Arakelov Inequalities for a Family of Surfaces Fibered by Curves

Abstract

The numerical invariants of a variation of Hodge structure over a smooth quasi-projective variety are a measure of complexity for the global twisting of the limit-mixed Hodge structure when it degenerates. These invariants appear in formulas that may have correction terms called Arakelov inequalities. We investigate numerical Arakelov-type equalities for a family of surfaces fibered by curves. Our method uses Arakelov identities in weight-one and weight-two variations of Hodge structure in a commutative triangle of two-step fibrations. Our results also involve the Fujita decomposition of Hodge bundles in these fibrations. We prove various identities and relationships between Hodge numbers and degrees of the Hodge bundles in a two-step fibration of surfaces by curves.

1. Introduction

In this paper, we consider identities involving numerical data associated with a geometric fibration of projective varieties and their Hodge structure. When families of algebraic manifolds come up in algebraic geometry, they usually have some singular fibers. We should distinguish between an abstract variation of Hodge structure and the case when the VHS arises from a geometric family f: $X\to S$. We refer to the latter as the geometric case, and we are particularly interested in understanding which results hold in the general (abstract) case or can only be established in the geometric case. We are interested in establishing identities involving Hodge numbers, degrees of the Hodge bundles, and any other algebraic geometric invariant associated with the fibration defined by f. These identities are considered in both cases: when there is a lack of singularities and when the singularities appear. This consideration is important in the analysis of them. They are useful in understanding or answering various geometric questions.

One of the important invariants of a variation of Hodge structure (VHS) that measure the complexity of their twist are the Chern classes of their Hodge bundles when they degenerate. A question in this line is how the degeneration of VHS affects these Chern classes. The numerical data of a geometric variation of Hodge structure also provide tools to study the iso-triviality of the family. A family $X\to S$ is isotrivial if it becomes a product over a Zariski open set of a branched covering of S. Arakelov inequalities also give bounds to determine global invariants associated with the singularities of the fibrations. For example, one can answer simple questions on fibrations in low dimensions, such as whether they must have singular fibers and even some bounds for the number of singular points. The Arakelov identities involve the degrees of the Hodge bundles in a VHS given by geometric fibration. The vanishing of the degrees of all the Hodge bundles gives a criterion to say whether the fibration is iso-trivial. These equalities have been systematically studied in low dimensions 1, 2, 3, and 4, cf. [1,2].

The Arakelov identities allow us to answer questions such as: Does a non-isotrivial VHS of weight n with specific Hodge numbers necessarily have degeneracies? For example, in the $n=1$ with $h^{1,0}=1$ case, it is not hard to deduce that a VHS without degeneracies is isotrivial. We call a variation of Hodge structure to be isotrivial if it becomes trivial on a finite branched covering of S. Equivalently, the global monodromy group is finite. Another example, in the case $n=3$ and $h^{3,0}=1$, of the type of VHS that arises from a family of Calabi–Yau threefolds, one may ask the same question. For instance, one can show that if there is a family of Calabi–Yau threefolds without singular fibers, then $h^{2,1}>h^{1,1}+12$, see [1].

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 $({\mathcal{H}}^{\left(k\right)}=R^k{f}_{*}\mathbb{C},F^p,\nabla )$ of weight k over S which is the Deligne extension of that over the smooth locus of the fibration. The graded sheaves $({\mathcal{H}}^{p,q}=F^p/F^{p-1},{\theta }_q:={\mathrm{Gr}}_F^p\nabla )$ define a Higgs bundle. Following [1], in a weight-one VHS ${\mathcal{H}}_{/S}$ obtained from the fibration by curves, we have the following exact sequence of sheaves

$$0\to {\mathcal{H}}_{0,e}^{1,0}\to {\mathcal{H}}_e^{1,0}\stackrel{\theta }{⟶}{\check{\mathcal{H}}}_e^{1,0}\otimes {\Omega }_S^1(logE)\to \mathcal{B}\to 0$$

(1)

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 $\delta =deg\left({\mathcal{H}}^{1,0}\right)$, (see [1]) gives the following general formula

$$\delta =\frac{1}{2}(h^{1,0}-h_0^{1,0})(2g\left(S\right)-2)+{\delta }_0-\frac{1}{2}\sum _{s\in S}{\nu }_s\left(\overline{\theta }\right)$$

(2)

where ${\delta }_0=deg\left({\mathcal{H}}_{0,e}^{1,0}\right)$, and ${\nu }_s\left(\overline{\theta }\right):=dim\mathrm{coker}\left\{{\mathcal{H}}_e^{1,0}/{\mathcal{H}}_{0,e}^{1,0}\stackrel{\overline{\theta }}{⟶}{({\mathcal{H}}_e^{1,0}/{\mathcal{H}}_{0,e}^{1,0})}^{\perp }\otimes {\Omega }_S^1\right\}$. The terms ${\nu }_s\left(\theta \right)$ are nonzero only if $s\in S$ is a critical value. Therefore, the last sum in (2) is a measure of degeneracies. For instance, forgetting the last term on the right side, one can say that $\delta$ is not bigger than the rest of the left side. In this form, we can also interpret (2) as an inequality. Similar identities can be obtained in higher weights by considering analogous short exact sequences (see Section 3 below). The above kinds 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 [1] for details.

Another approach to VHS and Arakelov identities is through the Fujita decomposition of the Hodge bundle, [2,3,4,5,6,7]. Let us sketch this. Assume we have the fibration f: $X\to S$ as before (see Section 4 for exact setup). Fujita decomposition is a splitting of the Hodge bundle of a VHS into an ample and a flat unitary subbundle. In general, the Fujita decomposition is of the form

$$f_{*}{\omega }_{X/S}=\mathcal{A}⨁{\mathcal{U}}^{\prime }⨁{\mathcal{O}}_S^{q_f},$$

(3)

where ${\omega }_{X/S}$ is the bundle of holomorphic differential forms ${\Omega }_X^{dimX}$, and $\mathcal{A}$ is an ample vector bundle, $\mathcal{U}={\mathcal{O}}_S^{q_f}⨁{\mathcal{U}}^{\prime }$ is a flat unitary vector bundle [see Section 4 for complete settings]. ${\mathcal{U}}^{\prime }$ is also flat but has no sections. The Gauss-Manin connection on ${\mathcal{H}}_e\otimes {\mathcal{O}}_S$ induces a connection ${\nabla }_{\mathcal{U}}:\mathcal{U}\to \mathcal{U}\otimes {\Omega }_S^1$. It is known by construction that $\mathcal{U}\subset ker\left(\theta \right)$.

