Hermitian-holomorphic (2)-Gerbes and tame symbols

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

                       aldrovandi@math.fsu.edu

Abstract

The tame symbol of two invertible holomorphic functions can be obtained by computing their cup product in Deligne cohomology, and it is geometrically interpreted as a holomorphic line bundle with connection. In a similar vein, certain higher tame symbols later considered by Brylinski and McLaughlin are geometrically interpreted as holomorphic gerbes and 2-gerbes with abelian band and a suitable connective structure.

In this paper we observe that the line bundle associated to the tame symbol of two invertible holomorphic functions also carries a fairly canonical hermitian metric, hence it represents a class in a Hermitian holomorphic Deligne cohomology group.

We put forward an alternative definition of hermitian holomorphic structure on a gerbe which is closer to the familiar one for line bundles and does not rely on an explicit “reduction of the structure group.” Analogously to the case of holomorphic line bundles, a uniqueness property for the connective structure compatible with the hermitian-holomorphic structure on a gerbe is also proven. Similar results are proved for 2-gerbes as well.

We then show the hermitian structures so defined propagate to a class of higher tame symbols previously considered by Brylinski and McLaughlin, which are thus found to carry corresponding hermitianholomorphic structures. Therefore we obtain an alternative characterization for certain higher Hermitian holomorphic Deligne cohomology groups.

Contents

1 Introduction 2 1.1 Background notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Statement of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Preliminaries 5 2.1 Notation and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Deligne cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Hermitian holomorphic Deligne cohomology 8 3.1 Metrized line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Hermitian holomorphic complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Explicit cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Tame symbol and hermitian structure 10 4.1 Cup product and Deligne torsor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.2 Heisenberg group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.3 Hermitian product structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.4 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5 Hermitian holomorphic gerbes and 2-gerbes 15

5.1 Higher tame symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.2 Gerbes with Hermitian structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.3 Hermitian connective structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.4 The symbol (f, L]_{h.h.} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.5 Hermitian 2-Gerbes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.6 The symbol (L, L′ ]_{h.h.} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.7 Comparisons and relations with other definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6 Concluding remarks 25

A Remarks on Hodge-Tate structures 25 A.1 A Mixed Hodge Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 A.2 The big period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 A.3 The extension class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1 Introduction

The aim of this work is two-fold. For an analytic manifold X we investigate geometric objects corresponding to the elements of certain low-degree Hermitian-Holomorphic Deligne cohomology groups. These groups, denoted here H_{𝒟_{h.h.}}^{k} (X, l), for two integers k and l, were defined in [9] and, in a slightly different fashion, later in [1]. It is already an observation by Deligne (cf. [14]) that H_{𝒟_{h.h.}}^{2} (X, 1) ≅ { \atop Pic X} , the group of isomorphism classes of holomorphic line bundles with hermitian fiber metric. Here we define an appropriate notion of hermitian structure on a gerbe (or 2-gerbe) bound by 𝒪_{X}^{×} and show that the corresponding (equivalence) classes are in bijective correspondence with the elements of H_{𝒟_{h.h.}}^{k} (X, 1), for k = 3, 4.

As a second result and application, we show that the torsors and (2-)gerbes underlying the cup products in ordinary Deligne cohomology studied by Brylinski-McLaughlin [10, 12] can be equipped in a rather natural way with the above mentioned hermitian structures, thus producing classes in the Hermitian-Holomorphic variant. More precisely, we modify the cup product at the level of Deligne complexes to land into a Hermitian-Holomorphic one. This modification is actually quite a natural one from the point of view of Mixed Hodge Structures.

1.1 Background notions

To explain things a little bit more, let X be an analytic manifold and let A ⊆ ℝ be a subring—typically A = ℤ, ℚ or ℝ. For any integer j, set A(j) = (2π\sqrt{−1})^{j}A and let A(j)_{𝒟}^{•} be the Deligne complex

A(j)_{X} ↪ 𝒪_{X} → \underline{Ω}_{X}^{1} → ⋯ → \underline{Ω}_{X}^{j−1} .

It is well known that (at the level of the derived category) there are maps A(j)_{𝒟}^{•} ⊗A(k)_{𝒟}^{•} → A(j +k)_{𝒟}^{•} inducing a cup product in cohomology

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

where we have used the notation H_{𝒟}^{p} (X, A(j)) = H^{p}(X, A(j)_{𝒟}^{•} ) for the Deligne cohomology groups, and H^{•}(X, −) denotes hypercohomology.

The question of obtaining a geometric picture of the cup product in cohomology is a very interesting one. A chief foundational example is the following. For A = ℤ the product

(1.1)  ℤ(1)_{𝒟}^{•} ⊗ ℤ(1)_{𝒟}^{•} ⟶ ℤ(2)_{𝒟}^{•}

corresponds to the morphism

(1.2)  𝒪_{X}^{×} ⊗ 𝒪_{X}^{×} ⟶ (𝒪_{X}^{×} ⟶^{d log} \underline{Ω}_{X}^{1})

via the quasi-isomorphisms ℤ(1)_{𝒟}^{•} →^{≃} 𝒪_{X}^{×}[−1] and ℤ(2)_{𝒟}^{•} →^{≃} (𝒪_{X}^{×} ⟶^{d log} \underline{Ω}_{X}^{1})[−1] . Deligne gave a geometric construction of (1.2)and the ensuing cup product

𝒪_{X}^{×}(X) ⊗ 𝒪_{X}^{×}(X) ⟶^{∪} H^{1}(X, 𝒪_{X}^{×} ⟶^{d log} \underline{Ω}_{X}^{1})

in his work on tame symbols, cf. [13]: If f and g are two invertible functions on X, namely two elements of 𝒪_{X}^{×}, their cup product corresponds to a 𝒪_{X}^{×}-torsor, denoted (f, g], equipped with an analytic connection.

Furthermore, if X is a Riemann surface, the complex (𝒪_{X}^{×} ⟶^{d log} \underline{Ω}_{X}^{1}) is quasi-isomorphic to ℂ^{×} and the product is interpreted as the holonomy of the connection. For X equal to a punctured disk D_{p} centered at p, if f and g are holomorphic on D_{p}, meromorphic at p, the holonomy of (f, g] computes the tame symbol

(f, g)_{p} = (−)^{v(f)v(g)}(f^{v(g)}/g^{v(f)})(p) ,

where v(f) is the valuation of f at p, cf. [2, 13, 20]. This justifies the use of the name tame symbol for (f, g].

A particularly pleasant property is that when f and 1 − f are both invertible a calculation [13] using the classical Euler’s dilogarithm Li_{2} shows that (f, 1 − f] is isomorphic to the trivial torsor equipped with the

trivial connection d, namely the unit element in the group H^{1}(X, 𝒪_{X}^{×} ⟶^{d log} \underline{Ω}_{X}^{1}) . From this one also builds an interpretation of the symbol associated to f and g in terms of Mixed Hodge Structures [13].

In this particular example there appear degree 1 and 2 Deligne cohomology groups: specifically, it is made use of the fact that invertible functions determine elements in the group H_{𝒟}^{1} (X, ℤ(1)) ≅ 𝒪_{X}^{×}(X) , and, given f

and g, the class of the torsor with connection (f, g] is an element of H_{𝒟}^{2} (X, ℤ(2)) ≅ H^{1}(X, 𝒪_{X}^{×} ⟶^{d log} \underline{Ω}_{X}^{1}) . It is therefore natural to investigate the geometric objects corresponding to similar cup products of higher degree. The case of (f, L], where f is again an invertible function and L is an 𝒪_{X}^{×}-torsor, so it determines a class in H_{𝒟}^{2} (X, ℤ(1)) ≅ H^{1}(X, 𝒪_{X}^{×}) , was already considered in ref. [13], where it is interpreted in terms of a gerbe 𝒢 over X.

This idea has been further pursued by Brylinski-McLaughlin, [10, 12]. In their study of degree 4 characteristic classes they considered the symbols (f, L] ∈ H_{𝒟}^{3} (X, ℤ(2)) and, for a pair of 𝒪_{X}^{×}-torsors, (L, L′] ∈ H_{𝒟}^{4} (X, ℤ(2)). The corresponding geometric objects are identified with a gerbe (resp. a 2-gerbe) both equipped with the appropriate analog of a connection. Furthermore, the obvious map ℤ(2)_{𝒟}^{•} → ℤ(1)_{𝒟}^{•} induces a corresponding map H_{𝒟}^{k} (X, ℤ(2)) → H_{𝒟}^{k} (X, ℤ(1)) which simply forgets the connection. Therefore elements in the groups H_{𝒟}^{k} (X, ℤ(1)) , for k = 3, 4 correspond to equivalence classes of (2-)gerbes bound by 𝒪_{X}^{×}, cf.[7, 10, 12]. Thus in the end several Deligne cohomology groups have a concrete interpretation in terms of geometric data.

Hermitian-Holomorphic Deligne cohomology, as defined by Brylinski, cf. [9], is an enhanced version of Deligne cohomology. For all positive integers l Brylinski introduces certain complexes C(l)^{•}, and defines the Hermitian-Holomorphic Deligne cohomology groups as the sheaf hypercohomology groups: H_{𝒟_{h.h.}}^{k} (X, l) = H^{k}(X, C(l)^{•}) . The complex C(l)^{•} has a map C(l)^{•} → ℤ(l)_{𝒟}^{•} , thus there is an obvious map H_{𝒟_{h.h.}}^{k} (X, l) → H_{𝒟}^{k} (X, ℤ(l)) forgetting the extra-structure.

A primary example is provided by Deligne’s observation mentioned before, cf. [14], that

(1.3)  { \atop Pic X} ≅ H^{2}(X, ℤ(1)_{X} → 𝒪_{X} → \underline{ℰ}_{X}^{0}) ,

where { \atop Pic X} is the set of isomorphism classes of 𝒪_{X}^{×}-torsors with hermitian metric, and \underline{ℰ}_{X}^{0} is the sheaf of smooth real-valued functions on X. The complex in (1.3) is quasi-isomorphic to C(1)^{•}, therefore

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

In fact, both complexes are quasi-isomorphic to the complex (𝒪_{X}^{×} ⊕ \underline{𝕋}_{X} → \underline{ℂ}_{X}^{×})[−1] , [12, 9], which encodes the reduction of the torsor structure from 𝒪_{X}^{×} to \underline{𝕋}_{X} afforded by the hermitian metric.

Concerning higher degrees, Brylinski-McLaughlin [12, 11] gave a geometric interpretation for some of the groups H_{𝒟_{h.h.}}^{k} (X, l), k = 3, 4 and l = 1, 2 in terms of classes of gerbes and 2-gerbes bound by \underline{𝕋}_{X} and equipped with a concept of connection valued in an appropriate Hodge filtration of the de Rham complex of X.

