}}),$
$\displaystyle=\frac{1}{12}\sum_{\begin{subarray}{c}(i,j)\in I^{2},\\\ i\prec
j\end{subarray}}(3-m_{i}/m_{j}-m_{j}/m_{i})i_{D_{ij}\ast}c_{N-2}(\Omega^{1\vee}_{D_{ij}}(\log
D_{ij}\cap D^{3})),$

hence pushing forward by $g$, the equality in
${\mathrm{CH}}_{0}(C)_{\mathbb{Q}}$:

$g_{\ast}\Big{[}c_{N-2}(\omega^{1\vee}_{Y/C})\Big{(}\frac{{\rm{Td}}([T_{g}])}{{\rm{Td}}(\omega^{1\vee}_{Y/C})}\Big{)}^{(2)}\Big{]}=\sum_{x\in\Delta}\alpha_{2,x}[x],$

where, for every point $x$ in $\Delta$, $\alpha_{2,x}$ is the rational number
given by:

$\displaystyle\alpha_{2,x}$
$\displaystyle=\frac{1}{12}\sum_{\begin{subarray}{c}(i,j)\in I_{x}^{2},\\\
i\prec
j\end{subarray}}(3-m_{i}/m_{j}-m_{j}/m_{i})\int_{D_{ij}}c_{N-2}(\Omega^{1\vee}_{D_{ij}}(\log
D_{ij}\cap D^{3})),$
$\displaystyle=\frac{1}{12}\sum_{\begin{subarray}{c}(i,j)\in I_{x}^{2},\\\
i\prec j\end{subarray}}(3-m_{i}/m_{j}-m_{j}/m_{i})\chi_{top}({\mathaccent
23{D}}_{ij})\textrm{ using again Theorem \ref{chi_top complement of DNC 2}.}$

Hence replacing in (4.2.4), we obtain the equality in
${\mathrm{CH}}_{0}(C)_{\mathbb{Q}}$:

$\sum_{n=0}^{2(N-1)}(-1)^{n-1}c_{1}(\mathcal{GK}_{C,-}(\mathbb{H}^{n}(Y_{\eta}/C_{\eta})))=\frac{1}{12}g_{\ast}(c_{1}c_{N-1})(\omega^{1\vee}_{Y/C})+\sum_{x\in\Delta}\alpha_{x}[x],$

where for every point $x$ in $\Delta$, the rational number $\alpha_{x}$ is
given by:

$\displaystyle\alpha_{x}=-\frac{N-1}{2}\alpha_{1,x}+\alpha_{2,x},$
$\displaystyle=\frac{N-1}{4}\sum_{i\in I_{x}}(m_{i}-1)\chi_{top}({\mathaccent
23{D}}_{i})+\frac{1}{12}\sum_{\begin{subarray}{c}(i,j)\in I_{x}^{2},\\\ i\prec
j\end{subarray}}(3-m_{i}/m_{j}-m_{j}/m_{i})\chi_{top}({\mathaccent
23{D}}_{ij}).$

For the other expression of the rational number $\alpha_{x}$, one reasons
similarly using the expression from (3.1.9) from Proposition 3.1.6 instead of
(3.1.10). ∎

##  5\. The alternating product of Griffiths line bundles associated to a
proper fibration with non-degenerate critical points

The objective of this section is to prove the following theorem.

######  Theorem 5.0.1.

Let $C$ be a connected smooth projective complex curve with generic point
$\eta$, $H$ be a smooth projective $N$-dimensional complex scheme, and let

$f:H\longrightarrow C$

be a morphism of complex schemes. Let us assume that there exists a finite
subset $\Sigma$ in $H$ such that $f$ is smooth on $H-\Sigma$ and admits a non-
degenerate critical point at any point of $\Sigma$.212121Namely the
differential of $f$ vanishes and its Hessian is non-degenerate at any point of
$\Sigma$; in other words, the only singularities of the fibers of $f$ are
ordinary double points.

Let $[\Sigma]$ be the reduced $0$-cycle in $H$ associated with the finite
subset $\Sigma$.

We have the equalities in ${\mathrm{CH}}_{0}(C)_{\mathbb{Q}}$:

(5.0.1)
$\sum_{n=0}^{2(N-1)}(-1)^{n-1}c_{1}\big{(}\mathcal{GK}_{C,-}(\mathbb{H}^{n}(H_{\eta}/C_{\eta}))\big{)}=\frac{1}{12}f_{\ast}\big{(}(c_{1}c_{N-1})([\Omega^{1}_{H/C}]^{\vee})\big{)}+u^{-}_{N}f_{\ast}[\Sigma],$

and

(5.0.2)
$\sum_{n=0}^{2(N-1)}(-1)^{n-1}c_{1}\big{(}\mathcal{GK}_{C,+}(\mathbb{H}^{n}(H_{\eta}/C_{\eta}))\big{)}=\frac{1}{12}f_{\ast}\big{(}(c_{1}c_{N-1}\big{)}([\Omega^{1}_{H/C}]^{\vee})\big{)}+u_{N}^{+}f_{\ast}[\Sigma],$

where $u^{-}_{N}$ and $u^{+}_{N}$ are the rational numbers defined by:

$u^{-}_{N}:=\begin{cases}(5N-3)/24&\text{if $N$ is odd}\\\ N/24&\text{if $N$
is even},\end{cases}$

and:

$u^{+}_{N}:=\begin{cases}-(7N-9)/24&\text{if $N$ is odd}\\\ N/24&\text{if $N$
is even}.\end{cases}$

Observe that, as the morphism of non-singular complex schemes $f:H\rightarrow
C$ is smooth on an open dense subset of $H$, we have the following equality in
$K^{0}(H)\simeq K_{0}(H)$:

$[T_{f}]=[\Omega^{1}_{H/C}]^{\vee},$

where $.^{\vee}$ denotes the duality involution on $K^{0}(H)$, by the same
reasoning as in Subsubsection 3.1.4.

###  5.1. Reduction of the proof of Theorem 5.0.1 to a computation in
${\mathrm{CH}}_{0}(H)$

As in Theorem 5.0.1, let $C$ be a connected smooth projective complex curve,
let $H$ be a smooth projective complex scheme of pure dimension $N\geq 1$, and
let

$f:H\longrightarrow C$

be a proper surjective morphism with a finite subset $\Sigma$ of critical
points, all of which are non-degenerate.

Let $\Delta:=f(\Sigma)$ be the set-theoretic image of $\Sigma$, that we see as
a reduced divisor in $C$. We can rephrase the hypothesis as the divisor in
$H$:

$H_{\Delta}:=f^{\ast}\Delta$

having only finitely many singularities, all of which are ordinary double
points.

Let

$\nu:\tilde{H}\longrightarrow H$

be the blow-up of $H$ at $\Sigma$, and let $E$ be the exceptional divisor. Let

$g:=f\circ\nu:\tilde{H}\longrightarrow C$

be the composition. Over the generic point $\eta$ of $C$, this morphism can be
identified with the morphism $f$.

For every point $P$ in $\Sigma$, let us define a subscheme in $\tilde{H}$ by:

$E_{P}:=\nu^{-1}(\\{P\\}),$

so that the $(E_{P})_{P\in\Sigma}$ are the connected components of the
exceptional divisor $E$. In particular, they are disjoint divisors, isomorphic
to the complex projective space $\mathbb{P}^{N-1}$; see for instance
Proposition 5.2.1 below.

Let us define a divisor in $\tilde{H}$ by:

$\tilde{H}_{\Delta}:=g^{\ast}\Delta=\nu^{\ast}H_{\Delta}.$

Since the divisor $H_{\Delta}$ has its only singularities at points of
$\Sigma$, and since all of these singularities are ordinary double points, its
pullback divisor $\tilde{H}_{\Delta}$ can be written as follows:

(5.1.1) $\tilde{H}_{\Delta}=2\sum_{P\in\Sigma}E_{P}+W$

where $W$ is the proper transform in $\tilde{H}$ of the divisor $H_{\Delta}$
in $H$: it is a non-singular divisor intersecting transversally the components
$(E_{P})_{P\in\Sigma}$ of the exceptional divisor, and for every point $P$,
the intersection $E_{P}\cap W$ is a smooth quadric in the projective space
$E_{P}$.

In particular, the divisor $\tilde{H}_{\Delta}$ is a divisor with strict
normal crossings, which allows us to define the relative logarithmic bundle
$\omega^{1}_{\tilde{H}/C}$.

We are going to apply Theorem 4.2.3 to the morphism $g$ and to the variations
of Hodge structures of relative cohomology:

$\mathbb{H}^{n}({H}_{\eta}/C_{\eta})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathbb{H}^{n}(\widetilde{H}_{\eta}/C_{\eta}).$

For this, we are going to prove the following lemma using the results of
Section 3.

######  Lemma 5.1.1.

With the above notation, the following formula holds in
${\mathrm{CH}}^{N}(H)$:

(5.1.2)
$\nu_{\ast}\big{(}(c_{1}c_{N-1})(\omega^{1\vee}_{\widetilde{H}/C})\big{)}=(c_{1}c_{N-1})([T_{f}])-(N-2)\frac{1-(-1)^{N}}{2}[\Sigma].$

Let us prove Theorem 5.0.1 using this lemma.

For every point $P$ in $\Sigma$, the complex scheme $E_{P}$ is isomorphic to
the projective space $\mathbb{P}^{N-1}$, hence its topological Euler
characteristic is given by:

$\chi_{top}(E_{P})=N.$

Furthermore, the intersection $E_{P}\cap W$ is isomorphic to a smooth quadric
in $\mathbb{P}^{N-1}$, hence its topological Euler characteristic is given by:

$\chi_{top}(E_{P}\cap W)=\frac{1}{2}[(-1)^{N}+2N-1].$

This follows for instance from equalities (5.2.5) and (5.2.3), from
Proposition 5.2.1 below, applied to the integers $m_{P}:=2$ and $r:=N-2$ and
the hypersurface $Q_{P}:=E_{P}\cap W$, and from equality (5.3.1) in
Proposition 5.3.3 below, applied to the integers $n:=N$ and $r:=2$ and the
complex number $a:=2$.

By additivity of the topological Euler characteristic, the characteristic of
the complement $E_{P}-E_{P}\cap W$ is given by:

$\chi_{top}(E_{P}-E_{P}\cap
W)=N-\frac{1}{2}[(-1)^{N}+2N-1]=\frac{1}{2}[1-(-1)^{N}].$

Consequently, for every point $x$ in $\Delta$, the expression (4.2.2) for the
rational number $\alpha_{x}$ introduced in Theorem 4.2.3 reduces to:

$\displaystyle\alpha_{x}$
$\displaystyle=\frac{N-1}{4}\sum_{P\in\Sigma_{x}}\chi_{top}(E_{P}-E_{P}\cap
W)+\frac{1}{12}\sum_{P\in\Sigma_{x}}\frac{1}{2}\chi_{top}(E_{P}\cap W),$
$\displaystyle=\frac{N-1}{8}[1-(-1)^{N}]|\Sigma_{x}|+\frac{1}{48}[(-1)^{N}+2N-1]|\Sigma_{x}|,$
$\displaystyle=\frac{1}{48}[6(N-1)(1-(-1)^{N})+(-1)^{N}+2N-1]|\Sigma_{x}|,$
$\displaystyle=\frac{1}{48}[(6N-7)(1-(-1)^{N})+2N]|\Sigma_{x}|.$

Hence replacing in Theorem 4.2.3 and using Lemma 5.1.1, we have the equality
in ${\mathrm{CH}}_{0}(C)_{\mathbb{Q}}$:

$\displaystyle\sum_{n=0}^{2(N-1)}(-1)^{n-1}c_{1}(\mathcal{GK}_{C,-}(\mathbb{H}^{n}(H_{\eta}/C_{\eta})))$
$\displaystyle=\frac{1}{12}g_{\ast}(c_{1}c_{N-1})(\omega^{1\vee}_{Y/C})+\sum_{x\in\Delta}\alpha_{x}[x],$
$\displaystyle=\frac{1}{12}f_{\ast}(c_{1}c_{N-1})([T_{f}])-\frac{1}{12}(N-2)\frac{1-(-1)^{N}}{2}f_{\ast}[\Sigma]$
$\displaystyle+\frac{1}{48}[(6N-7)(1-(-1)^{N})+2N]\sum_{x\in\Delta}|\Sigma_{x}|[x],$
$\displaystyle=\frac{1}{12}f_{\ast}(c_{1}c_{N-1})([T_{f}])+u_{N}^{-}f_{\ast}[\Sigma],$

where $u_{N}^{-}$ is the rational number given by:

$\displaystyle u_{N}^{-}$
$\displaystyle=-\frac{1}{24}(N-2)(1-(-1)^{N})+\frac{1}{48}[(6N-7)(1-(-1)^{N})+2N],$
$\displaystyle=\frac{1}{48}[-2(N-2)(1-(-1)^{N})+(6N-7)(1-(-1)^{N})+2N],$
$\displaystyle=\frac{1}{48}[(4N-3)(1-(-1)^{N})+2N],$
$\displaystyle=\begin{cases}(5N-3)/24&\text{if $N$ is odd}\\\ N/24&\text{if
$N$ is even},\end{cases}$

which shows equality (5.0.1).

