# Torus bundles and -forms on the universal family of Riemann surfaces

###### Abstract.

We revisit three results due to Morita expressing certain natural integral cohomology classes on the universal family of Riemann surfaces , coming from the parallel symplectic form on the universal jacobian, in terms of the Euler class and the Miller-Morita-Mumford class . Our discussion will be on the level of the natural -forms representing the relevant cohomology classes, and involves a comparison with other natural -forms representing , induced by the Arakelov metric on the relative tangent bundle of over . A secondary object called occurs, which was discovered and studied by Kawazumi around 2008. We present alternative proofs of Kawazumi’s (unpublished) results on the second variation of on . Also we review some results that were previously obtained on the invariant , with a focus on its connection with Faltings’s delta-invariant and Hain-Reed’s beta-invariant.

###### Key words and phrases:

Ceresa cycle, Deligne pairing, harmonic volume, jacobian, Johnson homomorphism, moduli space of curves, torus bundle###### 2010 Mathematics Subject Classification:

Primary 32G15, secondary 14H15.###### Contents

- 1 Introduction
- 2 Compact Riemann surfaces and their jacobians
- 3 Torus bundles and sections
- 4 Arakelov volume form and the invariant
- 5 Arakelov Green’s function and Arakelov metric
- 6 Deligne pairing
- 7 Proof of identities (K1) and (K3)
- 8 Pointed harmonic volume
- 9 Proof of identity (K2)
- 10 Faltings delta- and Hain-Reed beta-invariant

## 1. Introduction

Let be an integer. Let denote the moduli space of compact Riemann surfaces of genus and denote by the universal family of Riemann surfaces over . The second integral cohomology group of contains two well known -linearly independent classes , , where denotes the Euler class of the relative tangent bundle of , and where (fiber integral) is the first Miller-Morita-Mumford class in .

Let be a reference surface of genus and put , viewed as an -representation via the intersection product on , which is unimodular. In the usual way the -representation then gives rise to a local system of free abelian groups of rank on , and by pullback along on the universal family . Write . Then the intersection product on naturally induces a non-degenerate bilinear alternating pairing .

From the local system one constructs the Jacobian torus bundle , together with its higher versions for . The intersection pairing induces canonical maps and hence canonical de Rham cohomology classes in , which we denote by . It turns out that the class is not integral, but the class is. In a series of papers [29] [30] [32] S. Morita has studied the pullbacks of the integral classes along certain natural sections of over (and variants), and in particular expressed them as integral linear combinations of tautological classes.

In order to state Morita’s results, let for each denote the -fold fiber product of over . A first natural section that one considers is the section of the universal jacobian bundle given by the map . Here denotes the jacobian of the compact Riemann surface and is a canonical divisor on . A second natural section is the well known pointed harmonic volume studied by B. Harris [19] and M. Pulte [33] (cf. Section 8 below). Finally, one has the difference map given by . It is convenient to view as the universal Abel-Jacobi map sending a point on a pointed Riemann surface to its Abel-Jacobi class in .

With the above notation, Morita has obtained the following three identities [29, Theorem 2.1], [30, Theorem 1.7], [32, Theorem 5.1] in integral cohomology.

###### Theorem 1.1.

(Morita) Let denote the projections onto the first and second coordinate, respectively, and let be the diagonal on . Then the following relations:

hold in resp. .

We note that the formulation in Morita’s papers is slightly different: there, the relations (M1)–(M3) are presented as identities for characteristic classes of oriented surface bundles. In particular, one may view these relations as part of the topological study of oriented surfaces, and indeed, the methods employed in [29] [30] [32] are distinctively topological. For example, the pointed harmonic volume realizes the first extended Johnson homomorphism [31] on the mapping class group of pointed oriented surfaces.

In the present paper, as opposed to Morita’s [29] [30] [32], we would like to take a more analytical point of view. We start out by observing that each of the classes is represented by a canonical parallel -form, which we denote by . The form on for example is precisely the symplectic form associated to the intersection product. It follows that the left hand sides of Morita’s identities (M1)–(M3) are represented by canonical -forms , and .

Combining (M1) and (M2) we find that the -form

is a representative of the Euler class . It seems natural to compare the -form with other natural -forms on representing . The suggestion of N. Kawazumi [22] [24] is to consider the Arakelov metric on (cf. Section 5 below). Introduced by S.Y. Arakelov in [1] this is a second natural object on intimately related to the intersection form on and hence to the symplectic form on the jacobian bundle .