Assume the fibration f: $X\to S$ has a weight n, Hodge structure in the middle cohomology. Set

$${\kappa }^{n,0}:{\mathcal{H}}^{n,0}\stackrel{{\theta }^n}{⟶}{\mathcal{H}}^{n-1,1}\stackrel{{\theta }^{n-1}}{⟶}\dots \stackrel{\theta }{⟶}{\mathcal{H}}^{0,n}$$

(4)

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 ${\mathcal{H}}^{n,0}\ne 0$ and if ${\kappa }^{n,0}$ is an isomorphism, (cf. [2,8]). Similar to before we set ${\mathcal{H}}_0^{p,q}=ker\left({\theta }^p:{\mathcal{H}}^{p,q}\to {\mathcal{H}}^{p-1,q+1}\otimes {\Omega }_S^1(logE)\right)$, and $h_0^{p,q}=\mathrm{rank}{\mathcal{H}}_0^{p,q}$, with the same notation as previous section. We have the Arakelov inequality

$$deg\left({\mathcal{H}}^{n,0}\right)\le \frac{n}{2}.\mathrm{rank}\left({\mathcal{H}}^{n,0}\right).deg\left({\Omega }_S^1(logE)\right)$$

(5)

with equality if the Higgs field of $\mathcal{H}$ is maximal. Moreover if ${\theta }^n\ne 0$ then $deg\left({\mathcal{H}}^{n,0}\right)>0,deg\left({\Omega }_S^1(logE)\right)>0$. In particular, if $S={\mathbb{P}}^1$ then $♯E\ge 3$. The Fujita decomposition $f_{*}{\omega }_{X/S}=\mathcal{A}⨁\mathcal{U}$ where $\mathcal{A}$ is an ample sheaf and $\mathcal{U}$ is a flat unitary subsheaf, satisfies $deg\left({\mathcal{H}}^{n,0}\right)=deg\left(\mathcal{A}\right)$. Moreover, the Higgs field of $\mathcal{A}$ is strictly maximal, and $\mathcal{U}$ is a variation of polarized complex Hodge structure that is zero in the bidegree $(n, 0)$, [see [8] Lemma 4].

1.1. Problem Setup

Assume we have a VHS for surfaces f: $X\to S$ fibered by curves, i.e., a commutative triangle fibration

(6)

We call such a commutative triangle, a family of surfaces in X fibered over a family of curves in Y. Let $\mathcal{H},\mathcal{H}\left(1\right)$ and $\mathcal{H}\left(2\right)$ be the associated local systems of $H^1$ of fibers in the triangle for $f,g$ and h, respectively. Then, to each of the VHS $\mathcal{H}$ and $\mathcal{H}\left(2\right)$, we can associate an Arakelov identity similar to (2). Here, the situation for the fibration h is more delicate because the base of h: $X\to Y$ is two-dimensional. The point is that the fibers of f are also twisted along the fiber of g. It is natural to expect that the three weight-one Arakelov identities (2) for the three fibrations $f,g$, and h are globally related. Moreover, how are the weight-one identities related to the weight-two Arakelov identity of f: $X\to S$? Thus, we pose the following problem.

Problem 1.

Describe the Arakelov identities of the three fibrations in (6) relevant to each other. How the weight-one Arakelov identities in the triangle (6) are connected to the weight-two Arakelov identity of f.

To analyze the Problem 1, one needs to study the Kodaira–Spencer maps $\theta ,{\theta }^1$ and ${\theta }^2$ of the three fibrations $f,g$ and h (resp.) simultaneously and in a commutative diagram of short exact sequences similar to (1).

1.2. Contributions

Toward a solution of Problem 1, our first result, namely Proposition 1 and Theorem 1, which compares the relative de Rham complexes for the three fibrations f: $X\to S,g:Y\to S$ and h: $X\to Y$. Proposition 1 is a simple form of Theorem 1 in the absence of degeneracies (singularities). In Theorem 1, we prove the commutativity of the following diagram of sheaves and Kodaira–Spencer maps $\theta , {\theta }^1, g_{*}{\theta }^1$,

$$\begin{array}{ccccc}{g}_{*}{\Omega }_{Y/S}^1(logF)& \to & f_{*}{\Omega }_{X/S}^1(logG)& \to & f_{*}{\Omega }_{X/Y}^1(logD)\\ \downarrow {\theta }^1& & \downarrow \theta & & \phantom{g_{*}{\theta }^2}\downarrow g_{*}{\theta }^2\\ R^1{g}_{*}{\mathcal{O}}_Y\otimes {\Omega }_S^1(logE)& \to & R^1{f}_{*}{\mathcal{O}}_X\otimes {\Omega }_S^1(logE)& \to & g_{*}\left(R^1{h}_{*}{\mathcal{O}}_X\right)\otimes {\Omega }_Y^1(logE)\end{array}$$

(7)

where log we mean logarithmic sheaves with appropriate normal crossing divisors (assumed to be compatible with the fibration), see Appendix B. We denote the suffix ${\theta }_Y,{\theta }_{X/Y}$ to distinguish the KS-maps. The exactness of the rows in (7) is by the Grothendieck 5-term sequence, cf. Appendix B. The vertical maps (sequences) are induced from the relative sheaves of differentials.

To be able to compare the KS-maps of the three fibrations, we prove an auxiliary result, namely Lemma 1, which is a modification of a similar argument in ([9], p. 286). We show, there is a decomposition $\mathcal{H}={\mathcal{H}}_{\mathrm{fix}}⨁{\mathcal{H}}_{\mathrm{var}}$ into fixed and variable parts where ${\mathcal{H}}_{\mathrm{var}}:={\left\{\mathrm{image}(f_{*}{\Omega }_{X/S}^1\to f_{*}{\Omega }_{X/Y}^1)\right\}}^{\perp }$, and ${\mathcal{H}}_{\mathrm{fix}}$ is identified with the image. Then $g_{*}\left({\theta }^2\left({\mathcal{H}}_{\mathrm{var}}\right)\right)=0$. We need this result because the fibers in X are also twisted over fibers in Y in the commutative triangle (35). The Lemma 1 together with Proposition 1 and Theorem 1 allow us to compare the Fujita decomposition for the Hodge bundles of VHS over Y and S. We compare the degrees $\delta , {\delta }^1$, and ${\delta }^2$, and prove various relationships between them, in Proposition 2. The identities appearing in Proposition 2 are mainly a consequence of the Fujita decomposition and the fact that the commutative diagram induced from (7) on the corresponding local systems is split exact.

A crucial task toward a solution of Problem 1 is to connect the Kodaira–Spencer maps of the VHS in different degrees for (35). Our main result toward this is the Proposition 3. Here, the Kodaira–Spencer maps are induced by the Gauss–Manin connections. The proof of Proposition 3 uses a local product structure of the fibrations and the commutative diagram, which is a modified analog of a commutative diagram in [9]. The proposition deduces the existence of an induced map

$$\overline{{\vartheta }^{2,0}}:{\displaystyle \frac{g_{*}\left(R^1{h}_{*}{\Omega }_{X/Y,\mathrm{var}}^0\right)}{{\theta }_{\mathrm{var}}^2\left(g_{*}\left(R^0{h}_{*}{\Omega }_{X/Y,\mathrm{var}}^1\right)\right)}}⟶{\displaystyle \frac{R^2{f}_{*}{\Omega }_{X/S}^0\otimes {\left({\Omega }_S^1\right)}^{\otimes 2}}{{\theta }^{1,1}\left(R^1{f}_{*}{\Omega }_{X/S}^1\otimes {\Omega }_S^1\right)}}$$

(8)

which is injective. We use the existence of this map to obtain an interesting Arakelov-type inequality in a two-step fibration.

The numerical relationships we obtain are as follows.

1.

The first Arakelov inequality for the commutative triangle (35) is the following,

$${\delta }^{2,0}+2({\delta }^{1,0}-({\delta }^1+{\delta }^2))\ge (h^{1,1}-h^{2,0}){(2g-2)}^2-h^{1,0}\left(2\right)(2g-2)$$

(9)

It appears as Theorem 2. The relationship (9) follows from the structure of the map (8) computing the degrees of the different components in an exact sequence. In the way to prove (9), we prove several structural results associated with the VHS obtained from the fibrations in the triangle (6). They appear as Proposition 1, Theorem 1, and Proposition 2, which in their own are interesting from an algebraic geometric point of view.