Furthermore, for every integer $n$, equality (2.4.2) from Corollary 2.4.2
gives the equality in ${\mathrm{CH}}_{0}(C)_{\mathbb{Q}}$:

$c_{1}(\mathcal{GK}_{C,+}(\mathbb{H}^{n}(H_{\eta}/C_{\eta})))=c_{1}(\mathcal{GK}_{C,-}(\mathbb{H}^{n}(H_{\eta}/C_{\eta}))+\delta^{n,N-1}\eta_{N}\frac{N-1}{2}f_{\ast}[\Sigma],$

where $\eta_{N}$ is $1$ if $N$ is odd and $0$ if $N$ is even.

Taking the alternating sum over $n$ yields the equality in
${\mathrm{CH}}_{0}(C)_{\mathbb{Q}}$:

$\displaystyle\sum_{n=0}^{2(N-1)}(-1)^{n-1}c_{1}(\mathcal{GK}_{C,+}(\mathbb{H}^{n}(H_{\eta}/C_{\eta})))$
$\displaystyle=\sum_{n=0}^{2(N-1)}(-1)^{n-1}c_{1}(\mathcal{GK}_{C,-}(\mathbb{H}^{n}(H_{\eta}/C_{\eta})))$
$\displaystyle+(-1)^{N}\eta_{N}\frac{N-1}{2}f_{\ast}[\Sigma],$
$\displaystyle=\frac{1}{12}f_{\ast}(c_{1}c_{N-1})([T_{f}])+u_{N}^{-}f_{\ast}[\Sigma]-\eta_{N}\frac{N-1}{2}f_{\ast}[\Sigma],$
$\displaystyle=\frac{1}{12}f_{\ast}(c_{1}c_{N-1})([T_{f}])+u_{N}^{+}f_{\ast}[\Sigma],$

where $u_{N}^{+}$ is the rational number given by:

$u_{N}^{+}=u_{N}^{-}-\eta_{N}\frac{N-1}{2}=\begin{cases}-(7N-9)/24&\text{if
$N$ is odd}\\\ N/24&\text{if $N$ is even},\end{cases}$

which shows equality (5.0.2) and completes the proof of Theorem 5.0.1.

The rest of this section shall be devoted to proving Lemma 5.1.1.

###  5.2. Blowing up points in smooth schemes and Chern classes

We adopt the following notation. Let $k$ be a field, $H$ be a smooth
$k$-scheme of pure dimension $N\geq 1$, $\Sigma$ be a finite subset of $H$,
let

$\nu:\tilde{H}\longrightarrow H$

be the blow-up of $H$ at $\Sigma$, and let $E$ be the exceptional divisor.

For every point $P$ in $\Sigma$, let us define a subscheme in $\tilde{H}$ by:

$E_{P}:=\nu^{-1}(\\{P\\}),$

so that the $(E_{P})_{P\in\Sigma}$ are the connected components of the
exceptional divisor $E$. In particular, they are disjoint divisors.

For every point $P$ in $\Sigma$, let us denote the class of the divisor
$E_{P}$ by:

$\eta_{P}:=[E_{P}]=c_{1}({\mathcal{O}}_{\tilde{H}}(E_{P}))\in{\mathrm{CH}}^{1}(\tilde{H}).$

Observe that for every point $P$ in $\Sigma$, the restriction to $P$ of the
vector bundle $T_{H}$ can be trivialized, so we obtain, for every cycle
$\alpha$ in $H$ of positive codimension, the equality in
${\mathrm{CH}}^{\ast}(\tilde{H})$:

(5.2.1) $\eta_{P}\cdot\nu^{\ast}\alpha=0.$

Furthermore, for every couple $(P,P^{\prime})$ in $\Sigma^{2}$ such that
$P\neq P^{\prime}$, the divisors $E_{P}$ and $E_{P^{\prime}}$ in $\tilde{H}$
are disjoint, so we obtain the equality in ${\mathrm{CH}}^{2}(\tilde{H})$:

(5.2.2) $\eta_{P}\cdot\eta_{P^{\prime}}=0.$

######  Proposition 5.2.1.

Let $P$ be a point in $\Sigma$. The scheme $E_{P}$ is isomorphic to a
projective space of dimension $N-1$.

We have the equality in ${\mathrm{CH}}^{N}(H)$:

(5.2.3) $\nu_{\ast}\eta_{P}^{N}=(-1)^{N-1}[P].$

Moreover, for every non-negative integer $r$, we have the equality in
${\mathrm{CH}}^{r}(E_{P})$:

(5.2.4) $c_{r}(T_{E_{P}})=(-1)^{r}\binom{N}{r}\eta_{P|E_{P}}^{r}.$

Finally, let $m_{P}$ be a positive integer, and let $Q_{P}$ be a smooth
hypersurface of degree $m_{P}$ in $E_{P}$ and let $i_{Q_{P}}:Q_{P}\rightarrow
E_{P}$ be the inclusion morphism.

For every non-negative integer $r$, we have the equality in
${\mathrm{CH}}^{r+1}(E_{P})$:

(5.2.5)
$i_{Q_{P}\ast}c_{r}(T_{Q_{P}})=(-1)^{r+1}m_{P}\Big{[}\frac{(1+y)^{N}}{1+m_{P}y}\Big{]}^{[r]}\eta_{P|E_{P}}^{r+1},$

where $f(y)^{[r]}$ denotes the coefficient of $y^{r}$ in some formal series
$f(y)\in\mathbb{C}[[y]]$.

###### Proof.

Since $E_{P}$ is a connected component of the exceptional divisor $E$, there
is a canonical isomorphism of $k$-schemes:

$\varphi:E_{P}\longrightarrow\mathbb{P}_{k}(T_{P}H),$

and a canonical isomorphism of line bundles on $E_{P}$:

$\varphi^{\ast}{\mathcal{O}}_{T_{P}H}(-1)\simeq{\mathcal{O}}_{\tilde{H}}(E_{P})_{|E_{P}}.$

Consequently, we have the equality in ${\mathrm{CH}}^{1}(E_{P})$:

(5.2.6) $\eta_{P|E_{P}}=\varphi^{\ast}c_{1}({\mathcal{O}}_{T_{P}H}(-1)).$

Now, let us compute in ${\mathrm{CH}}^{N}(\tilde{H})$:

$\displaystyle\eta_{P}^{N}$ $\displaystyle=\eta_{P}^{N-1}[E_{P}],$
$\displaystyle=i_{E_{P}\ast}\eta_{P|E_{P}}^{N-1}\textrm{ by the projection
formula,}$
$\displaystyle=(-1)^{N-1}i_{E_{P}\ast}\varphi^{\ast}c_{1}({\mathcal{O}}_{T_{P}H}(1))^{N-1}.$

Hence pushing forward, we have the equality in ${\mathrm{CH}}^{N}(H)$:

$\displaystyle\nu_{\ast}\eta_{P}^{N}$
$\displaystyle=(-1)^{N-1}\nu_{\ast}i_{E_{P}\ast}\varphi^{\ast}c_{1}({\mathcal{O}}_{T_{P}H}(1))^{N-1},$
$\displaystyle=(-1)^{N-1}\Big{[}\int_{\mathbb{P}(T_{P}H)}c_{1}({\mathcal{O}}_{T_{P}H}(1))^{N-1}\Big{]}[P],$
$\displaystyle=(-1)^{N-1}[P],$

which shows equality (5.2.3).

Let $r$ be a non-negative integer. Since $\varphi$ is an isomorphism, we have
the equality in ${\mathrm{CH}}^{r}(E_{P})$:

$c_{r}(T_{E_{P}})=\varphi^{\ast}c_{r}(T_{\mathbb{P}(T_{P}H)}),$

using the exact sequence of vector bundles on $\mathbb{P}(T_{P}H)$ from
[Ful98, B.5.8]:

$0\longrightarrow{\mathcal{O}}_{\mathbb{P}(T_{P}H)}\longrightarrow
T_{P}H\otimes_{k}{\mathcal{O}}_{T_{P}H}(1)\longrightarrow
T_{\mathbb{P}(T_{P}H)}\longrightarrow 0,$

we can replace:

$c_{r}(T_{E_{P}})=\varphi^{\ast}[(1+c_{1}({\mathcal{O}}_{T_{P}H}(1)))^{N}]^{(r)}=\binom{N}{r}\varphi^{\ast}c_{1}({\mathcal{O}}_{T_{P}H}(1))^{r}=(-1)^{r}\binom{N}{r}\eta_{P|E_{P}}^{r}\textrm{
using \eqref{eq eta_P c_1(O(1))}, }$

which shows equality (5.2.4).

Now, let $m_{P}$ be a positive integer and $Q_{P}$ be an hypersurface of
degree $m_{P}$ in $E_{P}$.

We have the following exact sequence of vector bundles on $Q_{P}$:

$0\longrightarrow T_{Q_{P}}\longrightarrow
T_{E_{P}|Q_{P}}\longrightarrow{\mathcal{N}}_{Q_{P}}E_{P}\longrightarrow 0,$

hence the following equality in ${\mathrm{CH}}^{\ast}(Q_{P})$:

$\displaystyle c(T_{Q_{P}})$
$\displaystyle=\frac{c(T_{E_{P}|Q_{P}})}{c({\mathcal{N}}_{Q_{P}}E_{P})},$
$\displaystyle=\frac{\varphi_{|Q_{P}}^{\ast}(1+c_{1}({\mathcal{O}}_{T_{P}H}(1)))^{N}}{1+c_{1}({\mathcal{O}}_{E_{P}}(Q_{P})_{|Q_{P}})},$
$\displaystyle=\frac{\varphi_{|Q_{P}}^{\ast}(1+c_{1}({\mathcal{O}}_{T_{P}H}(1)))^{N}}{1+m_{P}\varphi_{|Q_{P}}^{\ast}c_{1}({\mathcal{O}}_{T_{P}H}(1))}\textrm{
because $Q_{P}$ is of degree $m_{P}$,}$
$\displaystyle=\varphi_{|Q_{P}}^{\ast}\Big{[}\frac{(1+c_{1}({\mathcal{O}}_{T_{P}H}(1)))^{N}}{1+m_{P}c_{1}({\mathcal{O}}_{T_{P}H}(1))}\Big{]}.$

Hence, for every integer $r$, we have the equality in
${\mathrm{CH}}^{r}(Q_{P})$:

(5.2.7)
$c_{r}(T_{Q_{P}})=\Big{[}\frac{(1+y)^{N}}{1+m_{P}y}\Big{]}^{[r]}\varphi_{|Q_{P}}^{\ast}c_{1}({\mathcal{O}}_{T_{P}H}(1))^{r}=(-1)^{r}\Big{[}\frac{(1+y)^{N}}{1+m_{P}y}\Big{]}^{[r]}\eta_{P|Q_{P}}^{r}\textrm{
using \eqref{eq eta_P c_1(O(1))}.}$

Pushing forward (5.2.7) by the inclusion map of $Q_{P}$ in $E_{P}$, and using
the projection formula and the equality in ${\mathrm{CH}}^{1}(E_{P})$:

$[Q_{P}]=\varphi^{\ast}{\mathcal{O}}_{T_{P}H}(m_{P})=-m_{P}\eta_{P|E_{P}},$

yields equality (5.2.5). ∎

Recall the following consequence of the “Riemann-Roch-Grothendieck without
denominators” formula:

######  Proposition 5.2.2 ([Ful98, Example 15.4.2, (c)]).

We have the equality in ${\mathrm{CH}}^{\ast}(\tilde{H})$:

(5.2.8)
$c(T_{\tilde{H}})=\nu^{\ast}c(T_{H})+\sum_{P\in\Sigma}[(1+\eta_{P})(1-\eta_{P})^{N}-1].$

The following proposition is a reformulation of Proposition 5.2.2, and we
leave its derivation from Proposition 5.2.2 to the reader.

