On hermitian-holomorphic classes related to uniformization, the dilogarithm, and the Liouville Action

Ettore Aldrovandi Department of Mathematics Florida State University Tallahassee, FL 32306-4510, USA

                          aldrovandi@math.fsu.edu

Abstract

Metrics of constant negative curvature on a compact Riemann surface are critical points of the Liouville action functional, which in recent constructions is rigorously defined as a class in a Čech-de Rham complex with respect to a suitable covering of the surface.

We show that this class is the square of the metrized holomorphic tangent bundle in hermitian-holomorphic Deligne cohomology. We achieve this by introducing a different version of the hermitian-holomorphic Deligne complex which is nevertheless quasi-isomorphic to the one introduced by Brylinski in his construction of Quillen line bundles. We reprove the relation with the determinant of cohomology construction.

Furthermore, if we specialize the covering to the one provided by a Kleinian uniformization (thereby allowing possibly disconnected surfaces) the same class can be reinterpreted as the transgression of the regulator class expressed by the Bloch-Wigner dilogarithm.

Contents

1 Introduction  2
1.1 Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Notation and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Deligne complexes  5
2.1 Cup products on cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Deligne complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Remarks on the dilogarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Constructions in hermitian holomorphic Deligne cohomology  9
3.1 Hermitian holomorphic Deligne cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Hermitian holomorphic line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Cup product of hermitian holomorphic line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Conformal metrics and the Liouville functional  14
4.1 Variational problem for conformal metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Direct construction of the Liouville cocycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.3 A cup product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.4 Two line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5 The Liouville functional and the Bloch-Wigner dilogarithm  19
5.1 Kleinian groups and fractional linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.2 Computation of the cocycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.3 Cross ratio, tetrahedra and the Block-Wigner function . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6 Conformal metrics and hermitian holomorphic cohomology  22
6.1 Comparison on a curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.2 Determinant of cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
A Cones  26
A.1 Cones and homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
A.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1 Introduction

Metrics of constant negative curvature play a very important role in uniformization problems for compact Riemann surfaces of genus g > 1. The condition that the scalar curvature associated to a conformal metric on a Riemann surface X be equal to −1 is equivalent to the fact that the associated conformal factor satisfies a nonlinear partial differential equation known as the Liouville equation.

The Liouville equation appears as early as in one of the approaches considered by Poincaré to attack the uniformization theorem [27]. In relatively recent times, it has received considerable attention in Theoretical and Mathematical Physics due to the key role it plays in Polyakov’s approach to String Theory [28], especially from the point of view of non-critical strings and two-dimensional quantum gravity. In this context one refers to the conformal factor of the metric as the Liouville “field.”

As usual in the context of differential equations with a physical motivation, one normally would like to formulate a variational principle to express the Liouville equation as an extremum condition. Namely, given a Riemann surface X and the space 𝒞 ℳ (X) of all conformal metrics on it, the metric of constant negative curvature should be a critical point of a functional defined over 𝒞 ℳ (X). This functional is the Liouville action. As it happens, action functionals may turn out to be even more relevant than the equations they are associated to. The Liouville action is no exception in this sense: it has deep connection with the geometry of Teichmüller spaces [34, 35], and in Physics it describes the conformal anomaly in String Theory.

Providing a rigorous mathematical definition of the Liouville action functional is however far from trivial. The very geometric properties of the Liouville field itself prevent expressing the corresponding functional as a plain integral of a 2-form on a Riemann surface. Correction terms are required, typically in the form of integration of lower degree forms over the 1-skeleton of an appropriate simplicial realization of X. (One should notice that this behavior is not specific to the Liouville equation, and it is by now possible to give a characterization, in terms of homological algebra, of these type of functionals, see ref. [1].)

It is possible to directly determine the necessary correction terms by requiring that the variational problem be well defined. This, however, is not completely satisfactory from the point of view of certain applications to deformation theory, where a consistent definition across a family of surfaces is required. Quite recently, a more systematic construction, based on the homological algebra techniques developed by the author and L. A. Takhtajan in [2], was carried out by L. A. Takhtajan and L.-P. Teo in ref. [31], generalizing the earlier results of [34, 35]. The authors of ref. [31] constructed a Čech cocycle with respect to the étale cover of X associated to a quasi-Fuchsian (and more generally Kleinian) uniformization. Since their construction works across (Kleinian) deformations, it could be exploited to obtain results of global nature on the analytic geometry of Kleinian deformation spaces. As a further result, the authors of loc. cit. were able to rigorously prove the validity of the “holography principle” for the Liouville action corresponding to a large class of Kleinian (in particular Fuchsian and quasi-Fuchsian) uniformizations. Specifically, they proved that given a second kind Kleinian group, the corresponding Liouville action can be obtained as the regularized limit of the hyperbolic volume of the corresponding associated 3-manifold. This extends to the general Kleinian case a previous formula obtained by Krasnov [24] for classical Schottky groups.

Our interest in this matter is two-fold. From the perspective of the newer methods adopted in [3], the covering map U → X associated to the uniformization by a discrete group Γ = Deck(U/X) is but one of the many possible covers comprising an appropriate category C of, say, local diffeomorphisms U → X — the most obvious choice being that of standard open cover 𝒰 = {U_{i}} with associated space U = ∐_{i} U_{i}. In particular one expects to be able to apply the methods of [2] and [3] uniformly on a class of reasonably behaved covers of X.

Second, the focus of ref. [3] was on the rigorous definition of a functional for quasi-conformal deformation of the Riemann surface X and its application to the study of projective structures. A main result is that the construction of the action is possible thanks to the vanishing of the “tame symbol” (see refs. [13] and [9] for the relevant definitions) (T_{X}, T_{X}], where T_{X} is the holomorphic tangent line bundle of X. The vanishing determines local choices (with respect to a cover) of a Bloch-type dilogarithm which then allow for a cohomological construction of the action. There are many indication that the Liouville action ought to be the hermitian square of a functional of the type studied in [3].^{1} Thus it is natural to ask whether there is an analogous mechanism as the one in loc. cit. to obtain a general construction of the Liouville action by replacing the holomorphic symbol maps and dilogarithms with corresponding real objects.

In this paper we answer this question in the affirmative. More precisely, we show that the Liouville action (up to the area term which is given by an ordinary 2-form) can be computed as a symbol map taking values in hermitian holomorphic Deligne cohomology, first introduced by Brylinski and McLaughlin in their study of degree four characteristic classes [9]. (By way of comparison, the tame symbols used in ref. [3] used holomorphic and smooth Deligne cohomology.) We will present a modified version of hermitian holomorphic cohomology that better suits our needs than the original definition—the resulting cohomology groups are however the same. In particular we show that the dilogarithm type terms are replaced here by the Bloch-Wigner function, the real valued counterpart of the dilogarithm (see refs. [7], and [23, 16] for a review.)

The appearance of the Bloch-Wigner function ties very well with the holography property of the Liouville function proved in [31] in the following sense. As mentioned before, the Liouville action (up to the area term) relative to a Kleinian uniformization^{2} U → X can also be computed as the “regularized volume” of the associated 3-manifold N = Γ\(U ∪ ℍ^{3}), where Γ = Deck(U/X) as before, U ⊂ ℙ^{1} is the domain of discontinuity for Γ, and ℍ^{3} is the standard hyperbolic 3-space. (To define the regularized volume would lead us too far afield. It suffices to mention that the conformal factor of a metric on X = ∂N can be used to select a compact submanifold N_{ϵ} whose volume is finite. One then subtracts from the volume of N_{ϵ} the areas of the boundary components and other carefully chosen constants independent of the metric structure, so that the resulting quantity will have a finite limit as ϵ → 0.) On the other hand, the hyperbolic volume in three dimensions corresponds to a three dimensional (purely imaginary) class on PSL_{2}(ℂ) expressible through the Bloch-Wigner dilogarithm, the so-called regulator class. We show that the regulator is precisely the obstruction to solving the cohomological descent conditions required to calculate the Liouville action for a covering map U → X with covering group a Kleinian group Γ. Indeed, since for a second kind Kleinian group the quotient ℍ^{3}/Γ — being non-compact — carries no cohomology in dimension three, the class represented by the Bloch-Wigner function, pulled back to Γ via the imbedding Γ ↪ PSL_{2}(ℂ), vanishes.

Returning to the cohomological interpretation of the construction of the Liouville action, it should be also noted that leaving aside the area term, our results show that the cohomologically non trivial part is indeed a square. Namely, for a conformal metric ρ ∈ 𝒞 ℳ (X) we consider the pair (T_{X}, ρ) as a holomorphic line bundle equipped with an hermitian metric. Then, using that hermitian holomorphic Deligne cohomology has a cup product, we show that the Liouville action is just the square of the class of (T_{X}, ρ). In fact this identification holds at the level of cocycles, rather than only for the corresponding classes.

Again leaving aside the area term, it immediately follows from the properties of hermitian holomorphic Deligne cohomology that most of the story carries over to the case of a pair of holomorphic line bundles L and L′ equipped with hermitian metrics ρ and ρ′, respectively. Furthermore, Brylinski shows in [11] that the pairing of two such holomorphic line bundles with metrics corresponds to the pairing defined by Deligne on the determinant of cohomology in [12]. Without introducing the machinery of 2-gerbes, we reobtain this result in our setting. Specifically, we directly obtain O. Gabber’s formula for the hermitian metric on the determinant line from the explicit cocycle for the cup product of two metrized line bundles. In turn this shows that the Liouville action is a multiple of the determinant of cohomology, thereby generalizing earlier results (cf. [33]) — without assuming criticality.

1.1 Organization of the paper

This paper is organized as follows. Section 2 is devoted to the explanation of the main feature of Deligne cohomology, paying special attention to the product structures and the cone constructions. We provide some examples and collect some facts about the dilogarithm from the point of view of Deligne cohomology. Due to the definition of hermitian holomorphic Deligne cohomology we have adopted, certain constructions available in the literature and recalled in section 2 need to be slightly modified in order to obtain a (graded) commutative product. The necessary arguments, being somewhat outside the line of development of the paper have been presented in detail in Appendix A. We present our approach to hermitian holomorphic Deligne cohomology in section 3. While our definition is computationally more complex, it has the advantage for us of keeping the metric structure explicit, while the definition in refs. [9] and [11] works from the point of view of reduction of the structural group. We have explicitly proved the isomorphism in Lemma 3.2. For the sake of completeness, we give an explicit description of the cocycle determined by a holomorphic line bundle with hermitian metric, and in sect. 3.3 we explicitly compute the cup product of two metrized line bundles for later usage.

1From a physical point of view this originates in the modular geometry approach to Conformal Field Theory advocated by Friedan and Shenker in [19]. Mathematically speaking, it is one of the many proposed forms of the holomorphic factorization property for determinant line bundles.

2Note that X is allowed to be disconnected.

Sections 4 and 6 form the core of the paper. The direct construction of the Liouville action according to the techniques of refs. [2, 3, 31] is presented in section 4. Since explicit calculations have been presented in great detail in ref. [31], and the calculation we need are quite straightforward, we keep details to a minimum. In Proposition 4.4 and Corollary 4.5 we show that the Liouville action functional computed via descent theory does solve the variational problem. These results have appeared also in ref. [31] and are presented for here completeness. We go into more details to present the connection with the dilogarithm, both for its own sake and also to facilitate the transition to the material of section 6, where our main results are contained. Before doing so, in section 5 we illustrate in some detail how the genuine Bloch-Wigner function appears as an obstruction to closing the descent equations relative to a cover associated to a Kleinian uniformization. We point out that this obstruction is in fact zero due to the non-compactness of the 3-manifold ℍ^{3}/Γ. This is the content of Theorem 5.2. Finally, in section 6 we compare the descent calculations in section 4 with the cup products computed in section 3 to conclude that the quadratic part of the Liouville action is indeed a square, see Theorem 6.1, Corollary 6.2, and Proposition 6.4. Finally, we prove the relation with the determinant of cohomology in Theorem 6.5 and Corollary 6.6.

1.2 Notation and conventions

If z is a complex number, then π_{p}(z) =^{def} \frac{1}{2}(z + (−1)^{p}z̄), and similarly for any other complex quantity, e.g. complex valued differential forms. If A is a subring of ℝ, we will use the “twist” A(j) = (2π\sqrt{−1})^{j} A.

If X is a complex manifold, \underline{A}_{X}^{•} and \underline{Ω}_{X}^{•} denote the de Rham complexes of smooth ℂ-valued and holomorphic forms, respectively. We denote by \underline{ℰ}_{X}^{•} the de Rham complex of sheaves of real valued differential forms and by \underline{ℰ}_{X}^{•}(j) the twist \underline{ℰ}_{X}^{•} ⊗_{ℝ} ℝ(j). We set 𝒪_{X} ≡ \underline{Ω}_{X}^{0} as usual. When needed, \underline{A}_{X}^{p,q} will denote the sheaf of smooth (p, q)-forms. We use the standard decomposition d = ∂ + ¯∂ according to types. Furthermore, we introduce the differential operator d^{c} = ∂ − ¯∂ (contrary to the convention, see, e.g. [25], we omit the factor 1/(4π\sqrt{−1})). We have 2∂ ¯∂ = d^{c}d. The operator d^{c} is an imaginary one, and accordingly, we have the rules

dπ_{p}(ω) = π_{p}(dω)
d^{c}π_{p}(ω) = π_{p+1}(d^{c}ω)

for any complex form ω.

An open cover of X will be denoted by 𝒰_{X}. If {U_{i}}_{i∈I} is the corresponding collection of open sets, we denote U_{ij} = U_{i} ∩ U_{j}, U_{ijk} = U_{i} ∩ U_{j} ∩ U_{k}, and so on. We can also consider more general covers 𝒰_{X} = {U_{i} → X}_{i∈I} where the maps are regular coverings in the appropriate category. In this case intersections are replaced by (n + 1)-fold fibered products U_{i_{0}i_{1}...i_{n}} = U_{i_{0}} ×_{X} ⋯ ×_{X} U_{i_{n}} . Open coverings fit this more general description, since if U_{i} and U_{j} are two open sets, then U_{i} ∩ U_{j} = U_{i} ×_{X} U_{j}, where the fiber product is taken with respect to the inclusion maps. As another example, one can consider regular covering maps U → X with Γ = Deck(U/X), and in this case, taking the fiber product over X (n + 1)-times, one gets U ×_{X} ⋯ ×_{X} U = U × Γ × ⋯ × Γ , where the group factor is repeated n-times. This includes the cases of Kleinian (and in particular Fuchsian) covers of Riemann surfaces.

The nerve of the cover 𝒰_{X} is the simplicial object n ↦ N_{n}(𝒰_{X}) = ∐ U_{i_{0}} ×_{X} ⋯ ×_{X} U_{i_{n}} where N_{n}(𝒰_{X}) maps into N_{n−1}(𝒰_{X}) in (n + 1) ways by forgetting in turn each factor. For open covers this just yields the expected inclusion maps.

If \underline{F}^{•} is a complex of sheaves on X, its Čech resolution with respect to a covering 𝒰_{X} → X is the double complex C^{p,q}(\underline{F}) =^{def} Č^{q}(𝒰_{X}, \underline{F}^{p}) , where the q-cochains with values in \underline{F}^{p} are given by ∏ \underline{F}^{p}(U_{i_{0}} ×_{X} ⋯ ×_{X} U_{i_{n}}) . The Čech coboundary operator is denoted δ̌. The sign convention we are going to use is that the index along the Čech resolution is the second one, so if we denote by d the differential in the complex \underline{F}^{•}, the total differential in the total simple complex of Č^{q}(𝒰_{X}, \underline{F}^{p}) will be D = d ± δ̌. For open covers we just get the familiar Čech (hyper)cohomology. The other interesting example is that of a regular covering map U → X: Čech cohomology with respect to this cover is the same as group cohomology for Γ = Deck(U/X) with coefficients in the Γ-module \underline{F}^{p}(U).