Denote by the first Chern form of the Arakelov metric on . In the paper [24] Kawazumi determined the difference between the forms and . Note that as this difference represents the zero class in cohomology, by general results it can be represented as the of a function on , which is pulled back from and uniquely defined up to an additive constant. The question is then: what ‘is’ this function?

Let be a compact Riemann surface of genus . In [22] [24] Kawazumi defines

(1.1) |

where is the Green operator with respect to the Arakelov volume form , and where is an orthonormal basis of the space of holomorphic -forms on (cf. Section 4). The invariant turns out to be a conformal invariant of , which is in fact positive real-valued. One obtains a natural function on the moduli space of Riemann surfaces, and Kawazumi proved in [24] that this function yields precisely the secondary object relating and .

###### Theorem 1.2.

As a variant on the form , consider the -form

(1.2) |

By (M1) and (M2) the form represents the tautological class . Let . Then we have the following variant on Theorem 1.2, also proved in [24].

###### Theorem 1.3.

Unfortunately, the paper [24] has not been published; both results above are stated in [22]. One aim of the present paper is to give another proof of Theorems 1.2 and 1.3. In fact, we will prove slightly more.

Recall that denotes the diagonal on . We let denote the first Chern form of the hermitian line bundle on , where the metric on is given by the Arakelov’s Green’s function (cf. Section 5). We have . Putting

one obtains a natural -form on representing the class .

Our main result is then the following.

###### Theorem 1.4.

Let denote the projections onto the first and second coordinate, respectively, and let be the diagonal on . Let be Kawazumi’s invariant (1.1). Then we have the equalities

of -forms on resp. .

Note that Theorem 1.2 follows as an immediate consequence of the equalities (K1) and (K2). We will give a short derivation of Theorem 1.3 from Theorem 1.4 at the end of Section 9.

The approach in [24] is based on the harmonic Magnus expansion on the universal family of Riemann surfaces. This map defines a flat connection on a vector bundle on the total space of minus the zero-section. The holonomy of this connection gives all the Johnson homomorphisms on the mapping class group . In particular, the first term of the connection form coincides with the first variation of the pointed harmonic volume [23, Section 8].

Our approach is rather different and more geometric in nature. An important ingredient in our proof of Theorem 1.4 is the use of the Deligne pairing for families of Riemann surfaces, a refined version of the Gysin map in cohomology (cf. Section 6 below). Using the Deligne pairing our proofs of especially (K1) and (K3) are remarkably straightforward. For the proof of (K2) we will use, besides the Deligne pairing, the connection discovered by Hain and Pulte [33] between the pointed harmonic volume and Ceresa’s cycle [6] in the jacobian of a pointed compact Riemann surface .

The proof we give of Theorem 1.4 is independent of Morita’s results (M1)–(M3) and in particular gives a new argument for them, by taking classes in cohomology. Another approach is given in the paper [14] by R. Hain and D. Reed. Because of its fundamental relationship with the cohomology of , we think that the function merits to be further studied. For example, in his paper [38] S. Zhang independently introduced the invariant and showed the relation of its strict positivity with some important conjectures in arithmetic geometry. In the papers [25], [26], [27], [28] the present author found a number of further properties of , including its asymptotics towards the boundary of in the Deligne-Mumford compactification, and closed expressions in the case of a hyperelliptic Riemann surface. We will review these results briefly in our final Section 10.

The starting point of that section is formed by a 1984 paper by G. Faltings [10] and a 2004 paper by R. Hain and D. Reed [13] where one finds a discussion of two other natural invariants in connected with the geometry of the universal family of (higher) jacobians. In both papers, the determinant of the Hodge bundle with its -metric plays a dominant role. In his paper [24], Kawazumi posed the question of determining the relationship of his invariant with the previously defined invariants by Faltings and by Hain-Reed. We will show in Section 10 that one can “eliminate” the role of the Hodge bundle, and arrive at a linear dependence relation

in . By combining the asymptotics towards the boundary of of from our paper [27] with previously established asymptotics of by J. Jorgenson [20] and R. Wentworth [37], we thus obtain asymptotics of . This will furnish an alternative proof of the main result of Hain-Reed’s paper [13].

*Conventions.—* For a complex manifold, a hermitian line bundle on consists of a holomorphic line bundle on together with a hermitian norm on each fiber of , varying in a manner over . More precisely, if is a local generating section of , then the functions are . The first Chern form of the hermitian line bundle is the real -form on given locally by the expression , where is a local generator of . Note that the -form is independent of the chosen local generator, and represents the first Chern class of in the cohomology group . We use the notation for the space of real valued differential -forms on . If is an analytic subvariety, we denote by the Dirac current associated to . The current is closed and positive. When are complex manifolds, we say that a surjective map is a family if is a proper holomorphic submersion.