######  Proposition 5.2.3.

For every integer $r\geq 1$, we have the equality in
${\mathrm{CH}}^{r}(\tilde{H})$:

$c_{r}(T_{\tilde{H}})=\nu^{\ast}c_{r}(T_{H})+(-1)^{r}\Bigg{[}\binom{N}{r}-\binom{N}{r-1}\Bigg{]}\sum_{P\in\Sigma}\eta_{P}^{r}.$

######  Corollary 5.2.4.

We have the equality in ${\mathrm{CH}}^{N}(\tilde{H})$:

(5.2.9)
$(c_{1}c_{N-1})(T_{\tilde{H}})=\nu^{\ast}(c_{1}c_{N-1})(T_{H})+(-1)^{N+1}N(N-1)(N-3)/2\sum_{P\in\Sigma}\eta_{P}^{N}$

and the equality in ${\mathrm{CH}}^{N-1}(\tilde{H})$:

(5.2.10)
$(c_{N-1}+c_{1}c_{N-2})(T_{\tilde{H}})=\nu^{\ast}(c_{N-1}+c_{1}c_{N-2})(T_{H})+a_{N}\sum_{P\in\Sigma}\eta_{P}^{N-1},$

where $a_{N}$ is the integer defined by:

$a_{N}:=\begin{cases}0&\text{if $N=1$,}\\\
(-1)^{N}N(N-3)/2+(-1)^{N}N(N-1)^{2}(N-5)/6&\text{if $N\geq 2$.}\end{cases}$

###### Proof.

Applying Proposition 5.2.3 to the integer $r:=1$ yields the equality in
${\mathrm{CH}}^{1}(\tilde{H})$:

(5.2.11)
$c_{1}(T_{\tilde{H}})=\nu^{\ast}c_{1}(T_{H})-\Bigg{[}\binom{N}{1}-\binom{N}{0}\Bigg{]}\sum_{P\in\Sigma}\eta_{P}=\nu^{\ast}c_{1}(T_{H})-(N-1)\sum_{P\in\Sigma}\eta_{P}.$

If the integer $N$ is at least $3$, applying the same proposition to $r:=N-2$
yields the equality in ${\mathrm{CH}}^{N-2}(\tilde{H})$:

$\displaystyle c_{N-2}(T_{\tilde{H}})$
$\displaystyle=\nu^{\ast}c_{N-2}(T_{H})+(-1)^{N-2}\Bigg{[}\binom{N}{N-2}-\binom{N}{N-3}\Bigg{]}\sum_{P\in\Sigma}\eta_{P}^{N-2},$
$\displaystyle=\nu^{\ast}c_{N-2}(T_{H})+(-1)^{N}[N(N-1)/2-N(N-1)(N-2)/6]\sum_{P\in\Sigma}\eta_{P}^{N-2},$
(5.2.12)
$\displaystyle=\nu^{\ast}c_{N-2}(T_{H})+(-1)^{N+1}N(N-1)(N-5)/6\sum_{P\in\Sigma}\eta_{P}^{N-2}.$

If the integer $N$ is at least $2$, applying the same proposition to $r:=N-1$
yields the equality in ${\mathrm{CH}}^{N-1}(\tilde{H})$:

$\displaystyle c_{N-1}(T_{\tilde{H}})$
$\displaystyle=\nu^{\ast}c_{N-1}(T_{H})+(-1)^{N-1}\Bigg{[}\binom{N}{N-1}-\binom{N}{N-2}\Bigg{]}\sum_{P\in\Sigma}\eta_{P}^{N-1},$
$\displaystyle=\nu^{\ast}c_{N-1}(T_{H})-(-1)^{N}[N-N(N-1)/2]\sum_{P\in\Sigma}\eta_{P}^{N-1},$
(5.2.13)
$\displaystyle=\nu^{\ast}c_{N-1}(T_{H})+(-1)^{N}N(N-3)/2\sum_{P\in\Sigma}\eta_{P}^{N-1}.$

Let us show equality (5.2.9). If the integer $N$ is $1$, then it follows
immediately from equality (5.2.11) (observe that the coefficient before
$\sum_{P}\eta_{P}$ vanishes in that case).

Let us assume that the integer $N$ is at least $2$.

Multiplying (5.2.11) and (5.2.13) yields the equality in
${\mathrm{CH}}^{N}(\tilde{H})$:

$\displaystyle(c_{1}c_{N-1})(T_{\tilde{H}})$
$\displaystyle=(\nu^{\ast}c_{1}(T_{H})-(N-1)\sum_{P\in\Sigma}\eta_{P})(\nu^{\ast}c_{N-1}(T_{H})+(-1)^{N}N(N-3)/2\sum_{P\in\Sigma}\eta_{P}^{N-1}),$
$\displaystyle=\nu^{\ast}(c_{1}c_{N-1})(T_{H})+(-1)^{N}N(N-3)/2\sum_{P\in\Sigma}\eta_{P}^{N-1}\,\nu^{\ast}c_{1}(T_{H})$
(5.2.14)
$\displaystyle-(N-1)\sum_{P\in\Sigma}\eta_{P}\,\nu^{\ast}c_{N-1}(T_{H})+(-1)^{N+1}N(N-1)(N-3)/2\sum_{P,P^{\prime}\in\Sigma}\eta_{P}\,\eta_{P^{\prime}}^{N-1}.$

Since we have assumed that $N$ was at least $2$, we can apply equality (5.2.1)
to the cycle $\alpha:=c_{N-1}(T_{H})$ of positive codimension to obtain, for
every point $P$ in $\Sigma$, the equality in ${\mathrm{CH}}^{N}(\tilde{H})$:

$\eta_{P}\,\nu^{\ast}c_{N-1}(T_{H})=0.$

Furthermore, we can apply this same equality to the cycle
$\alpha:=c_{1}(T_{H})$ to obtain, for every point $P$ in $\Sigma$, the
equality in ${\mathrm{CH}}^{N}(\tilde{H})$:

$\eta_{P}^{N-1}\,\nu^{\ast}c_{1}(T_{H})=\eta_{P}^{N-2}\big{(}\eta_{P}\,\nu^{\ast}c_{1}(T_{H})\big{)}=0.$

Finally, applying equality (5.2.2), we obtain the equality in
${\mathrm{CH}}^{N}(\tilde{H})$:

$\sum_{P,P^{\prime}\in\Sigma}\eta_{P}\,\eta_{P^{\prime}}^{N-1}=\sum_{P\in\Sigma}\eta_{P}^{N}.$

Hence replacing in (5.2.14):

$(c_{1}c_{N-1})(T_{\tilde{H}})=\nu^{\ast}(c_{1}c_{N-1})(T_{H})+(-1)^{N+1}N(N-1)(N-3)/2\sum_{P\in\Sigma}\eta_{P}^{N},$

which shows equality (5.2.9) if the integer $N$ is at least $2$. Consequently,
this equality holds for every positive integer $N$.

Now let us show equality (5.2.10). If the integer $N$ is $1$, it is trivial.
If the integer $N$ is $2$, the coefficient $a_{2}$ is equal to $-2$, so it
follows from multiplying equality (5.2.11) by $2$.

Let us assume that the integer $N$ is at least $3$.

Multiplying (5.2.11) and (5.2.12), we obtain the equality in
${\mathrm{CH}}^{N-1}(\tilde{H})$:

$(c_{1}c_{N-2})(T_{\tilde{H}})=(\nu^{\ast}c_{1}(T_{H})-(N-1)\sum_{P\in\Sigma}\eta_{P})(\nu^{\ast}c_{N-2}(T_{H})+(-1)^{N+1}\frac{N(N-1)(N-5)}{6}\sum_{P\in\Sigma}\eta_{P}^{N-2});$

developing and using equality (5.2.2), we get:

$\displaystyle(c_{1}c_{N-2})(T_{\tilde{H}})$
$\displaystyle=\nu^{\ast}(c_{1}c_{N-2})(T_{H})+(-1)^{N+1}N(N-1)(N-5)/6\sum_{P\in\Sigma}\eta_{P}^{N-2}\nu^{\ast}c_{1}(T_{H})$
(5.2.15)
$\displaystyle-(N-1)\sum_{P\in\Sigma}\eta_{P}\nu^{\ast}c_{N-2}(T_{H})+(-1)^{N}N(N-1)^{2}(N-5)/6\sum_{P\in\Sigma}\eta_{P}^{N-1}.$

Since we have assumed that the integer $N$ was at least $3$, then we can apply
equality (5.2.1) to the cycle $\alpha:=c_{N-2}(T_{H})$ of positive dimension
to obtain, for every point $P$ in $\Sigma$, the equality in
${\mathrm{CH}}^{N}(\tilde{H})$:

$\eta_{P}\,\nu^{\ast}c_{N-2}(T_{H})=0.$

Furthermore, we can apply this same equality to the cycle
$\alpha:=c_{1}(T_{H})$ to obtain, for every point $P$ in $\Sigma$, the
equality in ${\mathrm{CH}}^{N}(\tilde{H})$:

$\eta_{P}^{N-2}\,\nu^{\ast}c_{1}(T_{H})=\eta_{P}^{N-3}\big{(}\eta_{P}\,\nu^{\ast}c_{1}(T_{H})\big{)}=0.$

Hence replacing in (5.2.15), we obtain:

$(c_{1}c_{N-2})(T_{\tilde{H}})=\nu^{\ast}(c_{1}c_{N-2})(T_{H})+(-1)^{N}N(N-1)^{2}(N-5)/6\sum_{P\in\Sigma}\eta_{P}^{N-1},$

hence, adding equality (5.2.13), we obtain:

$(c_{N-1}+c_{1}c_{N-2})(T_{\tilde{H}})=\nu^{\ast}(c_{N-1}+c_{1}c_{N-2})(T_{H})\\\
+[(-1)^{N}N(N-3)/2+(-1)^{N}N(N-1)^{2}(N-5)/6]\sum_{P\in\Sigma}\eta_{P}^{N-1},$

which shows equality (5.2.10) if the integer $N$ is at least $3$.
Consequently, this equality holds for every positive integer $N$. ∎

###  5.3. Computation of the cycle classes $\nu_{\ast}(c_{1}c_{N-1})([T_{g}])$
and $\nu_{\ast}(c_{1}c_{N-1})(\omega^{1\vee}_{\widetilde{H}/C})$ in terms of
$(c_{1}c_{N-1})([T_{f}])$

In order to prove Lemma 5.1.1, we want to compare the cycles
$\nu_{\ast}(c_{1}c_{N-1})(\omega^{1\vee}_{\tilde{H}/C})$ and
$(c_{1}c_{N-1})([T_{f}])$ on $H$.

Using Corollary 5.2.4, we can compare the cycles
$(c_{1}c_{N-1})(T_{\tilde{H}})$ and $\nu^{\ast}(c_{1}c_{N-1})(T_{H})$ on
$\tilde{H}$. Then in Subsubsection 5.3.1, we shall combine Corollary 5.2.4 and
the definition of the relative tangent classes to compare the cycles
$\nu_{\ast}(c_{1}c_{N-1})([T_{g}])$ and $(c_{1}c_{N-1})([T_{f}])$ on $H$: this
will be the object of Proposition 5.3.2.

In Subsubsection 5.3.2, we shall apply equality (3.1.12) from Proposition
3.1.7 to compare the cycles $(c_{1}c_{N-1})(\omega^{1\vee}_{\tilde{H}/C})$ and
$(c_{1}c_{N-1})([T_{g}])$ on $\tilde{H}$ (this will be the object of
Proposition 5.3.5), then we shall push forward by $\nu$ and apply Proposition
5.3.2 to prove Lemma 5.1.1. This will complete the proof of Theorem 5.0.1.

####  5.3.1. Computation of $\nu_{\ast}(c_{1}c_{N-1})([T_{g}])$

######  Proposition 5.3.1.

Let $k$ be a field, $V$ be a smooth projective $k$-scheme of pure dimension
$N$, $C$ be a smooth projective $k$-curve, and

$h:V\longrightarrow C$

be a flat morphism.

Recall that the relative tangent class in $K$-theory of the morphism $h$ is
defined by:

$[T_{h}]:=[T_{V}]-h^{\ast}[T_{C}]\in K^{0}(V).$

We have the equality in ${\mathrm{CH}}^{N}(V)$:

$(c_{1}c_{N-1})([T_{h}])=(c_{1}c_{N-1})(T_{V})-(c_{N-1}+c_{1}c_{N-2})(T_{V})\,h^{\ast}c_{1}(T_{C}).$

###### Proof.

By definition of the class $[T_{h}]$, we have the equality in
${\mathrm{CH}}^{\ast}(V)$:

$\displaystyle c([T_{h}])$ $\displaystyle=c(T_{V})\,h^{\ast}c(T_{C})^{-1},$
$\displaystyle=c(T_{V})\,h^{\ast}(1+c_{1}(T_{C}))^{-1},$

since $C$ is a curve, the class $c_{1}(T_{C})^{2}$ vanishes, so that:

$c([T_{h}])=c(T_{V})h^{\ast}(1-c_{1}(T_{C})).$

Consequently, we have the equality in ${\mathrm{CH}}^{1}(V)$:

$c_{1}([T_{h}])=c_{1}(T_{V})-h^{\ast}c_{1}(T_{C}),$

and the equality in ${\mathrm{CH}}^{N-1}(V)$:

$c_{N-1}([T_{h}])=c_{N-1}(T_{V})-c_{N-2}(T_{V})\,h^{\ast}c_{1}(T_{C}).$

Hence multiplying, we have the equality in ${\mathrm{CH}}^{N}(V)$:

$(c_{1}c_{N-1})([T_{h}])=(c_{1}(T_{V})-h^{\ast}c_{1}(T_{C}))(c_{N-1}(T_{V})-c_{N-2}(T_{V})\,h^{\ast}c_{1}(T_{C})),$

and developing, and using that the class $c_{1}(T_{C})^{2}$ vanishes:

$(c_{1}c_{N-1})([T_{h}])=(c_{1}c_{N-1})(T_{V})-(c_{N-1}+c_{1}c_{N-2})(T_{V})\,h^{\ast}c_{1}(T_{C}),$

as wanted. ∎

Now let us show the main proposition of this subsubsection.