2.

The second Arakelov inequality is

$${\delta }_X^{2,0}\ge \frac{1}{2}(h_{X/Y}^{1,0}-h_{0,X/Y}^{1,0})(2g\left(S\right)-2)+h_{X/Y}^{1,0}$$

(10)

which appears in Theorem 3. The proof of (10) uses the context of middle convolution for VHS, [10]. Here, the VHS of the middle cohomology of the fibration $Y\to S$ appears as a middle convolution of the one for $X\to S$, [10]. The inequality (10) follows from a degree calculation in a middle convolution, followed by some standard Arakelov formulas.

3.

The third Arakelov relationship is a generalization of a result in [10] on a calculation of degrees in the left exact sequence of a Gauss–Manin connection followed by an application of the Riemann–Roch formula. In case we assume, the local system $\mathcal{V}=g_{*}\left(R^1{h}_{*}\mathbb{C}\right)$ is irreducible with regular singularities, then

$$\chi (S,\mathcal{V})={\delta }_{X/Y}^1+h^1\left(V\right)-{\delta }_{X/Y}^0+h^0\left(V\right)-\sum _s{\nu }_s\left({\overline{\theta }}_{X/Y}\right)\left(V\right).$$

(11)

The result in identity (11) appears in 4. The proof of the identity (11) is a result of mixing the techniques in this paper with those of [10].

All contributions appear in Section 3.

-

Related works:

An extensive explanation of the Arakelov inequalities can be found in [1]. In this case, we have used the results in [1] on numerical identities in algebraic geometry and Hodge theory and some Hodge theory techniques from [9]. In [1], the Arakelov inequalities have been studied in a single-step fibration in low weights. Our task in this work was to apply them in a double or two-step fibration. Some techniques of middle convolutions are presented in [10,11]. Other related works are [2,7,8,12,13].

-

Organization of the text:

Section 2 describes numerical invariants of variation of Hodge structures and closely follows [1]. The relevant references are [7,14,15,16,17,18,19,20,21,22,23,24,25]. In Section 3, we present all our contributions and calculations by the numerical invariants associated with the Problem 1. We have gathered all our main results in Section 3.

We have added two Appendices, namely Appendix A and Appendix B, for self-containment. Appendix A explains Fujita decomposition. The relevant references are [3,4,5,6,26,27,28,29]. Appendix B is classical, and we introduce the notation and concepts about the sheaf of relative differentials on algebraic varieties. References are [19,20,22,23,30,31].

2. Hodge–Arakelov Numerical Data

The reference for this section is [1,20]. 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 $X_s=f^{-1}\left(s\right)$. The variation of Hodge structure (VHS) associated with the k-th cohomology of the fibers forms a local system ${\mathcal{H}}_{/S^{*}}$ of vector spaces over $\mathbb{Q}$. Its complexification has a decreasing Hodge filtration ${\left(F^p\right)}_{0\le p\le k}$ and a decomposition ${\mathcal{H}}_{\mathbb{C}}={⨁}_{p+q=k}{\mathcal{H}}^{p,q}, {\mathcal{H}}^{p,q}:=F^p\cap \overline{F^q}$ where ${\mathcal{H}}^{p,q}\otimes {\mathcal{O}}_{S^{*}}$ is a $C^{\infty }$-bundle called Hodge bundle and is defined over the smooth locus $S^{*}\subset S$. By the Hironaka resolution of singularities, one may assume the degeneracy locus $E=S\setminus S^{*}$ is a normal crossing divisor. When $dimS=1$, this locus is a finite number of points. 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 $D=f^{-1}\left(E\right)$ is also a normal crossing. We may draw the picture of our fibration as follows:

$$\begin{array}{ccccc}D& \to & X& \leftarrow & X\setminus D\\ \downarrow & & \downarrow & & \downarrow \\ E& \to & S& \leftarrow & S\setminus E\end{array}$$

Then the sheaves $R^p{f}_{*}{\Omega }_{X/S}^{•}(logD)$ are locally free, and admit a (logarithmic) connection

$${\nabla }_e:R^p{f}_{*}{\Omega }_{X/S}^{•}(logD)\to {\Omega }_S^1(logE)\otimes R^p{f}_{*}{\Omega }_{X/S}^{•}(logD)$$

(12)

such that its residue is nilpotent, [16,18,19,20,22,23]. Moreover, there are isomorphisms

$$H^p(D,{\Omega }_{X/S}^{•}(logD))\otimes {\mathcal{O}}_D)\stackrel{\cong }{⟶}{H}^p(X_{\infty },\mathbb{C})$$

(13)

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

$${\delta }_p=deg(F_e^{n-p}/F_e^{n-p+1})$$

(14)

The better the quantity $\delta ={\delta }_0+{\delta }_1+...+{\delta }_p\ge 0$ which measures how far the VHS is 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

$${\Theta }_{{\mathcal{H}}^{p,q}}{=}^t{\theta }_q\wedge {\overline{\theta }}_q+{\overline{\theta }}_{q-1}{\wedge }^t{\theta }_{q-1}$$

(15)

where ${\theta }_q=G{r}_F^p\nabla$ are the Kodaira–Spencer maps and "^t" is the Hermitian adjoint. Here, we understand that $q=n-p$. The cohomology classes

$$\begin{array}{c}\hfill \begin{array}{c}\hfill c_1\left({\Theta }_{{\mathcal{H}}^{p,q}}\right)={\alpha }_p-{\alpha }_{p-1},\hspace{1em}{\int }_S{\alpha }_p=a_p\\ \hfill c_1({\Theta }_{{\mathcal{H}}_0^{p,q}})={\beta }_p-{\alpha }_{p-1},\hspace{1em}{\int }_S{\beta }_p=b_p\end{array}\end{array}$$

(16)

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

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

$$0\to {\mathcal{H}}_{0,e}^{1,0}\to {\mathcal{H}}_e^{1,0}\stackrel{{\theta }^{(1,0)}}{⟶}{\check{\mathcal{H}}}_e^{1,0}\otimes {\Omega }_S^1(logE)\to \mathrm{coker}\to 0$$

(17)

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 $h_0^{1,0}$. Please note that $\mathcal{B}$ is probably not a vector bundle; i.e., it 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 $\delta =deg\left({\mathcal{H}}^{p,q}\right)$ in terms of the other terms. For the map on the stalks, we have

$${\theta }_s\left({\mathcal{H}}_{e,s}^{1,0}\right)\subset {\left({\mathcal{H}}_{0,e,s}^{1,0}\right)}^{\perp }\otimes {\Omega }_{S,s}(logE)$$

(18)

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

$$\delta =\frac{1}{2}(h^{1,0}-h_0^{1,0})(2g-2)+{\delta }_0-\frac{1}{2}\sum _{s\in S}{\nu }_s\left(\overline{\theta }\right)$$

(19)

where ${\delta }_0=deg\left({\mathcal{H}}_{0,e}^{1,0}\right),{\nu }_s\left(\theta \right):=dim\left(\mathrm{coker}\left(\theta \right)\right)$ and $\overline{\theta }:{\mathcal{H}}_e^{1,0}/{\mathcal{H}}_{0,e}^{1,0}⟶{({\mathcal{H}}_e^{1,0}/{\mathcal{H}}_{0,e}^{1,0})}^{\vee }\otimes {\Omega }_S^1$. Alternatively, (19) can be written as

$$\begin{array}{cc}\hfill \delta & =\frac{1}{2}\left((h^{1,0}-h_0^{1,0})(2g-2)+\sum _{i}dim\left(\mathrm{image}{\overline{N}}_i\right)\right)-\hfill \\ \hfill & \left((-{\delta }_0)+\frac{1}{2}\left(\sum _{s\in S^{*}}{\nu }_s\left({\overline{\theta }}_s\right)+\sum _i{\nu }_{s_i}\left({\overline{A}}_i\right)\right)\right)\hfill \end{array}$$

