# Hodge-Arakelov inequalities for family of surfaces fibered by curves

Mohammad Reza Rahmati  Universidad De La Salle Bajío, Campestre - León,
Guanajuato, Mexico.
<EMAIL_ADDRESS>
###### Abstract.

The Hodge numerical invariants of a variation of Hodge structure over a smooth
quas–projective variety are a measure of complexity for the global twisting of
the limit mixed Hodge structure when it degenerates. These invariants appear
in inequalities which they may have correction terms, called Arakelov
inequalities. One may investigate the correction term to make them into
equalities, also called Arakelov equalities. We investigate numerical Arakelov
type (in)equilities for a family of surfaces fibered by curves. Our method
uses Arakelov identities in a weight 1 and also in a weight 2 variations of
Hodge structure (cf. [19]), in a commutative triangle of fibrations. We have
proposed to relate the degrees of the Hodge bundles in the two families. We
also compare the Fujita decomposition of Hodge bundles in these fibrations. We
examine various identities and relations between Hodge numbers and degrees of
the Hodge bundles in different levels.

###### Key words and phrases:

Variation of Hodge Structure, Kodaira-Spencer maps, Curvature of Hodge
bundles, Arakelov-inequality, Fujita decomposition

###### 1991 Mathematics Subject Classification:

14C17, 14D05, 14D06, 14D07, 14J27, 14J28, 14J32, 32G20

##  1\. Introduction

Of important invariants of a variation of Hodge structure (VHS) that measure
the complexity of their twist are the chern classes of their Hodge bundles. A
question in this line is how the degeration of VHS affect these chern classes.
The numerical data of a geometric variation of Hodge structure also provide
tools to study the iso-triviality of the family. They also give bounds to
determine global invariants associated to the singularities of the fibrations.
For example one can answer simple questions on fibrations in low dimensions
that, if they must have singular fibers and even some bounds for the number of
singular fibers. The Arakelov identities involve the degrees of the Hodge
bundles in VHS given by geometric fibration. In fact the vanishing of the
degrees of all the Hodge bundle give a criteria to say whether the fibration
is iso-trivial. These equalities have been systematically studied in low
dimensions 1, 2, 3 and 4, [19, 40].

Assume we have a fibration $f:X\to S$ of smooth projective varieties with $S$
being of dimension 1. For each $k$ it defines a variation of Hodge structure
$(\mathcal{H}^{(k)}=R^{k}f_{*}\mathbb{C},F^{p},\nabla)$ of weight $k$ over $S$
which is the Deligne extension of that over the smooth locus of the fibration.
Then the graded sheaves
$(\mathcal{H}^{p,q}=F^{p}/F^{p-1},\theta_{q}:=Gr_{F}^{p}\nabla)$ define a
Higgs bundle. Following [19], in a weight one VHS $\mathcal{H}_{/S}$ obtained
from the fibration by curves we have the following exact sequence of sheaves

(1)
$0\to\mathcal{H}_{0,e}^{1,0}\to\mathcal{H}_{e}^{1,0}\stackrel{{\scriptstyle\theta}}{{\longrightarrow}}\check{\mathcal{H}}_{e}^{1,0}\otimes\Omega_{S}^{1}(\log
E)\to\mathcal{B}\to 0$

where $\mathcal{H}_{e}$ is the extended Hodge bundle and
$\mathcal{H}_{0,e}^{1,0}:=\ker({\mathcal{H}}_{e}^{1,0}\to\check{\mathcal{H}}_{e}^{1,0}\otimes\Omega_{S}^{1})$.
We denote the rank of its generic fiber by $h_{0}^{1,0}$. The map $\theta$ is
induced by the Gauss-Manin connection and is called the Kodaira-Spencer map. A
calculation of the degree $\delta=\deg(\mathcal{H}^{1,0})$, [see [19]] gives
the following general formula

(2)
$\delta=\frac{1}{2}(h^{1,0}-h_{0}^{1,0})(2g(S)-2)+\delta_{0}-\frac{1}{2}\sum_{s\in
S}\nu_{s}(\bar{\theta})$

where $\delta_{0}=\deg(\mathcal{H}_{0,e}^{1,0})$, and
$\nu_{s}(\bar{\theta}):=\dim\text{coker}\left\\{\mathcal{H}_{e}^{1,0}/\mathcal{H}_{0,e}^{1,0}\stackrel{{\scriptstyle\bar{\theta}}}{{\longrightarrow}}(\mathcal{H}_{e}^{1,0}/\mathcal{H}_{0,e}^{1,0})^{\perp}\otimes\Omega_{S}^{1}\right\\}$.
Similar identities exists in higher weights. The above kind of identities are
called Arakelov identities. The Arakelov identities measure the complexity of
the global twist of the VHS, according to the existence of the degeneracies,
see [19] for details.

Another piece of information we encounter in our analysis is the Fujita
decomposition of the Hodge bundle, [3, 32, 39, 40, 6, 5]. Assume we have the
fibration $f:X\to S$ as before [see Section 4 for exact set-up]. In general
the Fujita decomposition is of the form

(3)
$f_{*}\omega_{X/S}=\mathcal{A}\bigoplus\mathcal{U}^{\prime}\bigoplus\mathcal{O}_{S}^{q_{f}},\qquad$

where $\omega_{X/S}$ is the bundle of holomorphic differential forms
$\Omega_{X}^{\dim X}$, and $\mathcal{A}$ is an ample vector bundle,
$\mathcal{U}=\mathcal{O}_{S}^{q_{f}}\bigoplus\mathcal{U}^{\prime}$ is a flat
unitary vector bundle. $\mathcal{U}^{\prime}$ is also flat but has no
sections. The Gauss-Manin connection on $\mathbb{H}$ induces a connection

(4) $\nabla_{\mathcal{U}}:\mathcal{U}\to\mathcal{U}\otimes\Omega_{S}^{1}$

It is known by construction that $\mathcal{U}\subset\ker(\theta)$. Let

(5)
$\kappa^{n,0}:\mathcal{H}^{n,0}\stackrel{{\scriptstyle\theta^{n}}}{{\rightarrow}}\mathcal{H}^{n-1,1}\stackrel{{\scriptstyle\theta^{n-1}}}{{\rightarrow}}...\stackrel{{\scriptstyle\theta}}{{\rightarrow}}\mathcal{H}^{0,n}$

be the Griffiths-Yukawa coupling, i.e the composition of the $n$ successive
KS-maps. The Griffiths-Yukawa coupling is said to be maximal, if
$\mathcal{H}^{n,0}\neq 0$ and if $\kappa^{n,0}$ is an isomorphism, [cf. [44]].
Similar to before we set

(6)
$\mathcal{H}_{0}^{n,0}=\ker\left(\theta^{p}:\mathcal{H}^{p,q}\to\mathcal{H}^{p-1,q+1}\otimes\Omega_{S}^{1}(\log
E)\right)$

and $h_{0}^{n,0}=\text{rank}\mathcal{H}_{0}^{n,0}$, with the same notation as
previous section. We have the Arakelov inequality

(7) $\deg(\mathcal{H}^{n,0})\leq\frac{n}{2}\
.\text{rank}(\mathcal{H}^{n,0}).\deg(\Omega_{S}^{1}(\log E))$

with equality if the Higgs field of $\mathcal{H}$ is maximal. Moreover if
$\theta^{n}\neq 0$ then $\deg(\mathcal{H}^{n,0})>0,\ \deg(\Omega_{S}^{1}(\log
E))>0$. In particular if $S=\mathbb{P}^{1}$ then $\sharp E\geq 3$. The Fujita
decomposition $f_{*}\omega_{X/S}=\mathcal{A}\bigoplus\mathcal{U}$ into an
ample sheaf $\mathcal{A}$ and a flat subsheaf $\mathcal{U}$, satisfies
$\deg(\mathcal{H}^{n,0})=\deg(\mathcal{A})$. Moreover the Higgs field of
$\mathcal{A}$ is strictly maximal and $\mathcal{U}$ is a variation of
polarized complex Hodge structure zero in the bidegree $(n,0)$, [see [44]
lemma 4, for instance].