We shall use the notation of Subsection 5.2, in a relative setting, over a
smooth projective $k$-curve $C$. Namely, let $H$ be a smooth $k$-scheme of
pure dimension $N\geq 1$, let:

$f:H\longrightarrow C$

be a flat morphism, let $\Sigma$ be a finite subset of $H$, let:

$\nu:\tilde{H}\longrightarrow H$

be the blow-up of $H$ at $\Sigma$, and let $E$ be the exceptional divisor.
Let:

$g:=f\circ\nu:\tilde{H}\longrightarrow C$

be the composition. Let $(E_{P})_{P\in\Sigma}$ be the connected components of
the exceptional divisor $E$.

For every point $P$ in $\Sigma$, let us denote the class of the divisor
$E_{P}$ by:

$\eta_{P}:=[E_{P}]=c_{1}({\mathcal{O}}_{\tilde{H}}(E_{P}))\in{\mathrm{CH}}^{1}(\tilde{H}).$

######  Proposition 5.3.2.

With the above notation, we have the equality in ${\mathrm{CH}}^{N}(H)$:

$\nu_{\ast}(c_{1}c_{N-1})([T_{g}])=(c_{1}c_{N-1})([T_{f}])+N(N-1)(N-3)/2[\Sigma].$

###### Proof.

Let us compute the class $(c_{1}c_{N-1})([T_{g}])$ in
${\mathrm{CH}}^{N}(\tilde{H})$. Applying Proposition 5.3.1 to the scheme
$V:=\tilde{H}$ and the morphism $h:=g$ yields the equality:

$\displaystyle(c_{1}c_{N-1})([T_{g}])=(c_{1}c_{N-1})(T_{\tilde{H}})-(c_{N-1}+c_{1}c_{N-2})(T_{\tilde{H}})g^{\ast}c_{1}(T_{C}),$
using (5.2.9) and (5.2.10):
$\displaystyle=\nu^{\ast}(c_{1}c_{N-1})(T_{H})+(-1)^{N+1}N(N-1)(N-3)/2\sum_{P\in\Sigma}\eta_{P}^{N}$
$\displaystyle-[\nu^{\ast}(c_{N-1}+c_{1}c_{N-2})(T_{H})+a_{N}\sum_{P\in\Sigma}\eta_{P}^{N-1}]g^{\ast}c_{1}(T_{C}),$
applying equality (5.2.1) to the cycle $\alpha:=f^{\ast}c_{1}(T_{C})$, we
obtain that for every point $P$ in $\Sigma$, the class
$\eta_{P}\,g^{\ast}c_{1}(T_{C})$ vanishes, hence replacing:
$\displaystyle=\nu^{\ast}((c_{1}c_{N-1})(T_{H}))+(-1)^{N+1}N(N-1)(N-3)/2\sum_{P\in\Sigma}\eta_{P}^{N}-\nu^{\ast}((c_{N-1}+c_{1}c_{N-2})(T_{H}))\,g^{\ast}c_{1}(T_{C}),$
$\displaystyle=\nu^{\ast}[(c_{1}c_{N-1})(T_{H}))-(c_{N-1}+c_{1}c_{N-2})(T_{H})f^{\ast}c_{1}(T_{C})]+(-1)^{N+1}N(N-1)(N-3)/2\sum_{P\in\Sigma}\eta_{P}^{N},$
applying Proposition 5.3.1 to the scheme $V:=H$ and the morphism $h:=f$:
$\displaystyle=\nu^{\ast}((c_{1}c_{N-1}([T_{f}]))+(-1)^{N+1}N(N-1)(N-3)/2\sum_{P\in\Sigma}\eta_{P}^{N}.$

Hence pushing forward and applying equality (5.2.3) from Proposition 5.2.1, we
obtain the equality in ${\mathrm{CH}}^{N}(H)$:

$\nu_{\ast}((c_{1}c_{N-1})([T_{g}]))=\nu_{\ast}\nu^{\ast}((c_{1}c_{N-1})([T_{f}]))+N(N-1)(N-3)/2\sum_{P\in\Sigma}[P];$

using the projection formula and the fact that the morphism $\nu$ is of
generic degree 1, we obtain:

$\nu_{\ast}((c_{1}c_{N-1})([T_{g}])=(c_{1}c_{N-1})([T_{f}])+N(N-1)(N-3)/2[\Sigma].$

This concludes the proof. ∎

####  5.3.2. Computation of
$\nu_{\ast}(c_{1}c_{N-1})(\omega^{1\vee}_{\widetilde{H}/C})$

We begin with a purely combinatorial result.

If $y$ denotes an indeterminate and $f(y)$ a formal series in
$\mathbb{C}[[y]]$, for every integer $p$, we shall denote by $f(y)^{[p]}$ its
coefficient of degree $p$. It vanishes if $p$ is negative.

######  Proposition 5.3.3.

For every $a$ in $\mathbb{C}^{\ast}$, and every $n$ and $r$ in $\mathbb{N}$,
we have the formulas:

(5.3.1) $\displaystyle\left[\frac{(1+y)^{n}}{1+ay}\right]^{[n-r]}$
$\displaystyle=\frac{(-1)^{n+r}}{a^{r}}[(a-1)^{n}-\sum_{0\leq k\leq
r-1}\binom{n}{k}(-1)^{n-k}a^{k}]=(-1)^{r}\sum_{r\leq k\leq
n}\binom{n}{k}(-1)^{k}a^{k-r},$ and:
$\displaystyle\left[\frac{(1+y)^{n+1}}{(1+ay)^{2}}\right]^{[n-r]}$
$\displaystyle=\frac{(-1)^{n+r}}{a^{r+1}}\Big{[}(r+(n+1-r)a)(a-1)^{n}$ (5.3.2)
$\displaystyle-\sum_{0\leq k\leq
r}\bigg{(}r\binom{n}{k}-(n+1-r)\binom{n}{k-1}\bigg{)}(-1)^{n-k}a^{k}\Big{]}$
$\displaystyle=\sum_{r+1\leq k\leq
n}\bigg{(}r\binom{n}{k}-(n+1-r)\binom{n}{k-1}\bigg{)}(-1)^{r-k}a^{k-r-1}.$

###### Proof.

The identities are proved in the same way, using residues and the change of
variables $u:=\frac{y}{1+y}$. We will only detail the proof of (5.3.1).

By definition of the residue in zero ${\rm Res}_{0},$ we have:

$\displaystyle\left[\frac{(1+y)^{n}}{1+ay}\right]^{[n-r]}$ $\displaystyle={\rm
Res}_{0}\frac{1}{y^{n-r+1}}\frac{(1+y)^{n}}{1+ay}\ \mathrm{d}y,$
$\displaystyle={\rm
Res}_{0}y^{r-1}\left(\frac{1+y}{y}\right)^{n}\frac{1}{1+ay}\ \mathrm{d}y.$

We apply the change of variables:

$u:=\frac{y}{1+y}\in y+y^{2}\mathbb{C}[[y]].$

Using it, we obtain:

$\displaystyle\left[\frac{(1+y)^{n}}{1+ay}\right]^{[n-r]}$ $\displaystyle={\rm
Res}_{0}\left(\frac{u}{1-u}\right)^{r-1}\frac{1}{u^{n}}\frac{1}{1+\frac{au}{1-u}}\
\frac{\mathrm{d}u}{(1-u)^{2}},$ $\displaystyle={\rm
Res}_{0}\frac{1}{u^{n-r+1}}\frac{1}{(1-u)^{r-1}}\frac{1-u}{1-u+au}\
\frac{\mathrm{d}u}{(1-u)^{2}},$ $\displaystyle={\rm
Res}_{0}\frac{1}{u^{n-r+1}}\ \frac{1}{1+(a-1)u}\
\frac{\mathrm{d}u}{(1-u)^{r}},$ $\displaystyle=\left[\frac{1}{1+(a-1)u}\
\frac{1}{(1-u)^{r}}\right]^{[n-r]}.$

We partially decompose this fraction using a simple induction:

$\frac{1}{1+(a-1)u}\ \frac{1}{(1-u)^{r}}=\sum_{0\leq i\leq
r-1}\frac{1}{a^{r-i}}\frac{(a-1)^{r-1-i}}{(1-u)^{i+1}}+\frac{1}{a^{r}}\frac{(a-1)^{r}}{1+(a-1)u}.$

Consequently:

$\displaystyle\left[\frac{(1+y)^{n}}{1+ay}\right]^{[n-r]}$
$\displaystyle=\sum_{0\leq i\leq
r-1}\frac{(a-1)^{r-1-i}}{a^{r-i}}\left[\frac{1}{(1-u)^{i+1}}\right]^{(n-r)}+\frac{(a-1)^{r}}{a^{r}}\left[\frac{1}{1+(a-1)u}\right]^{(n-r)},$
$\displaystyle=\sum_{0\leq i\leq
r-1}\frac{(a-1)^{r-1-i}}{a^{r-i}}\binom{n-r+i}{i}+\frac{(a-1)^{r}}{a^{r}}(1-a)^{n-r},$
$\displaystyle=\frac{(-1)^{n-r}}{a^{r}}\Big{[}\sum_{0\leq i\leq
r-1}(-1)^{n-r}a^{i}(a-1)^{r-1-i}\binom{n-r+i}{i}+(a-1)^{n}\Big{]},$
$\displaystyle=\frac{(-1)^{n-r}}{a^{r}}\Big{[}-\sum_{0\leq k\leq
r-1}m_{n,k,r}a^{k}+(a-1)^{n}\Big{]},$

where for every non-negative integer $k\leq r-1$, $m_{n,k,r}$ is some complex
number.

However, the number $\left[\frac{(1+y)^{n}}{1+ay}\right]^{[n-r]}$ can be
written as a polynomial in $a$, so we obtain that for every integer $k\leq
r-1$, the number $m_{n,k,r}$ is given by the term of degree $k$ in $a$ of the
polynomial $(a-1)^{n}$, i.e. by:

$m_{n,k,r}=(-1)^{n-k}\binom{n}{k}.$

So we obtain the wanted equality (5.3.1).

For the equality (5.3.2), we reason similarly using the partial decomposition:

$\frac{1}{(1+(a-1)u)^{2}}\frac{1}{(1-u)^{r}}=\sum_{0\leq i\leq
r-1}(r-i)\frac{1}{a^{r+1-i}}\frac{(a-1)^{r-1-i}}{(1-u)^{i+1}}\\\
+\frac{r}{a^{r+1}}\frac{(a-1)^{r}}{1+(a-1)u}+\frac{1}{a^{r}}\frac{(a-1)^{r}}{(1+(a-1)u)^{2}}.\qed$

Let us adopt the notation of Subsection 5.1.

For every point $P$ in $\Sigma$, let us denote the class of the divisor
$E_{P}$ by:

$\eta_{P}:=[E_{P}]=c_{1}({\mathcal{O}}_{\tilde{H}}(E_{P}))\in{\mathrm{CH}}^{1}(\tilde{H}).$

Observe that equalities (5.2.1) and (5.2.2) still hold in this setting.

######  Proposition 5.3.4.

With the above notation, for every non-negative integer $r$, we have the
equality in ${\mathrm{CH}}^{r}(\tilde{H})$:

$c_{r}(\omega^{1\vee}_{\tilde{H}/C})=c_{r}([T_{g}])+\alpha(N,r)\sum_{P\in\Sigma}\eta_{P}^{r},$

where $\alpha(N,r)$ is the integer given by:

$\alpha(N,r):=(-1)^{r-1}\Bigg{(}\binom{N}{r-1}-4\Big{[}\frac{(1+y)^{N}}{1+2y}\Big{]}^{[r-2]}\Bigg{)}.$

Observe that the integer $\alpha(N,1)$ is simply 1.

###### Proof.

Let us apply equality (3.1.12) from Proposition 3.1.7 to the scheme

$Y:=\tilde{H},$

and the morphism $g$, whose divisor of singular fibers $\tilde{H}_{\Delta}$ is
of the form given by (5.1.1). Observe that with the notation of that
proposition, the subscheme $D^{3}$ is empty.

