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: . 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
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
with
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
and
, 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
, see [
1].
Assume we have a fibration
f:
of smooth projective varieties with
S being of dimension 1. For each
k it defines a variation of Hodge structure
of weight
k over
S which is the Deligne extension of that over the smooth locus of the fibration. The graded sheaves
define a Higgs bundle. Following [
1], in a weight-one VHS
obtained from the fibration by curves, we have the following exact sequence of sheaves
where
is the extended Hodge bundle and
. We denote the rank of its generic fiber by
. The map
is induced by the Gauss–Manin connection and is called the Kodaira–Spencer map. A calculation of the degree
, (see [
1]) gives the following general formula
where
, and
. The terms
are nonzero only if
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
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:
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
where
is the bundle of holomorphic differential forms
, and
is an ample vector bundle,
is a flat unitary vector bundle [see
Section 4 for complete settings].
is also flat but has no sections. The Gauss-Manin connection on
induces a connection
. It is known by construction that
.
Assume the fibration
f:
has a weight
n, Hodge structure in the middle cohomology. Set
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
and if
is an isomorphism, (cf. [
2,
8]). Similar to before we set
, and
, with the same notation as previous section. We have the Arakelov inequality
with equality if the Higgs field of
is maximal. Moreover if
then
. In particular, if
then
. The Fujita decomposition
where
is an ample sheaf and
is a flat unitary subsheaf, satisfies
. Moreover, the Higgs field of
is strictly maximal, and
is a variation of polarized complex Hodge structure that is zero in the bidegree
, [see [
8] Lemma 4].
1.1. Problem Setup
Assume we have a VHS for surfaces
f:
fibered by curves, i.e., a commutative triangle fibration
We call such a commutative triangle, a family of surfaces in X fibered over a family of curves in Y. Let and be the associated local systems of of fibers in the triangle for and h, respectively. Then, to each of the VHS and , we can associate an Arakelov identity similar to (2). Here, the situation for the fibration h is more delicate because the base of h: 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 , and h are globally related. Moreover, how are the weight-one identities related to the weight-two Arakelov identity of f: ? 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 and of the three fibrations 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:
and
h:
. 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
,
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
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
into fixed and variable parts where
, and
is identified with the image. Then
. 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
, and
, 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
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,
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
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
appears as a middle convolution of the one for
, [
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
is irreducible with regular singularities, then
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].
- -
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:
is a fibration of smooth projective varieties over
. Hodge theory studies the variation of Hodge structure (VHS) constructed from the cohomologies of the smooth fibers
. The variation of Hodge structure (VHS) associated with the
k-th cohomology of the fibers forms a local system
of vector spaces over
. Its complexification has a decreasing Hodge filtration
and a decomposition
where
is a
-bundle called Hodge bundle and is defined over the smooth locus
. By the Hironaka resolution of singularities, one may assume the degeneracy locus
is a normal crossing divisor. When
, this locus is a finite number of points. The classical Hodge theory guarantees the existence of a canonical extension
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
is also a normal crossing. We may draw the picture of our fibration as follows:
Then the sheaves
are locally free, and admit a (logarithmic) connection
such that its residue is nilpotent, [
16,
18,
19,
20,
22,
23]. Moreover, there are isomorphisms
Among the invariants associated with VHS, s are the degrees of the Hodge bundles
The better the quantity
which measures how far the VHS is from trivial. Over the smooth locus, the Hodge bundles
are equipped with the Hodge metric induced from the polarization form. The invariants
may be calculated from the curvature of the associated metric connection. The curvature can be written as
where
are the Kodaira–Spencer maps and "
t" is the Hermitian adjoint. Here, we understand that
. The cohomology classes
are integrable closed (1,1)-forms and determine the Chern classes of the extended Hodge bundle;
.
In the weight-one VHS, as obtained from the fibration by curves, we have the following exact sequence of sheaves
where
. We denote the rank of its generic fiber by
. Please note that
is probably not a vector bundle; i.e., it may have torsion. It is also probable that the meromorphic
-matrix
drops rank at non-singular values
s. Because the degree map is additive on exact sequences, one can calculate the
in terms of the other terms. For the map on the stalks, we have
One considers a set of points where this inclusion is strict. This may also happen at nonsingular points. The calculation of the curvature gives
where
and
. Alternatively, (
19) can be written as
where
are the logarithms of the monodromies at the degeneracy point
and
are the induced map. The matrices
are defined via the matrix of the map
by
where
is the nonzero component of the matrix of
, [
1].
In the weight-two case, in the absence of degeneracy, we have the two short exact sequence
which are dual of one another. The Equation (
19) is replaced by the following:
where
and
. When there are degeneracies, the formula modifies as follows:
where we have set
and
are their Hodge numbers, cf. [
1] loc. cit.
Example 1 ([
1,
11]).