###  1.1. Contributions

We apply this formula for a VHS of surfaces $f:X\to S$ fibered by curves
$Y_{s}$. One obtains a triangle fibration

(8)
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{f}$$\scriptstyle{\circlearrowleft}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{S}$

Let $\mathcal{H},\ \mathcal{H}(1)$ and $\mathcal{H}(2)$ be the associated
local systems of $H^{1}$ of fibers in the triangle. Then an application of the
Grothendieck 5-term sequence gives the following diagram of sheaves with
Kodaira-Spencer maps

(9) $\begin{CD}g_{*}\Omega_{Y/S}^{1}(\log F)@>{}>{}>f_{*}\Omega_{X/S}^{1}(\log
G)@>{}>{}>f_{*}\Omega_{X/Y}^{1}(\log D)\\\
@V{}V{\theta^{1}}V@V{}V{\theta}V@V{}V{g_{*}{\theta^{2}}}V\\\
R^{1}g_{*}\mathcal{O}_{Y}\otimes\Omega_{S}^{1}(\log
E)@>{}>{}>R^{1}f_{*}\mathcal{O}_{X}\otimes\Omega_{S}^{1}(\log
E)@>{}>{}>g_{*}(R^{1}h_{*}\mathcal{O}_{X})\otimes\Omega_{Y}^{1}(\log E)\\\
\end{CD}$

With $\log$ we mean logarithmic sheaves with appropriate normal crossing
divisors, that assumed to be compatible in the fibration [see Section 2 for
the set up and Definitions]. In the text, we sometimes denote the suffix
$\theta_{Y},\theta_{X/Y}$ to distinguish the KS-map. We study how the
contributions in Arakelov equation (2) for the middle vertical KS-map splits
in the other two. We discuss the following

  * •

There is a decomposition
$\mathcal{H}=\mathcal{H}_{\text{fix}}\bigoplus\mathcal{H}_{\text{var}}$ into
fixed and variable parts where

(10) $\mathcal{H}_{\text{var}}:=\\{image(f_{*}\Omega_{X/S}^{1}\to
f_{*}\Omega_{X/Y}^{1})\\}^{\perp}$

and $\mathcal{H}_{\text{fix}}$ is identified with the image. Then
$g_{*}\left(\theta^{2}(\mathcal{H}_{var})\right)=0$

  * •

In the course to compare $\delta,\delta^{1}$, and $\delta^{2}$ there should be
a relation

(11) $\nu(\theta)=\nu(\theta^{1})+\nu(g_{*}\theta^{2}\mid_{\text{fix}})$

where $\nu$ is computed on the points where KS-map drops rank, rather they are
singularities or not.

  * •

The contribution for the terms $\delta_{0}$ ,s is additive

(12) $\delta_{0}=\delta_{0}^{1}+\delta_{0}(\mathcal{H}^{1,0}(2)_{\text{fix}})$

  * •

The $\mathcal{H}_{\text{var}}$ contribute to the similar invariants for the
weight 2 VHS in $H^{2}(X_{s},\mathbb{C})$.

We also compare the Fujita decomposition for the Hodge bundles of VHS over
${Y}$ and $S$. We have a similar decomposition

(13) $f_{*}\omega_{X/S}=f_{*}\omega_{Y/S}\oplus
f_{*}\omega_{X/S}^{\text{var}}$

One has the estimates for the ranks of the direct summands of Fujita
decomposition for the 3 fibrations. In a basic analysis we get that

  * •

$H_{e}^{1,0}=H_{e}^{1,0}(1)\bigoplus H_{e}^{1,0}(2)_{\text{fix}}$.

  * •

$\mathcal{U}_{f}=\mathcal{U}_{g}\bigoplus\mathcal{U}_{h}^{\text{fix}}$.

  * •

$h^{1,0}=h(1)^{1,0}+h_{\text{fix}}^{1,0}(2)$.

  * •

$h_{0}^{1,0}=h_{0}^{1,0}(1)+h_{0,\text{fix}}^{1,0}(2)$.

  * •

$\delta_{0}=\delta_{0}^{1}+\left(\delta_{0}^{2}\right)_{\text{fix}}$.

  * •

$\nu(\theta)=\nu(\theta^{1})+\nu(g_{*}\theta^{2}\mid_{\text{fix}})$.

  * •

$\delta^{1,0}=\delta^{1}+\delta^{2}_{\text{fix}}=\delta^{1}+\delta^{2}-\delta^{2}_{\text{var}}$.

  * •

$u_{f}=u_{g}+u_{h}^{\text{fix}}=u_{g}+u_{h}-u_{h}^{\text{var}}$.

By the local product structure of the double fibration in (8) and Kodaira-
Spencer maps induced from the Gauss-Manin connections we deduce the existence
of an induced map

(14)
$\overline{\vartheta^{2,0}}:\dfrac{g_{*}\left(R^{1}h_{*}\Omega_{X/Y,var}^{0}\right)}{\theta^{2}_{\text{var}}\left(g_{*}(R^{0}h_{*}\Omega_{X/Y,var}^{1})\right)}\
\longrightarrow\
\dfrac{R^{2}f_{*}\Omega_{X/S}^{0}\otimes(\Omega_{S}^{1})^{\otimes
2}}{\theta^{1,1}\left(R^{1}f_{*}\Omega_{X/S}^{1}\otimes\Omega_{S}^{1}\right)}$

which is injective. We shall use the existence of this map to obtain Arakelov
type identities in a two step fibration. We prove the following identity on
the degrees of the Hodge bundles in a family of surfaces fibered by curves

(15)
$\delta^{2,0}+2(\delta^{1,0}-(\delta^{1}+\delta^{2})\geq(h^{1,1}-h^{2,0})(2g-2)^{2}-h^{1,0}(2)(2g-2)$

By adjusting the existence of a local product structure on the fibred surface
we can obtain another inequality which relates the the Hodge numbers in weight
two HS to the one of weight one on the middle curves. Another inequality we
prove is

(16)
$\delta_{X}^{2,0}\geq\frac{1}{2}(h_{X/Y}^{1,0}-h_{0,X/Y}^{1,0})(2g(S)-2)+h_{X/Y}^{1,0}$

Moreover we have $\delta_{X}^{2,0}\ \geq\ \delta_{X/Y}^{1,0}$. The VHS of the
middle cohomology of the fibration $Y\to S$ appears as a middle convolution of
the one for $X\to S$, [37]. We give the following generalization of a result
in [37] on the invariants of the middle convolution of the local systems of HS
of surfaces. The result there was proved over $S=\mathbb{P}^{1}$ which is more
simpler. Our statement is as follows. If the local system
$\mathcal{V}=g_{*}\left(R^{1}h_{*}\mathbb{C}\right)$ is irreducible with
regular singularities, then

(17)
$\chi(S,\mathcal{V})=\delta_{X/Y}^{1}+h^{1}(V)-\delta_{X/Y}^{0}+h^{0}(V)-\sum_{s}\nu_{s}(\bar{\theta}_{X/Y})(V).$

Organization: Section 2 is classical and we just introduce the notation and
concepts we wish to employ. References are [21, 16, 31, 23, 24, 25]. Section 3
describes numerical invariants of variation of Hodge structures and closely
follows [19]. The relevant references are [33, 31, 39, 34, 22, 1, 2, 10, 15,
30, 35, 36, 23, 24, 25]. Section 4 explains Fujita decomposition. The relevant
references are [18, 3, 32, 26, 13, 12, 6, 5]. In section 5 we present some
contributions and calculations by the numerical invariants.

##  2\. Algebraic geometric background

We briefly review some basics on relative logarithmic differentials on schemes
used in later sections. This section is classical, and the reader can find
various texts and references describing logarithmic sheaves in slightly
different contexts (see [16, 31, 23, 24, 25]). Lets begin with a smooth
fibration of schemes $f:X\to S$ where $X$ and $S$ are smooth and of finite
type over an algebraically closed field of char=0. In this case the module of
relative q-differentials of $f$ on $X$ is defined by the short exact

(18) $0\to
f^{*}\Omega_{S}^{1}\wedge\Omega_{X}^{q-1}\to\Omega_{X}^{q}\to\Omega_{X/S}^{q}\to
0$

When you have three schemes

(19) $f:X\stackrel{{\scriptstyle h}}{{\rightarrow}}Y\stackrel{{\scriptstyle
g}}{{\rightarrow}}Z$

respectively, then we have the following exact sequence

(20) $0\to
h^{*}\Omega_{Y/S}^{1}\wedge\Omega_{X/S}^{q-1}\to\Omega_{X/S}^{q}\to\Omega_{X/Y}^{q}\to
0$

In case $X$ or $S$ not to be regular or $f$ fails to be smooth the above
definition is no longer valid. However one still may define it when the
singular locus is normal crossing, called logarithmic differentials. In this
case we replace our schemes with log-pairs $(X,D)$ and $(S,E)$, where $D$ and
$E$ are smooth normal crossing schemes or varieties and $f(D)=E$. According to
the Hironaka resolution of singularities theorem this is possible after
blowing up the singularities of $X,S$ along $f$ enough times. In this case one
can similarly choose

(21) $\Omega^{q}_{X/S}(\log D)=\frac{\Omega^{q}_{X}(\log
D)}{f^{*}\Omega^{1}_{S}(\log E)\wedge\Omega^{q-1}_{X}(\log D)}$

as the definition, [31, 23, 24, 25]. We shall consider three log-pairs

(22) $f:(X,D)\stackrel{{\scriptstyle
h}}{{\rightarrow}}(Y,F)\stackrel{{\scriptstyle g}}{{\rightarrow}}(S,E)$

Then we are assuming they are compatible log-smooth pairs in the category of
log-schemes. Then the sequence (20) is valid in the new category and by
definition reads as

(23) $0\to h^{*}\Omega_{Y/S}^{1}(\log F)\wedge\Omega_{X/S}^{q-1}(\log
D)\to\Omega_{X/S}^{q}(\log D)\to\Omega_{X/Y}^{q}(\log h^{*}F)\to 0$

We will be working with 1-forms and the global sections of the stalk of these
sheaves along fibers over $S$. In other word when pushforward by $f_{*}$ we
get

(24) $\displaystyle 0\to g_{*}\Omega_{Y/S}^{1}(\log F)\to
f_{*}\Omega_{X/S}^{1}(\log D)$ $\displaystyle\to f_{*}\Omega_{X/Y}^{1}(\log
h^{*}F)\to R^{1}g_{*}\Omega_{Y/S}^{1}(\log F)\to$ $\displaystyle\to
R^{1}f_{*}\Omega_{X/S}^{1}(\log D)\to R^{1}f_{*}\Omega_{X/Y}^{1}(\log h^{*}F)$

Note that varieties that the items in this exact sequence refer to are the
appropriate fibers in our triangle and the normal crossing divisors give
normal crossing sections inside the fibers. Regarding the situation that how
the relative singular fibers of the maps $f,g$ and $h$ behave geometrically
relative to each other, we hereby and later on are assuming that they are
located one above another so that the relative log-pairs are compatible as
mentioned above [cf. [31, 23, 24, 25] loc. cit.].