We obtain, for every integer $r$, the equality in
${\mathrm{CH}}^{r}(\tilde{H})$:

(5.3.3)
$c_{r}(\omega^{1\vee}_{\tilde{H}/C})=c_{r}([T_{g}])+\sum_{P\in\Sigma}i_{E_{P}\ast}c_{r-1}(T_{E_{P}})-2\sum_{P\in\Sigma}i_{E_{P}\cap
W\ast}c_{r-2}(T_{E_{P}\cap W}).$

Using (5.2.4) from Proposition 5.2.1, we have the equality in
${\mathrm{CH}}^{r-1}(E_{P})$:

$c_{r-1}(T_{E_{P}})=(-1)^{r-1}\binom{N}{r-1}\eta_{P|E_{P}}^{r-1},$

hence pushing forward by the inclusion in $\tilde{H}$ and using the projection
formula, we have the equality in ${\mathrm{CH}}^{r}(\tilde{H})$:

$i_{E_{P}\ast}c_{r-1}(T_{E_{P}})=(-1)^{r-1}\binom{N}{r-1}\eta_{P}^{r}.$

On the other hand, the intersection $Q_{P}:=E_{P}\cap W$ is a smooth
hypersurface of degree $m_{P}:=2$ in the projective space $E_{P}$, hence using
equality (5.2.5) from Proposition 5.2.1, we obtain the equality in
${\mathrm{CH}}^{r-1}(E_{P})$:

$i_{E_{P}\cap W\ast}c_{r-2}(T_{E_{P}\cap
W})=2(-1)^{r-1}\Big{[}\frac{(1+y)^{N}}{1+2y}\Big{]}^{[r-2]}\eta_{P|E_{P}}^{r-1},$

hence pushing forward by the inclusion in $\tilde{H}$ and using the projection
formula, we have the equality in ${\mathrm{CH}}^{r}(\tilde{H})$:

$i_{E_{P}\cap W\ast}c_{r-2}(T_{E_{P}\cap
W})=2(-1)^{r-1}\Big{[}\frac{(1+y)^{N}}{1+2y}\Big{]}^{[r-2]}\eta_{P}^{r}.$

Replacing in (5.3.3) yields the result. ∎

######  Proposition 5.3.5.

We have the equality in ${\mathrm{CH}}^{N}(\tilde{H})$:

$(c_{1}c_{N-1})(\omega^{1\vee}_{\tilde{H}/C})=(c_{1}c_{N-1})([T_{g}])+\beta(N)\sum_{P\in\Sigma}\eta_{P}^{N},$

where $\beta(N)$ is an integer given by:

$\beta(N)=\frac{(-1)^{N}}{2}[N^{3}-4N^{2}+4N-2+(-1)^{N+1}(N-2)].$

###### Proof.

If the integer $N$ is $1$, this follows directly from Proposition 5.3.4
applied to the integer $r:=1$ (observe that the combinatorial coefficient is
$1$).

Let us assume that the integer $N$ is at least $2$.

Applying Proposition 5.3.4 to the integer $r:=1$, we have the equality in
${\mathrm{CH}}^{1}(\tilde{H})$:

$\displaystyle c_{1}(\omega^{1\vee}_{\tilde{H}/C})$
$\displaystyle=c_{1}([T_{g}])+\alpha(N,1)\sum_{P\in\Sigma}\eta_{P},$
$\displaystyle=c_{1}([T_{g}])+\sum_{P\in\Sigma}\eta_{P},$

and applying it to the integer $r:=N-1$, we have the equality in
${\mathrm{CH}}^{N-1}(\tilde{H})$:

$c_{N-1}(\omega^{1\vee}_{\tilde{H}/C})=c_{N-1}([T_{g}])+\alpha(N,N-1)\sum_{P\in\Sigma}\eta_{P}^{N-1}.$

Hence multiplying, we have the equality in ${\mathrm{CH}}^{N}(\tilde{H})$:

$\displaystyle(c_{1}c_{N-1})(\omega^{1\vee}_{\tilde{H}/C})$
$\displaystyle=(c_{1}c_{N-1})([T_{g}])+\alpha(N,N-1)c_{1}([T_{g}]).\sum_{P\in\Sigma}\eta_{P}^{N-1}$
$\displaystyle+c_{N-1}([T_{g}])\sum_{P\in\Sigma}\eta_{P}+\alpha(N,N-1)\Big{(}\sum_{P\in\Sigma}\eta_{P}\Big{)}\Big{(}\sum_{P\in\Sigma}\eta_{P}^{N-1}\Big{)};$

using the projection formula, equality (5.2.2) and the hypothesis that $N$ is
at least $2$, we get:

$\displaystyle(c_{1}c_{N-1})(\omega^{1\vee}_{\tilde{H}/C})$
$\displaystyle=(c_{1}c_{N-1})([T_{g}])+\alpha(N,N-1)\sum_{P\in\Sigma}i_{E_{P}\ast}(c_{1}([T_{g}]_{|E_{P}})\eta_{P|E_{P}}^{N-2})$
(5.3.4)
$\displaystyle+\sum_{P\in\Sigma}i_{E_{P}\ast}c_{N-1}([T_{g}]_{|E_{P}})+\alpha(N,N-1)\sum_{P\in\Sigma}\eta_{P}^{N}.$

Using the definition of the relative tangent class $[T_{g}]$ and the fact that
for every $P$ in $\Sigma$, the line bundle $(g^{\ast}T_{C})_{|E_{P}}$ can be
trivialized on $E_{P}$, the middle two sums in (5.3.4) can be rewritten:

$\alpha(N,N-1)\sum_{P\in\Sigma}i_{E_{P}\ast}(c_{1}([T_{g}]_{|E_{P}})\eta_{P|E_{P}}^{N-2})+\sum_{P\in\Sigma}i_{E_{P}\ast}c_{N-1}([T_{g}]_{|E_{P}})\\\
=\alpha(N,N-1)\sum_{P\in\Sigma}i_{E_{P}\ast}(c_{1}(T_{\tilde{H}|E_{P}})\eta_{P|E_{P}}^{N-2})+\sum_{P\in\Sigma}i_{E_{P}\ast}c_{N-1}(T_{\tilde{H}|E_{P}}).$

For every point $P$ in $\Sigma$, we have an exact sequence of vector bundles
on $E_{P}$:

$0\longrightarrow T_{E_{P}}\longrightarrow
T_{\tilde{H}|E_{P}}\longrightarrow{\mathcal{N}}_{E_{P}}\tilde{H}\longrightarrow
0,$

hence for every integer $r$, we have the equality in
${\mathrm{CH}}^{r}(E_{P})$:

$c_{r}(T_{\tilde{H}|E_{P}})=c_{r}(T_{E_{P}})+c_{1}({\mathcal{N}}_{E_{P}}\tilde{H})c_{r-1}(T_{E_{P}})=c_{r}(T_{E_{P}})+\eta_{P|E_{P}}c_{r-1}(T_{E_{P}}).$

The middle two sums in (5.3.4) can be rewritten as follows, using notably the
equality (5.2.4) from Proposition 5.2.1:

$\displaystyle\alpha(N,N-1)$
$\displaystyle\sum_{P\in\Sigma}i_{E_{P}\ast}(c_{1}([T_{g}]_{|E_{P}})\eta_{P|E_{P}}^{N-2})+\sum_{P\in\Sigma}i_{E_{P}\ast}c_{N-1}([T_{g}]_{|E_{P}})$
$\displaystyle=\alpha(N,N-1)\sum_{P\in\Sigma}i_{E_{P}\ast}((c_{1}(T_{E_{P}})+\eta_{P|E_{P}})\eta_{P|E_{P}}^{N-2})$
$\displaystyle+\sum_{P\in\Sigma}i_{E_{P}\ast}(c_{N-1}(T_{E_{P}})+\eta_{P|E_{P}}c_{N-2}(T_{E_{P}})),$
$\displaystyle=\alpha(N,N-1)\sum_{P\in\Sigma}i_{E_{P}\ast}\Big{[}\Big{(}-\binom{N}{1}\eta_{P|E_{P}}+\eta_{P|E_{P}}\Big{)}\eta_{P|E_{P}}^{N-2}\Big{]}$
$\displaystyle+\sum_{P\in\Sigma}i_{E_{P}\ast}\Big{[}(-1)^{N-1}\binom{N}{N-1}\eta_{P|E_{P}}^{N-1}+(-1)^{N}\binom{N}{N-2}\eta_{P|E_{P}}^{N-1}\Big{]},$
$\displaystyle=\Big{[}\alpha(N,N-1)\Big{(}1-\binom{N}{1}\Big{)}+(-1)^{N-1}\binom{N}{N-1}+(-1)^{N}\binom{N}{N-2}\Big{]}\sum_{P\in\Sigma}i_{E_{P}\ast}\eta_{P|E_{P}}^{N-1},$
$\displaystyle=[-(N-1)\alpha(N,N-1)+(-1)^{N}(N(N-1)/2-N)]\sum_{P\in\Sigma}\eta_{P|E_{P}}^{N},$
$\displaystyle=[-(N-1)\alpha(N,N-1)+(-1)^{N}N(N-3)/2]\sum_{P\in\Sigma}\eta_{P|E_{P}}^{N}.$

By replacing in (5.3.4), we obtain the equality in
${\mathrm{CH}}^{N}(\tilde{H})$:

$(c_{1}c_{N-1})(\omega^{1\vee}_{\tilde{H}/C})=(c_{1}c_{N-1})([T_{g}])+\beta(N)\sum_{P\in\Sigma}\eta_{P}^{N},$

where $\beta(N)$ is the integer given by:

$\displaystyle\beta(N)$
$\displaystyle=-(N-1)\alpha(N,N-1)+(-1)^{N}N(N-3)/2+\alpha(N,N-1),$
$\displaystyle=-(N-2)\alpha(N,N-1)+(-1)^{N}N(N-3)/2.$

Now let us compute further the combinatorial coefficient $\beta(N)$.

By definition, the coefficient $\alpha(N,N-1)$ is given by:

$\alpha(N,N-1)=(-1)^{N}\Big{[}\binom{N}{N-2}-4\Big{[}\frac{(1+y)^{N}}{1+2y}\Big{]}^{[N-3]}\Big{]}.$

By using equality (5.3.1) of Proposition 5.3.3 applied to the integers $n:=N$
and $r:=3$, and to the complex number $a:=2$, we obtain:

$\displaystyle\alpha(N,N-1)$
$\displaystyle=(-1)^{N}[N(N-1)/2-1/2((-1)^{N+1}+2N(N-1)-2N+1)],$
$\displaystyle=\frac{(-1)^{N}}{2}[N(N-1)-(-1)^{N+1}-2N(N-1)+2N-1],$
$\displaystyle=\frac{(-1)^{N}}{2}[-N^{2}+3N-1-(-1)^{N+1}],$

hence the coefficient $\beta(N)$ is given by:

$\displaystyle\beta(N)$
$\displaystyle=-(N-2)\frac{(-1)^{N}}{2}[-N^{2}+3N-1-(-1)^{N+1}]+(-1)^{N}N(N-3)/2,$
$\displaystyle=\frac{(-1)^{N}}{2}[-(N-2)(-N^{2}+3N-1-(-1)^{N+1})+N(N-3)],$
$\displaystyle=\frac{(-1)^{N}}{3}[-(-N^{3}+2N^{2}+3N^{2}-6N-N+2)+(-1)^{N+1}(N-2)+N^{2}-3N],$
$\displaystyle=\frac{(-1)^{N}}{2}[N^{3}-4N^{2}+4N-2+(-1)^{N+1}(N-2)],$

as wanted. ∎

###### Proof of Lemma 5.1.1.

By Proposition 5.3.5, we have the equality in ${\mathrm{CH}}^{N}(\tilde{H})$:

$(c_{1}c_{N-1})(\omega^{1\vee}_{\tilde{H}/C})=(c_{1}c_{N-1})([T_{g}])+\frac{(-1)^{N}}{2}[N^{3}-4N^{2}+4N-2+(-1)^{N+1}(N-2)]\sum_{P\in\Sigma}\eta_{P}^{N},$

hence pushing forward by the morphism $\nu$, whose generic degree is one, and
using equality (5.2.3) from Proposition 5.2.1, we have the equality in
${\mathrm{CH}}^{N}(H)$:

$\nu_{\ast}(c_{1}c_{N-1})(\omega^{1\vee}_{\tilde{H}/C})=\nu_{\ast}(c_{1}c_{N-1})([T_{g}])-\frac{1}{2}[N^{3}-4N^{2}+4N-2+(-1)^{N+1}(N-2)][\Sigma].$

Combining this with Proposition 5.3.2 yields the equality in
${\mathrm{CH}}^{N}(H)$:

$\displaystyle\nu_{\ast}(c_{1}c_{N-1})(\omega^{1\vee}_{\tilde{H}/C})$
$\displaystyle=(c_{1}c_{N-1})([T_{f}])$
$\displaystyle\quad-\frac{1}{2}[N^{3}-4N^{2}+4N-2+(-1)^{N+1}(N-2)-N(N-1)(N-3)][\Sigma],$
$\displaystyle=(c_{1}c_{N-1})([T_{f}])-\frac{1}{2}[N-2+(-1)^{N+1}(N-2)][\Sigma],$
$\displaystyle=(c_{1}c_{N-1})([T_{f}])-(N-2)\frac{1-(-1)^{N}}{2}[\Sigma],$

as wanted. ∎

##  6\. The Griffiths height of the middle-dimensional cohomology of a pencil
of hypersurfaces

###  6.1. Families of ample divisors in a smooth pencil

The objective of this subsection is to prove the following proposition.