1.2 Statement of the results

In this work we take on the same question of a geometric interpretation for some Hermitian-Holomorphic Deligne cohomology groups from a holomorphic view-point which, we believe, is complementary to that of Brylinski-McLaughlin. We define a hermitian structure on a 𝒪_{X}^{×}-gerbe 𝒢 as the assignment of a \underline{ℰ}_{U,+}^{0}-torsor (the “+” denotes positive functions) to any object P of 𝒢_{U} subject to several conditions spelled out in Definition 5.2.1. We prove that classes of gerbes with hermitian structures in this sense correspond to elements of H_{𝒟_{h.h.}}^{3} (X, 1) ≅ H^{3}(X, ℤ(1)_{X} → 𝒪_{X} → \underline{ℰ}_{X}^{0}) , in complete analogy with (1.3). Moreover we can define a type (1, 0)-connective structure on 𝒢 by requiring that to any object P of 𝒢_{U} be assigned a F^{1}\underline{A}_{U}^{1}-torsor, essentially repeating the steps in ref. [12]. (Here \underline{A}_{U}^{•} is the smooth ℂ-valued de Rham complex, and F^{1} is the first Hodge filtration.) Then a notion of compatibility between the hermitian structure and the connective one is defined, and in fact we prove there is only one such type (1, 0) connective structure compatible with a given hermitian structure, up to equivalence. This result is analogous to the corresponding statement for hermitian holomorphic line bundles, that there is a unique connection — the canonical or Griffiths connection — compatible with both structures.

Similar results are available for 2-gerbes: we define a hermitian structure for a 𝒪_{X}^{×}-2-gerbe G as the assignment of a \underline{ℰ}_{U,+}^{0}-gerbe for each object P of G_{U}, subject to several conditions spelled out in Definition 5.5.1. Analogously to the simpler case of gerbes, we have a concept of type (1, 0) connectivity compatible with the hermitian structure and a uniqueness result up to equivalence.

A second line of results is more specific to the tame symbols we encountered before. Alongside with the map of complexes

ℤ(1)_{𝒟}^{•} ⊗ ℤ(1)_{𝒟}^{•} ⟶ ℤ(2)_{𝒟}^{•}

we define a companion map

(1.4)  ℤ(1)_{𝒟}^{•} ⊗ ℤ(1)_{𝒟}^{•} ⟶ 2π\sqrt{−1} ⊗ C(1)^{•}

so that it is possible to obtain a different cup product valued in Hermitian-Holomorphic Deligne cohomology:

H_{𝒟}^{i} (X, ℤ(1)) ⊗ H_{𝒟}^{j} (X, ℤ(1)) ⟶^{∪} 2π\sqrt{−1} ⊗ H_{𝒟_{h.h.}}^{i+j} (X, 1) .

An immediate consequence is that for f and g invertible, and L, L′ line bundles, the torsor (f, g] and the gerbe (f, L] support natural hermitian structures of the type discussed above, in addition to the analytic connection (or connective) ones associated with the cup product in standard Deligne cohomology. The same conclusions are valid for the 2-gerbe (L, L′]. It turns out that supporting both structures is an easy consequence of the commutativity of the following diagram:

H_{𝒟}^{i} (X, ℤ(1)) ⊗ H_{𝒟}^{j} (X, ℤ(1)) ⟶^{∪} 2π\sqrt{−1} ⊗ H_{𝒟_{h.h.}}^{i+j} (X, 1)
∪↓  ↓^{forget}
H_{𝒟}^{i+j}(X, ℤ(2))  ⟶^{forget}  H_{𝒟}^{i+j}(X, ℤ(1))

Indeed, forgetting either structure, brings us back to the same underlying object.

The map (1.4) has a rather natural definition from the point of view of Mixed Hodge Structures, whose role in the matter was mentioned in relation with the product (1.1), see [13]. Namely, there is a “universal” MHS ℳ^{(2)} corresponding to an iterated extension of ℤ(0) by ℤ(1) by ℤ(2), where in this case ℤ(n) denotes a Hodge-Tate structure. To ℳ^{(2)} we can associate a tensor — the “big period” — P (ℳ^{(2)}) ∈ ℂ ⊗_{ℚ} ℂ , cf. [17]. The period is in fact a multiple of the extension class of ℳ^{(2)}, and it belongs to the kernel ℐ = ker(m: ℂ⊗_{ℚ}ℂ → ℂ) of the multiplication map. We find the map (1.4) corresponds to the image of P (ℳ^{(2)}) under the “imaginary part” projection ℂ ⊗_{ℚ} ℂ → ℝ(1) given by a ⊗ b ↦ Im(a) Re(b). On the other hand, the standard one (1.1) involves the projection onto the Kähler differentials ℐ → ℐ /ℐ^{2} given by a ⊗ b ↦ a db.

Another consequence of the previous diagram is that (f, g], (f, L], and (L, L′] come equipped with two connection (or connective) structures. If the unitary connection in a line bundle L is also analytic, then L is flat. In the case of (f, g] we find there is an obstruction to this type of compatibility. This can be cast in cohomological terms, which allows to extend these considerations to 𝒪_{X}^{×}-gerbes and 2-gerbes. We find that the obstruction vanish, so compatibility can always be achieved.

1.3 Outline of the paper

This work is organized as follows. In section 2 we make some preliminaries observations about Deligne complexes and cohomology and collect a few needed facts. We recall the definition of Hermitian-Holomorphic Deligne cohomology and state some of its properties in section 3. Alongside Brylinski’s complex C(l)^{•}, we use a complex quasi-isomorphic to it, denoted D(l)_{h.h.}^{•}, which for a line bundle directly encodes the data defining the canonical connection.

In section 4 we recall the definition of the tame symbol (f, g] for two invertible functions and some of its properties. We define the modified product (1.4) and show that through it, the torsor associated to (f, g] also comes equipped with a hermitian structure. As mentioned before, the product (1.4) and its relation with the standard for Deligne complexes become more clear when analyzed in terms of Hodge Structures. In order to do this, we felt necessary to recall a few elementary facts and calculations concerning Hodge-Tate structures that are certainly well-known to experts. For this reason, and also because this development lies somewhat aside this work’s main lines, we present this material in appendix A. This presentation relies in part on the Heisenberg group picture of the Deligne torsor, which we have recalled in section 4.2.

Section 5 is the main part of this work. There we redefine the notion of hermitian structure (modeled after that of connective structure) and prove that equivalence classes of these are classified by the groups H_{𝒟_{h.h.}}^{k} (X, 1). We then apply this classification to the Hermitian structures and the product (1.4) for the higher versions of the tame symbols considered by Brylinski-McLaughlin.

The interplay between the analytic connection (or connective) structures arising from standard Deligne cohomology and their hermitian counterparts defined here is analyzed in sections 4.4 and 5.7.

Finally, section 6 is devoted to some concluding remarks.

Acknowledgments

Parts of the present work were written while visiting the Department of Mathematics, Aarhus University, Århus, Denmark; the International School for Advanced Studies (SISSA), Trieste, Italy; the Department of Mathematics, Instituto Superior Técnico, Lisbon, Portugal. It is a pleasure to thank all these institutions for hospitality, support, and for providing an excellent, friendly, and stimulating research environment. It is also a pleasure to thank the anonymous referee for raising important points and providing several stimulating comments leading to a much improved version of the paper.

2 Preliminaries

2.1 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. For a subring A of ℝ and an integer j, A(j) = (2π\sqrt{−1})^{j} A is the Tate twist of A. We identify ℂ/ℤ(j) ≅ ℂ^{×} via the exponential map z ↦ exp(z/(2π\sqrt{−1})^{j−1}), and ℂ/ℝ(j) ≅ ℝ(j − 1).

If X is a complex manifold, \underline{A}_{X}^{•} and \underline{Ω}_{X}^{•} denote the de Rham complexes of sheaves 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, 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 write U_{ij} = U_{i} ∩ U_{j}, U_{ijk} = U_{i} ∩ U_{j} ∩ U_{k}, and so on. More generally we can also have 𝔘_{X} = {U_{i} → X}_{i∈I}, where the maps are regular coverings in an 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}} .

If \underline{F}^{•} is a complex of abelian 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}···i_{n}}) . The Čech coboundary operator is denoted δ. The convention we use is to put the index along the Čech resolution in the second place, so if we denote by d the differential in the complex \underline{F}^{•}, the total differential is given by D = d + (−1)^{p}δ on the component Č^{q}(𝔘_{X}, \underline{F}^{p}) of the total simple complex. Furthermore, recall that the Koszul sign rule causes a sign being picked whenever two degree indices are formally exchanged. 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}} .

For a given complex of abelian objects, say C^{•}, the symbol σ^{i} denotes sharp truncation at the index i: σ^{i}C^{p} = 0 for p < i.

2.2 Deligne cohomology

There are several models for the complexes to use to compute Deligne cohomology [15, 2]. For A ⊂ ℝ and an integer j the latter is the hypercohomology:

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

Here A(p)_{𝒟}^{•} is the Deligne complex

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

⟶^{≃} Cone (A(j)_{X} ⊕ F^{j}\underline{Ω}_{X}^{•} ⟶^{ı−ȷ} \underline{Ω}_{X}^{•})[−1] , (2.2)

where F^{j}\underline{Ω}_{X}^{•} in eqn. (2.2) is the Hodge (“stupid”) filtration on the de Rham complex. The symbol ⟶^{≃} denotes a quasi-isomorphism. In view of Beĭlinson formula for the cup product on cones to be recalled below [3], Deligne complexes acquire a family of cup-products (depending on a real parameter α)

A(j)_{𝒟}^{•} ⊗ A(k)_{𝒟}^{•} ⟶^{∪_{α}} A(j + k)_{𝒟}^{•} .

Cup products related to different values of the parameter α are related by homotopy-commutative diagrams, hence they induce a well defined graded commutative cup-product in cohomology

(2.3)  H_{𝒟}^{p} (X, A(j)) ⊗ H_{𝒟}^{q} (X, A(k)) ⟶^{∪} H_{𝒟}^{p+q}(X, A(j + k)) .

In order to explicitly compute cup products, the model given by eq. (2.1) leads to simpler formulas (when it can be used). If f ∈ A(j)_{𝒟}^{•} and g ∈ A(k)_{𝒟}^{•} , then from ref. [15] we quote:

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

The following examples are well known and will frequently recur in the following.

Example 2.2.1. For A = ℤ it is immediately verified that 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.