Another fact, is that going to the long exact cohomology sequence of the short
exact sequence defining the relative differentials we obtain

(25) $0\to\Omega_{S}^{1}\to f_{*}\Omega_{X}^{1}\to
f_{*}\Omega_{X/S}^{1}\stackrel{{\scriptstyle\kappa=(.\kappa_{s})}}{{\longrightarrow}}R^{1}(f_{*}f^{*})\Omega_{S}^{1}=R^{1}f_{*}\mathcal{O}_{X}\otimes\Omega_{S}^{1}\to$

Then the connecting homomorphism $\kappa$ is the Kodaira-Spencer map, i.e
wedge with the Kodaira-Spencer class. For the last term in the sequence we
have used adjunction formula

(26) $\mathcal{G}\otimes
R^{i}f_{*}\mathcal{E}=R^{i}f_{*}f^{*}\mathcal{G}\otimes\mathcal{E}$

for suitable sheaves $\mathcal{G},\ \mathcal{E}$. This phenomenon maybe
studied in a triangle fibration of smooth varieties, or a compatible log-
smooth triple.

We use a basic property of Grothendieck spectral sequences, applied to a first
quadrant spectral sequence

(27) $E_{2}^{p,q}=R^{p}G\circ R^{q}F(A)\Rightarrow R^{p+q}(G\circ F)(A)$

for functors $F:\mathcal{A}\to\mathcal{B},\ G:\mathcal{B}\to\mathcal{C}$
between abelian categories. It is an exact sequence of terms of low degrees.

######  Proposition 2.1.

(Grothendieck 5-term sequence) If $E_{2}^{p,q}\Rightarrow H^{n}(A)$ is a first
quadrant spectral sequence, then there is an exact sequence

(28) $0\to E_{2}^{1,0}\to H^{1}(A)\to E_{2}^{0,1}\to E_{2}^{2,0}\to H^{2}(A)$

where $E_{2}^{0,1}\to E_{2}^{2,0}$ is the differential of the $E_{2}$-term.

When we have two successive fibrations $X\to Y\to S$ we get a form of 5-term
sequence as

(29) $0\to R^{1}g_{*}\mathcal{O}_{Y}\to R^{1}f_{*}\mathcal{O}_{X}\to
g_{*}R^{1}h_{*}\mathcal{O}_{X}\to R^{2}g_{*}\mathcal{O}_{Y}\to
R^{2}f_{*}\mathcal{O}_{X}$

We will employ a twist of this sequence in computation with Kodaira-Spencer
maps.

##  3\. Hodge-Arakelov numerical data

The reference for this section is [19], [24]. We follow [19] briefly to adjust
the formulas we use. Assume $f:X\to S$ is a fibration of smooth projective
varieties over $\mathbb{C}$. Hodge theory studies the variation of Hodge
structure (VHS)constructed from the cohomologies of the smooth fibers
$X_{s}=f^{-1}(s)$. The VHS,s associated to these fibrations form a local
system $\mathcal{H}_{/S^{*}}$ of vector spaces over $\mathbb{Q}$, such that

(30) $\mathcal{H}_{\mathbb{C}}=\bigoplus_{p+q=k}\
\mathcal{H}^{p,q},\qquad\mathcal{H}^{p,q}:=F^{p}\cap\overline{F^{q}}$

are the ($C^{\infty}$)-Hodge bundles defined over the smooth locus $S^{*}$. By
the Hironaka resolution of singularities one may assume the degeneracy locus
$E=S\setminus S^{*}$ is normal crossing divisor. The classical Hodge theory
guarantees the existence of a canonical extension $\mathcal{H}_{e}$ whose
sections have at worst logarithmic poles along the normal crossing divisor
$E$. A major task in Hodge theory is to study the degeneration of the Hodge
structure near the singular points. Assume $D=f^{-1}(E)$ is also normal
crossing. We may draw the picture of our fibration as follows

$\begin{CD}D@>{}>{}>X@<{}<{}<X\setminus D\\\ @V{}V{}V@V{}V{}V@V{}V{}V\\\
E@>{}>{}>S@<{}<{}<S\setminus E\end{CD}$

Then the sheaves $R^{p}f_{*}\Omega^{\bullet}_{X/S}(\log D)$ are locally free,
and admit a (logarithmic) connection

(31) $\nabla_{e}:R^{p}f_{*}\Omega^{\bullet}_{X/S}(\log
D)\to\Omega^{1}_{S}(\log E)\otimes R^{p}f_{*}\Omega^{\bullet}_{X/S}(\log D)$

such that its residue is nilpotent. Moreover there are isomorphisms

(32) $H^{p}(D,\Omega^{\bullet}_{X/S}(\log
D))\otimes\mathcal{O}_{D})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}H^{p}(X_{\infty},\mathbb{C})$

Among the invariants associated to VHS,s are the degrees of the Hodge bundles

(33) $\delta_{p}=\deg(F_{e}^{n-p}/F_{e}^{n-p+1})$

The better are the quantities

(34) $\delta_{0}+\delta_{1}+...+\delta_{p}\geq 0$

which measure how far the VHS is being from trivial. Over the smooth locus the
Hodge bundles $\mathcal{H}^{p,q}$ are equipped with the Hodge metric induced
from the polarization form. The invariants $\delta_{p}$ may be calculated from
the curvature of the associated metric connection. The curvature can be
written as

(35)
$\Theta_{\mathcal{H}^{p,q}}=^{t}\theta_{q}\wedge\bar{\theta}_{q}+\bar{\theta}_{q-1}\wedge^{t}\theta_{q-1}$

where $\theta_{q}=Gr_{F}^{p}\nabla$ are the Kodaira-Spencer maps and ”t” is
the Hermitian adjoint. Here we understand that $q=n-p$. The cohomology classes

(36) $\displaystyle\begin{aligned}
c_{1}(\Theta_{\mathcal{H}^{p,q}})=\alpha_{p}-\alpha_{p-1},\qquad\int_{S}\alpha_{p}=a_{p}\\\
c_{1}(\Theta_{\mathcal{H}_{0}^{p,q}})=\beta_{p}-\alpha_{p-1},\qquad\int_{S}\beta_{p}=b_{p}\end{aligned}$

are integrable closed (1,1)-forms and also determine the Chern classes of the
extended Hodge bundle; $\mathcal{H}_{e}^{p,q}$.

In the weight one VHS,s obtained from the fibration by curves we have the
following exact sequence of sheaves

(37)
$0\to\mathcal{H}_{0,e}^{1,0}\to\mathcal{H}_{e}^{1,0}\stackrel{{\scriptstyle\theta^{(1,0)}}}{{\longrightarrow}}\check{\mathcal{H}}_{e}^{1,0}\otimes\Omega_{S}^{1}(\log
E)\to\text{coker}\to 0$

where
$\mathcal{H}_{0,e}^{1,0}:=\ker({\mathcal{H}}_{e}^{1,0}\to\check{\mathcal{H}}_{e}^{1,0}\otimes\Omega_{S}^{1})$.
We denote the rank of its generic fiber by $h_{0}^{1,0}$. Note that
$\mathcal{B}$ is probably not a vector bundle; i.e may have torsion. It is
also probable that the meromorphic $h^{1,0}\times h^{1,0}$-matrix $\theta$
drops rank at non singular values $s$. Because the degree map is additive on
exact sequences one can calculate the $\delta=\deg(\mathcal{H}^{p,q})$ in
terms of the other terms. For the map on the stalks we have

(38)
$\theta_{s}(\mathcal{H}_{e,s}^{1,0})\subset(\mathcal{H}_{0,e,s}^{1,0})^{\perp}\otimes\Omega_{S,s}(\log
E)$

One considers set of points where this inclusion is strict. This may also
happen at nonsingular points. The calculation of the curvature gives

(39)
$\delta=\frac{1}{2}(h^{1,0}-h_{0}^{1,0})(2g-2)+\delta_{0}-\frac{1}{2}\sum_{s\in
S}\nu_{s}(\bar{\theta})$

where $\delta_{0}=\deg(\mathcal{H}_{0,e}^{1,0}),\
\nu_{s}(\theta):=\dim(\text{coker}(\theta))$ and

(40)
$\bar{\theta}:\mathcal{H}_{e}^{1,0}/\mathcal{H}_{0,e}^{1,0}\longrightarrow(\mathcal{H}_{e}^{1,0}/\mathcal{H}_{0,e}^{1,0})^{\vee}\otimes\Omega_{S}^{1}$

Alternatively (39) can be written as

(41)
$\begin{split}\delta&=\frac{1}{2}\Big{[}(h^{1,0}-h_{0}^{1,0})(2g-2)+\sum_{i}\dim({Im}\bar{N}_{i})\Big{]}-\\\
&\qquad\qquad\qquad\qquad\qquad\Big{[}(-\delta_{0})+\frac{1}{2}\left(\sum_{s\in
S^{*}}\nu_{s}(\bar{\theta}_{s})+\sum_{i}\nu_{s_{i}}(\bar{A}_{i})\right)\Big{]}\end{split}$

where $N_{i}$ are the logarithms of the monodromies at the degeneracy point
$s_{i}$ and since $N_{i}=\text{Res}_{s_{i}}\nabla$,

(42)
$\bar{N_{i}}:F_{e,s_{i}}^{1}/F_{0,e,s_{i}}^{1}\longrightarrow\left(F_{e,s_{i}}^{1}/F_{0,e,s_{i}}^{1}\right)^{\vee}$

are the induced map. The matrices $A_{i}$ are defined via the matrice of the
map $\bar{\theta}_{s_{i}}$ by

(43) $\bar{\theta}_{s_{i}}=\begin{pmatrix}\frac{B^{\prime}}{s}+...&*&*\\\
0&A^{\prime}(s)&*\\\ 0&0&0\end{pmatrix}$

where $B^{\prime}$ is the nonzero component of the matrix of $\theta_{s_{i}}$,
[19].