The Koszul sign rule that results in a sign being picked whenever two degree indices are formally exchanged is applied. In particular, for Čech resolutions of complexes of sheaves, it leads to the following conventions. If \underline{G}^{•} is a second complex of sheaves on X, then one defines the cup product

∪ : C^{p,q}(\underline{F}) ⊗ C^{r,s}(\underline{G}) ⟶ Č^{q+s}(𝒰_{X}, \underline{F}^{p} ⊗ \underline{G}^{r}) ⊂ C^{p+r,q+s}(\underline{F} ⊗ \underline{G})

of two elements \{f_{i_{0},...,i_{q}}\} ∈ C^{p,q}(\underline{F}) and \{g_{j_{0},...,j_{s}}\} ∈ C^{r,s}(\underline{G}) by

(−1)^{qr} f_{i_{0},...,i_{q}} ⊗ g_{i_{q},i_{q+1},...,i_{q+s}} .

Acknowledgments

Parts of this work were completed during visits at the International School for Advanced Studies (SISSA) in Trieste, Italy, and at the Department of Mathematics, Instituto Superior Técnico in Lisbon, Portugal. I would like to thank both institutions for support and for creating a friendly and stimulating research environment. I would also like to thank Paolo Aluffi, Phil Bowers, Ugo Bruzzo, Johan Dupont, Leon Takhtajan for illuminating discussions and/or patiently answering my many questions.

2 Deligne complexes

2.1 Cup products on cones

Recall that the cone of a map f : X^{•} → Y^{•} between two complexes is the complex C^{•}(f) = X^{•}[1] ⊕ Y^{•} with differential d_{f}(x, y) = (−d x, f(x) + d y), where [k] denotes the shift functor. The cone fits into the exact sequence

0 ⟶ Y^{•} ⟶ C^{•}(f) ⟶ X^{•}[1] ⟶ 0 .

The following constructions are a special case of those considered by Beĭlinson in ref. [5]. Suppose we are given complexes X_{i}^{•}, Y_{i}^{•}, and Z_{i}^{•} and maps f_{i} : X_{i}^{•} → Z_{i}^{•}, g_{i} : Y_{i}^{•} → Z_{i}^{•}, for i = 1, 2, 3. Suppose also that we have product maps X_{1}^{•} ⊗ X_{2}^{•} ⟶^{∪} X_{3}^{•}, and similarly for Y_{i}^{•}, and Z_{i}^{•}, strictly compatible with the f_{i}, g_{i} in the obvious sense. Then we can consider the cones

Cone (f_{i} − g_{i})[−1] ≡ Cone (X_{i}^{•} ⊕ Y_{i}^{•} ⟶^{f_{i}−g_{i}} Z_{i}^{•})[−1] .

For a real parameter α, there is a family of products

(2.1)  Cone (f_{1} − g_{1})[−1] ⊗ Cone (f_{2} − g_{2})[−1] ⟶^{∪_{α}} Cone (f_{3} − g_{3})[−1]

determined as follows. For (x_{i}, y_{i}, z_{i}) ∈ X_{i}^{•} ⊕ Y_{i}^{•} ⊕ Z_{i}^{•−1} , i = 1, 2 , one defines

\begin{matrix}(x_{1}, y_{1}, z_{1}) ∪_{α} (x_{2}, y_{2}, z_{2}) = ( & x_{1} ∪ x_{2}, y_{1} ∪ y_{2}, \\ (−1)^{deg(x_{1})}((1 − α)f_{1}(x_{1}) + αg_{1}(y_{1})) ∪ z_{2} \\ + z_{1} ∪ (αf_{2}(x_{2}) + (1 − α)g_{2}(y_{2}))) . \end{matrix} ; (2.2)

Note that deg(x_{1}) = deg(x_{1}, y_{1}, z_{1}). Checking that ∪_{α} is a map of complexes is a straightforward routine calculation. Different products for different values α and β of the real parameter are homotopic. Explicitly, we have

(x_{1}, y_{1}, z_{1}) ∪_{α} (x_{2}, y_{2}, z_{2}) − (x_{1}, y_{1}, z_{1}) ∪_{β} (x_{2}, y_{2}, z_{2}) = (d h_{α,β} + h_{α,β}d)((x_{1}, y_{1}, z_{1}) ⊗ (x_{2}, y_{2}, z_{2})) ,

where the homotopy

h_{α,β} : Tot ( Cone (f_{1} − g_{1})[−1] ⊗ Cone (f_{2} − g_{2})[−1])^{•} ⟶ Cone (f_{3} − g_{3})[−1]^{•−1}

is given by the formula

(2.3)  h_{α,β}((x_{1}, y_{1}, z_{1}) ⊗ (x_{2}, y_{2}, z_{2})) = (α − β) (−1)^{deg(x_{1})−1}(0, 0, z_{1} ∪ z_{2}) .

If the products X_{1}^{•} ⊗ X_{2}^{•} ⟶^{∪} X_{3}^{•} , etc., are graded commutative, then the swap functor on the tensor product maps the ∪_{α} product structure on the cones into the ∪_{1−α} structure. Using the homotopy (2.3) it follows at once that there is a well defined graded commutative product in cohomology.

If we do not assume the product structures X_{1}^{•} ⊗ X_{2}^{•} ⟶^{∪} X_{3}^{•}, etc., are strictly compatible with the maps f_{i}, g_{i}, some of the preceding constructions must be slightly modified. With an eye toward certain constructions to be carried out later in this paper, let us assume we have compatibility up to homotopy, namely there exist maps

h: (X_{1} ⊗ X_{2})^{•} ⟶ Z_{3}^{•−1}
k: (Y_{1} ⊗ Y_{2})^{•} ⟶ Z_{3}^{•−1}

such that

(2.4)  \begin{matrix}f_{3} ∘ ∪ − ∪ ∘ (f_{1} ⊗ f_{2}) = d h + h d \\ g_{3} ∘ ∪ − ∪ ∘ (g_{1} ⊗ g_{2}) = d k + k d , \end{matrix}

with obvious meaning of the symbols.

Lemma 2.1. Let X_{i}, Y_{i}, Z_{i} and the maps f_{i}, g_{i} be as above. Let α be a real parameter. We have a product of type (2.1) for the cones Cone(f_{i} − g_{i})[−1] defined by the following modification of formula (2.2):

\begin{matrix}(x_{1}, y_{1}, z_{1}) ∪_{α} (x_{2}, y_{2}, z_{2}) = ( & x_{1} ∪ x_{2}, y_{1} ∪ y_{2}, \\ (−1)^{deg(x_{1})}((1 − α)f_{1}(x_{1}) + αg_{1}(y_{1})) ∪ z_{2} \\ + z_{1} ∪ (αf_{2}(x_{2}) + (1 − α)g_{2}(y_{2})) \\ − h(x_{1} ⊗ x_{2}) + k(y_{1} ⊗ y_{2})) . \end{matrix} ; (2.5)

The product (2.5) is a map of complexes and two products ∪_{α} and ∪_{β} are related by the same homotopy formula (2.3).

Proof. Direct verification.

This modified framework carries over to the full structure considered by Beĭlinson in ref. [5]. We will still refer to this modified product as the Beĭlinson product. It is also necessary to relax the assumption that the products X_{1}^{•} ⊗ X_{2}^{•} ⟶^{∪} X_{3}^{•} , etc., be graded commutative. It is possible to complete all the diagrams so that the permutation of factors in the tensor products still yields a homotopy commutative product (2.5) for the cones. As a consequence the induced product in cohomology will still be graded commutative. Explicit formulas are not needed except to ensure this latter fact, therefore we shall not discuss this matter any further and refer the reader to the appendix, where a brief but explicit treatment can be found.

2.2 Deligne complexes

Let X be a complex manifold. Recall the standard Hodge filtration of \underline{Ω}_{X}^{•}:

(2.6)  F^{p}\underline{Ω}_{X}^{•} : 0 ⟶ ⋯ ⟶ \underline{Ω}_{X}^{p} ⟶ ⋯ ⟶ \underline{Ω}_{X}^{n} ,

where n = dim_{ℂ} X.

The corresponding filtration for the complex of smooth ℂ-valued forms is defined as follows: denote by F^{p}\underline{A}_{X}^{•} the subcomplex of \underline{A}_{X}^{•} comprising forms of type (r, s) where r is at least p, so that F^{p}\underline{A}_{X}^{k} = ⊕_{r≥p}\underline{A}_{X}^{r,k−r}. Then (cf. [14]) the inclusion \underline{Ω}_{X}^{•} ↪ \underline{A}_{X}^{•} is a quasi-isomorphism respecting the filtrations, namely F^{p}\underline{Ω}_{X}^{•} ↪ F^{p}\underline{A}_{X}^{•}, and the latter inclusion induces an isomorphism in cohomology.

If A is a subring of ℝ, and ı and ȷ denote the inclusions of A(p) and F^{p}\underline{Ω}_{X}^{•} into \underline{Ω}_{X}^{•} respectively, the p − th Deligne complex of sheaves is defined by

(2.7)  A(p)_{𝒟}^{•} = Cone (A(p)_{X} ⊕ F^{p}\underline{Ω}_{X}^{•} ⟶^{ı−ȷ} \underline{Ω}_{X}^{•})[−1] .

It is quasi-isomorphic to the complex:

(2.8)  Cone (A(p)_{X} ⊕ F^{p}\underline{A}_{X}^{•} ⟶^{ı−ȷ} \underline{A}_{X}^{•})[−1] ,

where ı and ȷ have the same meaning. A(p)_{𝒟}^{•} is quasi-isomorphic to:

(2.9)  \underline{A(p)}_{𝒟}^{•} = A(p)_{X} ⟶^{ı} 𝒪_{X} ⟶^{d} \underline{Ω}_{X}^{1} ⟶^{d} ⋯ ⟶^{d} \underline{Ω}_{X}^{p−1} ,

and an explicit quasi-isomorphism

ϕ_{p} : \underline{A(p)}_{𝒟}^{•} ⟶^{≃} A(p)_{𝒟}^{•}

is given in [18]. For completeness, we recall the explicit formula. We have

(2.10)  \begin{matrix}A(p)_{X} & ⟶^{ı} & 𝒪_{X} & ⟶^{d} & \underline{Ω}_{X}^{1} & ⟶^{d} & ⋯ & ⟶^{d} & \underline{Ω}_{X}^{p−1} \\ ↓^{ϕ_{0}} & & ↓^{ϕ_{1}} & & ↓^{ϕ_{2}} & & & & ↓^{ϕ_{p}} \\ A(p)_{X} & ⟶^{−ı} & 𝒪_{X} & ⟶^{−d} & \underline{Ω}_{X}^{1} & ⟶^{−d} & ⋯ & ⟶^{(0,−d)} & \underline{Ω}_{X}^{p} ⊕ \underline{Ω}_{X}^{p−1} & ⟶^{(d,ȷ−d)} & \underline{Ω}_{X}^{p+1} ⊕ \underline{Ω}_{X}^{p} & ⟶^{(d,ȷ−d)} & ⋯ \end{matrix}

with ϕ_{i}(x) = (−1)^{i}x, for 0 ≤ i < p and ϕ_{p}(x) = (−1)^{p}(dx, x). The complex (2.9) is perhaps more widely used (or known) as the complex used to define Deligne cohomology.

When A = ℝ there are further quasi-isomorphisms, namely

ℝ(p)_{𝒟}^{•} ⟶^{≃} Cone (F^{p}\underline{Ω}_{X}^{•} → \underline{ℰ}_{X}^{•}(p − 1))[−1] ⟶^{≃} Cone (F^{p}\underline{A}_{X}^{•} → \underline{ℰ}_{X}^{•}(p − 1))[−1]

since the maps

(ℝ(p) → \underline{Ω}_{X}^{•}) ⟶^{≃} ℝ(p) → ℂ ⟶^{≃} ℝ(p − 1) ⟶^{≃} \underline{ℰ}_{X}^{•}(p − 1)

are all quasi-isomorphisms in the derived category, cf. [18]. Here we have used ℂ ≅ ℝ(p) ⊕ ℝ(p − 1). Following op. cit., we set:

(2.11)  { \atop ℝ(p)}_{𝒟}^{•} =^{def} Cone (F^{p}\underline{A}_{X}^{•} ⟶^{−π_{p−1}} \underline{ℰ}_{X}^{•}(p − 1))[−1] .

Again, there is an explicit quasi-isomorphism ([18]):

(2.12)  \begin{matrix}ρ_{p} : ℝ(p)_{𝒟}^{•} ⟶^{≃} { \atop ℝ(p)}_{𝒟}^{•} \\ ρ_{p}|_{ℝ(p)}= 0 , ρ_{p}|_{F^{p}\underline{Ω}_{X}^{•}} = incl , ρ_{p}|\underline{Ω}_{X}^{•} = π_{p−1} \end{matrix}

The Deligne cohomology groups of X with coefficients in A(p) are the hypercohomology groups

H_{𝒟}^{•} (X, A(p)) = H^{•}(X, A(p)_{𝒟}^{•} ) .

and clearly, any complex quasi-isomorphic to A(p)_{𝒟}^{•} would do. In order to perform calculations with these cohomology groups we shall normally resort to a Čech resolution with respect to an open cover 𝒰_{X} of X or, later in the paper, an étale map 𝒰_{X} → X, e.g. a regular cover with deck group Γ. However, when A = ℝ, we can exploit the above quasi-isomorphisms and the fact that \underline{A}_{X}^{•} is a complex of fine sheaves to conclude that H_{𝒟}^{•} (X, ℝ(p)) can also be computed as the homology of the complex of global sections

Cone(F^{p}A^{•}(X) ⟶ ℰ^{•}(X)(p − 1))[−1] .

One of the important properties of Deligne cohomology is the existence of a graded commutative cup product

(2.13)  H_{𝒟}^{i} (X, A(p)) ⊗ H_{𝒟}^{j} (X, A(q)) ⟶^{∪} H_{𝒟}^{i+j}(X, A(p + q)) ,

which follows from the existence of the Beĭlinson cup product at the level of Deligne complexes whose construction was recalled above. There are products A(p) ⊗ A(q) → A(p + q) and F^{p}\underline{Ω}_{X}^{•} ⊗ F^{q}\underline{Ω}_{X}^{•} → F^{p+q}\underline{Ω}_{X}^{•}, plus the obvious (wedge) product on \underline{Ω}_{X}^{•}, thus it follows from the cone version (2.7) that the Deligne complexes come equipped with the Beĭlinson product, and therefore the Deligne cohomology groups inherit the graded commutative cup product (2.13). The explicit form, that is, the translation of (2.2) to the case at hand can be found in [18]. Furthermore, the product formula (cf. [10],[18])

(2.14)  f ∪ g = \begin{cases}f · g & deg f = 0 , \\ f ∧ dg & deg f > 0 and deg g = q , \\ 0 & otherwise, \end{cases}

on the complex (2.9) maps via (2.10) to the product ∪_{0} on the cone version (2.7). The explicit form of the cup product for the complex (2.11) as computed in [4] (see also [18]) will be needed in the sequel. Let (ω_{1}, η_{1}) be an element of degree n in { \atop ℝ(p)}_{𝒟}^{•} —this means that ω_{1} ∈ F^{p}\underline{Ω}_{X}^{n} and η_{1} ∈ \underline{ℰ}_{X}^{n−1}(p − 1)—and (ω_{2}, η_{2}) any element in ℝ̃(q)_{𝒟}^{•} . The product is defined by the formula:

(2.15)  (ω_{1}, η_{1}) ˜∪ (ω_{2}, η_{2}) = (ω_{1} ∧ ω_{2}, (−1)^{n} π_{p}ω_{1} ∧ η_{2} + η_{1} ∧ π_{q}ω_{2}) .