Example 2.2.2. ℤ(2)_{𝒟}^{•} →^{≃} (𝒪_{X}^{×} ⟶^{d log} \underline{Ω}_{X}^{1})[−1] . A fundamental observation by Deligne (see ref. [2]) is that H_{𝒟}^{2} (X, ℤ(2)) is identified with the group of isomorphism classes of holomorphic line bundles with (holomorphic) connection. This is easily understood from a Čech cohomology point of view. Using the cover 𝔘_{X} = {U_{i}}_{i∈I}, a class in

H_{𝒟}^{2} (X, ℤ(2)) ≅ H^{1}(X, 𝒪_{X}^{×} ⟶^{d log} \underline{Ω}_{X}^{1})

is represented by a pair (ω_{i}, g_{ij}) with ω_{i} ∈ \underline{Ω}_{X}^{1}(U_{i}) and g_{ij} ∈ 𝒪_{X}^{×}(U_{ij}) satisfying the relations

ω_{j} − ω_{i} = d log g_{ij} , g_{ij}g_{jk} = g_{ik} .

The Čech representative for the actual class in H_{𝒟}^{2} (X, ℤ(2)) is obtained (up to a multiplication by 2π\sqrt{−1}) by extracting local logarithms log g_{ij}, see ref. [15] for full details.

For real Deligne cohomology, i.e. when A = ℝ, other models quasi-isomorphic to those in eqs. (2.1) and (2.2) are available. Since the maps

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

are all quasi-isomorphisms in the derived category, cf. [15], we have

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

Moreover, we can use smooth forms thanks to the fact that the inclusion \underline{Ω}_{X}^{•} ↪ \underline{A}_{X}^{•} is a filtered quasiisomorphism with respect to the filtrations F^{j}\underline{Ω}_{X}^{•} ↪ F^{j}\underline{A}_{X}^{•}. Here F^{j}\underline{A}_{X}^{•} is the subcomplex of \underline{A}_{X}^{•} comprising forms of type (p, q) where p is at least j, so that F^{j}\underline{A}_{X}^{n} = ⊕_{p≥j}\underline{A}_{X}^{p,n−p}.

Let (ω_{1}, η_{1}) be an element of degree n in ℝ(j)_{𝒟}^{•} —this means that ω_{1} ∈ F^{j}\underline{Ω}_{X}^{n} and η_{1} ∈ \underline{ℰ}_{X}^{n−1}(j − 1)—and (ω_{2}, η_{2}) any element in ℝ(k)_{𝒟}^{•} . A product is given by the formula (cf. ref. [15]):

(2.6)  (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}) .

Example 2.2.3. H_{𝒟}^{1} (X, ℝ(1)) is the group of real valued functions η on X such that there exists a holomorphic one-form ω such that π_{0}ω = dη. In other words, it is the group of those real smooth functions η such that ∂η is holomorphic. In particular, if f is holomorphic and invertible, then the class in H_{𝒟}^{1} (X, ℝ(1)) determined by f is represented by (d log f, log |f|).

2.3 Cones

We recall here a variant of Beĭlinson’s formula for the cup product on certain diagrams of complexes. (For full details see refs. [1, 3, 15].)

For i = 1, 2, 3 consider the diagrams of complexes

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

and set

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

Suppose there are product maps X_{1}^{•} ⊗ X_{2}^{•} ⟶^{∪} X_{3}^{•}, and similarly for Y_{i}^{•}, and Z_{i}^{•}. We assume the products to be compatible with the f_{i}, g_{i} only 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

f_{3} ∘ ∪ − ∪ ∘ (f_{1} ⊗ f_{2}) = d h + h d , g_{3} ∘ ∪ − ∪ ∘ (g_{1} ⊗ g_{2}) = d k + k d ,

with obvious meaning of the symbols. The following lemma establishes a variant of Beĭlinson’s product formula [3].

Lemma 2.3.1. For (x_{i}, y_{i}, z_{i}) ∈ X_{i}^{•} ⊕ Y_{i}^{•} ⊕ Z_{i}^{•−1} , i = 1, 2 , and a real parameter α, the following formula:

\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} ; (2.8)

defines a family of products

C(𝒟_{1}) ⊗ C(𝒟_{2}) ⟶^{∪_{α}} C(𝒟_{3}) .

These products are homotopic to one another, and graded commutative up to homotopy. The homotopy formula is the same as that found in ref. [3].

Proof. Direct verification.

If the maps f_{i}, g_{i} above are strictly compatible with the products, namely the homotopies h and k are zero, (2.8) reduces to the formulas found in [3, 15]. Homotopy commutativity at the level of complexes ensures the corresponding cohomologies will have genuine graded commutative products.

3 Hermitian holomorphic Deligne cohomology

3.1 Metrized line bundles

Let X be a complex manifold. Consider a holomorphic line bundle L on X with hermitian fiber metric ρ or, equivalently, an invertible sheaf L equipped with a map ρ: L → \underline{ℰ}_{X,+}^{0} to (the sheaf of) positive real smooth

functions, see ref. [21] for the relevant formalism. Let { \atop Pic(X)} denote the group of isomorphism classes of line bundles with hermitian metric. A basic observation by Deligne (cf. [14]) is that { \atop Pic X} can be identified with the second hypercohomology group:

(3.1)  H^{2}(X, ℤ(1)_{X} ⟶^{ı} 𝒪_{X} ⟶^{−π_{0}} \underline{ℰ}_{X}^{0}) .

This is easy to see in Čech cohomology. Suppose s_{i} is a trivialization of L|_{U_{i}}, with transition functions g_{ij} ∈ 𝒪_{X}^{×}(U_{ij}) determined by s_{j} = s_{i}g_{ij}. Let ρ_{i} be the value of the quadratic form associated to ρ on s_{i}, namely ρ_{i} = ρ(s_{i}). Then we have ρ_{j} = ρ_{i} |g_{ij}|^{2}. Taking logarithms, we see that

(2π\sqrt{−1}c_{ijk}, log g_{ij}, \frac{1}{2} log ρ_{i}) ,

where 2π\sqrt{−1}c_{ijk} = log g_{jk} − log g_{ik} + log g_{ij} ∈ ℤ(1) , is a cocycle representing the class of the pair (L, ρ).

3.1.1 Canonical connection

Recall for later use that the canonical connection, [19] on a metrized line bundle (L, ρ) is the unique connection compatible with both the holomorphic and hermitian structures. In Čech cohomology with respect to the cover 𝔘_{X} as above, the canonical connection on (L, ρ) corresponds to a collection of (1, 0) forms ξ_{i} ∈ \underline{A}_{X}^{1,0}(U_{i}) satisfying the relations

(3.2)  ξ_{j} − ξ_{i} = d log g_{ij}
(3.3)  π_{0}(ξ_{i}) = \frac{1}{2}d log ρ_{i} .

The latter just means ξ_{i} = ∂ log ρ_{i} , in more familiar terms. The global 2-form

(3.4)  c_{1}(ρ) = η_{i} ≡ ¯∂∂ log ρ_{i}

represents the first Chern class of L in H^{2}(X, ℝ(1)). The class of c_{1}(ρ) is in fact a pure Hodge class in H^{1,1}(X)—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). It only depends on the class of (L, ρ) in { \atop Pic(X)}.

3.2 Hermitian holomorphic complexes

In ref. [9] Brylinski introduced the complexes

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

Definition 3.2.1. The hypercohomology groups

(3.6)  H_{𝒟_{h.h.}}^{p} (X, l) =^{def} H^{p}(X, C(l))

are the Hermitian holomorphic Deligne cohomology groups.

By the remark after eq. (2.5), the complex

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

also computes the real Deligne cohomology. Then consider the complex

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

In ref. [1] we prove Lemma 3.2.2. The complexes C(l)^{•} and D(l)_{h.h.}^{•} are quasi-isomorphic, hence we also have

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

Remark 3.2.3. The complex F^{l}\underline{A}_{X}^{•} ∩ σ^{2l}\underline{ℰ}_{X}^{•}(l) appearing in both (3.5) and (3.6) can be rewritten in terms of the complex G(l)^{•} of ref. [14]. Set

G(l)^{•} = 0 ⟶ ⋯ ⟶ 0 ⟶ \underline{A}_{X}^{(l,l)} ⟶^{d} \underline{A}_{X}^{(l+1,l)} ⊕ \underline{A}_{X}^{(l,l+1)} ⟶^{d} · · · .

Then we have F^{l}\underline{A}_{X}^{•} ∩ σ^{2l}\underline{ℰ}_{X}^{•}(l) = G(l)^{•} ∩ \underline{ℰ}_{X}^{•}(l) .

For certain ranges of values of the cohomology index the groups H_{𝒟_{h.h.}}^{p} (X, l) are fairly ordinary. Indeed we have the following easy

Lemma 3.2.4. For p ≤ 2l − 1 we have

H_{𝒟_{h.h.}}^{p} (X, l) ≅ H^{p−1}(X, ℝ(l)/ℤ(l)) .

Proof. Using either C(l)^{•} or D(l)_{h.h.}^{•}, we see that they are quasi-isomorphic to

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

which leads to the triangle

ℝ(l)/ℤ(l)[−1] ⟶ D(l)_{h.h.}^{•} ⟶ F^{l}\underline{A}_{X}^{•} ∩ σ^{2l}\underline{ℰ}_{X}^{•}(l) ⟶^{+1} .

The statement follows.

In general these groups are interesting when p ≥ 2l. The most important example is:

Lemma 3.2.5.

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

Proof. We have quasi-isomorphisms

ℤ(1)_{X} ⟶^{ı} 𝒪_{X} ⟶^{−π_{0}} \underline{ℰ}_{X}^{0} ⟶^{≃} D(1)_{h.h.}^{•} ⟶^{≃} C(1)^{•} .

Indeed, note that D(1)_{h.h.}^{•} can be rewritten as

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

and

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

By direct verification, the latter complex is quasi-isomorphic to \underline{ℰ}_{X}^{0}[−1]. Thus

D(1)_{h.h.}^{•} ⟶^{≃} Cone(ℤ(1)_{𝒟}^{•} → \underline{ℰ}_{X}^{0}[−1])[−1] ⟶^{≃} ℤ(1)_{X} → 𝒪_{X} → \underline{ℰ}_{X}^{0} .

Since hermitian holomorphic Deligne complexes can be expressed as cones of diagrams of the form (2.7), they admit cup products, and hence there is a cup product for hermitian holomorphic Deligne cohomology [9]:

H_{𝒟_{h.h.}}^{p} (X, l) ⊗ H_{𝒟_{h.h.}}^{q} (X, k) ⟶^{∪} H_{𝒟_{h.h.}}^{p+q} (X, l + k) .

3.3 Explicit cocycles

Use of the seemingly more complicated complex (3.7) in place of the one in (3.5) is justified by the fact that the data comprising the canonical connection can be characterized cohomologically, as follows:

Lemma 3.3.1. Let (L, ρ) be a metrized line bundle on X. Assume (L, ρ) to be trivialized with respect to the open cover 𝔘_{X} of X as before. The data:

ξ_{i} ∈ \underline{A}_{X}^{(1,0)}(U_{i}) , \frac{1}{2} log ρ_{i} ∈ \underline{ℰ}_{X}^{0}(U_{i}) , η_{i} ∈ \underline{A}_{X}^{(1,1)}(U_{i}) ,
2π\sqrt{−1}c_{ijk} ∈ ℤ(1)_{X}(U_{ijk}) , log g_{ij} ∈ 𝒪_{X}(U_{ij})

represent a degree 2 cocycle with values in Tot Č^{•}(𝔘_{X}, D(1)_{h.h.}^{•}) if and only if the relations (3.2), (3.3), (3.4), plus those in sect. 3.1, defining the canonical connection are satisfied.