(20)

where $N_i={\mathrm{Res}}_{s_i}\nabla$ are the logarithms of the monodromies at the degeneracy point $s_i$ and $\overline{N_i}:F_{e,s_i}^1/F_{0,e,s_i}^1⟶{\left(F_{e,s_i}^1/F_{0,e,s_i}^1\right)}^{\vee }$ are the induced map. The matrices $A_i$ are defined via the matrix of the map ${\overline{\theta }}_{s_i}$ by

$${\overline{\theta }}_{s_i}=\left(\begin{array}{ccc}\frac{B^{\prime }}{s}+...& *& *\\ 0& A^{\prime }\left(s\right)& *\\ 0& 0& 0\end{array}\right)$$

(21)

where $B^{\prime }$ is the nonzero component of the matrix of ${\theta }_{s_i}$, [1].

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

$$\begin{array}{c}\hfill \begin{array}{c}\hfill 0\to {\mathcal{H}}_{0,e}^{2,0}\to {\mathcal{H}}_e^{2,0}\stackrel{{\theta }^{2,0}}{⟶}{\mathcal{H}}_e^{1,1}\otimes {\Omega }_S^1\stackrel{{\theta }_1}{\to }{\check{\mathcal{H}}}_{0,e}^{1,1}\otimes {\Omega }_S^1\to 0\\ \hfill 0\to {\mathcal{H}}_{0,e}^{1,1}\to {\mathcal{H}}_e^{1,1}\stackrel{{\left({\theta }^{2,0}\right)}^{\vee }}{⟶}{\mathcal{H}}_e^{0,2}\otimes {\Omega }_S^1\to {\check{\mathcal{H}}}_e^{2,0}\otimes {\Omega }_S^1\to 0\end{array}\end{array}$$

(22)

which are dual of one another. The Equation (19) is replaced by the following:

$${\delta }^{2,0}=(h^{2,0}-h_0^{2,0})(2g-2)-\left(\delta \left(0\right)+\sum _{s\in S}{\nu }_s\left({\overline{\theta }}^{2,0}\right)\right)$$

(23)

where ${\overline{\theta }}^{2,0}:{\mathcal{H}}^{2,0}/{\mathcal{H}}_0^{2,0}⟶{\mathcal{H}}^{1,1}\otimes {\Omega }_S^1/ker{\theta }_1$ and $\delta \left(0\right)=a_0+b_0+b_1$. When there are degeneracies, the formula modifies as follows:

$${\delta }^{2,0}=(h^{2,0}-h_0^{2,0})(2g-2+N)+\left(\delta \left(0\right)+\sum _{s\in S}{\nu }_s(det\theta )+\sum _i({\widehat{h}}_i^{2,0}-h_i^{2,0})\right)$$

(24)

where we have set ${\widehat{H}}_i={\mathrm{Gr}}_q{\left(\mathrm{LMHS}\right)}_{s_i}$ and ${\widehat{h}}_i^{p,q}$ are their Hodge numbers, cf. [1] loc. cit.

Example 1

 ([1,11]). We consider family of elliptic curves

$$y^2=4x^3-g_2\left(t\right)x-g_3\left(t\right), t\in {\mathbb{P}}^1$$

(25)

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

$$\frac{d}{dt}\left[\begin{array}{c}\omega \\ \eta \end{array}\right]=\left(\begin{array}{cc}\frac{-1}{12}\frac{dlog\delta }{dt}& \frac{3\delta }{2\Delta }\\ \frac{-g_2\delta }{8\Delta }& \frac{1}{12}\frac{dlog\delta }{dt}\end{array}\right)\left[\begin{array}{c}\omega \\ \eta \end{array}\right], \delta =3g_3{g}_2^{\prime }-2g_2{g}_3^{\prime }$$

(26)

where $\omega ={\int }_{\gamma }\frac{dx}{y},\eta ={\int }_{\gamma }\frac{xdx}{y}$ and γ being a 1-cycle, [11]. The Equation (25) defines a fibration $\pi :X\to {\mathbb{P}}^1,X_t\mapsto t$ where 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

$$\begin{array}{cc}\hfill T\left(\alpha \right)& =\alpha \hfill \\ \hfill T\left(\beta \right)& =\beta +n_i\alpha \hfill \end{array}$$

(27)

A multi-valued section of ${\mathcal{H}}_e$ is of the form

$$\varphi \left(s\right)={\alpha }^{*}+\left(n_i\frac{logs}{2\pi \sqrt{-1}}+\psi \left(s\right)\right){\beta }^{*}$$

(28)

Then, the action of the Gauss–Manin connection can be written as

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

(29)

therefore $N_i={Res}_{s_i}\nabla =\left(\begin{array}{cc}0& n_i\\ 0& 0\end{array}\right)$. An easy calculation (cf. [1] loc cit.) gives

$$\delta =\frac{1}{12}\sum _i{n}_i=0-1+\frac{1}{2}(N-\sum _{s_i}{\nu }_s\left(\theta \right))=deg\left(J_t\right)$$

(30)

If we calculate N in the above formula we obtain $N=2\delta +1+{\sum }_i{\nu }_{s_i}\left(\theta \right)\ge 3$. It follows that a non-trivial elliptic fibration over ${\mathbb{P}}^1$ has at least three singular fibers.

Let us consider a family of $K3$ surfaces defined by

$$y^2=4x^3-G_2(t, s)x-G_3(t, s),\hspace{1em}{\pi }^2:X_{(t, s)}\to (t, s)$$

(31)

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, one still has $deg\left(f_{*}\left(w_{X/S}\right)\right)=deg{J}_{t,s}$ where J is the J-function of the fibers, cf. [1]. We have ${\mathcal{H}}_{\mathrm{var}}=0,{\mathcal{H}}_{\mathit{fix}}=\mathcal{H}$. In both fibrations, the matrix of $\overline{\theta }$ on the graded piece of middle cohomology is of the form (21). We observe that

$${\delta }^{1,0}\ne 0\Rightarrow {\delta }^{2,0}\ne 0\Rightarrow \mathit{Singular} \mathit{fibers} \mathit{exist}$$

(32)

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

$$\begin{array}{ccccc}{E}_t& \to & \mathcal{E}& \to & \mathbb{H}\\ \downarrow & & \downarrow & & \phantom{j\left(\tau \right)}\downarrow j\left(\tau \right)\\ t& \to & {\mathbb{P}}^1& \underset{j\left(t\right)}{\to }& {\mathbb{P}}_{j-\mathit{line}}^1\end{array}$$

(33)

One can apply the Riemann–Hurwitz ramification formula to obtain the interpretation of δ in terms of ramification numbers of the fibration [see [1] for details]. One may proceed inductively to 3-dimensional fibrations over surfaces, etc.

Remark 1.

F. Catanese [32,33,34] generalizes a theorem of Castelnuouvo–de Franchis for surfaces, so that; "if $dimX=n$ and there are one forms

$${\omega }_i\in H^0(X, {\Omega }^1), i=1, 2, \dots , k\hspace{1em}\mathit{such}\mathit{that} {\omega }_1\wedge {\omega }_2\wedge \dots \wedge {\omega }_k\ne 0,$$

(34)

Then X is fibered over a k-dimensional variety Y". This criterion can be used to extract certain inequalities involving Hodge numbers of fibrations and regularity (see also [35,36,37]).