The product ˜∪ is a morphism of complexes and (modulo the quasi-isomorphisms ρ_{p}) is homotopic to the Beĭlinson products ∪_{α} on the complexes ℝ(p)_{𝒟}^{•} . Specifically, if we denote an element of ℝ(p)_{𝒟}^{•} by the triple (r, f, ω), where r ∈ ℝ(p)_{X}, f ∈ F^{p}\underline{Ω}_{X}^{•}, and ω ∈ \underline{Ω}_{X}^{•}, the homotopy between ˜∪ ∘ (ρ_{p} ⊗ ρ_{q}) and ρ_{p+q} ∘ ∪_{α} is given by

(2.16)  h̃((r, f, ω) ⊗ (r′, f′, ω′)) = (−1)^{deg ω}(0, (1 − α) π_{p}ω ∧ π_{q−1}ω′ − α π_{p−1}ω ∧ π_{q}ω′) .

2.3 Examples

2.3.1

Let A = ℤ. Then ℤ(1)_{𝒟}^{•} ⟶^{≃} 𝒪_{X}^{×}[−1] via the standard exponential sequence, so that H_{𝒟}^{k} (X, ℤ(1)) ≅ H^{k−1}(X, 𝒪_{X}^{×}) . In particular H_{𝒟}^{1} (X, ℤ(1)) ≅ H^{0}(X, 𝒪_{X}^{×}), the global invertibles on X, and H_{𝒟}^{2} (X, ℤ(1)) ≅ Pic(X), the Picard group of line bundles over X.

If an open cover {U_{i}}_{i∈I} of X is chosen, the class of a line bundle L in H_{𝒟}^{2} (X, ℤ(1)) can be represented via a Čech resolution by the cocycle (f_{ij}, c_{ijk}), where f_{ij} ∈ 𝒪_{X}(U_{ij}), and c_{ijk} ∈ ℤ(1)_{X}(U_{ijk}). Here we have used the Deligne complex in the form (2.9). The functions f_{ij} should be interpreted as logarithms of the corresponding transition functions for L. Then, the collection c_{ijk} = (δ̌f)_{ijk} provides a representative for the first Chern Class c_{1}(L). Similarly, an invertible function f would be described by the collection f_{i} of its logarithms on each open U_{i}, plus the “integers” m_{ij} = f_{j} − f_{i} ∈ ℤ(1) on each U_{ij}.

2.3.2

Still using the exponential sequence, ℤ(2)_{𝒟}^{•} ⟶^{≃} (𝒪_{X}^{×} ⟶^{d log} \underline{Ω}_{X}^{1})[−1]. Thus H_{𝒟}^{2} (X, ℤ(2)) is the group of isomorphism classes of holomorphic line bundles with (holomorphic) connection. Using the (in fact, any) product ℤ(1)_{𝒟}^{•} ⊗ ℤ(1)_{𝒟}^{•} → ℤ(2)_{𝒟}^{•} , the cup product of two global invertible holomorphic functions f and g on X yields a line bundle with connection—the tame symbol—denoted by (f, g] whose class is in H_{𝒟}^{2} (X, ℤ(2)), see [13, 6, 29]. Higher cup products in this spirit have been studied in [9].

2.3.3

If A = ℝ, we have H_{𝒟}^{2p}(X, ℝ(p)) = H^{2p}(X, ℝ(p)) ∩ H^{p,p}(X) . The Čech representative (c_{ijk}, f_{ij}) of a class in H_{𝒟}^{2} (X, ℤ(1)) mentioned above maps to the cocycle (−df_{ij}, −|f_{ij}|) under (2.12). Taking into account that the f_{ij} are the logarithms of the transition functions, the corresponding (1, 1) class would be given by the associated canonical connection, see sec. 3.2.

2.3.4

H_{𝒟}^{1} (X, ℝ(1)) is the group of real valued functions f on X such that there exists a holomorphic one-form ω such that π_{0}ω = df. In other words it is the group of those smooth functions f such that ∂f is holomorphic, which amounts to say that such an f itself is harmonic. Going to higher values, we have H_{𝒟}^{2p−1}(X, ℝ(p)) = H^{2p−2}(X, ℝ(p − 1)) ∩ H^{p−1,p−1}(X) if X is the complex manifold corresponding to a projective variety, cf. [30]. In general, e.g. X quasi-projective, H_{𝒟}^{2p−1}(X, ℝ(p)) will correspond to ∂ ¯∂-closed ℝ(p − 1)-valued (p − 1, p − 1)-forms modulo those forms in Im ∂ + Im ¯∂.

2.4 Remarks on the dilogarithm

It is convenient to consider the case of the cup product of two invertible functions f and g in various complexes in more detail, and to introduce some related notions we shall need later.

As observed, ℤ(1)_{𝒟}^{•} ⟶^{≃} 𝒪_{X}^{×}[−1] and an invertible function f can be considered as an element of H_{𝒟}^{1} (X, ℤ(1)). Therefore, via (2.12), it induces ρ_{1}(f) ∈ H_{𝒟}^{1} (X, ℝ(1)) represented by (d log f, log |f|). (Note that log|f| fits the description of H_{𝒟}^{1} (X, ℝ(1)) in 2.3.4.) Given two such f and g, the expression for the cup product (2.15) gives the following element of H_{𝒟}^{2} (X, ℝ(2)):

(2.17)  ρ_{1}(f) ˜∪ ρ_{1}(g) = (d log f ∧ d log g, −π_{1}(d log f) log |g| + log |f| π_{1}(d log g)) .

The first term is obviously zero when X is a curve. Given f and g, invertible on X, let us define the imaginary 1-form:

(2.18)  r_{2}(f, g) = π_{1}(d log f) log |g| − log |f| π_{1}(d log g) .

On the other hand, the cup product of f and g as elements of H_{𝒟}^{1} (U, ℤ(1)) yields an element f ∪ g of H_{𝒟}^{2} (U, ℤ(2)) represented by (d log f ∧ d log g, log f d log g) (if we use the ∪_{0} product) and this maps via ρ_{2} to the element

(d log f ∧ d log g, −π_{1}(log f d log g)) .

This is equal to (2.18) only up to homotopy. Indeed, using π_{p+q−1}(a ∧ b) = π_{p}(a) ∧ π_{q−1}(b) + π_{p−1}(a) ∧ π_{q}(b) , we find the expression

(2.19)  r_{2}(f, g) = d(π_{1}(log f) log |g|) − π_{1}(log f d log g) ,

where the first term is just the explicit homotopy as computed from (2.16).

Recall that the tame symbol ([13, 6]) (f, g] associated to f and g is the line bundle with connection determined (up to isomorphism) by the class f ∪ g. A Bloch dilogarithm ([18]) is (the logarithm of) a horizontal trivializing section, namely a function L on U satisfying the equation

dL = − log f d log g .

Thus L realizes the isomorphism (f, g] ≅ 𝒪_{X}. The Steinberg relation satisfied by the Tame symbol says in particular this is true when g = 1−f: (1−f, f] is trivial [6, 18] in the sense specified above. The universal case corresponding to f = z on ℙ^{1} \ {0, 1, ∞} allows to identify L with the classical Euler dilogarithm Li_{2}, namely

Li_{2}(z) = − ∫_{0}^{z} log(1 − t)\frac{dt}{t} ,

see [13]. The classical dilogarithm has a single valued parter, denoted 𝒟_{2}, introduced by Bloch and Wigner:

(2.20)  𝒟_{2}(z) = arg(1 − z) log |z| + Im Li_{2}(z) .

𝒟_{2} is real-analytic on ℙ^{1} \ {0, 1, ∞} and extends continuously to ℙ^{1}. If we introduce

ℒ_{2}(z) = \sqrt{−1} 𝒟_{2}(z) ,

then clearly

dℒ_{2} = r_{2}(1 − z, z) .

More generally, if L trivializes (f, g] in the sense explained above, we can associate a function ℒ_{2}(f, g) such that

dℒ_{2}(f, g) = r_{2}(f, g)

via the position

(2.21)  ℒ_{2}(f, g) = π_{1}(log f) log |g| + Im L .

3 Constructions in hermitian holomorphic Deligne cohomology

In this section we construct a set of complexes to compute hermitian holomorphic Deligne cohomology, as in ref. [11]. The complexes we introduce here are tailored to the description of a metrized line bundle in terms of local representatives of the hermitian fiber metric, whereas those in loc. cit. work best from a structure group reduction point of view. The two constructions are related by a quasi-isomorphism, so the cohomology groups are the same, and therefore it is justified to retain the same name for the resulting cohomology groups.

3.1 Hermitian holomorphic Deligne cohomology

Brylinski introduced in ref. [11] the complexes C(l) in order to compare the Beĭlinson-Chern classes of a holomorphic vector bundle E with the Cheeger-Chern-Simons classes determined by (E, ∇), where ∇ is the canonical connection, namely the unique connection compatible with both the holomorphic and hermitian structures. The cohomology groups determined by these complexes were called Hermitian Holomorphic Deligne cohomology groups. We will recall Brylinski’s definition below; we introduce now a different set of complexes with a slightly different motivation in mind: as it has been mentioned above, and it will be expanded in further detail in sec. 3.2, a holomorphic line bundle L on X determines a class in H_{𝒟}^{2} (X, ℤ(1)), and this class has an image in real Deligne cohomology via the obvious map ℤ → ℝ inducing a map between the corresponding Deligne complexes. The image in real Deligne cohomology correspond to a class of pure Hodge type (1, 1). If L is equipped with a hermitian metric ρ, this class will be represented by a concrete 2-form built from the canonical connection associated to ρ, and furthermore this form is an imaginary (1, 1)-form. So we want a complex whose cohomology classes in degree 2p correspond to (p, p)-forms representing hodge classes of the corresponding pure type, coming from integer Deligne cohomology classes. To this end, consider the complex

(3.1)  D(l)_{h.h.}^{•} = Cone(ℤ(l)_{𝒟}^{•} ⊕ (F^{l}\underline{A}_{X}^{•} ∩ σ^{2l}\underline{ℰ}_{X}^{•}(l)) ⟶ ℝ̃(l)_{𝒟}^{•} )[−1] ,

where, as usual, σ^{p} denotes the (sharp) truncation in degree p, namely for a complex \underline{F}^{•}, σ^{p}\underline{F}^{•} is zero in degrees less than p. The map ℤ(l)_{𝒟}^{•} → ℝ̃(l)_{𝒟}^{•} is the composite of the obvious map ℤ(l)_{𝒟}^{•} → ℝ(l)_{𝒟}^{•} with the quasi-isomorphism ρ_{l} defined by (2.12). We will simply denote it by ρ_{l} in the sequel, suppressing the first morphism in the notation. The map (F^{l}\underline{A}_{X}^{•} ∩ σ^{2l}\underline{ℰ}_{X}^{•}(l)) → ℝ̃(l)_{𝒟}^{•} = Cone(F^{l}\underline{A}_{X}^{•} → \underline{ℰ}_{X}^{•}(l − 1))[−1] is induced by the inclusion of (F^{l}\underline{A}_{X}^{•} ∩ σ^{2l}\underline{ℰ}_{X}^{•}(l)) into F^{l}\underline{A}_{X}^{•}. In (3.1) we take the cone of the difference between these two maps. We have:

Definition 3.1. The complex (3.1) is the hermitian holomorphic Deligne complex. The hermitian holomorphic Deligne cohomology groups are the corresponding hypercohomology groups

(3.2)  H_{𝒟_{h.h.}}^{p} (X, l) =^{def} H^{p}(X, D(l)_{h.h.}^{•}) .

The specific choice of the complexes in Definition 3.1 is geared towards the computation of certain explicit cocycles representing cup products of metrized line bundles to be considered below. In order to justify our choice of the name, let us compare this definition with the one in ref. [11], where Brylinski introduces the complexes:

(3.3)  C(l)^{•} = Cone(ℤ(l)_{X} ⊕ (F^{l}\underline{A}_{X}^{•} ∩ σ^{2l}\underline{ℰ}_{X}^{•}(l)) ⟶ \underline{ℰ}_{X}^{•}(l))[−1] .

Lemma 3.2. The complexes C(l)^{•} and D(l)_{h.h.}^{•} are quasi-isomorphic.

Proof. By elementary manipulation of cones

D(l)_{h.h.}^{•} = Cone(F^{l}\underline{A}_{X}^{•} ∩ σ^{2l}\underline{ℰ}_{X}^{•}(l) → Cone(ℤ(l)_{𝒟}^{•} → ℝ̃(l)_{𝒟}^{•} ))[−1]

and clearly: Cone(ℤ(l)_{𝒟}^{•} → ℝ̃(l)_{𝒟}^{•} ) ⟶^{≃} Cone(ℤ(l)_{X} → ℝ(l)_{X}) ⟶^{≃} Cone(ℤ(l)_{X} → \underline{ℰ}_{X}^{•}(l)) , where all arrows are quasi-isomorphisms. Thus

D(l)_{h.h.}^{•} ⟶^{≃} Cone(F^{l}\underline{A}_{X}^{•} ∩ σ^{2l}\underline{ℰ}_{X}^{•}(l) → Cone(ℤ(l)_{X} → \underline{ℰ}_{X}^{•}(l)))[−1]
= Cone(ℤ(l)_{X} ⊕ (F^{l}\underline{A}_{X}^{•} ∩ σ^{2l}\underline{ℰ}_{X}^{•}(l)) → \underline{ℰ}_{X}^{•}(l))[−1]
≡ C(l) ,

as wanted.

It follows from Lemma 3.2 that the hypercohomology groups of the two complexes are the same and therefore the hermitian holomorphic Deligne cohomology groups as per Definition 3.1 are the same as those defined by Brylinski in ref. [11].

From (3.1) and the exact sequence satisfied by the cone construction, we can derive the exact sequence

(3.4)  ⋯ ⟶ H_{𝒟}^{2l−1}(X, ℝ(l)) ⟶ H_{𝒟_{h.h.}}^{2l} (X, l) ⟶ H_{𝒟}^{2l}(X, ℤ(l)) ⊕ A^{(l,l)}(X)_{ℝ(l)} ⟶ H_{𝒟}^{2l}(X, ℝ(l)) ⟶ ⋯

where A^{(l,l)}(X)_{ℝ(l)} denotes the space of smooth ℝ(l)-valued global (l, l)-forms on X. In view of the examples, we can rewrite the exact sequence as

⋯ ⟶ H_{𝒟}^{2l−1}(X, ℝ(l)) ⟶ H_{𝒟_{h.h.}}^{2l} (X, l) ⟶

⟶ H_{𝒟}^{2l}(X, ℤ(l)) ⊕ A^{(l,l)}(X)_{ℝ(l)} ⟶ H^{2l}(X, ℝ(l)) ∩ H^{l,l}(X) ⟶ ⋯

We see that the group H_{𝒟_{h.h.}}^{2l} (X, l) map onto those (l, l)-forms representing the Hodge classes corresponding to H_{𝒟}^{2l}(X, ℤ(l)).

Now recall that we have cup products for ℤ(l)_{𝒟}^{•} , ℝ̃(l)_{𝒟}^{•} , and F^{l}\underline{A}_{X}^{•} ∩ σ^{2l}\underline{ℰ}_{X}^{•}(l), the latter being induced by the standard wedge product. Since the complex D(l)_{h.h.}^{•} is a cone, we have the Beĭlinson family of products

(3.5)  D(l)_{h.h.}^{•} ⊗ D(k)_{h.h.}^{•} ⟶^{∪_{α}} D(l + k)_{h.h.}^{•} ,

and therefore a cup product

(3.6)  H_{𝒟_{h.h.}}^{i} (X, l) ⊗ H_{𝒟_{h.h.}}^{j} (X, k) ⟶^{∪} H_{𝒟_{h.h.}}^{i+j} (X, l + k) .