Proof. One need only unravel the cone defining D(1)_{h.h.}^{•} as follows:

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

and carefully chase the diagram.

On the other hand, the hermitian holomorphic Deligne complex in the form (3.5) corresponds to “reducing the structure group” from ℂ^{×} to 𝕋. This can be made explicit for l = 1 and a line bundle L → X by choosing sections t_{i} of the smooth bundle corresponding to L such that ρ(t_{i}) = 1. Clearly the resulting smooth transition functions will be sections of \underline{𝕋}_{X} over U_{ij}. See refs. [9] and [12] for more details.

4 Tame symbol and hermitian structure

Let X be a complex analytic manifold and U ⊂ X open. Let f and g two invertible holomorphic functions on U. The tame symbol [13] (f, g] associated to f and g is a 𝒪_{X}^{×}|_{U}-torsor equipped with an analytic connection.

4.1 Cup product and Deligne torsor

(See [13, 15].) We consider f and g as elements of H_{𝒟}^{1} (U, ℤ(1)). Then (f, g] = f ∪ g ∈ H_{𝒟}^{2} (U, ℤ(2)) . Consider the cover 𝔘_{X} of X so that U is covered by \{U ∩ U_{i}\}_{i∈I} and choose representatives (2π\sqrt{−1} m_{ij}, log_{i}f) and (2π\sqrt{−1} n_{ij}, log_{i}g) for f and g, respectively. Then, using (2.4), the cup product is represented by the cocycle:

(4.1)  ((2π\sqrt{−1})^{2}m_{ij}n_{jk} , −2π\sqrt{−1} m_{ij} log_{j}g , log_{i}f \frac{dg}{g} ) .

Under the quasi-isomorphism with the complex (𝒪_{X}^{×} → \underline{Ω}_{X}^{1}) (which essentially amounts to a division by 2π\sqrt{−1}) the cocycle (4.1) becomes

(4.2)  (g^{−m_{ij}} , −\frac{1}{2π\sqrt{−1}} log_{i}f \frac{dg}{g} ) .

In ref. [13] the trivializing section on U ∩ U_{i} corresponding to (4.2) is denoted {log_{i}f, g} . Two trivializations over U ∩ U_{i} and U ∩ U_{j} are related by {log_{j}f, g} = {log_{i}f, g} g^{−m_{ij}} . Furthermore, the analytic connection is defined by the rule:

(4.3)  ∇\{log_{i}f, g\} = −\{log_{i}f, g\} ⊗ \frac{1}{2π\sqrt{−1}} log_{i}f \frac{dg}{g} .

A general section s of (f, g] can be written as s = h_{i} {log_{i}f, g} , for some h_{i} ∈ 𝒪_{U}(U_{i}) , and therefore

(4.4)  ∇s = \{log_{i}f, g\} ⊗ (dh_{i} − \frac{1}{2π\sqrt{−1}} log_{i}f \frac{dg}{g} ) .

4.2 Heisenberg group

An equivalent approach to the Deligne symbol is via the complex three-dimensional Heisenberg group, see refs. [5, 20, 22]. Let H_{ℂ} denote the group of complex unipotent 3 × 3 lower triangular matrices. Let

H_{ℤ} = \{(\begin{matrix}1 \\ m_{1} & 1 \\ m_{2} & n_{1} & 1 \end{matrix}) | m_{1}, n_{1} ∈ ℤ(1) , m_{2} ∈ ℤ(2)\} ⊂ H_{ℂ} .

The quotient H_{ℂ}/H_{ℤ} is a ℂ/ℤ(2)-bundle over ℂ/ℤ(1) × ℂ/ℤ(1) via the projection map

p: [\begin{matrix}1 \\ x & 1 \\ z & y & 1 \end{matrix}] ↦ ([x], [y]) ,

where x, y, z ∈ ℂ, and the brackets denote the appropriate equivalence classes. (The ℂ/ℤ(2)-action is by multiplication with a matrix of the form ( \begin{matrix}1 \\ 0 & 1 \\ z & 0 & 1 \end{matrix} ) .)

The twisting of H_{ℂ}/H_{ℤ} is analogous to that of the Deligne torsor in sect. 4.1: the right action of H_{ℤ} on H_{ℂ} amounts to:

(4.5)  x ↦ x + m_{1} , y ↦ y + n_{1} , z ↦ z + m_{1} · y + m_{2} .

Moreover, the complex form

(4.6)  ω = \frac{1}{2π\sqrt{−1}}(dz − x dy)

is invariant under the action of H_{ℤ} and defines a ℂ/ℤ(2)-connection form on the total space H_{ℂ}/H_{ℤ}.

The invertible functions f and g on U define a map (f, g): U → ℂ^{×} × ℂ^{×}. Then the tame symbol (f, g] is obtained as the pull-back:

(f, g] = (f, g)^{∗}(H_{ℂ}/H_{ℤ}) ,

and the section {log_{i}f, g} corresponds to the class of the matrix

(\begin{matrix}1 \\ log_{i}f & 1 \\ 0 & log_{i}g & 1 \end{matrix}) .

Furthermore, the pull-back of the connection form ω on H_{ℂ}/H_{ℤ} along the section {log_{i}f, g} is the same form as the one in (4.1). More generally, a section s as given at the end of sect. 4.1 corresponds to the class of the matrix

(\begin{matrix}1 \\ log_{i}f & 1 \\ h_{i} & log_{i}g & 1 \end{matrix}) ,

Pulling back (4.6) along the section gives (4.4).

4.3 Hermitian product structure

Consider the “imaginary part” map

(4.7)  \begin{matrix}ℂ ⊗ ℂ & ⟶ ℝ(1) \\ a ⊗ b & ⟼ −π_{1}(a) π_{0}(b) ≡ −\sqrt{−1} Im(a) Re(b) , \end{matrix}

Similarly, we have:

(4.8)  𝒪_{X} ⊗ 𝒪_{X} ⟶ \underline{ℰ}_{X}^{0}(1) f ⊗ g ⟼ −π_{1}(f) π_{0}(g) .

Definition 4.3.1. Define the map

\begin{aligned}(ℤ(1)_{X} → 𝒪_{X}) ⊗ (ℤ(1)_{X} → 𝒪_{X}) & ⟶ (ℤ(2)_{X} → 𝒪_{X} ⟶^{−π_{1}} \underline{ℰ}_{X}^{0}(1)) \\ & ⟶^{≃} 2π\sqrt{−1} ⊗ (ℤ(1)_{X} → 𝒪_{X} ⟶^{−π_{0}} \underline{ℰ}_{X}^{0}) \end{aligned} ; (4.9)

by using (4.8) in place of the map 𝒪_{X} ⊗ 𝒪_{X} → \underline{Ω}_{X}^{1} , f ⊗ g ↦ fdg , in (2.4).

Proposition 4.3.2. The product map (4.9) is well defined, namely it is a map of complexes. Furthermore, it is homotopy graded commutative.

Proof. The fact that (4.9) is a map of complexes is a direct verification. After ref. [15], consider the map

h(f ⊗ g) = f g , f, g ∈ 𝒪_{X} ,

and zero otherwise. It provides the required homotopy.

The target complex of the product map in eq. (4.9) is the complex encoding hermitian structures appearing in sect. 3.1. In other words, up to quasi-isomorphism, we have a product:

ℤ(1)_{𝒟}^{•} ⊗ ℤ(1)_{𝒟}^{•} ⟶ 2π\sqrt{−1} ⊗ D(1)_{h.h.}^{•} .

Remark 4.3.3. The map (4.8) provides an explicit homotopy map for the homotopy commutative diagram

ℤ(1)_{𝒟}^{•} ⊗ ℤ(1)_{𝒟}^{•} ⟶ ℤ(2)_{𝒟}^{•}
↓  ↓
ℝ(1)_{𝒟}^{•} ⊗ ℝ(1)_{𝒟}^{•} ⟶ ℝ(2)_{𝒟}^{•}

where the model (2.5) for ℝ(k)_{𝒟}^{•} is used (see [15]).

Now, in view of Prop. 4.3.2, we have a graded commutative product at the level of cohomology groups. In particular, let f, g be two invertible holomorphic functions on U ⊂ X.

Proposition 4.3.4. The Deligne torsor underlying (f, g] admits a hermitian fiber metric.

Proof. View f and g as elements of H_{𝒟}^{1} (U, ℤ(1)). Taking the product according to (4.9) yields an element in

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

that is, a holomorphic line bundle with hermitian fiber metric (up to isomorphism).

Taking the image of the tame symbol (f, g] under the map H_{𝒟}^{•} (U, ℤ(2)) → H_{𝒟}^{•} (U, ℤ(1)) = Pic(U) induced by ℤ(2)_{𝒟}^{•} → ℤ(1)_{𝒟}^{•} forgets the analytic connection and retains just the line bundle. Similarly, the map H_{𝒟_{h.h.}}^{2} (U, 1) → H_{𝒟}^{•} (U, ℤ(1)) = Pic(U) induced by D(1)_{h.h.}^{•} → ℤ(1)_{𝒟}^{•} forgets the hermitian structure. Clearly both map to the same underlying line bundle.

Using a Čech cover we can represent f and g as in sect. 4.1. Then the cocycle corresponding to their product in H_{𝒟_{h.h.}}^{2} (U, 1) is:

(4.10)  (2π\sqrt{−1} m_{ij} n_{jk} , −m_{ij} log_{j}g , −\frac{1}{2π\sqrt{−1}} π_{1}(log_{i}f) log |g| ) .

This allows us to identify the representative of the hermitian metric, or rather its logarithm, as

(4.11)  \frac{1}{2} log ρ_{i} = −\frac{1}{2π\sqrt{−1}} π_{1}(log_{i}f) log |g| .

It follows that if s is the local section at the end of sect. 4.1 then

(4.12)  log ρ(s) = \frac{1}{2π\sqrt{−1}}(π_{1}(h_{i}) − π_{1}(log_{i}f) log |g|) .