In the weight 2 case in the absence of degeneracy, we have the two short exact
sequence

(44) $\displaystyle\begin{aligned}
0\to\mathcal{H}_{0,e}^{2,0}\to\mathcal{H}_{e}^{2,0}\stackrel{{\scriptstyle\theta^{2,0}}}{{\longrightarrow}}{\mathcal{H}}_{e}^{1,1}\otimes\Omega_{S}^{1}\stackrel{{\scriptstyle\theta_{1}}}{{\rightarrow}}\check{\mathcal{H}}_{0,e}^{1,1}\otimes\Omega_{S}^{1}\to
0\\\
0\to\mathcal{H}_{0,e}^{1,1}\to\mathcal{H}_{e}^{1,1}\stackrel{{\scriptstyle(\theta^{2,0})^{\vee}}}{{\longrightarrow}}{\mathcal{H}}_{e}^{0,2}\otimes\Omega_{S}^{1}\to\check{\mathcal{H}}_{e}^{2,0}\otimes\Omega_{S}^{1}\to
0\end{aligned}$

which are dual of one another. The equation (39) is replaced by the following

(45) $\delta^{2,0}=(h^{2,0}-h_{0}^{2,0})(2g-2)-[\delta(0)+\sum_{s\in
S}\nu_{s}(\bar{\theta}^{2,0})]$

where

(46)
${\bar{\theta}^{2,0}}:\mathcal{H}^{2,0}/\mathcal{H}_{0}^{2,0}{\longrightarrow}\mathcal{H}^{1,1}\otimes\Omega_{S}^{1}/\ker\theta_{1}$

and $\delta(0)=a_{0}+b_{0}+b_{1}$. When there are degeneracies the formula
modifies as follows

(47) $\delta^{2,0}=(h^{2,0}-h_{0}^{2,0})(2g-2+N)+[\delta(0)+\sum_{s\in
S}\nu_{s}(\det\theta)+\sum_{i}(\hat{h}_{i}^{2,0}-h_{i}^{2,0})]$

where we have set $\hat{H}_{i}=Gr_{q}(LMHS)_{s_{i}}$ and $\hat{h}_{i}^{p,q}$
are their Hodge numbers, cf. [19] loc. cit..

##  4\. Fujita decomposition

Fujita decomposition associated to a fibration $f:X\to S$ of smooth projective
schemes $X,S$ over $\mathbb{C}$ and where $\dim(S)=1,\ \dim(X)=n+1$ is a
decomposition of the sheaf $f_{*}\omega_{X/S}$ where $\omega_{X/S}$ is the
relative canonical sheaf of $f$. First denote

(48) $f:X^{*}\to S^{*},\qquad X^{*}=X\smallsetminus\text{Sing}(f)$

to be the same map on the smooth locus of $S$. Then the direct image
$\mathcal{H}=R^{n}f_{*}\mathbb{C}_{X^{*}}$ defines a local system on $S^{*}$.
By Riemann-Hilbert correspondence it is equivalent to the Gauss-Manin
connection

(49)
$\nabla:\mathcal{H}\otimes\mathcal{O}_{S}\to\mathcal{H}\otimes\Omega_{S}^{1}$

Its data can be encoded in a monodromy representation

(50) $\varrho:\pi_{1}(S^{*})\to GL(\mathcal{H}_{s_{0}})$

where $s_{0}\in S$. When studying this local system globally there appear two
type of monodromy representations. One is the monodromy around singularities
and second is the monodromy of $S$.

######  Theorem 4.1.

(Fujita, Catanese-Dettweiler) [32, 13, 12] There is a decomposition

(51) $f_{*}\omega_{X/S}=\mathcal{A}\bigoplus\mathcal{U},\qquad\text{(2nd
Fujita decomposition)}$

where $\mathcal{A}$ is an ample vector bundle on $S$, and $\mathcal{U}$ is
unitary flat sheaf whose restriction to $S^{*}$ is a holomorphic vector bundle
associated to a sub-local system of $\mathcal{H}$.

There is also a first Fujita decomposition.

######  Theorem 4.2.

(Fujita) [32, 13, 12] There exists a decomposition

(52)
$f_{*}\omega_{X/S}=\mathcal{E}\bigoplus\mathcal{O}_{S}^{q_{f}},\qquad\text{(1nd
Fujita decomposition)}$

with $\mathcal{E}$ a vector bundle, and
$q_{f}=h^{1}(\mathcal{O}_{X})-g(S)=q(X)-g(S)$ is the relative irregularity of
$f$.

The embedding $\mathcal{O}_{S}^{q_{f}}\hookrightarrow\mathcal{U}$ splits as

(53) $\mathcal{U}=\mathcal{O}_{S}^{q_{f}}\bigoplus\mathcal{U}^{\prime}$

where $\mathcal{U}^{\prime}$ is also flat but has no sections. The
decomposition (52) is compatible with that in (51), that we have

(54)
$f_{*}\omega_{X/S}=\mathcal{A}\bigoplus\mathcal{U}^{\prime}\bigoplus\mathcal{O}_{S}^{q_{f}}$

The Gauss-Manin connection on $\mathcal{H}$ induces a connection

(55) $\nabla_{\mathcal{U}}:\mathcal{U}\to\mathcal{U}\otimes\Omega_{S}^{1}$

We denote the corresponding local system by $\mathbb{U}$. We have
$\mathcal{U}=\mathbb{U}\otimes\mathcal{O}_{S}$. The data of the local system
is equivalent to a unitary representation

(56) $\tau:\pi_{1}(S)\to U(q_{f},\mathbb{C})$

That is $\mathbb{U}$ corresponds to the global monodromy of $S$. Fujita
decompositions can also be stated over a higher dimensional base, [3]. It is
known by construction that $\mathcal{U}\subset\ker(\theta)$. It follows that

(57) $u_{f}:=\text{rank}(\mathcal{U})\leq h_{0}^{n,0}$

Assume $f:X\to S$ is a fibration of $n$-folds over a smooth curve $S$ and
$\mathcal{H}=R^{n}f_{*}\mathbb{C}=\bigoplus_{p+q=n}\mathcal{H}^{p,q}$ is a
variation of polarized complex Hodge structure of weight $n$ with unipotent
monodromy over $S$. Let

(58)
$\kappa^{n,0}:\mathcal{H}^{n,0}\stackrel{{\scriptstyle\theta^{n}}}{{\rightarrow}}\mathcal{H}^{n-1,1}\stackrel{{\scriptstyle\theta^{n-1}}}{{\rightarrow}}...\stackrel{{\scriptstyle\theta}}{{\rightarrow}}\mathcal{H}^{0,n}$

be the Griffiths-Yukawa coupling, i.e the composition of the $n$ successive
KS-maps. The Griffiths-Yukawa coupling is said to be maximal, if
$\mathcal{H}^{n,0}\neq 0$ and if $\kappa^{n,0}$ is an isomorphism. The
coupling is said to be strictly maximal if the same conditions hold but all
the map stages $\theta^{i}$ in (56) are isomorphisms, [cf. [44]]. Similar to
before we set
$\mathcal{H}_{0}^{n,0}=\ker\left(\theta^{p}:\mathcal{H}^{p,q}\to\mathcal{H}^{p-1,q+1}\otimes\Omega_{S}^{1}(\log
E)\right)$ and $h_{0}^{n,0}=\text{rank}\mathcal{H}_{0}^{n,0}$, with the same
notation as previous section.

######  Theorem 4.3 ([44] proposition 1.1).

We have the Arakelov inequality

(59) $\deg(\mathcal{H}^{n,0})\leq\frac{n}{2}\
.\text{rank}(\mathcal{H}^{n,0}).\deg(\Omega_{S}^{1}(\log E))$

with equality if the Higgs field of $\mathcal{H}$ is maximal. Moreover if
$\theta^{n}\neq 0$ then $\deg(\mathcal{H}^{n,0})>0,\ \deg(\Omega_{S}^{1}(\log
E))>0$. In particular if $S=\mathbb{P}^{1}$ then $\sharp E\geq 3$.