Remark 3.3. A note of warning: in the definition (3.1) of D(l)_{h.h.}^{•} the complexes ℤ(l)_{𝒟}^{•} and ℝ̃(l)_{𝒟}^{•} have product structures that are compatible with the map ρ_{l} only up to homotopy (given by formula (2.16)), therefore the Beĭlinson product in (3.5) is to be intended in the modified version of formula (2.5). Similarly, the product structures of the individual cones in (3.1) are already graded commutative up to homotopy, thus that the product (3.5) is graded commutative up to homotopy follows from Proposition A.4 in the appendix. That allows to conclude that the induced product on cohomology (3.6) is genuinely graded commutative as intended.

To conclude this introduction, we briefly mention the case of a proper submersion of complex manifolds π: X → S. It follows from [20] that Deligne cohomology has an integration along the fiber ∫_{π}, hence in analogy with [11, 9] there is a mapping

Rπ_{∗}D(l)_{h.h.}^{•} ⟶ D(l − d)_{h.h.}^{•}[−2d]

where d is the complex dimension of the fiber, inducing

(3.7)  H_{𝒟_{h.h.}}^{i} (X, l) ⟶ H_{𝒟_{h.h.}}^{i−2d}(S, l − d)

and commutative diagrams analogous to [11, Theorem 5.1]. We shall be interested in the case of complex relative dimension 1 and degree 4, namely that of the map: H_{𝒟_{h.h.}}^{4} (X, 2) → H_{𝒟_{h.h.}}^{2} (S, 1) , inducing a complex hermitian line bundle on S (cf. below).

3.2 Hermitian holomorphic line bundles

A hermitian holomorphic line bundle or, equivalently, a metrized line bundle, cf. [25], is a holomorphic line bundle L over X together with a hermitian fiber metric ρ : L → ℝ_{≥0}. As a rule, we will not distinguish L and its sheaf of holomorphic sections. We will also use the alternate notation |s|_{ρ} to denote the length of a local section s of L with respect to ρ. Metrized line bundles can be tensor multiplied and an inverse is defined, see. op. cit. An isomorphism of metrized line bundles (L, ρ) and (L′, ρ′) is defined in the obvious way, namely it is a map ϕ : L → L′ such that |s|_{ρ} = |ϕ(s)|_{ρ′} for some local section s of L. We denote by { \atop Pic(X)} the group of isomorphism classes of metrized line bundles.

If L is trivialized over a Čech cover 𝒰_{X} = {U_{i}}_{i∈I} by sections s_{i}, then as usual we obtain the cocycle of transition functions g_{ij} ∈ 𝒪_{X}^{×}(U_{i} ∩ U_{j}) by writing s_{j} = s_{i} g_{ij}. Then if (L, ρ) is a metrized line bundle, we can define the positive function ρ_{i} = |s_{i}|_{ρ}^{2}, namely the local representative of the hermitian structure with respect to the given trivialization. It follows that the various local representatives satisfy

(3.8)  ρ_{j} = ρ_{i} |g_{ij}|^{2} .

Let us work out the local version of the isomorphism introduced above. Let s′_{i} be a local section of L′ over U_{i}. Introduce analogous (primed) quantities for L′ as we just did for L. Given the isomorphism ϕ : L → L′ we have ϕ(s_{i}) = s′_{i} f_{i}, for some f_{i} ∈ 𝒪_{X}^{×}(U_{i}). Then we find f_{i} g_{ij} = g′_{ij} f_{j} and ρ_{i} = ρ′_{i} |f_{i}|^{2}.

Still working with respect to the chosen cover 𝒰_{X}, a connection compatible with the holomorphic structure is the datum of a collection of (1, 0)-forms ξ_{i} ∈ \underline{A}_{X}^{1,0}(U_{i}) satisfying

(3.9)  ξ_{j} − ξ_{i} = d log g_{ij} .

Note for future reference that \underline{A}_{X}^{1,0} = F^{1}\underline{A}_{X}^{1}. The connection is compatible with the hermitian metric if

(3.10)  π_{0}(ξ_{i}) = \frac{1}{2}d log ρ_{i} .

Using d = ∂ + ¯∂ and decomposition with respect to (p, q)-types, we find the familiar relation

(3.11)  ξ_{i} = ∂ log ρ_{i}

for the unique connection compatible with both the complex and hermitian structures [22].

The global 2-form

(3.12)  c_{1}(ρ) = ¯∂∂ log ρ_{i}

represents the first Chern class of L in H^{2}(X, ℝ(1)). Actually, the class of c_{1}(ρ) is a pure Hodge class in H^{1,1}(X) and, according to the examples, it is the image of the first Chern class of L under the map H_{𝒟}^{2} (X, ℤ(1)) → H_{𝒟}^{2} (X, ℝ(1)) induced by ℤ(1) → ℝ(1). Observe that c_{1}(ρ) = c_{1}(ρ′) under the isomorphism considered above. In the same way as in [9, 11] we have the following:

Proposition 3.4.

{ \atop Pic(X)} ≅ H_{𝒟_{h.h.}}^{2} (X, 1)

Proof. As noted in the examples before, the class of L in H_{𝒟}^{2} (X, ℤ(1)) is represented by the Čech cocycle (log g_{ij}, c_{ijk}), where c_{ijk} = log g_{jk} − log g_{ik} + log g_{ij} represents c_{1}(L). Moreover, from (3.8) we have the relation

\frac{1}{2} log ρ_{j} − \frac{1}{2} log ρ_{i} = π_{0}(log g_{ij}) ≡ log |g_{ij}| ,

thus we obtain a degree 2 total cocycle (\frac{1}{2} log ρ_{i}, log g_{ij}, c_{ijk}) in the Čech resolution of the complex

(3.13)  0 ⟶ ℤ(1)_{X} ⟶^{ı} 𝒪_{X} ⟶^{−π_{0}} \underline{ℰ}_{X}^{0} ⟶ 0 .

By a standard computation isomorphic metrized line bundles determine equivalent cocycles, therefore a metrized line bundle (L, ρ) determines a class in the second (hyper)cohomology group of the complex (3.13). Furthermore, two such classes can be added by component-wise adding the corresponding cocycles. It is readily verified that for metrized bundles (L, ρ) and (L′, ρ′) this yields the cocycle corresponding to (L ⊗ L′, ρρ′). Thus, passing to classes, this corresponds to the group operation in { \atop Pic(X)}.

We now proceed to show that the complexes (3.13) and D(1)_{h.h.}^{•} are quasi-isomorphic. To begin with, note that the complex (3.13) is quasi-isomorphic to

(3.14)  Cone(ℤ(1)_{𝒟}^{•} ⟶^{π_{0}} \underline{ℰ}_{X}^{0}[−1])[−1] .

On the other hand, we want to consider the cone of the morphism

F^{1}\underline{A}_{X}^{•} ∩ σ^{2}\underline{ℰ}_{X}^{•}(1) ⟶^{−incl⊕0} { \atop ℝ(1)}_{𝒟}^{•} .

This morphism is mono, so its cone can be replaced (in the derived category) by

(3.15)  Cone(F^{1}\underline{A}_{X}^{•}/F^{1}\underline{A}_{X}^{•} ∩ σ^{2}\underline{ℰ}_{X}^{•}(1) ⟶^{−π_{0}} \underline{ℰ}_{X}^{•})[−1] ,

since

Cone(A^{•}/K^{•} → B^{•})[−1] ⟶^{≃} Cone(A^{•} → B^{•})[−1]/K^{•} ⟶^{≃} Cone(K^{•} → Cone(A^{•} → B^{•})[−1])

when K^{•} → A^{•} is mono. By direct inspection, the complex (3.15) is quasi-isomorphic to \underline{ℰ}_{X}^{0}[−1]. Therefore we have the following quasi-isomorphisms:

(3.14) ⟶^{≃} Cone(ℤ(1)_{𝒟}^{•} → Cone(F^{1}\underline{A}_{X}^{•}/F^{1}\underline{A}_{X}^{•} ∩ σ^{2}\underline{ℰ}_{X}^{•}(1) ⟶^{−π_{0}} \underline{ℰ}_{X}^{•})[−1])[−1]

⟶^{≃} Cone(ℤ(1)_{𝒟}^{•} → Cone(F^{1}\underline{A}_{X}^{•} ∩ σ^{2}\underline{ℰ}_{X}^{•}(1) → { \atop ℝ(1)}_{𝒟}^{•} ))[−1]

⟶^{≃} D(1)_{h.h.}^{•}

so that (3.13) and D(1)_{h.h.}^{•} are indeed quasi-isomorphic. Thus (3.13) or (3.14) also compute H_{𝒟_{h.h.}}^{•} (X, 1) and this proves the statement, since we have already seen that a metrized line bundle determines a degree two hypercohomology class with coefficients in (3.13).

There is another—indeed, more direct—proof to the above proposition, which is simply to exhibit a cocycle in a Čech resolution of D(1)_{h.h.}^{•}. Explicit representatives will be needed for the calculation of the cup products (3.6) and subsequent development, so we present this construction here.

Alternative proof of Proposition 3.4. Recall that D(1)_{h.h.}^{•} is the cone of the map ρ_{1} − ȷ, where ρ_{1} : ℤ(1)_{𝒟}^{•} → { \atop ℝ(1)}_{𝒟}^{•} , and ȷ : F^{1}\underline{A}_{X}^{•} ∩ σ^{2}\underline{ℰ}_{X}^{•}(1) → { \atop ℝ(1)}_{𝒟}^{•} . By unraveling the structure of all the cones involved we have:

ℤ(1)_{X} ⟶ \underline{Ω}_{X}^{1} ⊕ 𝒪_{X} ⟶ \underline{Ω}_{X}^{2} ⊕ \underline{Ω}_{X}^{1} ⟶ \underline{Ω}_{X}^{3} ⊕ \underline{Ω}_{X}^{2} ⟶ ⋯
↓^{ı⊕π_{0}}  ↓^{ı⊕π_{0}}  ↓^{ı⊕π_{0}}
F^{1}\underline{A}_{X}^{1} ⊕ \underline{ℰ}_{X}^{0} ⟶ F^{1}\underline{A}_{X}^{2} ⊕ \underline{ℰ}_{X}^{1} ⟶ F^{1}\underline{A}_{X}^{3} ⊕ \underline{ℰ}_{X}^{2} ⟶ ⋯
↑^{ȷ⊕0}  ↑^{ȷ⊕0}
F^{1}\underline{A}_{X}^{2} ∩ \underline{ℰ}_{X}^{2}(1) ⟶ F^{1}\underline{A}_{X}^{3} ∩ \underline{ℰ}_{X}^{3}(1) ⟶ ⋯

With respect to this diagram, an element of total degree 2 can be written in the form:

(3.16)  \begin{matrix}c_{ijk} & − d log g_{ij} ⊕ log g_{ij} & 0 \\ 0 & ξ_{i} ⊕ σ_{i} & X \\ 0 & 0 & η_{i} \end{matrix}

for ξ_{i} ∈ F^{1}\underline{A}_{X}^{1}(U_{i}), σ_{i} ∈ \underline{ℰ}_{X}^{0}(U_{i}) and η_{i} ∈ (F^{1}\underline{A}_{X}^{2} ∩ \underline{ℰ}_{X}^{2}(1))(U_{i}). To make sense out of (3.16), note that each entry is an element of the object in the corresponding position in the left 3 × 3 part of the previous diagram. Then since the total degree is 2, the degree of each element in the complex D(1)_{h.h.}^{•} is 2 minus the Čech degree as found in (3.16). The top line is the class of L in ℤ(1)_{𝒟}^{2} . Finally, the entry marked X means there is no applicable element—it would have degree 3.

A totally routine calculation shows that (3.16) is a degree 2 cocycle if and only if relations (3.9), (3.10) and η_{i} = dξ_{i} = 2 ¯∂∂σ_{i} are satisfied (with σ_{i} = \frac{1}{2} log ρ_{i}). Thus η_{i} = c_{1}(ρ)|_{U_{i}} and all the relations defining a metrized line bundle with its canonical connection are satisfied. The verification that adding an appropriate coboundary to the cocycle leads to an isomorphic metrized bundle in the sense explained above is also routine. Finally, the correspondence between component-wise addition of cocycles modulo coboundaries and the group operation in { \atop Pic(X)} is again a direct verification.

It follows from the arguments in the second proof of Proposition 3.4 that a cocycle representing (L, ρ) in the Čech resolution of D(1)_{h.h.}^{•} must have the expression

(3.17)

c_{ijk} − d log g_{ij} ⊕ log g_{ij}  0
0  ∂ log ρ_{i} ⊕ \frac{1}{2} log ρ_{i}  X
0  0  ¯∂∂ log ρ_{i}

Furthermore, if [L, ρ] and [L′, ρ′] are the classes corresponding to the metrized bundles (L, ρ) and (L′, ρ′), then we write [L ⊗ L′, ρρ′] = [L, ρ] + [L′, ρ′] .

3.3 Cup product of hermitian holomorphic line bundles

If L and L′ are two line bundles on X, their cup product in Deligne cohomology would produce a class in H_{𝒟}^{4} (X, ℤ(2)) denoted (L, L′]. Calculations were carried out in [9] where a geometric interpretation of (L, L′] as a 2-gerbe and its connection with the determinant of cohomology (when X is the total space of a family of Riemann surfaces) were put forward. The structure ensuing from the generalization to line bundles equipped with a hermitian metric was further analyzed in ref. [11] by means of hermitian holomorphic Deligne cohomology.

The cup product of two metrized line bundles (L, ρ) and (L′, ρ′) in hermitian holomorphic Deligne cohomology produces a class in H_{𝒟_{h.h.}}^{4} (X, 2). We explicitly compute a representative of this class in terms of the expression (3.17) for the class of a metrized line bundle and the modified Beĭlinson product (2.5) for

D(1)_{h.h.}^{•} ⊗ D(1)_{h.h.}^{•} ⟶^{∪_{α}} D(2)_{h.h.}^{•} .