4.3.1 Remarks on the Heisenberg bundle

The hermitian metric can be constructed from the more global point of view afforded by the use of the Heisenberg group recalled in sect. 4.2. The hermitian metric on the bundle H_{ℂ}/H_{ℤ} → ℂ^{×} × ℂ^{×} is given by the map ρ : H_{ℂ}/H_{ℤ} → ℝ_{+} defined by:

(4.13)  ρ: [\begin{matrix}1 \\ x & 1 \\ z & y & 1 \end{matrix}] ⟼ exp \frac{1}{2π\sqrt{−1}}(π_{1}(z) − π_{1}(x) π_{0}(y))

Indeed, using the explicit action (4.5), one checks (4.13) is invariant and provides the required quadratic form. In particular, the quantity

−\frac{1}{2π\sqrt{−1}} π_{1}(x) π_{0}(y)

is immediately shown to behave as the logarithm of the local representative of a hermitian metric. Thus the hermitian holomorphic line bundle represented by the cocycle (4.10) is the pull-back of (H_{ℂ}/H_{ℤ}, ρ) via the map (f, g): U → ℂ^{×} × ℂ^{×}.

4.3.2 Relations with Mixed Hodge Structures

Both structures, namely the standard cup product ℤ(1)_{𝒟}^{•} ⊗ ℤ(1)_{𝒟}^{•} → ℤ(2)_{𝒟}^{•} given by (2.4), and the modified one ℤ(1)_{𝒟}^{•} ⊗ℤ(1)_{𝒟}^{•} → 2π\sqrt{−1}⊗D(1)_{h.h.}^{•} of Definition 4.3.1, can be obtained by taking projections of a common object in two different ways.

Let s be a local section of the pull-back

(f, g] = (f, g)^{∗}(H_{ℂ}/H_{ℤ})

as at the end of sect. 4.1. (The local expression in terms of matrices is given at the end of sect. 4.2.) Equivalently, s can be considered as a (local) lift of the map (f, g) : X → ℂ^{×} × ℂ^{×} to H_{ℂ}/H_{ℤ}.

Let ℳ_{X}^{(2)} be the resulting variation of Mixed Hodge Structures on X obtained by pulling back the universal MHS ℳ^{(2)} on H_{ℂ}/H_{ℤ} via s.

Lemma 4.3.5 (See [17]). The period P (ℳ_{X}^{(2)}) ∈ 𝒪_{X} ⊗_{ℚ} 𝒪_{X} of ℳ_{X}^{(2)} is given by:

P (ℳ_{X}^{(2)}) = \frac{h}{(2π\sqrt{−1})^{2}} ⊗ 1 − 1 ⊗ \frac{h}{(2π\sqrt{−1})^{2}}

+ 1 ⊗ \frac{log f log g}{(2π\sqrt{−1})^{2}} − \frac{log g}{2π\sqrt{−1}} ⊗ \frac{log f}{2π\sqrt{−1}}

Proof. The expression is computed in the appendix for the universal case.

Notice that the period actually belongs to the kernel of the multiplication map a ⊗ b → ab.

Let us now use the map 𝒪_{X} ⊗_{ℚ} 𝒪_{X} → 𝒪_{X} ⊗_{ℂ} 𝒪_{X} . Let ℐ_{X} be the kernel of the multiplication map (over ℂ). Then \underline{Ω}_{X/ℂ}^{1} ≅ ℐ_{X}/ℐ_{X}^{2}. The calculations for the following proposition are done in the universal case in the appendix.

Proposition 4.3.6. The expressions (4.4) and (4.12) respectively correspond to the images of P (ℳ_{X}^{(2)}) under the projections ℐ_{X} ⊂ 𝒪_{X} ⊗_{ℂ} 𝒪_{X} → \underline{Ω}_{X/ℂ}^{1} , sending a ⊗ b − ab ⊗ 1 to a db, and ℐ_{X} ⊂ 𝒪_{X} ⊗_{ℂ} 𝒪_{X} → \underline{ℰ}_{X}^{0} given by (4.8).

4.4 Comparisons

In the previous sections we have shown that the Deligne torsor (f, g] associated to two invertible functions f and g naturally acquires two structures: the analytic connection ∇ described in section 4.1 via the standard cup product in Deligne cohomology, and the hermitian structure described in section 4.3 via the modified cup product (4.9). We wish to briefly compare the two structures.

First, observe that using the canonical connection (cf. section 3.1.1) a pair (L, ρ) can also be thought of as a triple (L, ρ, ∇^{ρ}), where ∇^{ρ} is the canonical connection determined by ρ. Equivalently, we can just consider the pair (L, ∇^{ρ}). Also, let us stress that the canonical connection is only a smooth connection and is in general far from being analytic (or algebraic).

Thus our question can be reformulated as follows: for a given line bundle L equipped with an analytic connection ∇ and a hermitian fiber metric ρ, how do the pairs (L, ∇) and (L, ∇^{ρ}) compare?

The answer is the following well-known

Lemma 4.4.1. Consider both ∇ and ∇^{h} as smooth connections. Then:

1. ∇ − ∇^{h} determines a global section of \underline{A}_{X}^{1,0}, and

2. this global section is zero, that is, ∇ = ∇^{h}, if and only if L is unitary flat, namely it defines an element of H^{1}(X, ℝ/ℤ).

Proof. It is a well-known fact that the difference of two connections is a global one-form. Working in a local setting, let s ∈ L|_{U} be a local section, and let ∥s∥ be its length with respect to the metric. Then ∇s = ω ⊗ s, for ω ∈ \underline{Ω}_{X}^{1}(U), whereas ∇^{ρ}s = ∂ log∥s∥ ⊗ s, and ∂ log∥s∥ gives a local (1, 0)-form representative of ∇^{ρ}, cf. section 3.1.1. Clearly, the difference ω − ∂ log∥s∥ gives a global section of \underline{A}_{X}^{1,0}.

As for the second point, one would have ¯∂∂ log∥s∥ = 0, but this represents c_{1}(L), hence the conclusion.

In the situation when the two connections agree, that is, the connection is simultaneously analytic and it is the canonical connection associated to a hermitian structure, we say they are compatible. The line bundle supporting it is necessarily flat.

Interestingly enough, the previous lemma can be recast into entirely cohomological terms. This is advantageous in dealing with the special case L = (f, g] of special interest to us, as well as to address the very same question in the case of gerbes later on in this paper.

In the previous lemma we have compared ∇ and ∇^{ρ} by mapping their respective local representatives in \underline{A}_{X}^{1,0}. It will be more convenient to use the sheaf of imaginary 1-forms instead, namely consider π_{1} : \underline{Ω}_{X}^{1} → \underline{ℰ}_{X}^{1}(1) and d : \underline{ℰ}_{X}^{0}(1) → \underline{ℰ}_{X}^{1}(1). Consider the complex

Λ(2)^{•} =^{def} (ℤ(2) →^{ı} 𝒪_{X} ⟶^{−π_{1}∘d} \underline{ℰ}_{X}^{1}(1)) ,

and the obvious maps of complexes

α : ℤ(2)_{𝒟}^{•} ⟶ Λ(2)^{•} and β : 2π\sqrt{−1} ⊗ D(1)_{h.h.}^{•} ⟶ Λ(2)^{•} .

As usual, the cone:

Γ(2)^{•} =^{def} Cone(α − β)[−1] ,

characterizes the elements in ℤ(2)_{𝒟}^{•} and 2π\sqrt{−1}⊗D(1)_{h.h.}^{•} which agree in Λ(2)^{•}. A tedious but straightforward direct verification yields:

Lemma 4.4.2. We have the quasi-isomorphism:

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

Dropping the last term in (4.14), we obtain the truncation

Γ̃(2)^{•} =^{def} (ℤ(2) →^{ı} 𝒪_{X} ⟶^{(d,−π_{1})} \underline{Ω}_{X}^{1} ⊕ \underline{ℰ}_{X}^{0}(1)) ,

which clearly characterizes the elements in ℤ(2)_{𝒟}^{•} and 2π\sqrt{−1} ⊗ D(1)_{h.h.}^{•} which agree in 2π\sqrt{−1} ⊗ ℤ(1)_{𝒟}^{•} . (In other words, Γ̃(2)^{•} can be obtained by replacing Λ(2)^{•} by ℤ(1)_{𝒟}^{•} in the previous paragraphs.) In particular, let us denote by Pic(X, ∇, h) the second hypercohomology group H^{2}(X, Γ̃(2)^{•}), namely the subgroup of H_{𝒟}^{2} (X, ℤ(2)) × { \atop Pic(X)} of classes of pairs (L, ∇) and (L, ρ) mapping to the same element of Pic(X) ≅ H_{𝒟}^{2} (X, ℤ(1)). Then lemma 4.4.1 has the following reformulation:

Lemma 4.4.3. There is an exact sequence:

(4.15)  0 ⟶ H^{1}(X, ℝ/ℤ) ⟶ Pic(X, ∇, h) ⟶ E^{1}(X)(1) ,

where E^{1}(X)(1) are the global sections of \underline{ℰ}_{X}^{1}(1). Thus compatible connections are necessarily flat.

Proof. The complex Γ̃(2)^{•} is a quotient of Γ(2)^{•}, namely we have the exact sequence:

0 ⟶ \underline{ℰ}_{X}^{1}(1)[−3] ⟶ Γ(2)^{•} ⟶ Γ̃(2)^{•} ⟶ 0 ,

and from the resulting long exact cohomology sequence:

0 → H^{2}(X, Γ(2)^{•}) → H^{2}(X, Γ̃(2)^{•}) → E^{1}(X)(1) → ⋯

It was noted above that H^{2}(X, Γ̃(2)^{•}) ≅ Pic(X, ∇, h) , whereas for Γ(2)^{•} we have

H^{2}(X, Γ(2)^{•}) ≅ H^{1}(X, ℝ(2)/ℤ(2)) .

The latter isomorphism follows either from a direct computation, or noticing that Γ(2)^{•} is a quotient of D(2)_{h.h.}^{•} (see eq. (3.7)) and

H^{2}(X, Γ(2)^{•}) ≅ H_{𝒟_{h.h.}}^{2} (X, 2)

and then using lemma 3.2.4.

4.4.1 Comparing (f, g] and (f, g]_{h.h.}

Suppose now L is the Deligne torsor determined by two invertible functions f and g. Clearly, the symbols (f, g] and (f, g]_{h.h.} taken together determine an element of Pic(X, ∇, h), since the underlying torsor in Pic(X) ≅ H_{𝒟}^{2} (X, ℤ(1)) is the same. This element can be represented by the cocycle

((2π\sqrt{−1})^{2}m_{ij}n_{jk} , −2π\sqrt{−1} m_{ij} log_{j}g , log_{i}f \frac{dg}{g} ⊕ −π_{1}(log_{i}f) log |g|)

with values in Γ̃(2)^{•}.

Following Goncharov ([18]) let us define for any two invertibles f and g the 1-form

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

This is clearly globally defined where f and g are invertible.

We finally obtain the following comparison.

Proposition 4.4.4. The analytic connection in (f, g] and the canonical one associated to the hermitian structure in (f, g]_{h.h.} are compatible if and only if r_{2}(f, g) = 0 in E^{1}(X)(1).