######  Proposition 6.1.1.

Let $C$ be a connected smooth projective complex curve with generic point
$\eta$, $X$ be a smooth projective complex scheme of pure dimension $N+1$, and
let

$\pi:X\longrightarrow C$

be a smooth surjective morphism of complex schemes. Let $H$ be a non-singular
hypersurface in $X$ such that the morphism

$\pi_{|H}:H\longrightarrow C$

is flat222222or equivalently, is non-constant on every connected component of
$H$, or has fibers of pure dimension $N-1$. and has a finite set $\Sigma$ of
critical points, all of which are non-degenerate.232323Equivalently, the only
possible singularities of the fibers $(H_{x})_{x\in C}$ are ordinary double
points.

If we denote by $L$ the line bundle ${\mathcal{O}}_{X}(H)$ on $X$, then the
following equality holds in ${\mathrm{CH}}_{0}(X)$:

(6.1.1) $[\Sigma]=\big{(}(1-c_{1}(L))^{-1}c(\Omega^{1}_{X/C})\big{)}^{(N+1)},$

If moreover the line bundle $L$ is ample relatively to $\pi$, then the
following equalities hold in ${\mathrm{CH}}_{0}(C)_{\mathbb{Q}}$:

(6.1.2) $c_{1}(\mathcal{GK}_{C,+}(\mathbb{H}^{N-1}(H_{\eta}/C_{\eta})))\\\
=c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{N-1}(X/C)))+c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{N+1}(X/C)))-c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{N}(X/C)))\\\
+\frac{1}{12}\pi_{\ast}\big{(}(1-c_{1}(L))^{-1}c_{1}(\Omega^{1}_{X/C})c(\Omega^{1}_{X/C})\big{)}^{(N+1)}-\frac{1}{12}\pi_{\ast}\big{(}c_{1}(L)c_{N}(\Omega^{1}_{X/C})\big{)}+v^{+}_{N}\,\pi_{\ast}[\Sigma],$

and:

(6.1.3) $c_{1}(\mathcal{GK}_{C,-}(\mathbb{H}^{N-1}(H_{\eta}/C_{\eta})))\\\
=c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{N-1}(X/C)))+c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{N+1}(X/C)))-c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{N}(X/C)))\\\
+\frac{1}{12}\pi_{\ast}\big{(}(1-c_{1}(L))^{-1}c_{1}(\Omega^{1}_{X/C})c(\Omega^{1}_{X/C})\big{)}^{(N+1)}-\frac{1}{12}\pi_{\ast}\big{(}c_{1}(L)c_{N}(\Omega^{1}_{X/C})\big{)}+v^{-}_{N}\,\pi_{\ast}[\Sigma],$

where:

$v^{+}_{N}:=\begin{cases}7(N-1)/24&\text{if $N$ is odd}\\\ (N+2)/24&\text{if
$N$ is even}.\end{cases}$

and:

$v^{-}_{N}:=\begin{cases}-5(N-1)/24&\text{if $N$ is odd}\\\ (N+2)/24&\text{if
$N$ is even}.\end{cases}$

####  6.1.1. Computation of $i_{\ast}[\Sigma]$ and of
$i_{\ast}(c_{1}c_{N-1})([T_{f}])$

Let $C$ be a connected smooth projective complex curve with generic point
$\eta$, let $X$ be a smooth projective complex scheme of pure dimension $N+1$,
and let

$\pi:X\longrightarrow C$

be a smooth surjective morphism of complex schemes. Let $H$ be a non-singular
hypersurface in $X$ such that the morphism

$f:=\pi_{|H}:H\longrightarrow C$

is flat and has a finite set $\Sigma$ of critical points, all of which are
non-degenerate.

Let

$i:H\longrightarrow X$

be the inclusion map. Let us also denote by $L$ the line bundle
${\mathcal{O}}_{X}(H)$ on $X$.

Observe that the line bundle $L^{\vee}_{|H}$ on $H$ can be identified with the
conormal line bundle ${\mathcal{N}}_{H}X^{\vee}$ of the hypersurface $H$ in
$X$.

Let us define a map $s$ of vector bundles on $H$ by the composition:

$s:L^{\vee}_{|H}\simeq{\mathcal{N}}_{H}X^{\vee}\longrightarrow\Omega^{1}_{X|H}\longrightarrow\Omega^{1}_{X/C|H}.$

We will also see $s$ as a section of the vector bundle
$\Omega^{1}_{X/C|H}\otimes L_{|H}$.

######  Lemma 6.1.2.

The section $s$ does not vanish on the open dense subset $H-\Sigma$ in $H$,
and for every point $P$ in $\Sigma$, there exists a system $(z_{1},...,z_{N})$
of local analytic coordinates on $H$ near $P$ such that, in some local frame
of the vector bundle $\Omega^{1}_{X/C|H}\otimes L_{|H}$, the section $s$ is
given by:

$s=(2z_{i})_{1\leq i\leq N}.$

###### Proof.

By definition of the subset $\Sigma$, the morphism of non-singular complex
schemes $f$ is smooth on the open dense subset $H-\Sigma$, so that by
elementary properties of the sheaf of Kähler differentials, the section $s$
does not vanish on $H-\Sigma$.

Now, let $P$ be a point in $\Sigma$, and let $y$ in $C$ be its image by $f$.
The hypersurfaces $H$ and $X_{y}$ in $X$ are both non-singular and meet in
$P$. By definition of $\Sigma$, the morphism $f$ has a critical point in $P$,
so we have the equality of hyperplanes in $T_{X,P}$:

(6.1.4) $T_{H,P}=T_{X_{y},P}$

Let $t$ be a local coordinate of $C$ in some analytic neighborhood of $y$.
Since the morphism $\pi$ is smooth, we can extend the function $\pi^{\ast}t$
into a local coordinate system $(\pi^{\ast}t,z_{1},...,z_{N})$ in some
analytic neighborhood $V$ of $P$.

Using (6.1.4) and the implicit function theorem, after possibly shrinking $V$,
there exists a diffeomorphism:

$\varphi:X_{y}\cap V\longrightarrow H\cap V$

which sends $P$ to $P$ and whose differential at $P$ is the identity map

${\mathrm{Id}}:T_{X_{y},P}\longrightarrow T_{H,P}.$

In other terms, there exists an analytic neighborhood $U$ of $0$ in
$\mathbb{C}^{N}$, and a function:

$u:U\longrightarrow\mathbb{C},$

vanishing in the origin, and whose derivative vanishes in the origin, and such
that $H\cap V$ is defined by the equation:

(6.1.5) $\pi^{\ast}t=u(z_{1},...,z_{N}).$

The morphism $\varphi$ and the function $u$ are related by the equality of
morphisms from $X_{y}$ to $\mathbb{C}^{N+1}$:

(6.1.6)
$(\pi^{\ast}t,z_{1},...,z_{N})\circ\varphi=(u(z_{1},...,z_{N}),z_{1},...,z_{N}).$

Observe that $H$ admits a local coordinate system given by
$(z_{1|H},...,z_{N|H})$.

By hypothesis, $y$ is a non-degenerate critical point of the morphism
$f=\pi_{|H}$ on $H$, hence it is also a non-degenerate critical point of the
morphism:

$\pi_{|H\cap V}\circ\varphi:X_{y}\cap V\longrightarrow C.$

Using (6.1.6), this morphism is given in local coordinates by:

$(\pi^{\ast}t)_{|H\cap V}\circ\varphi=u(z_{1},...,z_{N}),$

so we obtain that the Hessian at $0$ of the function $u$ is non-degenerate.
Consequently, after possibly shrinking the neighborhood $V$ and changing the
coordinates $z_{1},...,z_{N}$, we can assume that the function $u$ is given
by:

(6.1.7) $u(z_{1},...,z_{N})=z_{1}^{2}+...+z_{N}^{2}.$

The vector bundle $\Omega^{1}_{X}$ admits a local frame given by
$(\pi^{\ast}\mathrm{d}t,\mathrm{d}z_{1},...,\mathrm{d}z_{N})$. Using that the
hypersurface $H$ is defined by the equation (6.1.5), the conormal line bundle
$L_{|H}^{\vee}\simeq{\mathcal{N}}_{H}X^{\vee}$, as embedded in the vector
bundle $\Omega^{1}_{X|H}$, is generated by the non-vanishing section:

$s_{0}:=\frac{\partial u}{\partial z_{1}}\mathrm{d}z_{1}+...+\frac{\partial
u}{\partial
z_{N}}\mathrm{d}z_{N}-\pi^{\ast}\mathrm{d}t=2z_{1|H}\mathrm{d}z_{1}+...+2z_{N|H}\mathrm{d}z_{N}-\pi^{\ast}\mathrm{d}t\textrm{
using \eqref{u non-degenerate}}.$

The vector bundle $\Omega^{1}_{X/C}$ admits a local frame given by
$([\mathrm{d}z_{1}],...,[\mathrm{d}z_{N}])$. In this frame, the image by the
projection

$\Omega^{1}_{X|H}\longrightarrow\Omega^{1}_{X/C|H}$

of the section $s_{0}$ is given by:

$2z_{1|H}[\mathrm{d}z_{1}]+...+2z_{N|H}[\mathrm{d}z_{N}].$

This image is precisely the section $s$, which proves the lemma. ∎

######  Lemma 6.1.3.

We have an exact sequence of coherent sheaves on $H$:

(6.1.8) $0\longrightarrow
L^{\vee}_{|H}\overset{s}{\longrightarrow}\Omega^{1}_{X/C|H}\overset{{}^{t}(Di)}{\longrightarrow}\Omega^{1}_{H/C}\longrightarrow
0,$

where ${}^{t}(Di)$ is the map between sheaves of Kähler differentials
associated with the morphism of $C$-schemes $i$.

Furthermore, the subscheme of $H$ defined by the vanishing of $s$ is precisely
the reduced subscheme defined by the finite subset $\Sigma$.

###### Proof.

For the exactness of (6.1.8), everything but the injectivity of the morphism
$s$ is a standard property of sheaves of relative Kähler differentials. By
Lemma 6.1.2, the morphism $s$ does not vanish on the open dense subset
$H-\Sigma$ in $H$. Since both coherent sheaves $L^{\vee}_{|H}$ and
$\Omega^{1}_{X/C|H}$ are locally free on $H$, the morphism of coherent sheaves
$s$ is therefore injective, which completes the proof of the exactness of
(6.1.8).

Furthermore, by Lemma 6.1.2, the morphism $s$ vanishes with order $1$ in every
point of $\Sigma$. Consequently, the subscheme defined by the vanishing of the
morphism $s$ is precisely the reduced subscheme defined by $\Sigma$, as
wanted. ∎

######  Proposition 6.1.4.

We have the following equalities in ${\mathrm{CH}}_{0}(X)$:

(6.1.9) $i_{\ast}[\Sigma]=[(1-c_{1}(L))^{-1}c(\Omega^{1}_{X/C})]^{(N+1)},$

and:

(6.1.10) $i_{\ast}(c_{1}c_{N-1})([T_{f}])=(c_{1}c_{N})([T_{\pi}])\\\
+(-1)^{N}\big{(}[(1-c_{1}(L))^{-1}c_{1}(\Omega^{1}_{X/C})c(\Omega^{1}_{X/C})]^{(N+1)}+i_{\ast}[\Sigma]-c_{1}(L)c_{N}(\Omega^{1}_{X/C})\big{)}.$

###### Proof.

Using Lemma 6.1.3, the subscheme defined by the vanishing of the section $s$
of the vector bundle $\Omega^{1}_{X/C|H}\otimes L_{|H}$ on $H$ is exactly the
reduced subscheme defined by $\Sigma$, in particular, it is of dimension $0$,
hence of codimension $N$ in $H$. The rank of the vector bundle
$\Omega^{1}_{X/C|H}\otimes L_{|H}$ on $H$ is exactly $N$, so that $s$ is a
regular section in the sense of [Ful98, B.3.4].

Consequently, using [Ful98, Example 3.2.16, (ii)], we have the equality in
${\mathrm{CH}}_{0}(H)$:

$\displaystyle[\Sigma]$ $\displaystyle=c_{N}(\Omega^{1}_{X/C|H}\otimes
L_{|H}),$ $\displaystyle=i^{\ast}c_{N}(\Omega^{1}_{X/C}\otimes L),$
$\displaystyle=i^{\ast}\sum_{0\leq k\leq
N}c_{1}(L)^{N-k}c_{k}(\Omega^{1}_{X/C}),$

where the last expression follows from the formula for the top Chern class of
the tensor product of a vector bundle by a line bundle ([Ful98, Remark 3.2.3,
(b)]).

Pushing forward by the inclusion $i$ and applying the projection formula, we
obtain the equality in ${\mathrm{CH}}_{0}(X)$:

$\displaystyle i_{\ast}[\Sigma]$ $\displaystyle=[H]\cdot\sum_{0\leq k\leq
N}c_{1}(L)^{N-k}c_{k}(\Omega^{1}_{X/C}),$ $\displaystyle=c_{1}(L)\sum_{0\leq
k\leq N}c_{1}(L)^{N-k}c_{k}(\Omega^{1}_{X/C}),$ $\displaystyle=\sum_{0\leq
k\leq N}c_{1}(L)^{N+1-k}c_{k}(\Omega^{1}_{X/C}),$ $\displaystyle=\sum_{0\leq
k\leq N+1}c_{1}(L)^{N+1-k}c_{k}(\Omega^{1}_{X/C})\textrm{ because the vector
bundle $\Omega^{1}_{X/C}$ is of rank $N$,}$
$\displaystyle=[(1-c_{1}(L))^{-1}c(\Omega^{1}_{X/C})]^{(N+1)},$

which shows equality (6.1.9).