It is convenient to use the diagram for D(1)_{h.h.}^{•} displayed in the second proof to Proposition 3.4 and to use a corresponding diagram for the complex D(2)_{h.h.}^{•}, namely:

(3.18) \begin{aligned}ℤ(2)_{X} & ⟶ & 𝒪_{X} & ⟶ & \underline{Ω}_{X}^{2} ⊕ \underline{Ω}_{X}^{1} & ⟶ & \underline{Ω}_{X}^{3} ⊕ \underline{Ω}_{X}^{2} & ⟶ & \underline{Ω}_{X}^{4} ⊕ \underline{Ω}_{X}^{3} & ⟶ & ⋯ \\ & & ↓^{π_{1}} & & ↓^{ı⊕π_{1}} & & ↓^{ı⊕π_{1}} & & ↓^{ı⊕π_{1}} \\ & & \underline{ℰ}_{X}^{0}(1) & ⟶ & F^{2}\underline{A}_{X}^{2} ⊕ \underline{ℰ}_{X}^{1}(1) & ⟶ & F^{2}\underline{A}_{X}^{3} ⊕ \underline{ℰ}_{X}^{2}(1) & ⟶ & F^{2}\underline{A}_{X}^{4} ⊕ \underline{ℰ}_{X}^{3}(1) & ⟶ & ⋯ \\ ↑ & ȷ⊕0 \\ & & & & & & & & F^{1}\underline{A}_{X}^{4} ∩ \underline{ℰ}_{X}^{4}(2) & ⟶ & ⋯ \end{aligned}

The cup product of two elements of the form (3.17) in the Čech resolution of D(1)_{h.h.}^{•} would result in a 5 × 3 table. When computing the entries, the signs are determined by those in the Beĭlinson product (2.5) plus those arising from the Čech resolution as explained in sect. 1.2. The expression obtained is more manageable for certain specific values of the parameter α. For instance, for α = 0 we have:

(3.19)

c_{ijk}c′_{klm}  −c_{ijk} log g′_{kl}  −d log g_{ij} ∧ d log g′_{jk} ⊕ − log g_{ij} d log g′_{jk}  0  0

0 −π_{1} log g_{ij} log |g′_{jk}| −d log g_{ij} ∧ ∂ log ρ′_{j} ⊕(d^{c} log |g_{ij}| \frac{1}{2} log ρ′_{j} − log |g_{ij}| \frac{1}{2} d^{c} log ρ′_{j}) ∂ log ρ_{i} ∧ ¯∂∂ log ρ′_{i} ⊕ \frac{1}{2} log ρ_{i} ¯∂∂ log ρ′_{i} X

0  0  0  0  ¯∂∂ log ρ_{i} ∧ ¯∂∂ log ρ′_{i}

For general values of the real parameter α, the expression one obtains is rather unwieldy and not so easy to display

(0, 4)  c_{ijk}c′_{klm}
(1, 3)  −c_{ijk} log g′_{kl}
(2, 2)  (−d log g_{ij} ∧ d log g′_{jk} ⊕ − log g_{ij} d log g′_{jk}) ⨁ −π_{1} log g_{ij} log |g′_{jk}|

(3, 1)

(−(1 − α) d log g_{ij} ∧ ∂ log ρ′_{j} + α ∂ log ρ_{i} ∧ d log g′_{ij}) ⊕ \{(1 − α)(d^{c} log |g_{ij}| \frac{1}{2} log ρ′_{j} − log |g_{ij}| \frac{1}{2}d^{c} log ρ′_{j})
+α(−\frac{1}{2}d^{c} log ρ_{i} log |g′_{ij}| + \frac{1}{2} log ρ_{i} d^{c} log |g′_{ij}|)\}

(4, 0) ¯∂∂ log ρ_{i} ∧ ¯∂∂ log ρ′_{i} ⨁(α ¯∂∂ log ρ_{i} ∧ ∂ log ρ′_{i} + (1 − α) ∂ log ρ_{i} ∧ ¯∂∂ log ρ′_{i})

⊕ (α ¯∂∂ log ρ_{i} \frac{1}{2} log ρ′_{i} + (1 − α) \frac{1}{2} log ρ_{i} ¯∂∂ log ρ′_{i})

Table 1: Cup product of (L, ρ) with (L′, ρ′) for generic values of the parameter α.

in the same form as eq. (3.19). Therefore we have put it in table 1, where the corresponding bidegrees are explicitly indicated. (The first degree is the overall degree in the cone and the second is the Čech degree.) Only the nonzero terms have been explicitly written.

If π: X → S is a holomorphic fibration with compact connected Riemann surfaces as fibers, from eq. (3.7) we have that the cup product of (L, ρ) and (L′, ρ′) induces a metrized line bundle on S. From [9, Theorem 6.1] there is a corresponding map A^{(2,2)}(X)_{ℝ(2)} → A^{(1,1)}(S)_{ℝ(1)} . From the explicit cocycles we have computed the corresponding representative in H^{2}(S, ℝ(1)) ∩ H^{1,1}(S) of the first Chern class of the resulting metrized line bundle will be

∫_{π} c_{1}(ρ) ∧ c_{1}(ρ′) ,

as in [12], Proposition 6.6.1.

4 Conformal metrics and the Liouville functional

Let X be a compact Riemann surface of genus g ≥ 2. For simplicity we can assume X to be connected, although this is not necessary, and in fact this assumption will be dropped when dealing with Kleinian groups.

Let 𝒞 ℳ (X) be the space of conformal metrics on X. Locally on X, if z is a local analytic coordinate defined on an open set U, any metric ds^{2} can be represented as

ds^{2} = ρ |dz|^{2}

for a positive function ρ : U → ℝ_{>0}. According to sec. 3.2, a conformal metric corresponds to considering the metrized line bundle (T_{X}, ρ), where T_{X} is the holomorphic tangent bundle of X. With respect to a cover 𝒰_{X} of X, the conformal factors ρ_{i} and ρ_{j}, associated to U_{i} and U_{j} respectively, satisfy the relation

(4.1)  ρ_{j} = ρ_{i} |z′_{ij}|^{2} ,

where z′_{ij} = dz_{i}/dz_{j} and z_{i} is a local coordinate defined over U_{i}.

It follows from the uniformization theorem that there exists a unique conformal metric of scalar curvature equal to −1, the Poincaré metric. Locally on U_{i} ⊂ X, the condition for the metric to have curvature −1 is equivalent to the nonlinear PDE

(4.2)  \frac{∂^{2}}{∂z_{i}∂z̄_{i}} ϕ_{i} = \frac{1}{2} exp ϕ_{i} ,

known as the Liouville equation, for the smooth function ϕ_{i} = log ρ_{i}. Observe that equation (4.2) can be written in the form

c_{1}(ρ) = \sqrt{−1} ω_{ρ} ,

where we have used the Kähler form associated to the metric:

ω_{ρ}|_{U_{i}} = \frac{\sqrt{−1}}{2}ρ_{i} dz_{i} ∧ dz̄_{i} .

This representation makes it apparent that the Liouville equation is independent of the choice of the coordinate system. On the other hand, a direct verification of this fact is immediate using ϕ_{j} − ϕ_{i} = log|z′_{ij}|^{2}.

4.1 Variational problem for conformal metrics

It is well known that the Liouville equation has a local variational principle in the following sense. Let D be a region in the complex plane. Then equation (4.2) is the Euler-Lagrange equation for the variational problem defined by the action functional

(4.3)  S[ϕ] = \frac{\sqrt{−1}}{2} ∫_{D}(∂ϕ ∧ ¯∂ϕ + e^{ϕ} dz ∧ dz̄) ,

defined on smooth functions ϕ : D → ℝ, with the condition that variations \frac{d}{dα}(ϕ_{α})|_{α=0} of ϕ be zero on ∂D. However, it is easily seen that the functional (4.3) cannot be defined globally on X, since, as a consequence of (4.1), the first term under the integral sign would not yield a well-defined 2-form on X. (The second one would of course present no problems, it would just give the area A_{X}(ρ) of X with respect to the given metric ρ |dz|^{2} = e^{ϕ} |dz|^{2}.) Accordingly, it is convenient to write the integrand in (4.3) as

\sqrt{−1} ω[ϕ] + ω_{ρ}

where we have defined the 2-form^{3}

(4.4)  ω^{0}[ϕ] = \frac{1}{2}∂ϕ ∧ ¯∂ϕ ,

and restrict our considerations to the first term of (4.3) which we denote

(4.5)  Š[ϕ] = \sqrt{−1} ∫_{D} ω^{0}[ϕ] .

There is by now an established procedure on how to address the problem caused by the fact that (4.4) is not globally defined. In general terms, given the choice of a conformal metric ds^{2} ∈ 𝒞 ℳ (X) and a cover 𝒰_{X}, one works with the full Čech-de Rham complex Č^{•}(𝒰_{X}, \underline{ℰ}_{X}^{•}(1)) with respect to 𝒰_{X}, rather than with just differential forms. The 2-form (4.4) is then completed to a total degree 2-cocycle—to be denoted Ω[ϕ]. This results in a class in H^{2}(X, ℝ(1)) after taking cohomology. (Whether or not there also is a de Rham type theorem will depend on the acyclicity properties of 𝒰_{X}.)

This scheme has been previously carried out not quite for covers of X by open sets, but actually for different choices of planar coverings. For the covering associated to a Schottky uniformization of X a generalization of eq. (4.3) was written in ref. [35]. More recently, a detailed calculation of the cocycle for the general case of a covering associated to a Kleinian uniformization was carried out in ref. [31] by exploiting the homological methods developed in ref. [2]. Note, however, that from the point of view of ref. [3] these planar coverings are “étale” coordinates on X, so the group cohomology constructions required to work with the various kinds of uniformization coverings just follow from specializing the Čech formalism to the coverings at hand.

Finally, the integration in eqs. (4.3) or (4.5) should be replaced by the evaluation of Ω[ϕ] against an appropriate representative Σ of the fundamental class of X. The “appropriate” form for both Σ and the evaluation will be dictated by the chosen cover 𝒰_{X} and the cohomology theory being used. Typically Σ will be a cycle in a double complex of 𝒰_{X}-small simplices, where the differentials are the singular one and the one determined by the face maps induced by the cover. Thus, in the case of a Čech cover, it will be the complex determined by a triangulation of X subordinated to the open 𝒰_{X}, and in the same way, for a planar cover the singular complex of the planar domain 𝒰_{X} tensored with an appropriate bar resolution of the group of deck transformations. These issues have been discussed at length in refs. [2, 3, 31], so we will not repeat the discussion here. Whenever we have a cocycle extending (4.4) and a cycle Σ representing X we state Definition 4.1. The Liouville functional (without the area term) is given by the evaluation:

3Note that equation (4.4) defines an imaginary form. The reason for this choice will be apparent later.

(4.6)  Š[ϕ] = −\frac{1}{2π\sqrt{−1}}⟨Ω[ϕ] , Σ⟩ .

For the complete functional we add the area term

(4.7)  S[ϕ] = Š[ϕ] + \frac{1}{2π} ∫_{X} ω_{ρ} .

Remark 4.2. ⟨Ω[ϕ] , Σ⟩ ∈ ℝ(1), and Š[ϕ] (or S[ϕ]) is real. Division by 2π\sqrt{−1} is conventional, but note that ℝ(1) ⟶^{≃} ℝ_{+} via exp(·/2π\sqrt{−1}). In the sequel it will be more convenient to work with imaginary classes. (See also section 6.)

In next two subsections we examine these constructions in some detail. For definiteness, we initially make use of an ordinary open cover. First, we recall the direct construction of a cocycle generalizing (4.4), and we show that this way eq. (4.2) indeed is the resulting extremum condition. Then we emphasize the role played by Deligne cohomology and the tame symbol. These aspects will become important when introducing a Kleinian uniformization later in the paper, when we discuss connections with the dilogarithm function. In a later section we shall tackle the question of its geometrical significance by making full use of the hermitian-holomorphic version of Deligne cohomology presented in section 3, and we show that the cocycle constructed following refs. [35, 31] corresponds to the square of (T_{X}, ρ) in hermitian holomorphic Deligne cohomology.

4.2 Direct construction of the Liouville cocycle

4.2.1 Initial setup

Let X a compact Riemann surface of genus greater than 2. We shall not include the area term in our explicit calculations, therefore it makes sense to extend our considerations to a general metrized line bundle (L, ρ). Of course, whenever referring to a conformal metric or to the variational problem for the Liouville equation, it will be be assumed that L = T_{X} and that g_{ij} = z′_{ij}. Upon choosing a cover 𝒰_{X}, which for now we assume to be a Čech cover by open sets, the pair (L, ρ) is described in terms of the data expounded in section 3.2. Our starting point will be the 0-cochain

(4.8)  ω_{i}^{0}[log ρ_{i}] = \frac{1}{2} ∂ log ρ_{i} ∧ ¯∂log ρ_{i} = −\frac{1}{2}d log ρ_{i} ∧ \frac{1}{2}d^{c} log ρ_{i}

with values in \underline{ℰ}_{X}^{2}(1)(U_{i}).

Remark 4.3. A generalization for eq. (4.8) would be to consider a pair of metrized line bundles (L, ρ) and (L′, ρ′), and then the analogue of (4.8) would be

(4.9)  ω_{i}^{0}[log ρ_{i}, log ρ′_{i}] = \frac{1}{2}(−\frac{1}{2}d log ρ_{i} ∧ \frac{1}{2}d^{c} log ρ′_{i} + \frac{1}{2}d^{c} log ρ_{i} ∧ \frac{1}{2}d log ρ′_{i})

Note, however, that the expressions are quadratic. Moreover, L ⊗ L′ has metric ρρ′, so that there is a natural “polarization identity”

(4.10)  ω[log ρ, log ρ′] = \frac{1}{4}ω[log ρρ′] − \frac{1}{4}ω[log \frac{ρ}{ρ′}]

where we have omitted the indexes for simplicity of notation. We shall comment later on the significance of eq. (4.10).

4.2.2 Computation

Let us extend (4.8) to a degree 2 cocycle in the total simple complex associated to the double complex Č^{•}(𝒰_{X}, \underline{ℰ}_{X}^{•}(1)) of Čech cochains with values in the de Rham complex of imaginary smooth forms. This is accomplished in the usual fashion (see e.g. [8]) by finding a 1-cochain of 1-forms ω_{ij}^{1}[log ρ] on U_{ij} and a 2-cochain of 0-forms ω_{ijk}^{2}[log ρ] on U_{ijk} such that the relations

δ̌ω^{0} = −dω^{1} ,  δ̌ω^{1} = dω^{2} ,  δ̌ω^{2} = 0

are satisfied. Of course, the remaining one, namely dω = 0 is automatically satisfied for dimensional reasons. It turns out that to a great extent these relations are explicitly computable without further assumptions, such as that the cover 𝒰_{X} be good. The needed calculations are fairly standard, and they are presented in great detail in ref. [31], so we shall be brief. (The observation in [25] that on a Riemann surface for two smooth functions f and g one has df ∧ d^{c}g = dg ∧ d^{c}f is useful in carrying out the calculations.) The first two steps are as follows.

First, one has:

(4.11)

ω_{j}^{0}[ρ] − ω_{i}^{0}[ρ] = −d ω_{ij}^{1}[ρ]

ω_{ij}^{1}[ρ] = \frac{1}{2} log ρ_{i} d^{c} log |g_{ij}| + d^{c} log |g_{ij}| \frac{1}{2} log ρ_{j}

The next step yields:

δ̌(ω^{1}[log ρ])_{ijk} = ω_{jk}^{1}[ρ] − ω_{ik}^{1}[ρ] + ω_{ij}^{1}[ρ]
= log |g_{ij}| d^{c} log |g_{jk}| − d^{c} log |g_{ij}| log |g_{jk}|

and notice that d^{c} log |g_{ij}| = π_{1}d log g_{ij} , and π_{p+q−1}(a ∧ b) = π_{p}(a) ∧ π_{q−1}(b) + π_{p−1}(a) ∧ π_{q}(b) , so we have

(4.12)  δ̌(ω^{1}[log ρ])_{ijk} = π_{1}(log g_{ij} d log g_{jk}) − d(π_{1}(log g_{ij}) log |g_{jk}|) .

Observe that now the problem of continuing the descent becomes independent of the chosen metric ρ. The most direct way of proceeding is the following. If we assume the cover 𝒰_{X} to be acyclic for the de Rham complex \underline{ℰ}_{X}^{•}(1), then there exists ω_{ijk}^{2} ∈ \underline{ℰ}_{X}^{0}(1)(U_{ijk}) such that dω_{ijk}^{2} = δ̌(ω^{1}[log ρ])_{ijk}. Furthermore, consistency on a quadruple intersection requires that δ̌ω^{2} be a 3-cocycle with values in ℝ(1)_{X}. Since H^{3}(X, ℝ(1)) = 0, this cocycle must be a coboundary, therefore, up to readjusting the constants, there exists a choice of ω_{ijk}^{2} such that δ̌ω^{2} = 0, and furthermore, ω_{ijk}^{2} does not depend on the metric structure.