Proof. Let ω_{i} = log_{i}f dg/g and σ_{i} = −π_{1}(log_{i}f) log |g|. The connecting homomorphism from Γ̃(2)^{•} to \underline{ℰ}_{X}^{1}(1), that is the last map to the right in the sequence (4.15), amounts to computing π_{1}(ω_{i}) + dσ_{i}. A straightforward calculation yields

π_{1}(ω_{i}) + dσ_{i} = −r_{2}(f, g) .

5 Hermitian holomorphic gerbes and 2-gerbes

5.1 Higher tame symbols

Brylinski and McLaughlin considered higher degree versions of the tame symbol construction, [10, 12], namely cup products of higher degree Deligne cohomology classes: (f, L] for f a holomorphic invertible function and L a holomorphic line bundle, and (L, L′] for a pair of holomorphic line bundles. The geometric interpretation of the symbols so obtained, also put forward in refs. [10, 12], is that (f, L] is a gerbe on X with band (≡ lien) 𝒪_{X}^{×} and a holomorphic connective structure. A similar statement holds for the 2-gerbe (L, L′].

5.1.1 Cup products

From the point of view of cohomology classes, one computes the relevant cup products. Using (2.4), we find that (f, L] ∈ H_{𝒟}^{3} (X, ℤ(2)) is represented by the cocycle

(5.1)  (g_{jk}^{−m_{ij}} , −\frac{1}{2π\sqrt{−1}} log_{i}f d log g_{ij}) ,

having made the standard choices for log_{i} f and the transition functions g_{ij} of L with respect to the choice of a cover 𝔘_{X}. Similarly, if g′_{ij} are the transition functions of L′, and 2π\sqrt{−1}c_{ijk} represents c_{1}(L) with respect to the cover 𝔘_{X}, then (L, L′] ∈ H_{𝒟}^{4} (X, ℤ(2)) is represented by the cocycle

(5.2)  (g′_{kl}^{−c_{ijk}} , −\frac{1}{2π\sqrt{−1}} log g_{ij} d log g′_{jk}) .

5.1.2 Hermitian variant

If we use the product

ℤ(1)_{𝒟}^{•} ⊗ ℤ(1)_{𝒟}^{•} ⟶ D(1)_{h.h.}^{•}

introduced in sect. 4.3, for f, L and L′ as above we have

\begin{matrix}H_{𝒟}^{1} (X, ℤ(1)) & ⊗ H_{𝒟}^{2} (X, ℤ(1)) & ⟶ & H_{𝒟_{h.h.}}^{3} (X, 1) \\ f & ⊗ [L] & ⟼ & (f, L]_{h.h.} \end{matrix}

Using the same Čech data as before, the symbol (f, L]_{h.h.} is represented by the cocycle

(5.3)  (g_{jk}^{−m_{ij}} , −\frac{1}{2π\sqrt{−1}}π_{1}(log_{i}f) π_{0}(log g_{ij})) .

Similarly, with L and L′ we have the product

\begin{matrix}H_{𝒟}^{2} (X, ℤ(1)) & ⊗ H_{𝒟}^{2} (X, ℤ(1)) & ⟶ & H_{𝒟_{h.h.}}^{4} (X, 1) \\ [L] & ⊗ [L′] & ⟼ & (L, L′]_{h.h.} \end{matrix}

and the representing cocycle

(5.4)  (g′_{kl}^{−c_{ijk}} , −\frac{1}{2π\sqrt{−1}}π_{1}(log g_{ij}) π_{0}(log g′_{jk})) .

Similarly to the proof of prop. 4.3.4, the maps of complexes ℤ(2)_{𝒟}^{•} → ℤ(1)_{𝒟}^{•} and D(1)_{h.h.}^{•} → ℤ(1)_{𝒟}^{•} induce corresponding maps on the symbols (f, L] and (f, L]_{h.h.} , moreover their images agree in H_{𝒟}^{3} (X, ℤ(1)) . An identical statement holds for (L, L′] and (L, L′]_{h.h.} .

5.2 Gerbes with Hermitian structure

Let 𝒢 be a gerbe on X with band 𝒪_{X}^{×} ([16]). After [7, 8], its class is an element of H_{𝒟}^{3} (X, ℤ(1)) ≅ H^{2}(X, 𝒪_{X}^{×}) . Let \underline{ℰ}_{X,+}^{0} be the sheaf of real positive smooth functions on X.

Definition 5.2.1. A hermitian structure on 𝒢 consists of the following data:

1. For each object P in 𝒢_{U}, is assigned a \underline{ℰ}_{U,+}^{0}-torsor \underline{herm}(P) (a ℝ_{+}-principal bundle). The assignment must be compatible with the restriction functors i^{∗}: 𝒢_{U} → 𝒢_{V} arising from i: V ↪ U in the cover 𝔘_{X} of X.

2. For each morphism f: P → Q in 𝒢_{U} a corresponding morphism f_{∗}: \underline{herm}(P) → \underline{herm}(Q) of \underline{ℰ}_{U,+}^{0}-torsors.^{1} This map must be compatible with compositions of morphisms in 𝒢_{U} and with the restriction functors.

For an object P of 𝒢_{U}, an automorphism φ ∈ \underline{Aut}(P) is identified with a section of 𝒪_{X}^{×} over U. We then require that

(5.5)

φ_{∗}: \underline{herm}(P) ⟶^{≃} \underline{herm}(P)
h ⟼ h · |φ|^{2}

where the latter is the \underline{ℰ}_{U,+}^{0}-action on the torsor \underline{herm}(P).

Theorem 5.2.2. Equivalence classes of 𝒪_{X}^{×}-gerbes with hermitian structure are classified by the group

H^{3}(X, ℤ(1)_{X} → 𝒪_{X} → \underline{ℰ}_{X}^{0}) .

Proof. Let 𝒢 be an 𝒪_{X}^{×}-gerbe on X with hermitian structure as per definition 5.2.1. Choose a full decomposition (see [7]) with objects P_{i} of 𝒢_{U_{i}} and isomorphisms f_{ij}: P_{j}|_{U_{ij}} → P_{i}|_{U_{ij}} with respect to a cover 𝔘_{X} of X. By a standard procedure (see refs.[7, 8]) these data determine a cochain g_{ijk} ∈ \underline{Aut}(P_{i})|_{U_{ijk}} ≅ 𝒪_{X}^{×}|_{U_{ijk}} satisfying the cocycle condition and determining a class in H^{2}(X, 𝒪_{X}^{×}). Furthermore, choose sections r_{i} of the torsors \underline{herm}(P_{i}) above U_{i}. From condition 2 in definition 5.2.1 we have that there must exist ρ_{ij} ∈ \underline{ℰ}_{X,+}^{0}|_{U_{ij}} such that:

(5.6)  f_{ij}_{∗}(r_{j}) = r_{i} · ρ_{ij} .

On the 3-skeleton of the cover we have that on one hand

(5.7)  f_{ij}_{∗} ∘ f_{jk}_{∗}(r_{k}) = f_{ij}_{∗}(r_{j}) · ρ_{jk} = r_{i} · ρ_{ij} ρ_{jk} ,

1A \underline{ℰ} _{U,+}^{0}-torsor will in general be automatically trivializable. However, in this context it is convenient to “forget” the actual trivializing map.

whereas on the other hand, since f_{ij} ∘ f_{jk} = g_{ijk} ∘ f_{ik} , we have

(5.8)  (f_{ij} ∘ f_{jk})_{∗}(r_{k}) = g_{ijk}_{∗} ∘ f_{ik}_{∗}(r_{k}) = g_{ijk}_{∗}(r_{i} · ρ_{ik}) = r_{i} · |g_{ijk}|^{2} ρ_{ik} .

Equating the right hand sides of eqs. (5.7) and (5.8), and extracting the appropriate logarithms, we see we have obtained a Čech cocycle representing a class in

(5.9)  Ȟ^{3}(𝔘_{X}, ℤ(1)_{X} → 𝒪_{X} → \underline{ℰ}_{X}^{0}) .

Conversely, let a class in H_{𝒟_{h.h.}}^{3} (X, 1) be given, and assume we represent it via the choice of 𝔘_{X} by a degree 2 Čech cocycle with values in the complex

ℤ(1)_{X} → 𝒪_{X} → \underline{ℰ}_{X}^{0} ,

which we write as

(2π\sqrt{−1}c_{ijkl} , log g_{ijk} , \frac{1}{2} log ρ_{ij}) .

This cocycle determines, via the map D(1)_{h.h.}^{•} → ℤ(1)_{𝒟}^{•} , a cocycle {g_{ijk}} ∈ Č^{2}(𝔘_{X}, 𝒪_{X}^{×}) which can be used, according to refs. [7, 8], to glue the local stacks Tors(𝒪_{U_{i}}) into a global 𝒢, in fact a gerbe. Given a 𝒪_{U_{i}}^{×}-torsor P_{i}, namely an object of 𝒢_{U_{i}} ≅ Tors(𝒪_{U_{i}}), define a hermitian structure by:

\underline{herm}(P_{i}) = trivial \underline{ℰ}_{U_{i},+}^{0} − torsor

Then use ρ_{ij} to glue \underline{herm}(P_{i}) and \underline{herm}(P_{j}) over U_{ij}, namely define an isomorphism via eq. (5.6). Since the isomorphisms P_{k} → P_{i} and P_{k} → P_{j} → P_{i} differ by the equivalence determined by g_{ijk}, we see using (5.5) that the condition

ρ_{ij} ρ_{jk} = |g_{ijk}|^{2} ρ_{ik} ,

ensuing from the cocycle condition, ensures the compatibility of this definition over U_{ijk}.

Corollary 5.2.3. Using the quasi-isomorphism

D(1)_{h.h.}^{•} ⟶^{≃} (ℤ(1)_{X} → 𝒪_{X} → \underline{ℰ}_{X}^{0}) ,

the class of a gerbe with hermitian structure is in fact in H_{𝒟_{h.h.}}^{3} (X, 1) .

We will see (cf. sect. 5.3) this group also automatically classifies a special type of connective structure on 𝒢.

5.3 Hermitian connective structure

The structure defined in sect. 5.2 can be supplemented by a variant of Brylinski’s connective structure [8] by taking into account the first Hodge filtration as in ref. [9]. Let 𝒢 be an 𝒪_{X}^{×} gerbe over X.

Definition 5.3.1. A type (1, 0) connective structure on 𝒢 is the assignment to each object P of 𝒢_{U} of a F^{1}\underline{A}_{U}^{1}- torsor \underline{Co}(P) compatible with restriction functors and morphisms of objects. In particular, for φ ∈ \underline{Aut}(P), we require that

(5.10)  \begin{matrix}φ_{∗}: \underline{Co}(P) & ⟶^{≃} \underline{Co}(P) \\ ∇ & ⟼ ∇ + d log φ \end{matrix}

where ∇ is a section of \underline{Co}(P) over U.^{2}

Definition 5.3.2. Let 𝒢 be equipped with a hermitian structure. A type (1, 0) connective structure on 𝒢 is compatible with the hermitian structure if for each object P of 𝒢 there is an isomorphism of torsors

\underline{herm}(P) ⟶ \underline{Co}(P)
r ⟼ ∇_{r}

such that for a positive function ρ on U

r · ρ ⟼ ∇_{r} + ∂ log ρ .