If is a finite dimensional complex vector space, we shall view as a subspace of via the embedding . For each we have a canonical direct sum decomposition of vector spaces. This gives rise to a natural inclusion which we shall denote by .

*Acknowledgments.—* The author thanks Nariya Kawazumi for his many remarks and suggestions.

## 2. Compact Riemann surfaces and their jacobians

Let be a compact and connected Riemann surface of genus and write . Let and write . The Hodge -operator on the real cotangent bundle gives rise to a Hodge decomposition

(2.1) |

The complex vector space can be identified with the set of holomorphic -forms on , the vector space with the set of anti-holomorphic -forms on , and the total space with the set of harmonic -forms on . Write . The intersection pairing induces a non-degenerate bilinear alternating pairing on , which can be written explicitly as

for harmonic -forms on .

The jacobian of is defined to be the real torus . Let be an integer. We write for the canonically induced bilinear alternating map ; explicitly is given by

for . Note that we have . Let for be the “higher” intermediate jacobians. Then for each we define to be the canonical translation-invariant -form corresponding to .

Let be a symplectic basis of , that is

Let be the dual basis of . It is useful to have an expression for in terms of this basis.

###### Proposition 2.1.

We have the identity

in . Here we put for .

###### Proof.

This is a straightforward verification. ∎

Next, let be the natural map given by sending as in Section 1. Note that uniformizes the selfproduct . We can thus naturally interpret as an element of . Written out explicitly, by Proposition 2.1 we have

for . In particular, the restriction of to the diagonal gives back .

Note that we have a Hodge filtration

dual to the one in (2.1). The inclusion induces a natural isomorphism

(2.2) |

of real tori. The projection onto induces an isomorphism . Hence by the natural isomorphism (2.2) the jacobian of can be naturally viewed as a complex torus, whose tangent space at the origin is naturally identified with . More precisely we have as complex tori, where is embedded in by associating to the class of a -cycle on the linear functional given by on .

A similar description can be carried through for the higher jacobians . Using the Hodge filtration on one has a natural identification of with the complex torus , where is the space of harmonic -forms on . Here the homology group is embedded into by associating to the class of a -cycle in the functional given by on .

To finish this section we recall a construction due to P. Griffiths [11] that, restricted to our situation, associates to a homologically trivial cycle of dimension in a canonical class in the -th higher jacobian . First we write the cycle as the boundary of a -cycle in . Then we associate to the class in given by the functional on . Note that is well-defined: replacing by another -cycle such that , the functional changes by , which has zero class in , as yields an element of .

Griffiths’s construction generalizes the classical Abel-Jacobi map (where ) which assigns to a degree-zero divisor on a compact Riemann surface the class of the functional in , where is a -cycle in with . Of particular interest for us will be the class in associated by Griffiths’s construction to the so-called Ceresa cycle [6] on the pointed Riemann surface . This cycle is given as follows: let be the copy of embedded in given by all Abel-Jacobi images of degree-zero divisors on , where runs through , and let be the image of under the inversion on . Then the Ceresa cycle of is defined to be the cycle in . Since inversion acts by on , we see that the Ceresa cycle is homologically trivial in .

## 3. Torus bundles and sections

Let be a complex manifold, and let be a real torus bundle. Let be the associated local system of first homology groups. That is, the fiber of at is the free abelian group . Write , and assume that a zero section of has been given. Then there is a natural identification .

Let be a bilinear alternating pairing. As for each the vector space is identified with the relative tangent space to at the origin, the form naturally gives rise to a parallel -form on . It is clear that the pullback of along the zero section of is trivial, and that the restriction of to any of the fibers of is translation-invariant.

Let be a section of . Then associated to we have its first variation . It can be easily seen that is closed and that its cohomology class gives rise to an element of [12, Section 4.1]. Further, the pairing gives rise to a contraction map . A straightforward computation yields the identity

(3.1) |

of -forms on .

Now let be a family of compact Riemann surfaces of genus , with associated local system of first homology groups on . Then is a variation of Hodge structure of weight on with Hodge filtration

By our remarks in Section 2 the jacobian bundle can be naturally viewed as a bundle of complex tori.

Let be the nondegenerate bilinear alternating pairing derived from the intersection form, and for each write for the canonically induced bilinear alternating pairing . Let for be the higher jacobian bundles over . Then we define for to be the parallel -form corresponding to .

Let be a section. Then by equation (3.1) we can write

(3.2) |

in . In particular we can write our three main -forms of interest as:

Here, by a slight abuse of notation we view as a section of the universal family of jacobians over . Working locally over , let be a symplectic frame of , that is

Let be the dual frame of . Then Proposition 2.1 immediately translates into the following.