4.2.3 Solution to the variational problem

The previous preliminary calculation is sufficient from the point of view of finding the extrema. To this effect, we set L = T_{X}, for ρ a conformal metric in 𝒞 ℳ (X). Notice that the space of conformal metrics on X is affine over C^{∞}(X, ℝ) ≡ ℰ^{0}(X): if ds^{2} and ds′^{2} are two conformal metrics with local expressions ρ_{i} |dz_{i}|^{2} and ρ′_{i} |dz_{i}|^{2} respectively, then there exists σ ∈ C^{∞}(X, ℝ) such that log ρ′_{i} = σ|_{U_{i}} +log ρ_{i}. The change from Ω[log ρ] to Ω[log ρ+σ] can be exactly computed thanks to the fact that the last step in the determination of Ω[log ρ] is independent of ρ and the quadratic character of (4.4). Indeed we have:

Proposition 4.4.

S[log ρ + σ] − S[log ρ] = \frac{1}{2π\sqrt{−1}} ∫_{X}(\frac{1}{2}∂σ ∧ ¯∂σ + σ c_{1}(ρ) − \sqrt{−1}(e^{σ} − 1)ω_{ρ})

Proof. The change in ω_{i}^{0}[log ρ] is computed as

ω_{i}^{0}[log ρ + σ] − ω_{i}^{0}[log ρ] = −\frac{1}{2}dσ_{i} ∧ \frac{1}{2}d^{c}σ_{i} + σ_{i} \frac{1}{2}dd^{c} log ρ_{i} − \frac{1}{2}d(σ_{i} d^{c} log ρ_{i}) ,

where we set σ_{i} ≡ σ|_{U_{i}}. Note that the first two terms on the right hand side are globally well-defined 2-forms. On the other hand

ω_{ij}^{1}[log ρ + σ] − ω_{ij}^{1}[log ρ] = σ_{i} d^{c} log |g_{ij}| .

Letting χ_{i} = σ_{i} d^{c} log ρ_{i}, we see that

Ω[log ρ + σ] − Ω[log ρ] = −\frac{1}{2}dσ ∧ \frac{1}{2}d^{c}σ + σ \frac{1}{2}dd^{c} log ρ − Dχ ,

and taking the area terms into account, establishes the formula.

As a consequence, we obtain

Corollary 4.5. The Liouville equation (4.2) is the Euler-Lagrange equation for the Liouville functional (4.7) introduced in definition 4.1. The critical point is non-degenerate.

Proof. Replacing σ with t σ, t ∈ ℝ, in the previous proposition we find the infinitesimal change in Ω[log ρ] to be

\frac{d}{dt}|_{t=0} S[log ρ + tσ] = −\frac{1}{2π\sqrt{−1}} ∫_{X} σ(c_{1}(ρ) − \sqrt{−1} ω_{ρ}) ,

and it follows that S[log ρ] has an extremum if and only if the Liouville equation is satisfied. Non degeneracy follows from the quadratic part in the exact change formula in proposition 4.4.

Remark 4.6. The fact that the change in the cocycle is given by a pure 2-form term up to total coboundary can also be analyzed in terms of gluing properties of variational bicomplexes, cf. Theorem 1.2 in ref. [1] and the proof of Theorem 1 in ref. [3]. From this perspective, Corollary 4.5 is a direct consequence of the affine structure of the space 𝒞 ℳ (X) of conformal metrics.

4.3 A cup product

Formula (4.12) can be handled in a more geometric fashion as follows. From sec. 2.4 we can rewrite (4.12) as

(4.13)  δ̌(ω^{1}[log ρ])_{ijk} = −r_{2}(g_{ij}, g_{jk}) ,

and we have the collection of tame symbols (g_{ij}, g_{jk}] associated any triple intersection U_{ijk} in the cover 𝒰_{X}, [9]. These symbols glue to form a global symbol (L, L]. As a cohomology class on a curve X, however, (L, L] will be zero (that is, there is a global object in the associated 2-stack, cf. [11]), so that it will be possible to choose local functions L_{ijk} such that

dL_{ijk} = − log g_{ij} d log g_{jk} ,

as explained in detail in [3]. Note that we still need the cover 𝒰_{X} to be fine enough. Moreover, cf. loc. cit., the collection L_{ijk} can be chosen in a way that

δ̌L_{ijkl} = −c_{ijk} log g_{kl} + n_{ijkl} ,

where n_{ijkl} ∈ ℤ(2). Therefore from (2.21), (4.12) and (4.13), we can set

(4.14)  ω_{ijk}^{2} = −(π_{1}(log g_{ij}) log |g_{jk}|) − π_{1}L_{ijk} ,

namely according to sec. 2.4 we have set ω_{ijk}^{2} = −ℒ_{2}(g_{ij}, g_{jk}). Now, the last compatibility condition is satisfied, indeed we have:

δ̌ω_{ijkl}^{2} = −c_{ijk} log |g_{kl}| − π_{1}(δ̌L_{ijkl})
= −c_{ijk} log |g_{kl}| + π_{1}(c_{ijk} log g_{kl})
= 0 .

As a result, we obtain a 2-cocycle in the the total complex associated to Č^{•}(𝒰_{X}, \underline{ℰ}_{X}^{•}(1)) as before, with a more geometric interpretation of ω_{ijk}^{2} in terms of the trivialization of the symbol (L, L]. Moreover, notice that if X is a curve then \underline{ℰ}_{X}^{•}(1)[1] ⟶^{≃} { \atop ℝ(2)}_{𝒟}^{•} thus we may interpret the class so determined by Ω[log ρ] = ω^{0}[log ρ] + ω^{1}[log ρ] + ω^{2} as a (degree 3) class in H_{𝒟}^{3} (X, ℝ(2)).

4.4 Two line bundles

For a pair (L, ρ), (L′, ρ′) of metrized line bundles we can complete (4.9) to a cocycle Ω[log ρ, log ρ′] via an analogous procedure to the one presented in sects. 4.2 and 4.3. The relevant calculations being entirely similar, we limit ourselves to quoting the relevant expressions. Starting from (4.9), which we rewrite in the form

(4.15)  ω_{i}^{0}[log ρ_{i}, log ρ′_{i}] = \frac{1}{2}d^{c} log ρ_{i} ∧ \frac{1}{2}d log ρ′_{i} ,

the corresponding expression for the degree (1, 1) term is:

(4.16)  ω_{ij}^{1}[log ρ, log ρ′] = \frac{1}{2} log ρ_{i} d^{c} log |g′_{ij}| + \frac{1}{2}d^{c} log |g_{ij}| log ρ′_{j} .

Computing the Čech coboundary we find:

(4.17)

(δ̌ω^{1}[log ρ, log ρ′])_{ijk} = log |g_{ij}| d^{c} log |g′_{jk}| − d^{c} log |g_{ij}| log |g′_{jk}|
= −r_{2}(g_{ij}, g′_{jk}) ,

from which ω_{ijk}^{2} (now independent of ρ and ρ′) can be obtained as −ℒ_{2}(g_{ij}, g′_{jk}) by looking at a collection L_{ijk} such that

dL_{ijk} = − log g_{ij} d log g′_{jk} ,

from the triviality of the symbol (L, L′].

5 The Liouville functional and the Bloch-Wigner dilogarithm

In this section we specialize some of the constructions in sect. 4, and in particular those in sect. 4.3 to the case of the cover 𝒰_{X} of X given by a Kleinian uniformization. The reader should consult ref. [31] for a complete treatment. Our task here is to point out that the analog of the dilogarithm used in section 4.3 becomes literally the Bloch-Wigner function (2.20), which appears as the obstruction to closing the descent equations for the cocycle defining the Liouville functional. This clarifies the vanishing of the cohomological obstructions in the calculations in loc. cit. The methods of sect. 6 however, sidestep the problem entirely.

5.1 Kleinian groups and fractional linear transformations

As a reference, the reader can consult, among many others, the book [26]. We consider here a finitely generated purely loxodromic non-elementary Kleinian group Γ. We assume the group to be a Kleinian group of the second kind, namely to have a non empty discontinuity region U_{Γ} ⊂ ℙ^{1}. The limit set is L_{Γ} = ℙ^{1} \ U_{Γ}. According to Ahlfors’ finiteness theorem the quotient X = U_{Γ}/Γ is a finite union of analytically finite Riemann surfaces. Thus:

U_{Γ}/Γ = U_{1}/Γ_{1} ⊔ ⋯ ⊔ U_{n}/Γ_{n} ,

where U_{1}, …, U_{n} are the inequivalent components of U_{Γ} and Γ_{1}, …, Γ_{n} their stabilizers. By way of example, a Schottky group has just one component, whereas a Fuchsian or quasi-Fuchsian group will have exactly two components, and the resulting Riemann surfaces are mirror images of one another.

In the language of [3] the map U_{Γ} → X is an étale cover and Čech cohomology with respect to it translates into group cohomology for the group Γ where the coefficient modules are sections over U_{Γ} of the relevant sheaves. The group action is by pull-back. Still according to the conventions in loc. cit. we will write the coboundary operation in the following slightly non-standard way. Assume c is an n-cochain with values in some right Γ-module A. Rather than turning A into a left Γ-module via the inversion γ a: = aγ^{−1}, we use the following expression for the coboundary:

(5.1)  (δ̌c)_{γ_{1},...,γ_{n}} = c_{γ_{2},...,γ_{n}} + ∑_{i=1}^{n−1}(−1)^{i}c_{γ_{1},...,γ_{i}γ_{i+1},...,γ_{n}} + (−1)^{n}(c_{γ_{1},...,γ_{n−1}}) · γ_{n} .

As explained at length in ref. [3], the right action is more suited to handle the case where group acts on geometric objects by pull-back, which is a right action if the group Γ acts on U_{Γ} from the left, as we implicitly assume. Correspondingly, (5.1) is the Čech coboundary applied to the nerve of the cover U_{Γ} → X. The convention in eq. (5.1) avoids a host of inversions in the group in order to have a more conformant left action.

We assume that Γ is normalized, namely the point ∞ belongs to the limit set L_{Γ}. Finally, let us set some more conventions. Let γ be an element of Γ represented as a fractional linear transformation:

ℙ^{1} ∋ z ⟼ γ(z) = \frac{az + b}{cz + d} .

Then we can write the derivative with respect to z as:

(5.2)  γ′(z) = \frac{det γ}{c^{2}(z − z_{γ})^{2}}

where z_{γ} = −\frac{d}{c} ≅ γ^{−1}(∞). We will set

c(γ) ≡ c_{γ} =^{def} \frac{det γ}{c^{2}} .

The following properties are easily verified. If z_{0} and z_{∞} are the attracting and repelling fixed points, respectively, then

c_{γ} = \frac{(z_{0} − z_{∞})^{2}λ_{γ}}{(1 − λ_{γ})^{2}} ,

where λ_{γ} is the dilating factor. For γ_{i} and γ_{j} two elements of Γ, denote:

z_{i} = γ^{−1}(∞) , z_{ij} = (γ_{i}γ_{j})^{−1}(∞) = γ_{j}^{−1}(z_{i}) , c_{i} = c_{γ_{i}} , c_{ij} = c_{γ_{i}γ_{j}} .

Then the following relation is easily checked:

(5.3)  c_{γ_{1}γ_{2}} = \frac{c_{γ_{1}}}{c_{γ_{2}}} (z_{12} − z_{2})^{2} .

Finally, given four points z_{1}, z_{2}, z_{3}, z_{4} ∈ ℙ^{1}, we define their cross-ratio by:

(5.4)  [z_{1} : z_{2} : z_{3} : z_{4}] = \frac{(z_{1} − z_{2})(z_{3} − z_{4})}{(z_{1} − z_{4})(z_{3} − z_{2})} .

5.2 Computation of the cocycle

A conformal metric ρ on X = X_{1} ⊔ ⋯ ⊔ X_{n} appears as an automorphic function on U_{Γ}:

(5.5)  \frac{1}{2} log ρ − \frac{1}{2} log ρ ∘ γ = log |γ′|

for γ ∈ Γ. Eq. (5.5) is the direct translation of (3.8) following the principles of [3]. Accordingly, on has that the first two terms of the Liouville cocycle computed by applying the procedure explained in sect. 4 are:

(5.6) ω^{0}[log ρ] = −\frac{1}{2}d log ρ ∧ \frac{1}{2}d^{c} log ρ

(5.7) ω_{γ}^{1}[log ρ] = (\frac{1}{2} log ρ + \frac{1}{2} log ρ ∘ γ) d^{c} log |γ′|

Computing the coboundary of the term in (5.7) according to:

(δ̌ω^{1})_{γ_{1},γ_{2}} = ω_{γ_{2}}^{1} − ω_{γ_{1}γ_{2}}^{1} + (γ_{2})^{∗}ω_{γ_{1}}^{1} ,

where we have suppressed the dependency on log ρ for simplicity of notation, yields, analogously to what we have seen in sect. 4.3:

(5.8)  δ̌(ω^{1}[log ρ])_{γ_{1},γ_{2}} = −r_{2}((γ_{1}γ_{2})′, γ′_{2}) ,

where r_{2} has been introduced in eqs. (2.18) and (2.19). According to sect. 2, and in particular sect. 2.4, the coboundary of ω^{1} is the cup product of the two real Deligne cohomology classes (d log(γ_{1}γ_{2})′, log |(γ_{1}γ_{2})′|) and (d log γ′_{2}, log |γ′_{2}|) associated to the rational functions (γ_{1}γ_{2})′ and γ′_{2}. Thus we will have to consider a double complex of group cochains on Γ with values in Deligne complexes on the region of discontinuity U_{Γ}. Notice that from this point on only rational functions with singularities at certain prescribed points will appear. Hence, following ref. [21], we can consider the Deligne complex on the generic point η_{ℙ^{1}} of ℙ^{1}. Now, as before, the automorphy factor describing the metric has disappeared and the problem of finding a function ω^{2} such that its derivative equals −r_{2} involves only objects related to the geometry of Γ.

As in sect. 5.1, for γ_{i}, γ_{j} ∈ Γ, consider the points z_{j}, z_{ij}—the inverse images of ∞—and define the element T_{ij} ∈ PSL_{2}(ℂ) by

(5.9)  z ⟼ T_{i,j}(z) = [z: z_{ij}: z_{j}: ∞] .

A fairly straightforward calculation, using (5.3) and the notation γ_{12} ≡ γ_{1}γ_{2}, shows that

r_{2}((γ_{1}γ_{2})′, γ′_{2}) = 4 T_{1,2}^{∗}(r_{2}(1 − z, z)) + 2 log|γ′_{12}(z_{2})| π_{1}d log(z − z_{2})

− 2 log|γ′_{2}(z_{12})| π_{1}d log(z − z_{12}) , with a slight abuse of notation in the first term on the right hand side. Therefore from eqs. (2.20) and (2.21) we have

r_{2}((γ_{1}γ_{2})′, γ′_{2}) = 4 d(T_{1,2}^{∗}ℒ_{2})(z) + 2 log|γ′_{12}(z_{2})| π_{1}d log(z − z_{2})

− 2 log|γ′_{2}(z_{12})| π_{1}d log(z − z_{12}) . (5.10)

In analogy with ref. [31] we introduce the 1-cochain on Γ with values in \underline{ℰ}_{ℙ^{1}}^{1}(1)(η_{ℙ^{1}}):

(5.11)  κ_{γ} = log |c_{γ}| π_{1}d log γ′ .

Obviously dκ_{γ} = 0, and moreover we have:

Lemma 5.1. δ̌κ_{γ_{1},γ_{2}} = 2 log|γ′_{12}(z_{2})| π_{1}d log(z − z_{2}) − 2 log|γ′_{2}(z_{12})| π_{1}d log(z − z_{12}) .

Proof. A straightforward calculation exploiting relation (5.3).

Using the lemma, from eq. (5.8) we have:

δ̌(ω^{1}[log ρ])_{γ_{1},γ_{2}} = −4 d(T_{1,2}^{∗}ℒ_{2})(z) − δ̌κ_{γ_{1},γ_{2}} ,

so that, upon redefining ω_{γ}^{1} → ω_{γ}^{1} + κ_{γ} , we have the last relation δ̌ω^{1} = dω^{2} with:

(5.12)  ω_{γ_{1},γ_{2}}^{2} = −4 (T_{1,2}^{∗}ℒ_{2}) .