Now let us show equality (6.1.10). Using exact sequence (6.1.8) and the
multiplicativity of total Chern classes, we have the equality in
${\mathrm{CH}}^{\ast}(H)$:

$c([\Omega^{1}_{H/C}])=c(\Omega^{1}_{X/C|H})c(L^{\vee}_{|H})^{-1}=i^{\ast}[(1-c_{1}(L))^{-1}c(\Omega^{1}_{X/C})].$

Taking the terms of codimension $1$, we obtain the equality in
${\mathrm{CH}}^{1}(H)$:

(6.1.11)
$c_{1}([\Omega^{1}_{H/C}])=i^{\ast}[c_{1}(\Omega^{1}_{X/C})+c_{1}(L)],$

and taking the terms of codimension $N-1$, we obtain the equality in
${\mathrm{CH}}^{N-1}(H)$:

(6.1.12) $c_{N-1}([\Omega^{1}_{H/C}])=i^{\ast}\sum_{0\leq k\leq
N-1}c_{1}(L)^{N-1-k}c_{k}(\Omega^{1}_{X/C}).$

Multiplying (6.1.11) and (6.1.12), we obtain the equality in
${\mathrm{CH}}^{N}(H)\simeq{\mathrm{CH}}_{0}(H)$:

$\displaystyle(c_{1}c_{N-1})([\Omega^{1}_{H/C}])$
$\displaystyle=i^{\ast}\Big{[}\sum_{0\leq k\leq
N-1}c_{1}(L)^{N-k}c_{k}(\Omega^{1}_{X/C})+\sum_{0\leq k\leq
N-1}c_{1}(L)^{N-1-k}(c_{1}c_{k})(\Omega^{1}_{X/C})\Big{]}.$

Pushing forward by $i$ and applying the projection formula, we obtain the
equality in ${\mathrm{CH}}_{0}(X)$:

$\displaystyle i_{\ast}(c_{1}c_{N-1})([\Omega^{1}_{H/C}])$
$\displaystyle=c_{1}(L).\Big{[}\sum_{0\leq k\leq
N-1}c_{1}(L)^{N-k}c_{k}(\Omega^{1}_{X/C})+\sum_{0\leq k\leq
N-1}c_{1}(L)^{N-1-k}(c_{1}c_{k})(\Omega^{1}_{X/C})\Big{]},$ (6.1.13)
$\displaystyle=\sum_{0\leq k\leq
N-1}c_{1}(L)^{N+1-k}c_{k}(\Omega^{1}_{X/C})+\sum_{0\leq k\leq
N-1}c_{1}(L)^{N-k}(c_{1}c_{k})(\Omega^{1}_{X/C}).$

Using that the vector bundle $\Omega^{1}_{X/C}$ on $X$ is of rank $N$, the
first sum in (6.1.13) is given by:

$\displaystyle\sum_{0\leq k\leq N-1}c_{1}(L)^{N+1-k}c_{k}(\Omega^{1}_{X/C})$
$\displaystyle=\sum_{0\leq k\leq
N+1}c_{1}(L)^{N+1-k}c_{k}(\Omega^{1}_{X/C})-c_{1}(L)c_{N}(\Omega^{1}_{X/C}),$
$\displaystyle=[(1-c_{1}(L))^{-1}c(\Omega^{1}_{X/C})]^{(N+1)}-c_{1}(L)c_{N}(\Omega^{1}_{X/C}),$
$\displaystyle=i_{\ast}[\Sigma]-c_{1}(L)c_{N}(\Omega^{1}_{X/C})\textrm{ using
equality \eqref{eq Sigma for hyp}}.$

Similarly, the second sum in (6.1.13) is given by:

$\displaystyle\sum_{0\leq k\leq
N-1}c_{1}(L)^{N-k}(c_{1}c_{k})(\Omega^{1}_{X/C})$ $\displaystyle=\sum_{0\leq
k\leq
N}c_{1}(L)^{N-k}(c_{1}c_{k})(\Omega^{1}_{X/C})-(c_{1}c_{N})(\Omega^{1}_{X/C}),$
$\displaystyle=[(1-c_{1}(L))^{-1}c_{1}(\Omega^{1}_{X/C})c(\Omega^{1}_{X/C})]^{(N+1)}-(c_{1}c_{N})(\Omega^{1}_{X/C}).$

Hence replacing in (6.1.13), we obtain the equality in ${\mathrm{CH}}_{0}(X)$:

$i_{\ast}(c_{1}c_{N-1})([\Omega^{1}_{H/C}])=i_{\ast}[\Sigma]-c_{1}(L)c_{N}(\Omega^{1}_{X/C})+[(1-c_{1}(L))^{-1}c_{1}(\Omega^{1}_{X/C})c(\Omega^{1}_{X/C})]^{(N+1)}-(c_{1}c_{N})(\Omega^{1}_{X/C}).$

Since the morphism of non-singular complex schemes $f$ is smooth on an open
dense subset of $H$, similarly to Subsubsection 3.1.4, we have the following
equality in $K^{0}(H)\simeq K_{0}(H)$:

$[T_{f}]=[\Omega^{1}_{H/C}]^{\vee},$

where $.^{\vee}$ denotes the duality involution on $K^{0}(H)$.

Consequently, we have the equality in ${\mathrm{CH}}_{0}(X)$:

$\displaystyle i_{\ast}(c_{1}c_{N-1})([T_{f}])$
$\displaystyle=(-1)^{N}i_{\ast}(c_{1}c_{N-1})([\Omega^{1}_{H/C}]),$
$\displaystyle=(c_{1}c_{N})([T_{\pi}])$
$\displaystyle+(-1)^{N}\big{(}[(1-c_{1}(L))^{-1}c_{1}(\Omega^{1}_{X/C})c(\Omega^{1}_{X/C})]^{(N+1)}+i_{\ast}[\Sigma]-c_{1}(L)c_{N}(\Omega^{1}_{X/C})\big{)},$

which shows equality (6.1.10). ∎

####  6.1.2. An application of Lefschetz’s weak theorem

We can also prove the following consequence of Lefschetz’s weak theorem and of
Poincaré duality.

######  Proposition 6.1.5.

If the line bundle $L$ is ample relatively to the morphism $\pi$, then for
every integer $n$ such that $n<N-1$, the local monodromy of
$\mathbb{H}^{n}(H_{\eta}/C_{\eta})$ is unipotent and we have the equality in
${\mathrm{CH}}_{0}(C)$:

(6.1.14)
$c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{n}(H_{\eta}/C_{\eta})))=c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{n}(X/C))),$

and for every integer $n$ such that $n>N-1$, the local monodromy of
$\mathbb{H}^{n}(H_{\eta}/C_{\eta})$ is unipotent and we have the equality in
${\mathrm{CH}}_{0}(C)_{\mathbb{Q}}$:

(6.1.15)
$c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{n}(H_{\eta}/C_{\eta})))=c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{n+2}(X/C))).$

A minor variant of the proof below would actually show that the local
monodromy of $\mathbb{H}^{n}(H_{\eta}/C_{\eta})$ is trivial when $n\neq N-1$.

###### Proof.

Let us assume that the line bundle $L$ is ample relatively to the morphism
$\pi$. Let $\Delta$ be the set-theoretic image:

$\Delta:=f(\Sigma),$

and let:

${\mathaccent 23{C}}:=C-\Delta$

be its complement in $C$.

Let $n$ be an integer such that $n<N-1$.

Since the line bundle $L={\mathcal{O}}_{X}(H)$ is ample relatively to the
morphism $\pi$, using Lefschetz’s weak theorem, we obtain that the pullback
morphism of variations of Hodge structures on ${\mathaccent 23{C}}$:

$\mathbb{H}^{n}(i^{\ast}):\mathbb{H}^{n}(X-X_{\Delta}/{\mathaccent
23{C}})\longrightarrow\mathbb{H}^{n}(H-H_{\Delta}/{\mathaccent 23{C}})$

is an isomorphism.

Since $X$ is smooth over $C$, the VHS
$\mathbb{H}^{n}(X-X_{\Delta}/{\mathaccent 23{C}})$ on ${\mathaccent 23{C}}$
can be extended into the VHS $\mathbb{H}^{n}(X/C)$ on $C$, hence its local
monodromy at every point of $\Delta$ is trivial.

Consequently, the local monodromy of the VHS
$\mathbb{H}^{n}(H-H_{\Delta}/{\mathaccent 23{C}})$ at every point of $\Delta$
is trivial, in particular unipotent, and the isomorphism of VHS
$\mathbb{H}^{n}(i^{\ast})$ extends into an isomorphism on the (upper or lower)
Deligne extension.

Consequently, it induces an isomorphism of line bundles on $C$:

$\mathcal{GK}_{C}(\mathbb{H}^{n}(i^{\ast})):\mathcal{GK}_{C}(\mathbb{H}^{n}(X/C))\longrightarrow\mathcal{GK}_{C}(\mathbb{H}^{n}(H_{\eta}/C_{\eta})).$

Taking Chern classes, we obtain immediately equality (6.1.14).

Now, let $n$ be an integer such that $n>N-1$. Applying Proposition 2.5.2 to
the blow-up $Y$ of $H$ at every point of $\Sigma$, satisfying that the divisor
$Y_{\Delta}$ is a divisor with strict normal crossings, and to the integer

$n^{\prime}:=2(N-1)-n<N-1,$

and using that the local monodromy at every point of $\Delta$ of the VHS
$\mathbb{H}^{n^{\prime}}(Y-Y_{\Delta}/C-\Delta)$ is unipotent (which we showed
above), we obtain that the line bundle on $C$:

$\mathcal{GK}_{C}(\mathbb{H}^{n}(Y-Y_{\Delta}/{\mathaccent
23{C}}))\otimes\mathcal{GK}_{C}(\mathbb{H}^{2(N-1)-n}(Y-Y_{\Delta}/\mathbb{C}))^{\vee}\\\
\simeq\mathcal{GK}_{C}(\mathbb{H}^{n}(H_{\eta}/C_{\eta}))\otimes\mathcal{GK}_{C}(\mathbb{H}^{2(N-1)-n}(H_{\eta}/C_{\eta}))$

is of $2$-torsion. Consequently, we have the equality in
${\mathrm{CH}}_{0}(C)_{\mathbb{Q}}$,

$c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{n}(H_{\eta}/C_{\eta})))=c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{2(N-1)-n}(H_{\eta}/C_{\eta}))).$

Applying equality (6.1.14) to the integer $n^{\prime}:=2(N-1)-n<N-1$, we
obtain:

$c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{n}(H_{\eta}/C_{\eta})))=c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{2(N-1)-n}(X/C))).$

Applying Proposition 2.5.2 to $X$, with the set of critical values being
empty, we obtain:

$c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{n}(H_{\eta}/C_{\eta})))=c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{2N-(2(N-1)-n)}(X/C)))=c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{n+2}(X/C)),$

which shows equality (6.1.15). ∎

Now we can conclude the proof of Proposition 6.1.1. Let us assume that the
line bundle $L$ is ample relatively to the morphism $\pi$.

Applying equality (5.0.1) from Theorem 5.0.1 to the $C$-scheme $H$, then
applying equality (6.1.10) from Proposition 6.1.4, we have the equalities in
${\mathrm{CH}}_{0}(C)_{\mathbb{Q}}$:

$\displaystyle\sum_{n=0}^{2(N-1)}(-1)^{n-1}c_{1}\big{(}\mathcal{GK}_{C,-}(\mathbb{H}^{n}(H_{\eta}/C_{\eta}))\big{)}$
$\displaystyle=\frac{1}{12}f_{\ast}\big{(}(c_{1}c_{N-1})([\Omega^{1}_{H/C}]^{\vee})\big{)}+u_{N}^{-}f_{\ast}[\Sigma],$
$\displaystyle=\frac{1}{12}\pi_{\ast}(c_{1}c_{N})([T_{\pi}])+(u_{N}^{-}+\frac{(-1)^{N}}{12})f_{\ast}[\Sigma]$
(6.1.16)
$\displaystyle+\frac{(-1)^{N}}{12}\pi_{\ast}\big{(}[(1-c_{1}(L))^{-1}c_{1}(\Omega^{1}_{X/C})c(\Omega^{1}_{X/C})]^{(N+1)}-c_{1}(L)c_{N}(\Omega^{1}_{X/C})\big{)}.$

where $u_{N}^{-}$ is the rational number defined by:

$u_{N}^{-}:=\begin{cases}(5N-3)/24&\text{if $N$ is odd}\\\ N/24&\text{if $N$
is even}.\end{cases}$

Applying Theorem 5.0.1 to the smooth $C$-scheme $X$ yields the equality in
${\mathrm{CH}}_{0}(C)_{\mathbb{Q}}$:

(6.1.17)
$\sum_{n=0}^{2N}(-1)^{n-1}c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{n}(X/C)))=\frac{1}{12}\pi_{\ast}(c_{1}c_{N})([T_{\pi}]).$

Finally, it follows from Proposition 6.1.5 that we have the equality in
${\mathrm{CH}}_{0}(C)_{\mathbb{Q}}$:

(6.1.18)
$\sum_{n=0}^{2(N-1)}(-1)^{n-1}c_{1}\big{(}\mathcal{GK}_{C,-}(\mathbb{H}^{n}(H_{\eta}/C_{\eta}))\big{)}-\sum_{n=0}^{2N}(-1)^{n-1}c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{n}(X/C)))\\\
=(-1)^{N}[c_{1}(\mathcal{GK}_{C,-}(\mathbb{H}^{N-1}(H_{\eta}/C_{\eta})))-c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{N-1}(X/C)))\\\
+c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{N}(X/C)))-c_{1}(\mathcal{GK}_{C}(\mathbb{H}^{N+1}(X/C)))].$

Equality (6.1.3) follows by combining equalities (6.1.16), (6.1.17) and
(6.1.18). Equality (6.1.2) is proved in the same way, using equality (5.0.2)
from Theorem 5.0.1 instead of equality (5.0.1).