We consider family of elliptic curveswhere and are polynomials of degrees at most 4 and 6, respectively. Set where are Weierestrass coefficients, discriminant and J-function. The Picard-Fuchs equation for the family is given bywhere and γ being a 1-cycle, [11]. The Equation (25) defines a fibration where the local system has a two-step Hodge filtration . In this case, the monodromy near can be written in terms of a canonical basis of asA multi-valued section of is of the form Then, the action of the Gauss–Manin connection can be written astherefore . An easy calculation (cf. [1] loc cit.) gives If we calculate N in the above formula we obtain . It follows that a non-trivial elliptic fibration over has at least three singular fibers.
Let us consider a family of surfaces defined bywhere are polynomials of degree at most in the affine coordinate s and such that they are also polynomials in t. The periods are calculated via the integral and the local systems has a weight 2 Hodge filtration . In this case, one still has where J is the J-function of the fibers, cf. [1]. We have . In both fibrations, the matrix of on the graded piece of middle cohomology is of the form (21). We observe that 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 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 and there are one forms 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
One recovers the Leray–Serre spectral sequence from above. Apply the Grothendieck spectral sequence (Proposition A1) to the following spectral sequence
where
. It is convergent. One obtains
Applying
we obtain the following
This sequence fits with the long exact cohomology sequence of
and gives the following proposition.
Proposition 1. In the absence of singularities (degenerations), we have a commutative diagram.
Proof. We have the short exact sequence downstairs because
(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
and
does not guarantee that the fibers of
are all nonsingular. In general, the exact sequence must be written in the form
where
D is a normal crossing. However,
is a line bundle, i.e., of rank one. Therefore
The stalks of the horizontal row upstairs are
For the last term we have , where are relative fibers in . 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
where D is a union of curves or points. Proof. We set the singular locus to be
in
, 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
where
Let us, for simplicity, consider the case where no degeneracies appear in the fibrations
and
. Working locally over
, we can choose coordinates as
in the fibrations. Then
is singular if and only if the matrix
drops rank, i.e., has rank
. 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 decompositioninto fixed and variable parts where Proof. We can illustrate the decomposition
where
For
the VHS of
is a sub-HS of the one for
. Then, by the semisimplicity of the monodromy, it has a complement that is also invariant by the monodromy action. Therefore, we have
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
produces a Fujita decomposition, cf. Theorems A1 and A2 in
Appendix A. Thus, we have the three decompositions
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
and their ranks. One has the existence of a decomposition
where
. Because the component
A is a maximal ample subbundle therefore, we must have
and similar for the last terms in the decompositions (50). We have the exact sequence
which states that the third identity is a quotient of the first one. Again, this criterion applies componentwise. It follows that
That is .
Remark 2. The proof of decomposition (51) is the same as the Lemma 1.
Remark 3. The proof of the Lemma 1 shows that .
Using Theorem 1 and the Fujita decompositions (50), we can deduce various identities.
Proposition 2. We have
.
.
.
.
.
.
.
.
Proof. (Sketch) By Proposition 1 and Theorem 1 we have the following commutative diagram with short exact rows
The above diagram of sheaves induces the following split short exact diagram on the graded part of the local systems
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 trianglewhich is injective. Proof. The proof uses a modification of an argument in ([
9], p. 287, for product family). We will use the local variables
for the maps in (35). If we take a small enough neighborhood of
such that the two fibrations
X and
Y trivialize over, then locally we have
and
, as topological spaces, where
are fibers of
, (here is different from [
9], there the fibered surfaces
are self-product of
). Thus,
. We have the standard exact sequence
Then
and
. Thus,
When
s varies the whole sequence (57) varies to give VMHS’s. This shows (cf. [
9]) the existence of a GM-connection
We can illustrate all the GM-maps in the triangle by the following commutative diagram with short exact columns,
By [
9] from the data of the above diagram and
, one can deduce the existence of the map
from which we obtain the map (56). □
Remark 4. 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 curveswhere g is the genus of S. Proof. The map produced in (56) can be written as a short exact sequence
By the additivity of the degree function on the category of coherent sheaves, we obtain
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).
By substituting
from Proposition 2 we obtain
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 Moreover we have where the sub-indices denote the corresponding fibration.
Proof. We have
, where
is a curve and
is a surface. The variety
is also fibered over
. A generic fiber over a point
is a curve, namely
. Therefore, apart from a finite number of points of degeneracies on
S we have
where
is open in
. In both of the directions of
s and
there are finitely many singular fibers. The local system
obtained from the middle cohomology of the fibration
is given as
, where
and
are
HS. It follows that
Now we calculate the degrees of the associated Hodge bundles (denoted by the same symbols)
from which (
67) follows. Because
, it follows that
. □
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 is irreducible with regular singularities, thenwhere and . Proof. Set
and
. Consider the resolution
Then by our assumption its only nonzero cohomology
is a HS of
, and
. We know that the spectral sequence
degenerates at
. Computing the Euler characteristics on
of (71), gives
Applying the Riemann–Roch theorem, we obtain
where we have used
. □
Remark 6. A version of Theorem 4 over for the tensor product of a (single) VHS is proved in [10]. Formulating similar formulas for the fibrations separately is also possible.