(In other words, ∇_{r·ρ} = ∇_{r} + ∂ log ρ .)

2Note that d log φ is holomorphic, hence of type (1, 0).

Connective structures of type (1, 0) are classified as follows.

Theorem 5.3.3. Let again D(1)_{h.h.}^{•} be the complex given by (3.7) for l = 1. Equivalence classes of connective structures on a 𝒪_{X}^{×}-gerbe 𝒢 compatible with a given hermitian structure are classified by the group

H^{3}(X, D(1)_{h.h.}^{•}) .

We have the following analog of the existence and uniqueness of the canonical connection on an invertible sheaf.

Corollary 5.3.4. A connective structure compatible with a hermitian structure on a gerbe 𝒢 is uniquely determined up to equivalence.

Proof. It is an immediate consequence of the fact that the groups in Theorems 5.2.2 and 5.3.3, being computed from quasi-isomorphic complexes, are actually the same (and equal to H_{𝒟_{h.h.}}^{3} (X, 1).)

Remark 5.3.5. The group H^{3}(X, D(1)_{h.h.}^{•}) ≅ H_{𝒟_{h.h.}}^{3} (X, 1) is not equal to Brylinski’s

H^{3}(X, ℤ(1) → \underline{ℰ}_{X}^{0}(1) → \underline{ℰ}_{X}^{1}(1)) ,

cf. ref. [9, Proposition 6.9 (1)]. (In fact there is an epimorphism C(1)^{•} → (ℤ(1) → \underline{ℰ}_{X}^{0}(1) → \underline{ℰ}_{X}^{1}(1)) with non-trivial kernel.) It follows that the notion of “hermitian gerbes with hermitian connective structure” in loc. cit. is not identical to our notion of 𝒪_{X}^{×}-gerbe with hermitian structure and compatible type (1, 0) connective structure.

Proof of Theorem 5.3.3. Choose a cover 𝔘_{X} as usual and let (P_{i}, f_{ij}, r_{i}) be a decomposition of 𝒢 and its hermitian structure as in the proof of Theorem 5.2.2.

If 𝒢 has a compatible type (1, 0) connective structure, we have a map \underline{herm}(G_{U_{i}}) ∋ r_{i} ↦ ∇_{i} ∈ \underline{herm}(G_{U_{i}}). For every isomorphism f_{ij} the compatibility condition from Definition 5.3.2 determines a form

ξ_{ij} = ∂ log ρ_{ij} ∈ F^{1}\underline{A}_{X}^{1}(U_{ij})

satisfying the condition

(5.11)  ξ_{jk} − ξ_{ik} + ξ_{ij} = d log g_{ijk} .

The imaginary 2-form η_{ij} =^{def} ¯∂ξ_{ij} = ¯∂∂ log ρ_{ij} then is a cocycle with values in F^{1}\underline{A}_{X}^{2} ∩ \underline{ℰ}_{X}^{2}(1). Altogether, g_{ijk}, \frac{1}{2} log ρ_{ij}, ξ_{ij} and η_{ij} determine a cocycle of total degree 3 in the Čech resolution Č^{•}(𝔘_{X}, D(1)_{h.h.}^{•}). Conversely, given a degree 3 cocycle with values in D(1)_{h.h.}^{•}, a gerbe 𝒢 with hermitian structure can be

obtained by gluing trivial 𝒪_{U_{i}}^{×}-torsors and \underline{ℰ}_{U_{i},+}^{0} torsors as in Theorem 5.2.2. Furthermore, define a map by assigning the trivial F^{1}\underline{A}_{U_{i}}^{1}-torsor to the trivial \underline{ℰ}_{U_{i},+}^{0}-torsor by

r ⟼ ∇_{r} ≡ ∂ log r .

Clearly, this defines a type (1, 0) connective structure compatible with the hermitian structure on 𝒢.

Remark 5.3.6. Note the proof of Theorem 5.3.3 that dη_{ij} = 0, hence we obtain a class

[η_{ij}] ∈ H^{3}(X, F^{1}\underline{A}_{X}^{•} ∩ σ^{2}\underline{ℰ}_{X}^{•}(1))

which can be associated to 𝒢 via the obvious map

D(1)_{h.h.}^{•} ⟶ F^{1}\underline{A}_{X}^{•} ∩ σ^{2}\underline{ℰ}_{X}^{•}(1) .

This class plays the same role for 𝒢 as the (global) imaginary form c_{1}(ρ) = ¯∂∂ log ρ_{i} for a metrized line bundle (L, ρ). Remark 5.3.7 ( Hermitian curving). An equivalent degree 3 cocycle can be obtained by introducing the cochain K_{i} ∈ \underline{A}_{X}^{1,1} ∩ \underline{ℰ}_{X}^{2}(1)(U_{i}) of imaginary 2-forms such that

¯∂∂ log ρ_{ij} = K_{j} − K_{i} ,

and the imaginary 3-form Ω_{i} ≡ Ω|_{U_{i}} such that

dK_{i} = Ω|_{U_{i}} ,

where Ω ∈ F^{1}A^{3}(X) ∩ E^{3}(X)(1) (global sections). We can regard K_{i} as the hermitian curving and Ω as the hermitian 3-curvature, respectively, of the type (1, 0) hermitian connection.

5.4 The symbol (f, L]_{h.h.}

Given an invertible function f and a line bundle L we have seen there is a product (f, L]_{h.h.} ∈ H_{𝒟_{h.h.}}^{3} (X, 1). We briefly give a geometric construction of the corresponding hermitian-holomorphic gerbe.

We need to recall from [12] the construction of the gerbe 𝒞 underlying (f, L]. 𝒞 is the stackification of the following pre-stack 𝒞^{0}. For U ↪ X objects of the category 𝒞_{U}^{0} are non vanishing sections of L|_{U}. If s ∈ L|_{U}, and non vanishing, it is denoted (f, s] as an object of 𝒞_{U}^{0}. Given another non vanishing section s′ of L over U, there is g ∈ 𝒪_{U}^{×} such that s′ = sg. Morphisms from (f, s′] to (f, s] are given by sections of the Deligne torsor (f, g] over U. For a third non vanishing section s′′, with s′′ = s′g′ = sgg′, composition of morphisms in the category 𝒞_{U}^{0} corresponds to the K-theoretic property of the Deligne torsor:

(f, gg′] ≅ (f, g] ⊗ (f, g′] .

Given a trivialization of L by a collection {s_{i}} relative to a cover 𝔘_{X} = {U_{i}}_{i∈I}, with transition functions g_{ij} ∈ 𝒪_{X}^{×}(U_{ij}), the objects (f, s_{i}] and the morphisms

ϕ_{ij} = {log_{i} f, g_{ij}}: (f, s_{j}] → (f, s_{i}]

provide a decomposition of 𝒞 in the sense of [7]. It follows that the automorphisms

(5.12)  h_{ijk} = ϕ_{ij} ⊗ ϕ_{jk} ⊗ ϕ_{ik}^{−1} = g_{jk}^{−m_{ij}} ∈ \underline{Aut}((f, s_{i}]|_{U_{ijk}}) ≅ 𝒪_{X}^{×}(U_{ijk})

represent the cohomology class of 𝒞 in H_{𝒟}^{3} (X, ℤ(1)) ≅ H^{2}(X, 𝒪_{X}^{×}).

Now define a hermitian structure on 𝒞 as follows. To an object (f, s] of 𝒞_{U} we assign

(5.13)  (f, s] ⇝ \underline{herm}((f, s]) = trivial \underline{ℰ}_{U,+}^{0}-torsor.

Then, given a morphism (f, g] ∋ ϕ: (f, s′] → (f, s] in 𝒞_{U}, with s′ = sg as above, we use the hermitian structure on the Deligne torsor underlying (f, g] defined in sect. 4.3, Proposition 4.3.4. Namely

(5.14) ϕ_{∗}: \underline{herm}((f, s′]) ⟶ \underline{herm}((f, s]) h ⟼ h · ∥ϕ∥^{2}

where h is a local section of \underline{herm}((f, s′]), to be identified with one of \underline{ℰ}_{U,+}^{0} and ∥ϕ∥ is the length of the non-vanishing section ϕ. We have the following analog of Proposition 4.3.4:

Proposition 5.4.1. The class of the gerbe 𝒞 underlying the symbol (f, L] with hermitian structure defined by eqs. (5.13) and (5.14) is given by the product (f, L]_{h.h.} in the group H^{3}(X, ℤ(1)_{X} → 𝒪_{X} → \underline{ℰ}_{X}^{0}) ≅ H_{𝒟_{h.h.}}^{3} (X, 1) .

Proof. We need to find the class of the 𝒞 as in the proof of Thm. 5.2.2 and show it coincides with (f, L]_{h.h.} as computed in eq. (5.3). To this end, let us use the decomposition of 𝒞 given by the objects (f, s_{i}] and morphisms ϕ_{ij} = {log_{i} f, g_{ij}}: (f, s_{j}] → (f, s_{i}] for non vanishing sections s_{i} ∈ L|_{U_{i}}, as before. The class of 𝒞 (without extra structures) is represented by the cochain g_{jk}^{−m_{ij}} already appearing in eq. (5.12).

Furthermore, in the hermitian Deligne torsor (f, g_{ij}] over U_{ij} the logarithm of the length of the section ϕ_{ij} = {log_{i} f, g_{ij}} is given by

σ_{ij} ≡ \frac{1}{2} log∥ϕ_{ij}∥^{2} ≡ \frac{1}{2} log ρ_{ij} = −\frac{1}{2π\sqrt{−1}} π_{1}(log_{i}f) log |g_{ij}| ,

cf. eq. (4.11). Thus we have found the total cocycle representing (f, L]_{h.h.} as in eq. (5.3). Indeed, by computing the Čech coboundary we find

σ_{ij} − σ_{ik} + σ_{jk} = −m_{ij} log |g_{jk}| ,

as desired.

5.5 Hermitian 2-Gerbes

Let us briefly extend the considerations outlined in the previous sections to 2-gerbes over X bound by 𝒪_{X}^{×}. (An extended exposition of the local geometry of 2-gerbes is to be found in ref. [7]. See also [10] for the abelian case.)