###### Proposition 3.1.

Locally over , the equality

holds in . Here we put for .

Note that in particular can be interpreted as an element of . Let be the natural map induced by sending as in Section 1. As the local system uniformizes the selfproduct of jacobian bundles we can naturally interpret each as an element of . Written out explicitly by Proposition 3.1 we have

for , locally over . In particular, the restriction of to the diagonal gives the -form .

To finish this section, we discuss a useful differential property of the Griffiths Abel-Jacobi construction in the family of complex tori . Let and let be a topological cycle in whose restriction to each fiber of is homologically trivial of dimension . Note that for each there exists an open neighborhood of in and a topological cycle in such that . Hence, applying fiberwise the Griffiths Abel-Jacobi construction from Section 2 we obtain a natural section associated to . The following proposition describes the canonical element of induced by and as a fiber integral.

###### Proposition 3.2.

Let be a topological cycle in whose restriction to each fiber of is homologically trivial of dimension . Let be the section obtained by fiberwise applying Griffiths’s Abel-Jacobi construction to . Let be the projection map. Then the equalities

hold in .

###### Proof.

The first equality is just equation (3.2). To see the second equality, let be any element of . Note that we can interpret as a parallel -form on . By contraction we find an element . Working locally over , let be a cycle on such that . An application of Stokes’s theorem then yields that

in , as is closed. It follows that for we have

in . The proposition follows. ∎

## 4. Arakelov volume form and the invariant

Let be an integer. Let be a compact and connected Riemann surface of genus . The purpose of this section is to define the Arakelov volume form and the conformal invariant of introduced by Kawazumi. Our exposition here is based on the discussion in [24, Section 1].

As before let and let be the complex vector space of harmonic -forms on with its Hodge decomposition

Let be an orthonormal basis of , that is,

Then

Let be the map sending a linear form to the harmonic -form that corresponds to it. It is convenient to see this map as an -valued real harmonic -form . Letting be the dual basis of

Regarding the as holomorphic -forms on , this naturally leads to considering the -form

on , where we note that . By the Riemann-Roch theorem the canonical linear system has no base points, and hence the -form is a volume form on . We call the Arakelov volume form.

Let for be the space of complex valued -currents on . In order to introduce the invariant of , we recall the Green operator with respect to the Arakelov volume form . It is defined uniquely by the following two properties:

(4.1) |

for all . If is a smooth form, then is a smooth function. Moreover, we have the symmetry relation

(4.2) |

for all .

###### Definition 4.1.

We have the following positivity result for .

###### Proposition 4.2.

(cf. [24, Corollary 1.2]) We have , and if and only if .

###### Proof.

Let . Then we have

and hence . Applying this to for we obtain the first part of the proposition. Assume that . Then both and vanish and hence is a constant. From (4.1) we infer that is a scalar multiple of . We find that if , then each is a scalar multiple of . It follows that . Vice versa we have that vanishes if . ∎

We mention that the strict positivity for for has also been obtained by S. Zhang in [38, Proposition 2.5.3]. He uses this result in [38] to give a surprising application in arithmetic geometry: for a smooth, projective and geometrically connected curve of genus defined over a number field, the truth of an arithmetic version of a standard conjecture of Hodge index type due to Grothendieck implies the Bogomolov conjecture.

## 5. Arakelov Green’s function and Arakelov metric

In this section we introduce the Arakelov-Green’s function on a compact Riemann surface of genus . References for this section are Arakelov’s original paper [1, Sections 3-4] and [37, Section 2].

For denote by the Dirac current supported at . Let denote the Green operator with respect to the Arakelov volume form . The Arakelov Green’s function is then defined to be the function on given by

The square of is a -function on , vanishing only at . From (4.1) we obtain that the Arakelov Green’s function satisfies the conditions

(5.1) |

and in fact these conditions uniquely determine as an element of . Further, applying (4.2) to we obtain

(5.2) |

for all .

Let be the diagonal on . We use to put a hermitian metric on , as follows. Let denote the canonical meromorphic section of . Then we demand that . As the normal bundle to the diagonal equals the tangent bundle of , we obtain a natural metric on . We call this the Arakelov metric. We put

It follows that the identity of currents

holds on . In [1, Proposition 3.1] one finds the explicit formula

(5.3) |

for the -form . In particular we find that

(5.4) |

The following proposition expresses explicitly in terms of and .

###### Proposition 5.1.

The formula

holds.

###### Proof.

By (