5.3 Cross ratio, tetrahedra and the Block-Wigner function

Relation eq. (5.12) is suggestive, in that the true Bloch-Wigner function, as presented in sect. 2.4 appears explicitly. However, we need to check the closure of the descent condition, or in other words, to compute the coboundary of ω^{2} to ensure that we have obtained a cocycle.

For three elements γ_{1}, γ_{2}, γ_{3} ∈ Γ we have the following calculation:

δ̌ω_{γ_{1},γ_{2},γ_{3}}^{2} = ω_{γ_{2},γ_{3}}^{2} − ω_{γ_{1}γ_{2},γ_{3}}^{2} + ω_{γ_{1},γ_{2}γ_{3}}^{2} − γ_{3}^{∗}ω_{γ_{1},γ_{2}}^{2}
= −4(ℒ_{2}([z: z_{23} : z_{3} : ∞]) − ℒ_{2}([z: z_{123} : z_{3} : ∞])
+ ℒ_{2}([z: z_{123} : z_{23} : ∞]) − ℒ_{2}([γ_{3}(z): z_{12} : z_{2} : ∞]))

By the PSL_{2}(ℂ) invariance property of the cross-ratio, in the last term in the previous relation we have

[γ_{3}(z): z_{12} : z_{2} : ∞] = [z: z_{123} : z_{23} : z_{3}] .

Recall the 5-term relation satisfied by the Bloch-Wigner dilogarithm 𝒟_{2} [16, 23]: for five points a_{0}, …, a_{4} ∈ ℙ^{1} we have the relation

(5.13)  ∑_{i=0}^{4}(−1)^{i}𝒟_{2}([a_{0} : ⋯: â_{i} : ⋯: a_{4}]) = 0 ,

where the hat sign denotes omission. As a consequence we have:

δ̌ω_{γ_{1},γ_{2},γ_{3}}^{2} = −4 ℒ_{2}([z_{123} : z_{23} : z_{3} : ∞]) ,

and again by the invariance of the cross ratio, we finally obtain:

(5.14)  δ̌ω_{γ_{1},γ_{2},γ_{3}}^{2} = −4 ℒ_{2}([∞: γ_{1}∞: γ_{1}γ_{2}∞: γ_{1}γ_{2}γ_{3}∞]) .

The right hand side of eq. (5.14) defines an ℝ(1)-valued 3-cocycle on PSL_{2}(ℂ), restricted to Γ, thanks to the five-term relation. We can summarize the foregoing as follows:

Theorem 5.2. The obstruction to closing the descent equations relative to the cover U_{Γ} → X associated to a Kleinian uniformization, is given by the universal volume class. This obstruction is zero for Γ of the second kind.

Proof. First, it is clear from the calculations above that for the total coboundary we have

D(ω^{0} + ω^{1} + ω^{2}) = −4 ℒ_{2}([∞: γ_{1}∞: γ_{1}γ_{2}∞: γ_{1}γ_{2}γ_{3}∞]) .

The only thing left to do is to collect the necessary results from the literature. From refs. [17, 15] we have that the hyperbolic volume of a totally ideal tetrahedron is computed by evaluating 𝒟_{2} at the cross ratio of its vertices. Now consider the four points in ℙ^{1} obtained by applying three elements of PSL_{2}(ℂ) to ∞ ∈ ℙ^{1} as in eq. (5.14). From[15], we have that computing the Bloch-Wigner dilogarithm at their cross ratio yields the imaginary part of the second Cheeger-Simons universal secondary class which generates the third cohomology group of PSL_{2}(ℂ) with values in ℝ(1). Pulling back this class along Γ ↪ PSL_{2}(ℂ) we obtain zero, since the cohomology of Γ is computed by the 3-manifold ℍ^{3}/Γ, which, being non-compact has zero three-dimensional cohomology.

6 Conformal metrics and hermitian holomorphic cohomology

In sect. 4 we presented a construction of a degree 2, ℝ(1)-valued class corresponding to a conformal metric ρ ∈ 𝒞 ℳ (X), represented by the cocycle Ω[log ρ]. Supplemented by the area of X computed with respect to ρ, it provides a global functional for the variational problem associated to the Liouville equation (4.2). We now show that it coincides with the square of the class of (T_{X}, ρ) in hermitian holomorphic Deligne cohomology introduced in sect. 3. Moreover, we show that this equality holds at the cocycle level. More generally, without considering the area term, we show that the cup product of (L, ρ) and (L′, ρ′) in hermitian holomorphic Deligne cohomology coincides with the class of Ω[log ρ, log ρ′], and again the equality in fact holds at the cocycle level.

6.1 Comparison on a curve

Let X be a compact Riemann surface. From the results of sect. 3.3, the cup product of the classes of (L, ρ) and (L′, ρ′) in hermitian holomorphic Deligne cohomology yields a class in H_{𝒟_{h.h.}}^{4} (X, 2), and on a curve we only capture the 2-dimensional part of this class. Indeed, in the exact sequence (3.4), the cohomology class corresponding to the symbol (L, L′] ∈ H_{𝒟}^{4} (X, ℤ(2)) is zero, and A^{(2,2)}(X)_{ℝ(2)} is also zero for obvious dimensional reasons, so we have:

⋯ ⟶ H_{𝒟}^{3} (X, ℝ(2)) ⟶ H_{𝒟_{h.h.}}^{4} (X, 2) ⟶ 0 .

It follows that the class [L, ρ] ∪ [L′, ρ′] must come from an element in H_{𝒟}^{3} (X, ℝ(2)). As already remarked, on a curve we have { \atop ℝ(2)}_{𝒟}^{•} ⟶^{≃} \underline{ℰ}_{X}^{•}(1)[−1], thus H_{𝒟}^{3} (X, ℝ(2)) ≅ H^{3}(X, \underline{ℰ}_{X}^{•}(1)[−1]) ≅ H^{2}(X, \underline{ℰ}_{X}^{•}(1)) ≅ H^{2}(X, ℝ(1)), in agreement with the calculations performed in sect. 4.

More in detail, in complex dimension 1 the second hermitian holomorphic Deligne complex D(1)_{h.h.}^{•} simplifies considerably and diagram (3.18) reduces to

(6.1)  \begin{aligned}ℤ(2) & ⟶^{−ı} & 𝒪_{X} & ⟶^{−d} & \underline{Ω}_{X}^{1} \\ & & ↓^{π_{1}} & & ↓^{π_{1}} \\ & & \underline{ℰ}_{X}^{0}(1) & ⟶^{−d} & \underline{ℰ}_{X}^{1}(1) & ⟶^{−d} & \underline{ℰ}_{X}^{2}(1) \end{aligned}

so that D(2)_{h.h.}^{•} becomes just the cone of the morphism ℤ(2)_{𝒟}^{•} ⟶^{π_{1}} \underline{ℰ}_{X}^{•}(1)[−1]. In other words, on a curve X we have that D(2)_{h.h.}^{•} is given by the complex

(6.2)  ℤ(2)_{X} ⟶^{−ı} 𝒪_{X} ⟶^{(−d,−π_{1})} \underline{Ω}_{X}^{1} ⊕ \underline{ℰ}_{X}^{0}(1) ⟶^{−π_{1}+d} \underline{ℰ}_{X}^{1}(1) ⟶^{d} \underline{ℰ}_{X}^{2}(1) ,

where the differentials have been written explicitly. We can see the complex \underline{ℰ}_{X}^{•} appears as a subcomplex in (6.2) and the shift of two positions to the right clearly accounts for the cohomology degree shift from 2 to 4.

Our main result is the following comparison

Theorem 6.1. Let X be a compact Riemann surface of genus g > 1. Let (L, ρ) and (L′, ρ′) be two hermitian holomorphic line bundles. The class of [L, ρ] ∪ [L′, ρ′] in H_{𝒟_{h.h.}}^{4} (X, 2) ≅ H^{2}(X, ℝ(1)) coincides with the one represented by the cocycle Ω[log ρ, log ρ′] constructed in section 4.

Proof. We have observed above that H_{𝒟_{h.h.}}^{4} (X, 2) ≅ H^{2}(X, ℝ(1)), and by construction the class of Ω[log ρ, log ρ′] is in H^{2}(X, \underline{ℰ}_{X}^{•}(1)) ≅ H^{2}(X, ℝ(1)). Note that for X connected they must coincide up to a proportionality factor, since H^{2}(X, ℝ(1)) ≅ ℝ(1) in this case. In general, we compute the proportionality factor from the explicit cocycles from sects. 3.3 and 4.2 to 4.4. (Since Ω[log ρ, log ρ′] is computed under suitable acyclicity assumptions on the cover, so we will use such a cover to establish the comparison.)

Let us assume L and L′ and their respective hermitian metric structures are represented by cocycles of type (3.17) with respect to the chosen cover 𝒰_{X}. Specializing the general expression in table 1 in sect. 3.3 to the case at hand we obtain, with reference to (6.2):

(6.3)

(0, 4)  c_{ijk}c′_{klm}
(1, 3)  −c_{ijk} log g′_{kl}
(2, 2)  − log g_{ij} d log g′_{jk} ⊕ −π_{1} log g_{ij} log |g′_{jk}|
(3, 1) (1 − α)(d^{c} log |g_{ij}| \frac{1}{2} log ρ′_{j} − log |g_{ij}| \frac{1}{2}d^{c} log ρ′_{j}) + α(−\frac{1}{2}d^{c} log ρ_{i} log |g′_{ij}| + \frac{1}{2} log ρ_{i} d^{c} log |g′_{ij}|)
(4, 0)  α ¯∂∂ log ρ_{i} \frac{1}{2} log ρ′_{i} + (1 − α) \frac{1}{2} log ρ_{i} ¯∂∂ log ρ′_{i}

where we have followed the convention explained in the introduction for the bidegrees. Let us denote by θ^{i} the term of bidegree (4 − i, i) in (6.3) and by Θ the total cocycle. (For simplicity, we suppress ρ and ρ′ from the notation.) A direct calculation shows that

θ_{i}^{0} = ω_{i}^{0} + dλ_{i}^{0} , θ_{ij}^{1} = ω_{ij}^{1} − δ̌λ_{ij}^{0} ,

where ω_{i}^{0} is given by eq. (4.15), ω^{1} is given by eq. (4.16), and

λ_{i}^{0} = α \frac{1}{2}d^{c} log ρ_{i} log ρ′_{i} + (1 − α) \frac{1}{2} log ρ_{i} d^{c} log ρ′_{i} .

Note that at this point we could simply define Ω = Θ − Dλ^{0}. Furthermore, note that Ω does not explicitly depend on the parameter α from the Beĭlinson product (2.5). To finish the comparison, let us assume the cover 𝒰_{X} allows us to find a collection L_{ijk} ∈ 𝒪_{X}(U_{ijk}) such that

(6.4) dL_{ijk} = − log g_{ij} d log g′_{jk} δ̌L_{ijkl} = −c_{ijk} log g′_{kl} + n_{ijkl} ,

as in sect. 4.3. In this way we have Ω = (ω^{0}, ω^{1}, ω^{2}) with ω_{ijk}^{2} = −ℒ_{2}(g_{ij}, g′_{jk}). Ω is a cocycle of total degree 2 in Tot Č^{•}(𝒰_{X}, \underline{ℰ}_{X}^{•}), and it injects (via the exact sequence of the cone) into Tot Č^{•}(𝒰_{X}, D(2)_{h.h.}^{•}) as

(ω^{0}, ω^{1}, 0 ⊕ ω^{2}) .

Then via equations (6.4) it is easily seen that

θ_{ijk}^{2} = d_{D(2)_{h.h.}^{•}}(−L_{ijk}) + 0 ⊕ ω_{ijk}^{2} ,

where d_{D(2)_{h.h.}^{•}} is the differential in D(2)_{h.h.}^{•}, and therefore

Θ = (ω^{0}, ω^{1}, 0 ⊕ ω^{2}) + Dλ^{0} + D(−L, n) ,

where we have put D = d_{D(2)_{h.h.}^{•}} ± δ̌ for the total differential. Thus the two cocycles constructed via the direct method of sect. 4.2 and the cup product of metrized bundles define the same class. By direct comparison, the proportionality factor is 1.

In light of the previous theorem, the polarization identity in Remark 4.3 is now easily explained. Using [L ⊗ L′, ρρ′] = [L, ρ] + [L′, ρ′] and [L ⊗ L′^{∨}, ρ/ρ′] = [L, ρ] − [L′, ρ′] , and the (graded) commutativity of the cup product

H_{𝒟_{h.h.}}^{2} (X, 1) ⊗ H_{𝒟_{h.h.}}^{2} (X, 1) ⟶^{∪} H_{𝒟_{h.h.}}^{4} (X, 2) ,

we obtain the polarization identity

4[L, ρ] ∪ [L′, ρ′] = [L ⊗ L′, ρρ′]^{2} − [L ⊗ L′^{∨}, ρ/ρ′]^{2} ,

where the squares in the right hand side refer to ∪. A polarization identity at the level of representative cocycles, and hence the one in Remark 4.3, follow by applying Thm. 6.1 to the latter identity.

By choosing L = L′ = T_{X}, the holomorphic tangent bundle of X, we immediately obtain:

Corollary 6.2. Let ρ ∈ 𝒞 ℳ (X) be a conformal metric. The Liouville functional without area term (4.6) is given by the (evaluation of) the square [T_{X}, ρ] ∪ [T_{X}, ρ] with respect to the cup product in hermitian holomorphic Deligne cohomology. The full-fledged Liouville functional is obtained by adding the area term \frac{1}{2π} ∫_{X} ω_{ρ} to (4.6).

Remark 6.3. Due to the specific form of the differential in the complex D(2)_{h.h.}^{•} the descent equations are explicit and close automatically. Therefore the cocycle resulting from the calculation of the cup product sidesteps the problem of the explicit calculation of the last term, unlike the more direct version from sect. 4. Thus, thanks to the explicit character of the calculation, specific assumptions on the nature of the cover 𝒰_{X} are not required.

It follows from Thm 6.1, corollary 6.2 and the previous remark that definition 4.1 applies to any (étale) cover 𝒰_{X} → X. Indeed, proposition 4.4 from sect. 4.2 can be reformulated at the cocycle level as follows:

Proposition 6.4. Let X be a compact, genus g > 1 Riemann surface and let 𝒰_{X} → X be a cover. For a conformal metric ρ ∈ 𝒞 ℳ (X) and σ ∈ C^{∞}(X, ℝ), there is a cocycle Ω̂_{𝒰_{X}} [log ρ] solving the variational problem for the Liouville equation.

Proof. If ρ is a conformal metric, let the pair (T_{X}, ρ) be represented, as an hermitian line bundle, by a cocycle c(T_{X}, ρ) with respect to the cover 𝒰_{X}. We set

Ω[log ρ] = c(T_{X}, ρ) ∪ c(T_{X}, ρ) ,

and a simple calculation starting from eq. (6.3) yields

Ω[log ρ + σ] − Ω[log ρ] = σ c_{1}(ρ) + \frac{1}{2} σ ∂ ¯∂σ + Dχ ,

where χ_{i} = \frac{1}{2}σ \frac{1}{2}d^{c} log ρ_{i} − \frac{1}{2}d^{c}σ \frac{1}{2} log ρ_{i} . Now define

Ω̂_{𝒰_{X}} [log ρ] = Ω[log ρ] − \sqrt{−1}ω_{ρ} .