Assume $A^{n,0}$ be a subsheaf of $\mathcal{H}^{n,0}$ such that we have the
equality above when $\mathcal{H}^{n,0}$ is replaced by $A^{n,0}$. Then we have
a decomposition $\mathcal{H}=\mathcal{A}\bigoplus\mathcal{U}$ and subsheaves
$A^{p,n-p}$ of $\mathcal{H}^{p,n-p}$ such that the Higgs bundle of
$\mathcal{A}$ is $(\bigoplus A^{p,q},\bigoplus\tau^{p})$ and the Higgs field
$\tau$ is strictly maximal.

The Fujita decomposition $f_{*}\omega_{X/S}=\mathcal{A}\bigoplus\mathcal{U}$
into an ample sheaf $\mathcal{A}$ and a flat subsheaf $\mathcal{U}$, satisfies

(60) $\deg(\mathcal{H}^{n,0})=\deg(\mathcal{A})$

Moreover the Higgs field of $\mathcal{A}$ is strictly maximal and
$\mathcal{U}$ is a variation of polarized complex Hodge structure zero in the
bidegree $(n,0)$, [see [44] lemma 4, for instance].

##  5\. Family of Surfaces fibered by Curves

We consider a system of fibration of surfaces over families of curves as

(61)
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{f}$$\scriptstyle{\circlearrowleft}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{S}$

One recovers the Leray-Serre spectral sequence from above. Apply the
Grothendieck spectral sequence to the following spectral sequence

(62) $R^{p}g_{*}(R^{q}h_{*}\mathcal{F})\Rightarrow R^{n}f_{*}\mathcal{F}$

where $\mathcal{F}\in Coh(X)$. It is obviously convergent. One obtains

(63) $0\to R^{1}g_{*}\mathcal{O}_{Y}\to R^{1}f_{*}\mathcal{O}_{X}\to
g_{*}R^{1}h_{*}\mathcal{O}_{X}\to R^{2}g_{*}\mathcal{O}_{Y}\to
R^{2}f_{*}\mathcal{O}_{X}$

Applying $\otimes\Omega_{S}^{1}$ we get the following

(64) $0\to R^{1}g_{*}\mathcal{O}_{Y}\otimes\Omega_{S}^{1}\to
R^{1}f_{*}\mathcal{O}_{X}\otimes\Omega_{S}^{1}\to
g_{*}R^{1}h_{*}\mathcal{O}_{X}\otimes\Omega_{S}^{1}\\\ \to
R^{2}g_{*}\mathcal{O}_{Y}\otimes\Omega_{S}^{1}\to
R^{2}f_{*}\mathcal{O}_{X}\otimes\Omega_{S}^{1}$

This sequence fits with the long exact cohomology sequence of

(65) $0\to g_{*}\Omega_{Y/S}^{1}\to f_{*}\Omega_{X/S}^{1}\to
f_{*}\Omega_{X/Y}\to 0$

and gives

######  Proposition 5.1.

In the absence of singularities (degenerations) we have a commutative diagram

$\begin{CD}\Omega_{S}^{1}\Omega_{S}^{1}\Omega_{Y}^{1}\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\
g_{*}\Omega_{Y}^{1}f_{*}\Omega_{X}^{1}f_{*}\Omega_{X}^{1}\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\
0@>{}>{}>g_{*}\Omega_{Y/S}^{1}@>{}>{}>f_{*}\Omega_{X/S}^{1}@>{}>{}>f_{*}\omega_{X/Y}@>{}>{}>R^{1}g_{*}\Omega_{Y/S}^{1}\\\
@V{}V{\theta_{Y}}V@V{}V{\theta_{X}}V@V{}V{g_{*}{\theta_{X/Y}}}V\\\
0@>{}>{}>R^{1}g_{*}\mathcal{O}_{Y}\otimes\Omega_{S}^{1}@>{}>{}>R^{1}f_{*}\mathcal{O}_{X}\otimes\Omega_{S}^{1}@>{}>{}>g_{*}R^{1}h_{*}\mathcal{O}_{X}\otimes\Omega_{S}^{1}@>{}>{}>R^{2}g_{*}\mathcal{O}_{Y}\\\
@V{}V{}V@V{}V{}V@V{}V{}V\end{CD}$

###### Proof.

We have the short exact sequence downstairs because
$R^{2}g_{*}\mathcal{O}_{Y}=0$ (Kodaira vanishing). The fact that the
connecting homomorphism in the vertical directions is the Kodaira-Spencer map
is a classical known fact. The only thing that remains to be proved is that
the third term in the horizontal row upstairs is well defined and correct.
First note that in our case assuming that no singularity may appear in the
fibers of $X\to S$ nor $Y\to S$, it is not guaranteed that the fibers of
$X_{s}\to Y_{s}$ are all non singular. In general the exact sequence must be
written in the form

(66) $0\to g_{*}\Omega_{Y/S}^{1}\to f_{*}\Omega_{X/S}^{1}\to
f_{*}\Omega_{X/Y}^{1}(\log D)\to R^{1}g_{*}\Omega_{Y/S}^{1}\\\ $

where $D$ is normal crossing. However $\Omega_{X/Y}^{1}(\log D)$ is a line
bundle, i.e of rank one. Therefore

(67) $\displaystyle\Omega_{X/Y}^{1}(\log D)$
$\displaystyle=\det(\Omega_{X/Y}^{1}(\log D))$
$\displaystyle=\det(\Omega_{X}^{1}(\log
D))\otimes\det(h^{*}\Omega_{Y}^{1}(\log F))^{\vee}$
$\displaystyle=\omega_{X/Y}$

The stalks of the horizontal row upstairs are

(68) $H^{0}(Y_{s},\Omega_{Y_{s}}^{1})\to H^{0}(X_{s},\Omega_{X_{s}}^{1})\to
H^{0}(X_{s},\omega_{X/Y}|_{X_{s}})$

For the last term we have

(69)
$H^{0}(X_{s},\omega_{X/Y}|_{X_{s}})=H^{0}(Y_{s},h_{*}\omega_{X_{s}/Y_{s}})=H^{0}(W_{t},\Omega_{W_{t}}^{1})$

where $W_{t}$ are relative fibers in $X_{s}\to Y_{s}$. The proposition
follows. ∎

######  Theorem 5.2.

If we have degeneracies then the aforementioned digram modifies as

$\begin{CD}g_{*}\Omega_{Y}^{1}f_{*}\Omega_{X}^{1}f_{*}\Omega_{X}^{1}\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\ g_{*}\Omega_{Y/S}^{1}(\log
F)@>{}>{}>f_{*}\Omega_{X/S}^{1}(\log G)@>{}>{}>f_{*}\omega_{X/Y}(\log
D)@>{}>{}>\\\ @V{}V{\theta_{Y}}V@V{}V{\theta_{X}}V@V{}V{g_{*}\theta_{X/Y}}V\\\
R^{1}g_{*}\mathcal{O}_{Y}\otimes\Omega_{S}^{1}(\log
E)@>{}>{}>R^{1}f_{*}\mathcal{O}_{X}\otimes\Omega_{S}^{1}(\log
E)@>{}>{}>g_{*}R^{1}h_{*}\mathcal{O}_{X}\otimes\Omega_{S}^{1}(\log
E)@>{}>{}>\\\ @V{}V{}V@V{}V{}V@V{}V{}V\end{CD}$

where $D$ is a union of curves or points.

###### Proof.

We set the singular locus to be $E,F,G$ in $S,Y,X$ respectively and assume we
have sufficiently blown up that all being normal crossing and no
multiplicities. We shall assume all the monodromies being unipotent.

(70) $\begin{CD}G@>{}>{}>F@>{}>{}>E\\\ @V{}V{}V@V{}V{}V@V{}V{}V\\\
X@>{}>{}>Y@>{}>{}>S\end{CD}$

We have the following exact sequence of logarithmic sheaves

(71) $0\to g_{*}\Omega_{Y/S}^{1}(\log F)\to f_{*}\Omega_{X/S}^{1}(\log G)\to
f_{*}\Omega_{X/Y}^{1}(\log D)\to R^{1}g_{*}\Omega_{Y/S}^{1}(\log F)\\\ $

where

(72) $D=\bigcup_{t\in Y}W_{t}\qquad(h^{-1}(t)\ \text{is Singular})$

Lets for simplicity consider the case where no degeneracies appear in the
fibrations $X\to S$ and $Y\to S$. Working locally over $s\in S$, we can choose
coordinates as

$x=(v,u,s)\mapsto y=(u,s)\mapsto s$

in the fibrations. Then $p\in W_{t}$ is singular if and only if the matrix

(73) $Dh(p)=\left(\begin{array}[]{cccc}\partial h_{s}/\partial x(p)&\partial
h_{s}/\partial u(p)&\partial h_{s}/\partial s(p)\\\ 0&0&1\end{array}\right)$

drops rank, i.e. has rank $\leq 1$. Thus $D$ is closed in $X$. Also its image
in $Y$ is a closed proper subvariety, i.e. a union of curves and isolated
points. ∎

######  Lemma 5.3 (cf. [20] page 286).

There is a decomposition

(74)
$\mathcal{H}:=f_{*}\omega_{X/S}=\mathcal{H}_{\text{fix}}\bigoplus\mathcal{H}_{\text{var}}$

into fixed and variable parts where

(75) $\displaystyle\begin{aligned}
\mathcal{H}_{\text{var}}:&=\\{image(f_{*}\Omega_{Y/S}^{1}\to
f_{*}\omega_{X/S})\\}^{\perp}\\\
\mathcal{H}_{\text{fix}}:&=\\{image(f_{*}\Omega_{Y/S}^{1}\to
f_{*}\omega_{X/S})\\}\end{aligned}$