3. Family of Surfaces Fibered by Curves–Main Results

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

(35)

One recovers the Leray–Serre spectral sequence from above. Apply the Grothendieck spectral sequence (Proposition A1) to the following spectral sequence

$$R^p{g}_{*}\left(R^q{h}_{*}\mathcal{F}\right)\Rightarrow R^n{f}_{*}\mathcal{F}$$

(36)

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

$$0\to R^1{g}_{*}{\mathcal{O}}_Y\to R^1{f}_{*}{\mathcal{O}}_X\to g_{*}{R}^1{h}_{*}{\mathcal{O}}_X\to R^2{g}_{*}{\mathcal{O}}_Y\to R^2{f}_{*}{\mathcal{O}}_X$$

(37)

Applying $\otimes {\Omega }_S^1$ we obtain the following

$$0\to R^1{g}_{*}{\mathcal{O}}_Y\otimes {\Omega }_S^1\to R^1{f}_{*}{\mathcal{O}}_X\otimes {\Omega }_S^1\to g_{*}{R}^1{h}_{*}{\mathcal{O}}_X\otimes {\Omega }_S^1\to R^2{g}_{*}{\mathcal{O}}_Y\otimes {\Omega }_S^1\to R^2{f}_{*}{\mathcal{O}}_X\otimes {\Omega }_S^1$$

(38)

This sequence fits with the long exact cohomology sequence of

$$0\to g_{*}{\Omega }_{Y/S}^1\to f_{*}{\Omega }_{X/S}^1\to f_{*}{\Omega }_{X/Y}\to 0$$

(39)

and gives the following proposition.

Proposition 1.

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

$\begin{array}{ccccccc} & & g_{*}{\Omega }_Y^1& & f_{*}{\Omega }_X^1& & f_{*}{\Omega }_X^1\\ & & \downarrow & & \downarrow & & \downarrow \\ 0& \to & g_{*}{\Omega }_{Y/S}^1& \to & f_{*}{\Omega }_{X/S}^1& \to & f_{*}{\omega }_{X/Y}& \to & R^1{g}_{*}{\Omega }_{Y/S}^1\\ & & \phantom{{\theta }_Y}\downarrow {\theta }_Y& & \phantom{{\theta }_X}\downarrow {\theta }_X& & \phantom{g_{*}{\theta }_{X/Y}}\downarrow g_{*}{\theta }_{X/Y}\\ 0& \to & R^1{g}_{*}{\mathcal{O}}_Y\otimes {\Omega }_S^1& \to & R^1{f}_{*}{\mathcal{O}}_X\otimes {\Omega }_S^1& \to & g_{*}{R}^1{h}_{*}{\mathcal{O}}_X\otimes {\Omega }_S^1& \to & R^2{g}_{*}{\mathcal{O}}_Y\\ & & \downarrow & & \downarrow & & \downarrow \end{array}$

Proof. 

We have the short exact sequence downstairs because $R^2{g}_{*}{\mathcal{O}}_Y=0$ (Kodaira vanishing). The connecting homomorphism in the vertical directions is the Kodaira–Spencer map. The only thing that remains to be proved is that the third term in the horizontal row upstairs is well-defined and correct. Please note that assuming no singularity exists in the fibers of $X\to S$ and $Y\to S$ does not guarantee that the fibers of $X_s\to Y_s$ are all nonsingular. In general, the exact sequence must be written in the form

$$0\to g_{*}{\Omega }_{Y/S}^1\to f_{*}{\Omega }_{X/S}^1\to f_{*}{\Omega }_{X/Y}^1(logD)\to R^1{g}_{*}{\Omega }_{Y/S}^1$$

(40)

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

$$\begin{array}{cc}\hfill {\Omega }_{X/Y}^1(logD)& =det\left({\Omega }_{X/Y}^1(logD)\right)\hfill \\ \hfill & =det\left({\Omega }_X^1(logD)\right)\otimes det{\left(h^{*}{\Omega }_Y^1(logF)\right)}^{\vee }\hfill \\ & ={\omega }_{X/Y}\hfill \end{array}$$

(41)

The stalks of the horizontal row upstairs are

$$H^0(Y_s, {\Omega }_{Y_s}^1)\to H^0(X_s, {\Omega }_{X_s}^1)\to H^0(X_s, {\omega }_{X/Y}{|}_{X_s})$$

(42)

For the last term we have $H^0(X_s, {\omega }_{X/Y}{|}_{X_s})=H^0(Y_s, h_{*}{\omega }_{X_s/Y_s})=H^0(W_t, {\Omega }_{W_t}^1)$, where $W_t$ are relative fibers in $X_s\to Y_s$. The proposition follows. □

When we have singular fibers we must modify Proposition 1 as follows.

Theorem 1.

If we have degeneracies, then the aforementioned diagram is modified as

$$\begin{array}{ccccc}{g}_{*}{\Omega }_Y^1& & f_{*}{\Omega }_X^1& & f_{*}{\Omega }_X^1\\ \downarrow & & \downarrow & & \downarrow \\ g_{*}{\Omega }_{Y/S}^1(logF)& \to & f_{*}{\Omega }_{X/S}^1(logG)& \to & f_{*}{\omega }_{X/Y}(logD)& \to \\ \phantom{{\theta }_Y}\downarrow {\theta }_Y& & \phantom{{\theta }_X}\downarrow {\theta }_X& & \phantom{g_{*}{\theta }_{X/Y}}\downarrow g_{*}{\theta }_{X/Y}\\ R^1{g}_{*}{\mathcal{O}}_Y\otimes {\Omega }_S^1(logE)& \to & R^1{f}_{*}{\mathcal{O}}_X\otimes {\Omega }_S^1(logE)& \to & g_{*}{R}^1{h}_{*}{\mathcal{O}}_X\otimes {\Omega }_S^1(logE)& \to \\ \downarrow & & \downarrow & & \downarrow \end{array}$$

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 are normal crossings with no multiplicities. We shall assume all the monodromies are unipotent. We have the following exact sequence of logarithmic sheaves

$$0\to g_{*}{\Omega }_{Y/S}^1(logF)\to f_{*}{\Omega }_{X/S}^1(logG)\to f_{*}{\Omega }_{X/Y}^1(logD)\to R^1{g}_{*}{\Omega }_{Y/S}^1(logF)$$

(43)

where

$$D=\bigcup _{t\in Y}{W}_t\left(h^{-1}\left(t\right)\mathrm{is}\mathrm{Singular}\right)$$

(44)

Let us, 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

$$Dh\left(p\right)=\left(\begin{array}{ccc}\partial h_s/\partial x\left(p\right)& \partial h_s/\partial u\left(p\right)& \partial h_s/\partial s\left(p\right)\\ 0& 0& 1\end{array}\right)$$

(45)

drops rank, i.e., has rank $\le 1$. 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. □

We need the following lemma as proof of our next results. The lemma is a modification of a similar argument in ([9], p. 286). Here, the difference is that the triangle fibration (35) is generally far from a self-product of a curve. For this, we have included a short proof.

Lemma 1.