This concludes the proof of Proposition 6.1.1.

###  6.2. Pencils of hypersurfaces in the projective space

In this subsection, we shall use Proposition 6.1.1 to compute the Griffiths
height of the middle -dimensional cohomology of pencils of projective
hypersurfaces. Before stating our results, we introduce some notation and
recall some basic facts concerning projective bundles over a projective curve,
their Chow groups, and their hypersurfaces.

####  6.2.1. Preliminary: projective bundles over a curve and horizontal
hypersurfaces

Let $C$ be a connected smooth projective complex curve with generic point
$\eta$, let $E$ be a vector bundle of rank $N+1\geq 1$ over $C$, and let:

$\pi:\mathbb{P}(E):={\rm Proj\,}S^{\bullet}E^{\vee}\longrightarrow C,$

be the associated projective bundle over $C$.

We denote by ${\mathcal{O}}_{E}(1)$ the tautological quotient line bundle over
$\mathbb{P}(E)$; it is the dual of the tautological subbundle of rank $1$ of
$\pi^{\ast}E$.

We shall use the fact (see for instance [Ful98, B.5.8]) that the relative
tangent bundle

$T_{\mathbb{P}(E)/C}:=T_{\pi}$

fits into a short exact sequence:

(6.2.1)
$0\longrightarrow{\mathcal{O}}_{\mathbb{P}(E)}\longrightarrow\pi^{\ast}E\otimes{\mathcal{O}}_{E}(1)\longrightarrow
T_{\mathbb{P}(E)/C}\longrightarrow 0.$

Let $H$ be an hypersurface, namely an effective Cartier divisor, in
$\mathbb{P}(E)$. It is a projective complex scheme of dimension $N$. Its
generic fiber $H_{\eta}$ is an hypersurface in the projective space
$\mathbb{P}(E)_{\eta}$ of dimension $N$ over $\mathbb{C}(C)$. We shall denote
its degree by $d$.

We shall also denote by:

$i:H\longrightarrow\mathbb{P}(E)$

the inclusion morphism of $H$ in the projective bundle $\mathbb{P}(E)$, and
by:

$f:=\pi\circ i:H\longrightarrow C$

its projection to the curve $C$.

As is well-known, the hypersurface $H_{\eta}$ in the projective space
$\mathbb{P}(E)_{\eta}$ may be defined, as a subscheme, by the vanishing of an
homogeneous polynomial of degree $d$ — namely by a non-zero element of
$S^{d}E^{\vee}_{\eta}$ — unique up to the multiplication by an element of
$\mathbb{C}(C)^{\ast}.$

This description of $H_{\eta}$ extends to a description of the “relative
hypersurface” $H$ over $C$ as follows.

Since $\mathbb{P}(E)$ is a regular scheme, its Picard group is isomorphic to
its Chow group of codimension one ${\mathrm{CH}}^{1}(\mathbb{P}(E))$.
According to the structure of the Chow groups of a projective bundle (see
[Ful98, Th. 3.3 (b)], with $k=\dim(\mathbb{P}(E))-1$), there is an
isomorphism:

$\mathbb{Z}\oplus{\mathrm{CH}}^{1}(C)\overset{\sim}{\longrightarrow}{\mathrm{CH}}^{1}(\mathbb{P}(E)),\quad(\delta,\alpha)\longmapsto\delta
c_{1}({\mathcal{O}}_{E}(1))+\pi^{\ast}\alpha.$

Consequently the line bundle ${\mathcal{O}}_{\mathbb{P}(E)}(H)$ is isomorphic
to ${\mathcal{O}}_{E}(\delta)\otimes\pi^{\ast}M$ for some integer $\delta$ and
some line bundle $M$ over $C$. By considering the restriction of this
isomorphism to $\mathbb{P}(E)_{\eta}$, we see that $\delta$ coincides with the
generic degree $d$ of $H$. Moreover the line bundle $M$ is unique up to
isomorphism.

We will denote by:

(6.2.2)
$\sigma:{\mathcal{O}}_{\mathbb{P}(E)}(H)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{O}}_{E}(d)\otimes\pi^{\ast}M$

the isomorphism obtained by this construction. This isomorphism may be seen as
a morphism of line bundles over $\mathbb{P}(E)$:

(6.2.3) $\sigma:\pi^{\ast}M^{\vee}\longrightarrow{\mathcal{O}}_{E}(d),$

the vanishing of which defines $H$ scheme-theoretically. In turn, the data of
the morphism (6.2.3) is equivalent to the data of a morphism of sheaves of
${\mathcal{O}}_{C}$-modules:

(6.2.4) $\tau:M^{\vee}\longrightarrow\pi_{\ast}{\mathcal{O}}_{E}(d)\simeq
S^{d}E^{\vee},$

which is clearly injective.

For every point $x$ of $C$, the image $\tau_{x}(y)$ of any non-zero element
$y$ of $M^{\vee}_{x}$ is an element of $S^{d}E^{\vee}_{x}$, which is a
(scheme-theoretic) equation for the fiber $H_{x}$ of $H$ in the projective
space $\mathbb{P}(E)_{x}$ over $\mathbb{C}$. This shows that the hypersurface
$H$ is horizontal — namely, that no fiber of $\pi$ is a component of $H$, or
equivalently that the morphism $f:H\rightarrow C$ is flat — if and only if
$\tau_{x}$ is non-zero for every point $x$ of $C$, or equivalently if and only
if $\tau$ is an isomorphism onto a locally direct summand of $S^{d}E^{\vee}$.

In the sequel, we shall use the following notation for the first Chern classes
of $E$ and $M$, and of ${\mathcal{O}}_{E}(1)$, in ${\mathrm{CH}}^{1}(C)$ and
${\mathrm{CH}}^{1}(\mathbb{P}(E))$ respectively:

$e:=c_{1}(E),\quad m:=c_{1}(M),\quad\mbox{and}\quad
h:=c_{1}({\mathcal{O}}_{E}(1)).$

Using the isomorphism (6.2.2), we obtain the equality in
${\mathrm{CH}}^{1}(\mathbb{P}(E))$:

(6.2.5) $[H]=d\,h+\pi^{\ast}m.$

Using the definition of Segre classes and their relation to Chern classes
([Ful98, 3.1, 3.2]), and the fact that $C$ is one-dimensional, we obtain the
following equality, for every $i\in\mathbb{N}$:

(6.2.6) $\pi_{\ast}h^{i}=:s_{i-N}(E)=\begin{cases}0&\quad\mbox{if }i<N,\\\
[C]&\quad\mbox{if }i=N,\\\ -e&\quad\mbox{if }i=N+1,\\\ 0&\quad\mbox{if
}i>N+1.\end{cases}$

From the relations above and the projection formula, we deduce:

(6.2.7) $\pi_{\ast}(h^{N-1}\cap[H])=d[C],$

and

(6.2.8) $\pi_{\ast}(h^{N}\cap[H])=-de+m.$

In computations, we will use that ${\mathrm{CH}}^{i}(C)$ vanishes for $i>1$;
in particular, we have:

(6.2.9) $m\cdot m=e\cdot e=m\cdot e=0.$

######  Remark 6.2.1.

The classes $m$ and $e$ depend on the choice of the bundle $E$, and not only
on the $C$-scheme $\mathbb{P}{}(E)$.

More specifically, if $L$ is a line bundle on $C$, and $E$ is replaced by
$E^{\prime}:=E\otimes L$, then we may identify the $C$-schemes
$\mathbb{P}(E^{\prime})$ and $\mathbb{P}(E)$, but $e$ is replaced by:

$e^{\prime}:=c_{1}(E^{\prime})=c_{1}(E\otimes L)=e+(N+1)\textrm{ }c_{1}(L).$

The line bundle ${\mathcal{O}}_{E}(1)$ is replaced by:

${\mathcal{O}}_{E^{\prime}}(1)\simeq{\mathcal{O}}_{E}(1)\otimes\pi^{\ast}L^{\vee},$

and the class $h$ is replaced by:

$h^{\prime}:=c_{1}({\mathcal{O}}_{E^{\prime}}(1))=h-\pi^{\ast}c_{1}(L).$

Moreover, the bundle $S^{d}E^{\prime\vee}$ may be identified with
$(S^{d}E^{\vee})\otimes L^{-d}$. The morphism of vector bundles:

$\tau^{\prime}:=\tau\otimes{\mathrm{Id}}_{L^{-d}}:M^{\vee}\otimes
L^{-d}\hookrightarrow S^{d}E^{\prime\vee}$

clearly represents the same equation as $\tau$, hence it also defines the
hypersurface $H$ in $\mathbb{P}(E^{\prime})$.

Finally the line bundle $M$ is replaced by $M^{\prime}:=M\otimes L^{d}$, and
consequently the class $m$ is replaced by:

$m^{\prime}:=c_{1}(M^{\prime})=m+d\textrm{ }c_{1}(L).$

Observe that this implies the equality of classes in
${\mathrm{CH}}^{1}(C)_{\mathbb{Q}}$:

$(N+1)m^{\prime}-de^{\prime}=(N+1)m-de.$

We shall denote the relative canonical sheaf of the smooth morphism

$\pi:\mathbb{P}(E)\longrightarrow C$

by:

(6.2.10) $\omega_{\mathbb{P}(E)/C}\simeq\Omega^{N}_{\mathbb{P}(E)/C}\simeq\det
T_{\mathbb{P}(E)/C}^{\vee}.$

######  Definition and Proposition 6.2.2.

Let $h^{\ast}\in{\mathrm{CH}}^{1}(\mathbb{P}(E))_{\mathbb{Q}}$ be the class
defined by:

$h^{\ast}:=-\frac{1}{N+1}c_{1}(\omega_{\mathbb{P}(E)/C}),$

and let ${\mathrm{ht}}_{int}(H/C)$ be the rational number defined by:

${\mathrm{ht}}_{int}(H/C):=\int_{\mathbb{P}(E)}h^{\ast N}\cap[H].$

They satisfy the following relations:

(6.2.11) $h^{\ast}=h+\frac{1}{N+1}\pi^{\ast}e,$ (6.2.12)
$\displaystyle{\mathrm{ht}}_{int}(H/C)$
$\displaystyle=\int_{\mathbb{P}(E)}h^{N}\cap[H]+\frac{dN}{N+1}\deg E,$
(6.2.13) $\displaystyle=\deg M-\frac{d}{N+1}\deg E.$

In particular, the rational number $(N+1){\mathrm{ht}}_{int}(H/C)$ is an
integer.

###### Proof.

The equality (6.2.11) comes from taking the determinant of the exact sequence
(6.2.1). This gives isomorphisms:

$\det
T_{\mathbb{P}(E)/C}\simeq\det\big{(}\pi^{\ast}E\otimes{\mathcal{O}}_{E}(1)\big{)}\simeq{\mathcal{O}}_{E}(N+1)\otimes\pi^{\ast}\det
E.$

Together with (6.2.10), this establishes (6.2.11).

By using successively (6.2.9), (6.2.5) and (6.2.9) again, we have:

$\displaystyle h^{\ast N}\cap[H]$
$\displaystyle=\left(h+\frac{1}{N+1}\pi^{\ast}e\right)^{N}\cap[H],$
$\displaystyle=\left(h^{N}+\frac{N}{N+1}h^{N-1}\pi^{\ast}e\right)\cap[H],$
$\displaystyle=h^{N}\cap[H]+\frac{N}{N+1}h^{N-1}\pi^{\ast}e\textrm{ }.\textrm{
}(dh+\pi^{\ast}m),$ (6.2.14)
$\displaystyle=h^{N}\cap[H]+\frac{dN}{N+1}h^{N}\pi^{\ast}e.$

After pushing forward by $\pi$ and taking the degree, using (6.2.6), this