We see that it yields the formula in Proposition 4.4. In particular we have that

\frac{d}{dt}|_{t=0} Ω̂_{𝒰_{X}} [log ρ + tσ] ≡ σ(c_{1}(ρ) − \sqrt{−1}ω_{ρ}) ,

where ≡ means “up to total coboundary.”

6.2 Determinant of cohomology

Let again L and L′ be two holomorphic line bundles with hermitian metrics ρ and ρ′, respectively, on the compact Riemann surface X. Brylinski proves in [11] that the cup product of L and L′ in hermitian holomorphic Deligne cohomology yields the (logarithm) of the metric ∥·∥ on the Deligne pairing ⟨L , L′⟩ defined in [12].

It follows, via Thm 6.1 and the isomorphism 3.2 between our version of hermitian holomorphic cohomology and Brylinski’s, that the class of Ω[log ρ, log ρ′] is also equal to log∥⟨L , L′⟩∥. It is worthwhile to provide a direct proof of this fact starting from the explicit cocycle given in (6.3).

First, we need to recall a few definitions from [12]. A complex line ⟨L , L′⟩ is assigned to the pair (L, L′) as follows. Let D and D′ be divisors on X corresponding to L and L′, and assume they have disjoint supports. Consider two rational sections, s and s′ such that (s) = D and (s′) = D′. To this datum one assigns a copy of the complex line generated by the symbol ⟨s , s′⟩ subject to the relations:

(6.5)  \begin{aligned}⟨fs , s′⟩ & = f(D′)⟨s , s′⟩ \\ ⟨s , gs′⟩ & = g(D)⟨s , s′⟩ \end{aligned}

whenever f is a rational function with divisor (f) disjoint from D′, and similarly for g. The Weil reciprocity relation f(div(g)) = g(div(f)) (cf. ref. [22]) for two rational functions f and g with disjoint divisors implies that the relations (6.5) are consistent and the complex line depends only on the pair L, L′. When the line bundles are equipped with hermitian metrics, generically denoted by ∥·∥, the assignment^{4}

(6.6)  log ∥⟨s , s′⟩∥^{2} = \frac{1}{2π\sqrt{−1}} ∫_{X} ∂ ¯∂log∥s∥^{2} log∥s′∥^{2} + log∥s∥^{2}[D′] + log∥s′∥^{2}[D]

is compatible with the relations (6.5) and defines an hermitian metric on the complex line ⟨L , L′⟩. In formula (6.6) the operator ∂ ¯∂ is to be computed in the sense of distributions.

Having covered the main definitions, we can now state Theorem 6.5. The cup product of (L, ρ) with (L′, ρ′) in hermitian holomorphic Deligne cohomology corresponds to the logarithm of the norm (6.6) on the Deligne pairing ⟨L , L′⟩. The proportionality factor is −π \sqrt{−1}.

4We write the square explicitly, whereas the symbol ∥·∥ used in ref. [12] denotes the square of the norm.

Proof. Let D and D′ be divisors with disjoint support on X corresponding to L and L′, respectively. Using the same technique as in refs. [11, 12], consider two C^{∞} positive real functions f_{1} and f_{2} such that f_{1} + f_{2} = 1 and f_{1} (resp. f_{2}) vanishes in a neighborhood of the support of D′ (resp. D). Also, set U_{1} = X \ supp(D′) and U_{2} = X \ supp(D). Thus {f_{1}, f_{2}} is just a partition of unity subordinated to the cover 𝒰_{X} = {U_{1}, U_{2}}.

The only two terms different from zero in the cocycle Θ in (6.3) representing the class [L, ρ]∪[L′, ρ′] with respect to this cover are θ_{i}^{0} and θ_{i}^{1}, with the Čech index i ∈ {1, 2}. Thus the class we are after is equivalently given by the integral

(6.7)  ∫_{X} f_{1}θ_{1}^{0} + f_{2}θ_{2}^{1} + df_{2} ∧ θ_{21}^{1} ,

which is arrived at by applying in the standard homotopy operator based on the partition of unity {f_{1}, f_{2}}: from δ̌θ_{12}^{1} = 0 we obtain that θ_{12}^{1} is the coboundary of the cochain j → ∑_{i=1,2} f_{i}θ_{ij}^{1} and then we use θ_{2}^{0} − θ_{2}^{0} = −dθ_{12}^{1}. Observe that the 2-form in (6.7) is globally well defined over X: θ_{21}^{1} is defined only on U_{1} ∩ U_{2}, but df_{2} has support on U_{1} ∩ U_{2}, so their wedge product is defined everywhere; similarly, f_{i} θ_{i}^{0} is everywhere defined thanks to the fact that f_{i} has support in U_{i}, i = 1, 2.

Consider rational sections s and s′ of L and L′ such that div(s) = D and div(s′) = D′ as above. With respect to the two-element cover 𝒰_{X} = {U_{1}, U_{2}}, the section s corresponds to the pair {s_{1}, s_{2}}, and similarly for s′ = {s′_{1}, s′_{2}}. Since supp(D) is contained in U_{1} but not in U_{2}, and the other way around for D′, it follows that s_{2} and s′_{1} are actually invertible functions over their respective domains. Following [25], we can assume that s and s′ are in fact the rational section 1, so that s_{2} = 1 and s′_{1} = 1, and therefore:

|s|_{ρ}^{2} = \begin{cases}log ρ_{1} + log|s_{1}|^{2} & on U_{1} , \\ log ρ_{2} & on U_{2} , \end{cases}

and

|s′|_{ρ′}^{2} = \begin{cases}log ρ′_{1} & on U_{1} , \\ log ρ′_{2} + log |s′_{2}|^{2} & on U_{2} . \end{cases}

Furthermore, log |g′_{21}| = log |s′_{2}| on U_{1} ∩ U_{2}. Let us denote by ∥·∥ = |·|_{ρ} = |·|_{ρ′} for simplicity. Using the relevant entries from (6.3) we have:

θ_{1}^{0} = c_{1}(ρ) log∥s′∥ ,

θ_{2}^{0} = ¯∂∂ log ρ_{2} \frac{1}{2} log ρ′_{2} = dd^{c} log ∥s∥ \frac{1}{2} log ρ′_{2} .

An elementary integration by parts leads to:

∫_{X} f_{2}θ_{2}^{0} = ∫_{X} f_{2} log ∥s∥ c_{1}(ρ′)

+ ∫_{X} log ∥s∥ df_{2} ∧ \frac{1}{2}d^{c} log ρ′_{2} − ∫_{X} \frac{1}{2} log ρ′_{2} df_{2} ∧ d^{c} log ∥s∥ .

On the other hand, we have

∫_{X} df_{2} ∧ θ_{12}^{1} = ∫_{X} log ∥s∥ df_{2} ∧ d^{c} log |s′_{2}| − ∫_{X} log |s′_{2}| df_{2} ∧ d^{c} log ∥s∥ ,

and putting all terms together we obtain

∫_{X} f_{1}θ_{1}^{0} + f_{2}θ_{2}^{1} + df_{2} ∧ θ_{21}^{1} = ∫_{X} f_{1} c_{1}(ρ) log∥s′∥ + ∫_{X} f_{2} log ∥s∥ c_{1}(ρ′)
+ ∫_{X} log ∥s∥ df_{2} ∧ d^{c} log ∥s′∥ + ∫_{X} log ∥s′∥ df_{1} ∧ d^{c} log ∥s∥

(6.8) which, if expressed in terms of the squares of the norms, is (up to a factor) the logarithm of ∥⟨s , s′⟩∥, as it is found in [12, formula 6.5.1]. This version is due to O. Gabber. Via the Poincaré-Lelong lemma (see, e.g. [22])

¯∂∂ log ∥s∥ = c_{1}(ρ) + 2π\sqrt{−1} [D] ,

where [D] is the delta-current supported at the divisor of s, and similarly for s′, formula (6.8) can be recast into:

∫_{X} f_{1}θ_{1}^{0} + f_{2}θ_{2}^{1} + df_{2} ∧ θ_{21}^{1} = ∫_{X} dd^{c} log ∥s∥ log ∥s′∥ − 2π\sqrt{−1} log ∥s∥[D′] − 2π\sqrt{−1} log ∥s′∥[D]
= −2π\sqrt{−1} log ∥⟨s , s′⟩∥

(6.9)

which is what we wanted to show.

This allows us to recast the Liouville functional for conformal metrics on X in the following form.

Corollary 6.6. The exponential of the Liouville functional defines an hermitian metric on the complex line ⟨T_{X} , T_{X}⟩, namely for a conformal metric ρ ∈ 𝒞 ℳ (X) we have:

exp S[log ρ] = ∥⟨T_{X} , T_{X}⟩∥ exp \frac{1}{2π} A_{X}(ρ) .

Remark 6.7. The above corollary justifies the choice made in Definition 4.1 for the various factors 2π\sqrt{−1}.

A similar result has been obtained in ref. [33] by considering the Liouville action functional defined on the Schottky space, and in fact the results in loc. cit. are formulated in terms of a Schottky family.

Indeed, the statement in Corollary 6.6 can be immediately reformulated for a family π: X → S with base parameter space S by considering the relative holomorphic tangent line bundle T_{X/S} with an hermitian fiber metric ρ. (Thus ρ_{s} ∈ 𝒞 ℳ (X_{s}) for every fiber X_{s} , s ∈ S.) Notice that the fiber metric ρ needs not be critical (i.e. satisfying the fiberwise constant negative curvature condition).

A Cones

In the main body of the paper we have used iterated cones to define the hermitian holomorphic Deligne complexes. One technical problem one has to face concerns the homotopy (graded) commutativity of the modified Beĭlinson product defined in eq. (2.5). A problem arises because the factors in the cones are cones themselves and therefore they have multiplication structures which are graded commutative up to homotopy to begin with. We want to show that even in this situation the final resulting product on cones is again homotopy graded commutative. This ensures that on cohomology the product will be genuinely graded commutative, so that in particular hermitian holomorphic Deligne cohomology as defined in section 3 has the correct product structure.

A.1 Cones and homotopies

We consider the following situation. For i = 1, 2, 3 we have maps of complexes: f_{i}: A_{i}^{•} → B_{i}^{•} , and for i < j maps a_{ji}: A_{i}^{•} → A_{j}^{•} and b_{ji}: B_{i}^{•} → B_{j}^{•} . Also, let C^{•}(f_{i}) = Cone(f_{i} : A_{i}^{•} → B_{i}^{•}) , for i = 1, 2, 3.

First, consider the homotopy commutative diagram:

(A.1)  [figure: A_{j}^{•}; f_{j}; a_{ij}; B_{j}^{•}; b_{ij}; s_{ij}; A_{i}^{•}; f_{i}; B_{i}^{•}]

where s_{ij}: A_{j}^{•} → B_{i}^{•−1} is the homotopy map of complexes:

f_{i}a_{ij} − b_{ij}f_{j} = d s_{ij} + s_{ij}d .

An immediate verification yields Lemma A.1. The diagram (A.1) can be extended to

[figure: A_{j}^{•}; f_{j}; a_{ij}; B_{j}^{•}; b_{ij}; s_{ij}; C^{•}(f_{j}); c_{ij}; [1]; A_{j}^{•}; a_{ij}; A_{i}^{•}; f_{i}; B_{i}^{•}; C^{•}(f_{i}); [1]; A_{i}^{•}]

where the map c_{ij} is given by ( \begin{matrix}a_{ij} & 0 \\ −s_{ij} & b_{ij} \end{matrix} ) and the squares containing the cones are in fact strictly commutative.

Remark A.2. This lemma is nothing other than the statement that any homotopy commutative diagram of the form (A.1) in the category of complexes in an abelian category can be extended to a (homotopy) commutative diagram of distinguished triangles, that is, one of the axioms defining a triangulated category, see, e.g. [32].

For k < j < i consider the homotopy commutative triangle

[figure: A_{j}^{•}; a_{ij}; α_{ijk}; A_{k}^{•}; a_{ik}; a_{jk}; A_{i}^{•}]

where a_{ik} − a_{ij}a_{jk} = dα_{ijk} + α_{ijk}d , and similarly for the complexes B_{i}^{•}, with a corresponding homotopy β_{ijk}. Thus α_{ijk}: A_{k}^{•} → A_{i}^{•−1} and β_{ijk}: B_{k}^{•} → B_{i}^{•−1} . Now consider the diagram:

(A.2)  [figure: A_{k}^{•}; a_{jk}; a_{ik}; f_{k}; A_{j}^{•}; a_{ij}; f_{j}; A_{i}^{•}; f_{i}; B_{k}^{•}; b_{jk}; b_{jk}; B_{j}^{•}; b_{ij}; B_{i}^{•}]

The faces in (A.2) are homotopy commutative, however we assume that composing the faces is strictly commutative, namely the two possible homotopies b_{ij} b_{jk} f_{k} ⟹ f_{i} a_{ik} must be equal. Concretely, this corresponds to the relation

(A.3)  s_{ik} + β_{ijk} f_{k} = f_{i} α_{ijk} + s_{ij} a_{jk} + b_{ij} s_{jk} .

We have:

Lemma A.3. The map

( \begin{matrix}−α_{ijk} & 0 \\ 0 & β_{ijk} \end{matrix} ): C^{•}(f_{k}) ⟶ C^{•−1}(f_{i})

realizes the homotopy

[figure: C^{•}(f_{j}); c_{ij}; C^{•}(f_{k}); c_{jk}; c_{ik}; C^{•}(f_{i})]

Proof. It is an elementary calculation based on writing c_{ik} − c_{ik} c_{jk} explicitly via the matrix representation given in Lemma A.1 and using eq. (A.3).

A.2 Applications

Consider the same setup as in section 2.1, with the same complexes X_{i}^{•}, etc., and diagrams:

𝒟_{i} =^{def} X_{i}^{•} ⟶^{f_{i}} Z_{i}^{•} ⟵^{g_{i}} Y_{i}^{•}

from which we construct the cones

C(𝒟_{i}) = Cone(X_{i}^{•} ⊕ Y_{i}^{•} ⟶^{f_{i}−g_{i}} Z_{i}^{•})[−1] , i = 1, 2, 3 .

Moreover, following ref. [5], define 𝒟_{i} ⊗ 𝒟_{j} by taking the tensor product component-wise. Thus

𝒟_{1} ⊗ 𝒟_{2} = X_{1}^{•} ⊗ X_{2}^{•} ⟶^{f_{1}⊗f_{2}} Z_{1}^{•} ⊗ Z_{2}^{•} ⟵^{g_{1}×g_{2}} Y_{1}^{•} ⊗ Y_{2}^{•} .

Assuming as in section 2.1 that the product maps are compatible with the f_{i}, etc., the diagram

𝒟_{1} ⊗ 𝒟_{2} → 𝒟_{3}

is of the same type as (A.1), and therefore lemma A.1 implies lemma 2.1.

Now, let the multiplication maps X_{1}^{•} ⊗ X_{2}^{•} → X_{3}^{•} be graded commutative up to homotopy and similarly for the Y_{i}^{•} and the Z_{i}^{•}. We are interested in the commutativity properties of multiplication map given by the Beĭlinson product (2.5).

Proposition A.4. The multiplication map ∪_{α}: C(𝒟_{1}) ⊗ C(𝒟_{2}) ⟶ C(𝒟_{3}) given by (2.5) is homotopy graded commutative.

Proof. The permutation operation on tensor products induces the diagram

\xymatrix{𝒟_{1} ⊗ 𝒟_{2} \ar[dr] \ar[rr] &  & 𝒟_{2} ⊗ 𝒟_{1} \ar[dl] \\  & 𝒟_{3}}

which is of type (A.2) and we can apply lemma A.3.

It follows from the proposition that the cohomology inherits a well defined graded commutative product. This in particular applies to the definition of hermitian holomorphic Deligne cohomology that uses the cone (3.1). Therefore we conclude that the cup product (3.6) is graded commutative, as wanted.

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