There is a decomposition

$$\mathcal{H}:=f_{*}{\omega }_{X/S}={\mathcal{H}}_{\mathit{fix}}⨁{\mathcal{H}}_{\mathrm{var}}$$

(46)

into fixed and variable parts where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{H}}_{\mathit{var}}:& ={\left\{\mathit{image}(f_{*}{\Omega }_{Y/S}^1\to f_{*}{\omega }_{X/S})\right\}}^{\perp }\hfill \\ \hfill {\mathcal{H}}_{\mathit{fix}}:& =\left\{\mathit{image}(f_{*}{\Omega }_{Y/S}^1\to f_{*}{\omega }_{X/S})\right\}\hfill \end{array}\end{array}$$

(47)

Proof. 

We can illustrate the decomposition $H^1(X_s,\mathbb{C})=H^1(Y_s,\mathbb{C})\oplus H_{\mathrm{var}}^1(X_s,\mathbb{C})$ where

$$H_{\mathrm{var}}^1(X_s,\mathbb{C})=\mathrm{image}{\{H^1(X_s,\mathbb{C})\to H^1(X_s,\mathbb{C})\}}^{\perp }$$

(48)

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

$$\begin{array}{cc}\hfill R^1{f}_{*}\mathbb{C}& =R^1{g}_{*}\mathbb{C}\oplus {\left(R^1{f}_{*}\mathbb{C}\right)}^{\mathrm{var}}\hfill \\ \hfill R^1{f}_{*}{\mathcal{O}}_X& =R^1{g}_{*}{\mathcal{O}}_Y\oplus {\left(R^1{f}_{*}{\mathcal{O}}_X\right)}^{\mathrm{var}}\hfill \end{array}$$

(49)

This proves the lemma. □

We can apply Proposition 1 and Theorem 1 to the Fujita decompositions for the three fibrations involved in the triangle (35). Each of the three fibrations $h,g,f$ produces a Fujita decomposition, cf. Theorems A1 and A2 in Appendix A. Thus, we have the three decompositions

$$\begin{array}{cc}\hfill f_{*}{\omega }_{X/S}& ={\mathcal{A}}_X⨁{\mathcal{U}}_X^{\prime }⨁{\mathcal{O}}_S^{q_f},\hfill \\ \hfill f_{*}{\omega }_{Y/S}& ={\mathcal{A}}_Y⨁{\mathcal{U}}_Y^{\prime }⨁{\mathcal{O}}_S^{q_g},\hfill \\ \hfill f_{*}{\omega }_{X/Y}& ={\mathcal{A}}_{X/Y}⨁{\mathcal{U}}_{X/Y}^{\prime }⨁{\mathcal{O}}_S^{q_h}\hfill \end{array}$$

(50)

We have used a version of Theorems A1 and A2 over a higher-dimensional base in the third identity, cf. [5]. We investigate the relationship between unitary sheaves ${\mathcal{U}}_f^{\prime },{\mathcal{U}}_g^{\prime },{\mathcal{U}}_h^{\prime }$ and their ranks. One has the existence of a decomposition

$$f_{*}{\omega }_{X/S}=f_{*}{\omega }_{Y/S}\oplus f_{*}{\omega }_{X/S}^{\mathrm{var}}$$

(51)

where $f_{*}{\omega }_{X/S}^{\mathrm{var}}={\left\{\mathrm{image}\left(f_{*}{\omega }_{Y/S}⟶f_{*}{\omega }_{X/S}\right)\right\}}^{\perp }$. 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 terms in the decompositions (50). We have the exact sequence

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

(52)

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

$$\begin{array}{cc}\hfill rank\left({\mathcal{A}}_X\right)={\mathcal{A}}_Y+{\mathcal{A}}_X^{var}& \le rank\left({\mathcal{A}}_Y\right)+rank\left({\mathcal{A}}_{X/Y}\right)\hfill \\ \hfill rank\left({\mathcal{U}}_X\right)={\mathcal{U}}_Y+{\mathcal{U}}_X^{var}& \le rank\left({\mathcal{U}}_Y\right)+rank\left({\mathcal{U}}_{X/Y}\right)\hfill \\ \hfill q_f=q_g+q_f^{var}& \le q_g+q_h\hfill \end{array}$$

(53)

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

Remark 2.

The proof of decomposition (51) is the same as the Lemma 1.

Remark 3.

The proof of the Lemma 1 shows that $g_{*}\left({\theta }^2{\mathcal{H}}_{\mathit{fix}}\right)=0$.

Using Theorem 1 and the Fujita decompositions (50), we can deduce various identities.

Proposition 2.

We have

• $H_e^{1,0}=H_e^{1,0}\left(1\right)⨁H_e^{1,0}{\left(2\right)}_{\mathit{fix}}$.
• ${\mathcal{U}}_f={\mathcal{U}}_g⨁{\mathcal{U}}_h^{\mathit{fix}}$.
• $h^{1,0}=h{\left(1\right)}^{1,0}+h_{\mathit{fix}}^{1,0}\left(2\right)$.
• $h_0^{1,0}=h_0^{1,0}\left(1\right)+h_{0,\mathit{fix}}^{1,0}\left(2\right)$.
• ${\delta }_0={\delta }_0^1+{\left({\delta }_0^2\right)}_{\mathit{fix}}$.
• $\nu \left(\theta \right)=\nu \left({\theta }^1\right)+\nu \left(g_{*}{\theta }^2{\mid }_{\mathit{fix}}\right)$.
• ${\delta }^{1,0}={\delta }^1+{\delta }_{\mathit{fix}}^2={\delta }^1+{\delta }^2-{\delta }_{\mathit{var}}^2$.
• $u_f=u_g+u_h^{\mathit{fix}}=u_g+u_h-u_h^{\mathit{var}}$.

Proof. 

(Sketch) By Proposition 1 and Theorem 1 we have the following commutative diagram with short exact rows

$$\begin{array}{ccccc}{g}_{*}{\Omega }_{Y/S}^1(logF)& \to & f_{*}{\Omega }_{X/S}^1(logG)& \to & f_{*}{\Omega }_{X/Y}^1{(logD)}_{\mathrm{fix}}\\ \phantom{{\theta }^1}\downarrow {\theta }^1& & \phantom{\theta }\downarrow \theta & & \phantom{g_{*}{\theta }^2}\downarrow g_{*}{\theta }^2\\ R^1{g}_{*}{\mathcal{O}}_Y\otimes {\Omega }_S^1(logE)& \to & R^1{f}_{*}{\mathcal{O}}_X\otimes {\Omega }_S^1(logE)& \to & g_{*}\left(R^1{h}_{*}{\mathcal{O}}_X\right)\otimes {\Omega }_Y^1{(logE)}_{\mathrm{fix}}\end{array}$$

(54)

The above diagram of sheaves induces the following split short exact diagram on the graded part of the local systems

$$\begin{array}{ccccc}{H}_e^{1,0}\left(1\right)& \to & H_e^{1,0}& \to & H_e^{1,0}{\left(2\right)}_{\mathrm{fix}}\\ \phantom{{\theta }^1}\downarrow {\theta }^1& & \phantom{\theta }\downarrow \theta & & \phantom{{\theta }^2}\downarrow {\theta }^2\\ H_e^{0,1}\left(1\right)\otimes {\Omega }_S^1(logE)& \to & H_e^{0,1}\otimes {\Omega }_S^1(logE)& \to & H_e^{0,1}{\left(2\right)}_{\mathrm{fix}}^{\vee }\otimes {\Omega }_S^1(logE)\end{array}$$

(55)

The diagram (55) is split exactly for short exact sequences in rows. This proves that the kernels and cokernels also split, from which the identities in the lemma follow. □

In the next proposition, we connect the different KS-maps in the commutative diagram (35).