###### Proof.

We can illustrate the decomposition
$H^{1}(X_{s},\mathbb{C})=H^{1}(Y_{s},\mathbb{C})\oplus
H_{\text{var}}^{1}(X_{s},\mathbb{C})$ where

(76)
$H_{\text{var}}^{1}(X_{s},\mathbb{C})=\text{im}\\{H^{1}(X_{s},\mathbb{C})\to
H^{1}(X_{s},\mathbb{C})\\}^{\perp}$

For $s\in S$ the VHS of $H^{1}(X_{s},\mathbb{C})$ is a sub-HS of the one for
$H^{1}(X_{s},\mathbb{C})$. Then by the semisimplicity of the monodromy, it has
a complement that is also invariant by the monodromy action. Therefore we have

(77) $\displaystyle R^{1}f_{*}\mathbb{C}$
$\displaystyle=R^{1}g_{*}\mathbb{C}\oplus\left(R^{1}f_{*}\mathbb{C}\right)^{\text{var}}$
$\displaystyle R^{1}f_{*}\mathcal{O}_{X}$
$\displaystyle=R^{1}g_{*}\mathcal{O}_{Y}\oplus\left(R^{1}f_{*}\mathcal{O}_{X}\right)^{\text{var}}$

This proves the lemma. ∎

In surfaces fibered by curves one may consider the Fujita decompositions for
the 3 fibrations involved. If we have a fibration of surfaces fibered by
curves as in (8) then each of the 3 fibrations $h,\ g,\ f$ produce 3 Fujita
decompositions as in (54). Thus we have 3 decompositions

(78) $\displaystyle f_{*}\omega_{X/S}$
$\displaystyle=\mathcal{A}_{X}\bigoplus\mathcal{U}_{X}^{\prime}\bigoplus\mathcal{O}_{S}^{q_{f}},$
$\displaystyle f_{*}\omega_{Y/S}$
$\displaystyle=\mathcal{A}_{Y}\bigoplus\mathcal{U}_{Y}^{\prime}\bigoplus\mathcal{O}_{S}^{q_{g}},$
$\displaystyle f_{*}\omega_{X/Y}$
$\displaystyle=\mathcal{A}_{X/Y}\bigoplus\mathcal{U}_{X/Y}^{\prime}\bigoplus\mathcal{O}_{S}^{q_{h}}$

where we have used a version of Theorem 4.1 and 4.2 over a higher dimensional
base in the third identity, cf. [3]. We investigate the relation between
unitary sheaves $\mathcal{U}_{f}^{\prime},\ \mathcal{U}_{g}^{\prime},\
\mathcal{U}_{h}^{\prime}$ and their ranks. In the same way one can prove the
existence of a decomposition

(79) $f_{*}\omega_{X/S}=f_{*}\omega_{Y/S}\oplus
f_{*}\omega_{X/S}^{\text{var}}$

where

(80)
$f_{*}\omega_{X/S}^{\text{var}}=\\{image\left(f_{*}\omega_{Y/S}\longrightarrow
f_{*}\omega_{X/S}\right)\\}^{\perp}$

Assume for the moment we deduce that there is an image of the second row in
the first row. Because the component $A$ is a maximal ample subbundle
therefore we must have $\mathcal{A}_{Y}\subset\mathcal{A}_{X},\
\mathcal{U}_{Y}^{\prime}\subset\mathcal{U}_{X}^{\prime}$ and similar for the
last term. We also have the exact sequence

(81) $h^{*}\left(f_{*}\omega_{Y/S}\right)\to f_{*}\omega_{X/S}\to
f_{*}\omega_{X/Y}\to 0$

which states that the third identity is a quotient of the first one. Again
this criterion applies componentwise. It follows that

(82)
$\displaystyle{rank}(\mathcal{A}_{X})=\mathcal{A}_{Y}+\mathcal{A}_{X}^{var}$
$\displaystyle\leq{rank}(\mathcal{A}_{Y})+{rank}(\mathcal{A}_{X/Y})$
$\displaystyle{rank}(\mathcal{U}_{X})=\mathcal{U}_{Y}+\mathcal{U}_{X}^{var}$
$\displaystyle\leq{rank}(\mathcal{U}_{Y})+{rank}(\mathcal{U}_{X/Y})$
$\displaystyle q_{f}=q_{g}+q_{f}^{var}$ $\displaystyle\leq q_{g}+q_{h}$

That is $A_{X}^{var}\subset A_{X/Y},\
\mathcal{U}_{X}^{var}\subset\mathcal{U}_{X/Y}$.

######  Remark 5.4.

The proof of decomposition (79) is exactly the same as the lemma.

######  Remark 5.5.

The proof of the Lemma shows that
$g_{*}\left(\theta^{2}\mathcal{H}_{\text{fix}}\right)=0$.

######  Proposition 5.6.

We have

  * •

$H_{e}^{1,0}=H_{e}^{1,0}(1)\bigoplus H_{e}^{1,0}(2)_{\text{fix}}$.

  * •

$\mathcal{U}_{f}=\mathcal{U}_{g}\bigoplus\mathcal{U}_{h}^{\text{fix}}$.

  * •

$h^{1,0}=h(1)^{1,0}+h_{\text{fix}}^{1,0}(2)$.

  * •

$h_{0}^{1,0}=h_{0}^{1,0}(1)+h_{0,\text{fix}}^{1,0}(2)$.

  * •

$\delta_{0}=\delta_{0}^{1}+\left(\delta_{0}^{2}\right)_{\text{fix}}$.

  * •

$\nu(\theta)=\nu(\theta^{1})+\nu(g_{*}\theta^{2}\mid_{\text{fix}})$.

  * •

$\delta^{1,0}=\delta^{1}+\delta^{2}_{\text{fix}}=\delta^{1}+\delta^{2}-\delta^{2}_{\text{var}}$.

  * •

$u_{f}=u_{g}+u_{h}^{\text{fix}}=u_{g}+u_{h}-u_{h}^{\text{var}}$.

###### Proof.

By Proposition (5.1) and Theorem (5.2) we have the following commutative
diagram with short exact rows

(83) $\begin{CD}g_{*}\Omega_{Y/S}^{1}(\log
F)@>{}>{}>f_{*}\Omega_{X/S}^{1}(\log G)@>{}>{}>f_{*}\Omega_{X/Y}^{1}(\log
D)_{\text{fix}}\\\
@V{}V{\theta^{1}}V@V{}V{\theta}V@V{}V{g_{*}{\theta^{2}}}V\\\
R^{1}g_{*}\mathcal{O}_{Y}\otimes\Omega_{S}^{1}(\log
E)@>{}>{}>R^{1}f_{*}\mathcal{O}_{X}\otimes\Omega_{S}^{1}(\log
E)@>{}>{}>g_{*}(R^{1}h_{*}\mathcal{O}_{X})\otimes\Omega_{Y}^{1}(\log
E)_{\text{fix}}\\\ \end{CD}$

Which induces the following split short exact sequence on the graded part of
the local systems

(84)
$\begin{CD}H_{e}^{1,0}(1)@>{}>{}>H_{e}^{1,0}@>{}>{}>H_{e}^{1,0}(2)_{\text{fix}}\\\
@V{}V{\theta^{1}}V@V{}V{\theta}V@V{}V{\theta^{2}}V\\\
H_{e}^{0,1}(1)\otimes\Omega_{S}^{1}(\log
E)@>{}>{}>H_{e}^{0,1}\otimes\Omega_{S}^{1}(\log
E)@>{}>{}>H_{e}^{0,1}(2)_{\text{fix}}^{\vee}\otimes\Omega_{S}^{1}(\log
E)\end{CD}$

The diagram (84) is split exact for short exact sequences in rows. This proves
that the kernels and cokernels also split, from which the identities in the
lemma follows. ∎

######  Proposition 5.7 (see [20] page 287).

There is an induced map by the Gauss-Manin connections in the triangle

(85)
$\dfrac{g_{*}\left(R^{1}h_{*}\Omega_{X/S,var}^{0}\right)}{\theta^{2}_{\text{var}}\left(g_{*}(R^{0}h_{*}\Omega_{X/S,var}^{1})\right)}\
\stackrel{{\scriptstyle\overline{\theta_{\log}^{2,0}}}}{{\longrightarrow}}\
\dfrac{R^{2}f_{*}\Omega_{X/S}^{0}\otimes(\Omega_{S}^{1})^{\otimes
2}}{\theta^{1,1}\left(R^{1}f_{*}\Omega_{X/S}^{1}\otimes\Omega_{S}^{1}\right)}$

which is injective.