Proposition 3.

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

$${\displaystyle \frac{g_{*}\left(R^1{h}_{*}{\Omega }_{X/S,\mathit{var}}^0\right)}{{\theta }_{\mathit{var}}^2\left(g_{*}\left(R^0{h}_{*}{\Omega }_{X/S,\mathit{var}}^1\right)\right)}}\stackrel{\overline{{\theta }_{log}^{2,0}}}{⟶}{\displaystyle \frac{R^2{f}_{*}{\Omega }_{X/S}^0\otimes {\left({\Omega }_S^1\right)}^{\otimes 2}}{{\theta }^{1,1}\left(R^1{f}_{*}{\Omega }_{X/S}^1\otimes {\Omega }_S^1\right)}}$$

(56)

which is injective.

Proof. 

The proof uses a modification of an argument in ([9], p. 287, for product family). We will use the local variables $x=(v,u,s)⟼y=(u,s)\mapsto s$ for the maps in (35). If we take a small enough neighborhood of $s\in S$ such that the two fibrations X and Y trivialize over, then locally we have $X_s=W_s\times F_s,$ and $F_s\cong \mathrm{disc}$, as topological spaces, where $W_s$ are fibers of $X\to Y$, (here is different from [9], there the fibered surfaces $X_s$ are self-product of $Y_s$). Thus, $U_s=X_s\setminus W_s\cong \mathrm{tubular}\mathrm{neighborhood}\mathrm{of}{W}_s\mathrm{in}{X}_s$. We have the standard exact sequence

$$0\to H^0\left(W_s\right)\to H^2\left(X_s\right)\to H^2\left(U_s\right)\stackrel{\mathrm{Res}}{\to }{H}^1\left(W_s\right)\to H^3\left(X_s\right)\to 0$$

(57)

Then $\mathrm{Res}{H}^2\left(U_s\right)=H_{\mathrm{var}}^1\left(X_s\right),$ and $\mathrm{Res}{F}^i{H}^2\left(U_s\right)=F^{i-1}{H}_{\mathrm{var}}^1\left(X_s\right)$. Thus,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{F^1{H}^2\left(U_s\right)}{F^2{H}^2\left(U_s\right)}& =H^1({\Omega }_{X_s}^1(log{W}_s))\hfill \\ \hfill \frac{F^0{H}^2\left(U_s\right)}{F^1{H}^2\left(U_s\right)}& =H^2\left({\mathcal{O}}_{X_s}\right)\hfill \end{array}\end{array}$$

(58)

When s varies the whole sequence (57) varies to give VMHS’s. This shows (cf. [9]) the existence of a GM-connection

$${\nabla }_{log}:R^1{f}_{*}{\Omega }_{X/S}^1(log{W}_s)\otimes {\Omega }_S^1⟶R^2{f}_{*}{\Omega }_{X/S}^0(log{W}_s)\otimes {\left({\Omega }_S^1\right)}^{\otimes 2}$$

(59)

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

$$\begin{array}{ccccc}{R}^0{f}_{*}{\Omega }_{X/S}^2& \stackrel{\nabla }{\to }& R^1{f}_{*}{\Omega }_{X/S}^1\otimes {\Omega }_S^1& \stackrel{\nabla }{\to }& R^2{f}_{*}{\Omega }_{X/S}^0\otimes {\left({\Omega }_S^1\right)}^{\otimes 2}\\ \downarrow & & \downarrow & & \downarrow \\ R^0{f}_{*}{\Omega }_{X/S}^2(log{W}_s)& \stackrel{{\nabla }_{log}}{\to }& R^1{f}_{*}{\Omega }_{X/S}^1(log{W}_s)\otimes {\Omega }_S^1& \stackrel{{\nabla }_{log}}{\to }& R^2{f}_{*}{\Omega }_{X/S}^0(log{W}_s)\otimes {\left({\Omega }_S^1\right)}^{\otimes 2}\\ {\mathrm{Res}}_{h=y}\downarrow \phantom{{\mathrm{Res}}_{h=y}}& & {\mathrm{Res}}_{h=y}\downarrow \phantom{{\mathrm{Res}}_{h=y}}\\ g_{*}{R}^0{h}_{*}{\Omega }_{X/S,\mathrm{var}}^1& \stackrel{\nabla }{\to }& g_{*}{R}^1{h}_{*}{\Omega }_{X/S,\mathrm{var}}^0\end{array}$$

(60)

By [9] from the data of the above diagram and ${\nabla }_{log}^2=0$, one can deduce the existence of the map

$$\overline{{\nabla }_{log}}:{\displaystyle \frac{g_{*}\left(R^1{h}_{*}{\Omega }_{X/S,\mathrm{var}}^0\right)}{{\nabla }_X\left(g_{*}{R}^0{h}_{*}{\Omega }_{X/S,\mathrm{var}}^1\right)}}⟶{\displaystyle \frac{R^2{f}_{*}{\Omega }_{X/S}^0\otimes {\left({\Omega }_S^1\right)}^{\otimes 2}}{{\nabla }_X\left(R^1{f}_{*}{\Omega }_{X/S}^1\otimes {\Omega }_S^1\right)}}$$

(61)

from which we obtain the map (56). □

Remark 4.

$\nu \left(g_{*}{\theta }^2{\mid }_{\mathit{var}}\right)\le \nu \left(\theta \right)$ follows from the (injectivity) of the map in (56).

We can now express our first Arakelov-type inequality concerning the numerical invariants of the VHS in the commutative triangle (35).

Theorem 2.

We have the following inequality on the degrees of the Hodge bundles in a family of surfaces fibered by curves

$${\delta }^{2,0}+2({\delta }^{1,0}-({\delta }^1+{\delta }^2))\ge (h^{1,1}-h^{2,0}){(2g-2)}^2-h^{1,0}\left(2\right)(2g-2)$$

(62)

where g is the genus of S.

Proof. 

The map produced in (56) can be written as a short exact sequence

$$\begin{array}{cc}\hfill 0\to g_{*}\left(R^0{h}_{*}{\Omega }_{X/S,\mathrm{var}}^1\right)\stackrel{{\theta }_{\mathrm{var}}^2}{⟶}& g_{*}\left(R^1{h}_{*}{\Omega }_{X/S,\mathrm{var}}^0\right)\otimes {\Omega }_S^1\stackrel{\overline{{\theta }_{log}^{2,0}}}{⟶}\hfill \\ \hfill & {\displaystyle \frac{R^2{f}_{*}{\Omega }_{X/S}^0\otimes {\left({\Omega }_S^1\right)}^{\otimes 2}}{{\theta }^{1,1}\left(R^1{f}_{*}{\Omega }_{X/S}^1\otimes {\Omega }_S^1\right)}}\to \mathrm{coker}\left(\overline{{\theta }_{log}^{2,0}}\right)\to 0\hfill \end{array}$$

(63)

By the additivity of the degree function on the category of coherent sheaves, we obtain

$$\begin{array}{c}deg\left(g_{*}{R}^0{h}_{*}{\Omega }_{X/S,\mathrm{var}}^1\right)-deg(g_{*}{R}^1{h}_{*}{\Omega }_{X/S,\mathrm{var}}^0\otimes {\Omega }_S^1)+\hfill \\ \hfill deg(R^2{f}_{*}{\Omega }_{X/S}^0\otimes {\left({\Omega }_S^1\right)}^{\otimes 2})-deg({\theta }^{1,1}{R}^1{f}_{*}{\Omega }_{X/S}^1\otimes {\Omega }_S^1)-deg\left(\mathrm{coker}\left(\overline{{\theta }_{log}^{2,0}}\right)\right)=0\end{array}$$