###### Proof.

The proof is a modification of the argument in [[20] page 287, for product
family]. However because we also need the details with other relations, we
repeat the part loc. cit.. We will use the local variables
$x=(v,u,s)\longmapsto y=(u,s)\mapsto s$ for the maps in (61). If we take a
small enough neighborhood of $s\in S$ such that the two fibrations $X$ and $Y$
trivialize over the open set, then locally we have

(86) $X_{s}=W_{s}\times F_{s},\qquad F_{s}\cong\text{disc}$

as topological spaces, where $W_{s}$ are fibers of $X\to Y$, (here is
different from [20], where there the fibered surfaces $X_{s}$ are self product
of $Y_{s}$). In this way

(87) $U_{s}=X_{s}\diagdown W_{s}\qquad\cong\text{ tubular neighborhood of}\
W_{s}\ \text{in}\ X_{s}$

We have a standard exact sequence

(88) $0\to H^{0}(W_{s})\to H^{2}(X_{s})\to
H^{2}(U_{s})\stackrel{{\scriptstyle\text{Res}}}{{\rightarrow}}H^{1}(W_{s})\to
H^{3}(X_{s})\to 0$

Then $\text{Res}H^{2}(U_{s})=H_{\text{var}}^{1}(X_{s})$. Moreover

(89) $\text{Res}F^{i}H^{2}(U_{s})=F^{i-1}H_{\text{var}}^{1}(X_{s})$

Thus

(90) $\displaystyle\begin{aligned}
\frac{F^{1}H^{2}(U_{s})}{F^{2}H^{2}(U_{s})}&=H^{1}(\Omega_{X_{s}}^{1}(\log
W_{s})\\\
\frac{F^{0}H^{2}(U_{s})}{F^{1}H^{2}(U_{s})}&=H^{2}(\mathcal{O}_{X_{s}})\end{aligned}$

When $s$ varies the whole sequence (85) varies to give VMHS’s. This shows [cf.
[20] the existence of a GM-connection

(91) $\nabla_{\log}:R^{1}f_{*}\Omega_{X/S}^{1}(\log
W_{s})\otimes\Omega_{S}^{1}\longrightarrow R^{2}f_{*}\Omega_{X/S}^{0}(\log
W_{s})\otimes(\Omega_{S}^{1})^{\otimes 2}$

We can illustrate all the GM-maps in the triangle by the following commutative
diagram with short exact columns,

(92)
$\begin{CD}R^{0}f_{*}\Omega_{X/S}^{2}@>{\nabla}>{}>R^{1}f_{*}\Omega_{X/S}^{1}\otimes\Omega_{S}^{1}@>{\nabla}>{}>R^{2}f_{*}\Omega_{X/S}^{0}\otimes(\Omega_{S}^{1})^{\otimes
2}\\\ @V{}V{}V@V{}V{}V@V{}V{}V\\\ R^{0}f_{*}\Omega_{X/S}^{2}(\log
W_{s})@>{{\nabla}_{\log}}>{}>R^{1}f_{*}\Omega_{X/S}^{1}(\log
W_{s})\otimes\Omega_{S}^{1}@>{{\nabla}_{\log}}>{}>R^{2}f_{*}\Omega_{X/S}^{0}(\log
W_{s})\otimes(\Omega_{S}^{1})^{\otimes 2}\\\
@V{\text{Res}_{h=y}}V{}V@V{\text{Res}_{h=y}}V{}V\\\
g_{*}R^{0}h_{*}\Omega_{X/S,var}^{1}@>{\nabla}>{}>g_{*}R^{1}h_{*}\Omega_{X/S,var}^{0}\end{CD}$

As it is explained in [20] from the data of the above diagram and
$\nabla_{\log}^{2}=0$, one can deduce the existence of the map

(93)
$\overline{\nabla_{\log}}:\dfrac{g_{*}\left(R^{1}h_{*}\Omega_{X/S,var}^{0}\right)}{\nabla_{X}\left(g_{*}R^{0}h_{*}\Omega_{X/S,var}^{1}\right)}\longrightarrow\dfrac{R^{2}f_{*}\Omega_{X/S}^{0}\otimes(\Omega_{S}^{1})^{\otimes
2}}{\nabla_{X}\left(R^{1}f_{*}\Omega_{X/S}^{1}\otimes\Omega_{S}^{1}\right)}$

From which we obtain the map (85). ∎

######  Remark 5.8.

$\nu(g_{*}\theta^{2}\mid_{\text{var}})\leq\nu(\theta)$ follows from the
(injectivity) of the map in (85).

######  Theorem 5.9.

we have the following inequality on the degrees of the Hodge bundles in a
family of surfaces fibred by curves

(94)
$\delta^{2,0}+2(\delta^{1,0}-(\delta^{1}+\delta^{2})\geq(h^{1,1}-h^{2,0})(2g-2)^{2}-h^{1,0}(2)(2g-2)$

###### Proof.

The map produced in (85) can be written as a short exact sequence

(95) $\displaystyle 0\to
g_{*}(R^{0}h_{*}\Omega_{X/S,var}^{1})\stackrel{{\scriptstyle{\theta}^{2}_{\text{var}}}}{{\longrightarrow}}$
$\displaystyle
g_{*}\left(R^{1}h_{*}\Omega_{X/S,var}^{0}\right)\otimes\Omega_{S}^{1}\
\stackrel{{\scriptstyle\overline{\theta_{\log}^{2,0}}}}{{\longrightarrow}}$
$\displaystyle\dfrac{R^{2}f_{*}\Omega_{X/S}^{0}\otimes(\Omega_{S}^{1})^{\otimes
2}}{\theta^{1,1}\left(R^{1}f_{*}\Omega_{X/S}^{1}\otimes\Omega_{S}^{1}\right)}\to
Coker(\overline{\theta_{\log}^{2,0}})\to 0$

Using the additivity of the degree function on the category of coherent
sheaves, we get

(96)
$\deg(g_{*}R^{0}h_{*}\Omega_{X/S,var}^{1})-\deg(g_{*}R^{1}h_{*}\Omega_{X/S,var}^{0}\otimes\Omega_{S}^{1})+\\\
\deg(R^{2}f_{*}\Omega_{X/S}^{0}\otimes(\Omega_{S}^{1})^{\otimes
2})-\deg(\theta^{1,1}R^{1}f_{*}\Omega_{X/S}^{1}\otimes\Omega_{S}^{1})-\deg\left(Coker(\overline{\theta_{\log}^{2,0}})\right)=0$

Then calculating the degrees using the degree formula of the product gives the
result (we are using the argument [19] page 505-506 on calculation of degree
in a general sequence).

(97)
$2\delta^{2}_{\text{var}}-(h^{1,0}(2))_{\text{var}}(2g-2)=\delta^{2,0}+(h^{2,0}-h^{1,1})(2g-2)^{2}+\sum_{s}\nu_{s}(\overline{\theta_{\log}^{2,0}})$

By substituting $\delta^{2}_{\text{var}}$ from proposition 5.6 we get

(98)
$\delta^{2,0}+2(\delta^{1,0}-(\delta^{1}+\delta^{2})=(h^{1,1}-h^{2,0})(2g-2)^{2}-h^{1,0}(2)_{\text{var}}(2g-2)+\sum_{s}\nu_{s}(\overline{\theta_{\log}^{2,0}})$

from which the inequality of the Theorem follows. ∎

######  Example 5.10.

[11, 19] We consider family of elliptic curves

(99) $y^{2}=4x^{3}-g_{2}(t)x-g_{3}(t),\qquad t\in\mathbb{P}^{1}$

where $g_{2}$ and $g_{3}$ are polynomials of degrees at most $4$ and $6$
respectively. Set $\Delta=g_{2}^{3}-27g_{3}^{2},\ \ J=g_{2}^{3}/\Delta$ where
$g_{2},\ g_{3},\ \Delta,\ J$ are Weierestrass coefficients, discriminant and
$J$-function. The Picard-Fuchs equation for the family is given by

(100) $\frac{d}{dt}\begin{bmatrix}\omega\\\
\eta\end{bmatrix}=\begin{pmatrix}\frac{-1}{12}\frac{d\log\delta}{dt}&\frac{3\delta}{2\Delta}\\\
\frac{-g_{2}\delta}{8\Delta}&\frac{1}{12}\frac{d\log\delta}{dt}\end{pmatrix}\begin{bmatrix}\omega\\\
\eta\end{bmatrix},\qquad\delta=3g_{3}g_{2}^{\prime}-2g_{2}g_{3}^{\prime}$

where $\omega=\int_{\gamma}\frac{dx}{y},\ \eta=\int_{\gamma}\frac{xdx}{y}$ and
$\gamma$ being a $1$-cycle, [11]. The equation (99) defines fibration

(101) $\pi:X\to\mathbb{P}^{1},\qquad X_{t}\mapsto t$

where the the local system $\mathcal{H}=R^{1}\pi_{*}\mathbb{C}$ has a two step
Hodge filtration $F^{0}=H^{1}(X_{t},\mathbb{C})\supset F^{1}$. In this case
the monodromy near $s_{i}$ can be written in terms of a canonical basis
$\alpha,\ \beta$ of $H_{1}(X_{t},\mathbb{C})$ as

(102) $\displaystyle T(\alpha)$ $\displaystyle=\alpha$ $\displaystyle
T(\beta)$ $\displaystyle=\beta+n_{i}\alpha$

A multi-valued section of $\mathcal{H}_{e}$ is

(103) $\phi(s)=\alpha^{*}+\left(n_{i}\frac{\log
s}{2\pi\sqrt{-1}}+\psi(s)\right)\beta^{*}$

where the action of the Gauss-Manin connection can be written as

(104)
$\nabla\phi(s)=\left(n_{i}\frac{ds}{2\pi\sqrt{-1}s}+\psi^{\prime}(s)ds\right)\beta^{*}$

therefore

(105) $N_{i}=Res_{s_{i}}\nabla=\begin{pmatrix}0&n_{i}\\\ 0&0\end{pmatrix}$

An easy calculation [cf. [19] loc cit.] gives

(106)
$\delta=\frac{1}{12}\sum_{i}n_{i}=0-1+\frac{1}{2}(N-\sum_{s_{i}}\nu_{s}(\theta))=\deg(J_{t})$

If we calculate $N$ in the above formula we get

(107) $N=2\delta+1+\sum_{i}\nu_{s_{i}}(\theta)\geq 3$

It follows that a non-trivial elliptic fibration over $\mathbb{P}^{1}$ has at
least three singular fibers. Now we go to a family of $K3$ surfaces defined by

(108) $y^{2}=4x^{3}-G_{2}(t,s)x-G_{3}(t,s),\qquad\pi^{2}:X_{(t,s)}\to(t,s)$

where $G_{2},\ G_{3}$ are polynomials of degree at most $8,\ 12$ in the affine
coordinate $s$ and such that they are also polynomials in $t$. The periods are
calculated via the integral $\int_{\gamma}ds\wedge\frac{dx}{y}$ and the local
systems $\mathcal{H}^{2}=R^{2}\pi_{*}^{2}\mathbb{C}$ has a weight $2$ Hodge
filtration $F^{0}=H^{2}(X_{(t,s)},\mathbb{C})\supset F^{1}\supset F^{2}$. In
this case still one has

(109) $\deg(f_{*}(w_{X/S}))=\deg J_{t,s}$

where $J$ is the $J$-function of the fibers, cf. [19]. We have
$\mathcal{H}_{var}=0,\ \mathcal{H}_{fix}=\mathcal{H}$. The formulas in
proposition 5.6 get simplified. In both of the fibrations the matrix of
$\bar{\theta}$ on the graded piece of middle cohomology is of the form (43).
We easily note that

(110) $\delta^{1,0}\neq 0\Rightarrow\delta^{2,0}\neq
0\Rightarrow\text{Singular fibers exist}$

By what was said, any family of elliptic curves parametrized by a complete
curve must have at least 3 singular fibers. An illustration for the second
family is

(111) $\begin{CD}E_{t}@>{}>{}>\mathcal{E}@>{}>{}>\mathbb{H}\\\
@V{}V{}V@V{}V{}V@V{}V{j(\tau)}V\\\
t@>{}>{}>\mathbb{P}^{1}@>{}>{j(t)}>\mathbb{P}_{j-line}^{1}\\\ \end{CD}$

One can apply the Riemann-Hurwitz ramification formula to obtain the
interpretation of $\delta$ in terms of remification numbers of the
fibration[see [19] for details]. One may proceed inductively to 3-dimensional
fibrations over surfaces, etc.

######  Remark 5.11.

A triangle fibration can also be studied in higher dimensional fibrations,
when suitable configuration is settled.

######  Remark 5.12.

F. Catanese [7, 27, 29] generalizes a theorem of Castelnuouvo-de Franchis for
surfaces, so that if $\dim X=n$ and there are one forms

(112) $\omega_{i}\in H^{0}(X,\Omega^{1}),\ i=1,2,...,k\qquad\text{such that}\
\omega_{1}\wedge\omega_{2}\wedge...\wedge\omega_{k}\neq 0,$

Then $X$ is fibered over a $k$-dimensional variety $Y$, The theorem can be
used to extract certain inequalities involving Hodge numbers of fibrations and
regularity, [see also [28, 14, 8]].

######  Theorem 5.13.

In a commutative triangle fibration of surfaces fibred by curves we have

(113)
$\delta_{X/S}^{2,0}\geq\frac{1}{2}(h_{X/Y}^{1,0}-h_{0,X/Y}^{1,0})(2g(S)-2)+h_{X/Y}^{1,0}$

Moreover we have $\delta_{X}^{2,0}\ \geq\ \delta_{X/Y}^{1,0}$ where the sub-
indices denote the corresponding fibration.

###### Proof.

In a fibration of the form (8) for a generic fiber we shall have $X_{s}\mapsto
Y_{s}\mapsto s\in S$, where $Y_{s}$ is a curve and $X_{s}$ is a surface. The
variety $X_{s}$ is also fibred over $Y_{s}$. Again a generic fiber over a
point $p_{s}\in Y_{s}$ is a curve namely $W_{p_{s}}$. Therefore apart from a
finite number of points of degeneracies on $S$

(114) $X_{s}=U_{s}\times W_{p_{s}},\qquad s\in T\subset S.$

where $U_{s}$ is open in $Y_{s}$. In both of the directions of $s$ and $Y_{s}$
there are finitely many singular fibers. By (114) the local system
$\mathcal{H}$ obtained from the middle cohomology of the fibration $X\to S$ is
given as

(115) $\mathcal{H}=\ R^{2}f_{*}\mathbb{C}=\ \mathcal{V}\ \otimes\
\mathcal{L},\qquad\mathcal{V}=g_{*}\left(R^{1}h_{*}\mathbb{C}\right)$

where $\mathcal{V}$ and $\mathcal{L}$ are $wt=1$ HS. It follows that

(116) $\mathcal{H}^{2,0}=\mathcal{V}^{1,0}\otimes\ \mathcal{L}^{1,0}$

Now we calculate the degrees of the associated Hodge bundles (denoted by the
same symbols)

(117) $\displaystyle\delta^{2,0}(\mathcal{H})$
$\displaystyle=\delta^{1,0}(\mathcal{V})rank(\mathcal{L}^{1,0})+h^{1,0}(\mathcal{V})\deg(\mathcal{L}^{1,0})$
$\displaystyle=\left(\frac{1}{2}(h_{X/Y}^{1,0}-h_{0,X/Y}^{1,0})(2g(S)-2)+\delta_{0,X/Y}-\frac{1}{2}\sum_{s\in
S}\nu_{s}(\overline{\theta_{X/Y}})\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times
rk(\mathcal{L}^{1,0})+h^{1,0}(\mathcal{V})\deg(\mathcal{L}^{1,0})$

from which (113) follows. Because $\deg(\mathcal{L})\geq 0$, it follows that
$\delta^{2,0}\geq\delta^{1,0}(\mathcal{V})$. ∎

######  Theorem 5.14.

If the local system $\mathcal{V}=g_{*}\left(R^{1}h_{*}\mathbb{C}\right)$ is
irreducible with regular singularities, then

(118)
$h^{p}(H^{1}(S,\mathcal{V}))=-\delta^{p}(V)-\left(h^{p}(V)-h^{p-1}(V)\right)(2g(S)-2)+\delta^{p-1}+\sum_{s}\nu_{s}(\bar{\theta})(V)$

where $V=\mathcal{V}\otimes\mathcal{O}_{S}$ and $h^{i}(V)=\dim Gr_{F}^{i}(V)$.

###### Proof.

Set $\mathcal{V}=g_{*}\left(R^{1}h_{*}\mathbb{C}\right)$ and
$V=\mathcal{V}\otimes\mathcal{O}_{S}$. Consider the resolution

(119)
${H}^{\bullet}:0\to\mathcal{V}\longrightarrow\mathcal{V}\otimes\mathcal{O}_{S}\stackrel{{\scriptstyle\nabla}}{{\longrightarrow}}\mathcal{V}\otimes\Omega_{S}^{1}$

Then by our assumption its only non-zero cohomology $H^{1}(S,\mathcal{V})$ is
a HS of $wt=2$,

(120)
$F^{p}(\mathcal{V}\otimes\Omega_{S}^{\bullet})=F^{p-\bullet}\mathcal{V}\otimes\Omega_{S}^{\bullet}$

We know that the spectral sequence

(121)
$\mathbb{H}^{r+s}(S,Gr_{F}^{r}(\Omega_{S}^{\bullet}\otimes\mathcal{V}))\Rightarrow\mathbb{H}^{r+s}(S,\mathcal{V})$

degenerates at $E_{1}$. Computing the Euler characteristics on $Gr_{F}^{p}$ of
(119), gives

(122) $\displaystyle h^{p}\left(H^{1}(S,\mathcal{V})\right)$
$\displaystyle=\chi(Gr_{F}^{p}\mathbb{H}^{\bullet}(S,V))=\chi(\mathbb{H}^{\bullet}(S,Gr_{F}^{p}V)),\qquad\text{degeneration
at}\ E_{1}$ $\displaystyle=-\chi(S,Gr_{F}^{p}V)+\chi(S,\Omega_{S}^{1}\otimes
Gr_{F}^{p-1}V)$

Applying the Riemann-Roch theorem we get

(123)
$h^{p}(H^{1}(S,\mathcal{V}))=-\delta^{p}(V)-h^{p}(V)(2g(S)-2)+\delta^{p-1}+h^{p-1}(V)(2g(S)-2)+\sum_{s}\nu_{s}(\bar{\theta})(V)$

where we have used
$\deg(gr_{F}^{p-1}V)=\delta^{p-1}(V)+\sum_{s}\nu_{s}(\bar{\theta})(V)$.

∎

######  Remark 5.15.

A version of Theorem 5.14 over $\mathbb{P}^{1}$ for tensor product of a
(single) VHS with a line bundle is proved in [37]. It is also possible to
formulate similar formulas for the fibrations $X/S,\ Y/S$ separately.

Declaration of interest: There is no declaration of interests by authors.

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