(64)

Calculating the degrees using the degree formula of the product gives the result (we are using the argument of [1] pages 505–506 on a calculation of degrees in a general exact sequence).

$$2{\delta }_{\mathrm{var}}^2-{\left(h^{1,0}\left(2\right)\right)}_{\mathrm{var}}(2g-2)={\delta }^{2,0}+(h^{2,0}-h^{1,1}){(2g-2)}^2+\sum _s{\nu }_s\left(\overline{{\theta }_{log}^{2,0}}\right)$$

(65)

By substituting ${\delta }_{\mathrm{var}}^2$ from Proposition 2 we obtain

$${\delta }^{2,0}+2({\delta }^{1,0}-({\delta }^1+{\delta }^2))=(h^{1,1}-h^{2,0}){(2g-2)}^2-h^{1,0}{\left(2\right)}_{\mathrm{var}}(2g-2)+\sum _s{\nu }_s\left(\overline{{\theta }_{log}^{2,0}}\right)$$

(66)

from which the inequality of the Theorem follows. □

Remark 5.

A triangle fibration can also be studied in higher-dimensional fibrations when a suitable configuration is settled. This case can be considered to be a further and future study in this direction.

To obtain stronger identities, we have compared our construction with the results in [10]. The following is a sample result.

Theorem 3.

In a commutative triangle fibration of surfaces fibered by curves (35), we have

$${\delta }_{X/S}^{2,0}\ge \frac{1}{2}(h_{X/Y}^{1,0}-h_{0,X/Y}^{1,0})(2g\left(S\right)-2)$$

(67)

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

Proof. 

We 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 fibered over $Y_s$. 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 we have $X_s=U_s\times W_{p_s},$$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. The local system $\mathcal{H}$ obtained from the middle cohomology of the fibration $X\to S$ is given as $\mathcal{H}=R^2{f}_{*}\mathbb{C}=\mathcal{V}\otimes \mathcal{L},\mathcal{V}=g_{*}\left(R^1{h}_{*}\mathbb{C}\right)$, where $\mathcal{V}$ and $\mathcal{L}$ are $wt=1$ HS. It follows that

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

(68)

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

$$\begin{array}{cc}\hfill {\delta }^{2,0}\left(\mathcal{H}\right)& ={\delta }^{1,0}\left(\mathcal{V}\right)\mathrm{rank}\left({\mathcal{L}}^{1,0}\right)+h^{1,0}\left(\mathcal{V}\right)deg\left({\mathcal{L}}^{1,0}\right)\hfill \\ \hfill & =\left(\frac{1}{2}(h_{X/Y}^{1,0}-h_{0,X/Y}^{1,0})(2g\left(S\right)-2)+{\delta }_{0,X/Y}-\frac{1}{2}\sum _{s\in S}{\nu }_s\left(\overline{{\theta }_{X/Y}}\right)\right)\hfill \\ \hfill & \times \mathrm{rank}\left({\mathcal{L}}^{1,0}\right)+h^{1,0}\left(\mathcal{V}\right)deg\left({\mathcal{L}}^{1,0}\right)\hfill \end{array}$$

(69)

from which (67) follows. Because $deg\left(\mathcal{L}\right)\ge 0$, it follows that ${\delta }^{2,0}\ge {\delta }^{1,0}\left(\mathcal{V}\right)$. □

The operation of taking the tensor product of local systems in (68) is called the middle convolution. A systematic study of the behavior of numerical Hodge data in a middle convolution process is given in [10]. The following theorem generalizes a result in [10] (see Propositions 2.3.2 and 2.3.3 there).

Theorem 4.

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

$$h^p\left(H^1(S,\mathcal{V})\right)=-{\delta }^p\left(V\right)-\left(h^p\left(V\right)-h^{p-1}\left(V\right)\right)(2g\left(S\right)-2)+{\delta }^{p-1}+\sum _s{\nu }_s\left(\overline{\theta }\right)\left(V\right)$$

(70)

where $V=\mathcal{V}\otimes {\mathcal{O}}_S$ and $h^i\left(V\right)=dim{\mathrm{Gr}}_F^i\left(V\right)$.

Proof. 

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

$$H^{•}:0\to \mathcal{V}⟶\mathcal{V}\otimes {\mathcal{O}}_S\stackrel{\nabla }{⟶}\mathcal{V}\otimes {\Omega }_S^1$$

(71)

Then by our assumption its only nonzero cohomology $H^1(S,\mathcal{V})$ is a HS of $wt=2$, and $F^p(\mathcal{V}\otimes {\Omega }_S^{•})=F^{p-•}\mathcal{V}\otimes {\Omega }_S^{•}$. We know that the spectral sequence

$${\mathbb{H}}^{r+s}(S,{\mathrm{Gr}}_F^r({\Omega }_S^{•}\otimes \mathcal{V}))\Rightarrow {\mathbb{H}}^{r+s}(S,\mathcal{V})$$

(72)

degenerates at $E_1$. Computing the Euler characteristics on ${\mathrm{Gr}}_F^p$ of (71), gives

$$\begin{array}{cc}\hfill h^p\left(H^1(S,\mathcal{V})\right)& =\chi \left({\mathrm{Gr}}_F^p{\mathbb{H}}^{•}(S,V)\right)=\chi \left({\mathbb{H}}^{•}(S,{\mathrm{Gr}}_F^{p}V)\right),\mathrm{degeneration}\mathrm{at}{E}_1\hfill \\ \hfill & =-\chi (S,{\mathrm{Gr}}_F^{p}V)+\chi (S,{\Omega }_S^1\otimes {\mathrm{Gr}}_F^{p-1}V)\hfill \end{array}$$

(73)

Applying the Riemann–Roch theorem, we obtain

$$h^p\left(H^1(S,\mathcal{V})\right)=-{\delta }^p\left(V\right)-h^p\left(V\right)(2g\left(S\right)-2)+{\delta }^{p-1}+h^{p-1}\left(V\right)(2g\left(S\right)-2)+\sum _s{\nu }_s\left(\overline{\theta }\right)\left(V\right)$$

(74)

where we have used $deg\left({\mathrm{Gr}}_F^{p-1}V\right)={\delta }^{p-1}\left(V\right)+{\sum }_s{\nu }_s\left(\overline{\theta }\right)\left(V\right)$. □

Remark 6.

A version of Theorem 4 over ${\mathbb{P}}^1$ for the tensor product of a (single) VHS is proved in [10]. Formulating similar formulas for the fibrations $X/S,Y/S$ separately is also possible.

4. Conclusions

The analysis of the degrees of the Hodge bundles in a two-step fibration by curves allows us to obtain Arakelov-type inequalities for the VHS associated with a commutative triangle (35). This process is required to relate the three relative logarithmic de Rham complexes in such a triangle and investigate how the different KS-maps of these families are related. In this process, we used standard Arakelov identities that appeared in [1] for weight-one VHS associated with each of the three fibrations. Further results are obtained by applying the technique of middle convolution borrowed from [10].

The aforementioned investigation can be applied to families of algebraic varieties of higher dimensions fibered in a similar commutative triangle. However, one needs to consider Arakelov identities in higher-weight VHS. This question is a natural motivation for further studies in this direction and appears in possible future work.