Recall that a 2-gerbe G over X bound by a sheaf of abelian groups \underline{H} is a fibered 2-category over X which satisfies the 2-descent condition for objects, and such that for any two objects P and Q in the fiber 2-category G_{U} over U ⊂ X the fibered category \underline{Hom}(P, Q) is a stack. If fact, this fibered category turns out to be an \underline{H}-gerbe equivalent to the neutral one Tors(\underline{H}). The properties of interest to us are the following: G is locally non-empty, namely there is a cover 𝔘_{X} of X such that for U ⊂ X in the cover, the object set of G_{U} is non-empty; G is locally connected, namely any two objects can be connected by a weakly invertible 1-arrow (that is, invertible up to a 2-arrow); any two 1-arrows can be (locally) joined by a 2-arrow; finally, for every 1-arrow its automorphism group is isomorphic in a specified way to \underline{H}.

Once the appropriate notion of isomorphism for 2-gerbes is introduced, isomorphism classes of 2-gerbes bound by \underline{H} are classified by the sheaf cohomology group H^{3}(X, \underline{H}) , see, e.g. refs. [7, 10].

In what follows, we shall set \underline{H} = 𝒪_{X}^{×}. Hence we can rephrase the previous statement by saying that isomorphism classes of 2-gerbes bound by 𝒪_{X}^{×} are classified by the group

H^{3}(X, 𝒪_{X}^{×}) ≅ H_{𝒟}^{4} (X, ℤ(1)) .

We shall need the local calculation leading to the classification, so we recall it here. Given a 2-gerbe G, let us choose a decomposition by selecting a cover 𝔘_{X} of X and a collection of objects P_{i} in G_{U_{i}}. There is a 1-arrow

f_{ij}: P_{j} → P_{i}

between their restrictions to G_{U_{ij}}. Furthermore, from the axioms there is a 2-arrow

α_{ijk}: f_{ij} ∘ f_{jk} ⟹ f_{ik} .

Further restricting over a 4-fold intersection U_{ijkl}, we have two 1-arrows f_{ij} ∘f_{jk} ∘f_{kl}: P_{l} → P_{i} and f_{il}: P_{l} → P_{i} and between them two 2-arrows, namely α_{ijl} ∘ (Id_{f_{ij}} ∗α_{jkl}) and α_{ikl} ∘ (α_{ijk} ∗ Id_{f_{kl}}) . Since 2-arrows are strictly invertible, it follows again from the axioms that there exists a section h_{ijkl} of 𝒪_{X}^{×} over U_{ijkl} such that

(5.15)  α_{ijl} ∘ (Id_{f_{ij}} ∗α_{jkl}) = h_{ijkl} ∘ α_{ikl} ∘ (α_{ijk} ∗ Id_{f_{kl}}) .

This section is a 3-cocycle and the assignment G ↦ [h] gives the classification isomorphism.

In analogy with what was previously done for gerbes, we are going to define a notion of hermitian structure and of type (1, 0) connectivity for 2-gerbes on X bound by 𝒪_{X}^{×}. Brylinski and McLaughlin defined a concept of connectivity on a 2-gerbe G over X to be the datum of a compatible class of connective structures on the gerbes \underline{Hom}_{U}(P, Q) for two objects P, Q in the fiber G_{U}. It is possible to introduce several variants of this notion, as done in refs. [10, 12]. Thus a type (1, 0) connectivity will just be the requirement that these connective structures take their values in F^{1}\underline{A}_{X}^{1} − torsors.

Let us model the concept of hermitian structure on a 2-gerbe after the one for gerbes given above in definition 5.2.1.

Definition 5.5.1. A hermitian structure on a 𝒪_{X}^{×}-2-gerbe G over X consists of the following data.

1. To each object P in the fiber 2-category G_{U} over U ⊂ X we assign a \underline{ℰ}_{U,+}^{0}-gerbe \underline{herm}(P) over U. (As before, \underline{ℰ}_{U,+}^{0} is the sheaf of real positive functions on U.)

2. This assignment must be compatible with the inverse image 2-functors i^{∗}: G_{U} → G_{V} , natural transformations φ_{i,j}: j^{∗}i^{∗} ⇒ (ij)^{∗} and modifications α_{i,j,k}: φ_{ij,k} ∘ (h^{∗} ∗ φ_{i,j}) ⇛ φ_{i,jk} ∘ (φ_{j,k} ∗ i^{∗}) arising from the inclusions i: V ↪ U, j: W ↪ V, and k: Z ↪ W, in the cover 𝔘_{X}.

3. For each 1-arrow f: P → Q in G_{U} a corresponding equivalence f_{∗}: \underline{herm}(P) → \underline{herm}(Q) of \underline{ℰ}_{U,+}^{0}-gerbes. For each 2-arrow α: f ⇒ f′ a corresponding natural transformation α_{∗}: f_{∗} ⇒ f′_{∗} between equivalences. We ask that this correspondence be compatible with compositions of 1- and 2-arrows. Namely, for 1arrows f, f′: P → Q and g, g′: Q → R and for 2-arrows α: f ⇒ f′ and β: g ⇒ g′ in G_{U}, which we compose as β ∗ α: g ∘ f ⇒ g′ ∘ f′, we find a diagram of natural transformations

(5.16)  \xymatrix{g_{∗} ∘ f_{∗} \ar@{=>}[d]_{β_{∗}∗α_{∗}} \ar@{=>}[r]^{ε(f,g)} & (g ∘ f)_{∗} \ar@{=>}[d]^{(β∗α)_{∗}} \\ g′_{∗} ∘ f′_{∗} \ar@{=>}[r]_{ε(f′,g′)} & (g′ ∘ f′)_{∗}}

of equivalences between the \underline{ℰ}_{U,+}^{0}-gerbes \underline{herm}(P) and \underline{herm}(R) on U ⊂ X.

4. From the axioms, the group of automorphisms of a 1-arrow f: P → Q in G_{U} is identified with 𝒪_{U}^{×}. It follows that such an automorphism α (that is, a 2-arrow from f to itself) can be identified with a section a ∈ 𝒪_{U}^{×}. We then require that the induced natural isomorphism

α_{∗}: f_{∗} ⟹ f_{∗} , where f_{∗}: \underline{herm}(P) ⟶ \underline{herm}(Q)

be identified with a section of \underline{ℰ}_{U,+}^{0} via the map

(5.17)  a ⟼ |a|^{2}

and an appropriate labeling of \underline{herm}(P) and \underline{herm}(Q) by objects r and s, respectively. In more detail, given an arrow f_{∗}(r) → s in \underline{herm}(Q), the action of α via α_{∗} will amount to an automorphism of s. We require that it be |a|^{2}.

Remark 5.5.2. The abstract nonsense of definition 5.5.1 could have more succinctly characterized by saying that the correspondence \underline{herm}(·) realizes a Cartesian 2-functor between G and the 2-gerbe Gerbes(\underline{ℰ}_{X,+}^{0}) on X, shifting to the reader the burden of unraveling the diagrams.

We have the following analog of theorem 5.2.2:

Theorem 5.5.3. Isomorphism classes of 𝒪_{X}^{×}-2-gerbes with hermitian structure in the sense of definition 5.5.1 are classified by the group

H^{4}(X, ℤ(1)_{X} → 𝒪_{X} → \underline{ℰ}_{X}^{0}) ≅ H_{𝒟_{h.h.}}^{4} (X, 1) .

Proof. Let G be a 𝒪_{X}^{×}-2-gerbe on X with hermitian structure as per definition 5.5.1. Forgetting the hermitian structure, G will determine a class in the group H_{𝒟}^{4} (X, ℤ(1)) ≅ H^{3}(X, 𝒪_{X}^{×}), and we have briefly recalled before — cf. eq. (5.15) — how to obtain a 3-cocycle representing the class of G.

To obtain the rest of the cocycle with values in the complex ℤ(1)_{X} → 𝒪_{X} → \underline{ℰ}_{X}^{0} let us make the same choice for a decomposition of G with respect to the cover 𝔘_{X}: a collection of objects P_{i} in G_{U_{i}}, 1-arrows f_{ij}: P_{j} → P_{i} between their restrictions and 2-arrows α_{ijk}: f_{ij} ∘ f_{jk} ⇒ f_{ik} .

We shall also need a decomposition of the \underline{ℰ}_{U_{i},+}^{0}-gerbes \underline{herm}(P_{i}): to this end let us choose objects r_{i} over U_{i} and arrows ξ_{ij}: (f_{ij})_{∗}(r_{j}) → r_{i} between their restriction to U_{ij}.

Let us consider a triple of objects P_{i}, P_{j}, P_{k} over U_{ijk}. (we are implicitly restricting to the fiber 2-category G_{U_{ijk}}.) We obtain the following diagram in \underline{herm}(P_{i})|_{U_{ijk}}:

(5.18)  [figure: (f_{ij})_{∗}(f_{jk})_{∗}(r_{k}); (f_{ij})_{∗}(ξ_{jk}); (f_{ij})_{∗}(r_{j}); ξ_{ij}; (f_{ik})_{∗}(r_{k}); ξ_{ik}; r_{i}; ρ_{ijk}]

The left vertical arrow in (5.18) results from the composition of two-arrows

\xymatrix{(f_{ij})_{∗} ∘ (f_{jk})_{∗} \ar@{=>}[r]^{ε_{ijk}} & (f_{ij} ∘ f_{jk})_{∗} \ar@{=>}[r]^{(α_{ijk})_{∗}} & (f_{ik})_{∗}}

resulting from diagram (5.16) in definition 5.5.1. At the level of objects in the gerbe \underline{herm}(P_{i}) diagram (5.16) is of course not commutative, so we obtain a section ρ_{ijk} ∈ \underline{Aut}(r_{i}), which we can identify with a section of the sheaf \underline{ℰ}_{U,+}^{0} over U_{ijk}.

Now consider a four-fold intersection U_{ijkl}: we have a cube determined by the objects r_{i}, …, r_{l} whose faces are built from copies of (5.18). Since this cube brings in the relation (5.15), using the mapping of the 𝒪_{X}^{×} action spelled out in the last point in definition 5.5.1, we get the relation

(5.19)  ρ_{jkl} ρ_{ikl}^{−1} ρ_{ijl} ρ_{ijk}^{−1} = |h_{ijkl}|^{2}

which, after taking the appropriate logarithms, defines a Čech cocycle representing a class in

Ȟ^{4}(𝔘_{X}, ℤ(1)_{X} → 𝒪_{X} → \underline{ℰ}_{X}^{0}) .

Details (and diagram chasing) are straightforward and left to the reader.

Conversely, let us be given a class in

H^{4}(X, ℤ(1)_{X} → 𝒪_{X} → \underline{ℰ}_{X}^{0}) ≅ H^{3}(X, 𝒪_{X}^{×} →^{|·|} \underline{ℰ}_{X,+}^{0}) ,

and let us assume it is represented by the (multiplicative) Čech cocycle (h_{ijkl}, ρ_{ijk}) . Let just explain the construction of a corresponding 2-gerbe with hermitian structure (up to equivalence). Again, details will be left to the reader.

We first apply the map

(ℤ(1)_{X} → 𝒪_{X} → \underline{ℰ}_{X}^{0}) ⟶ (ℤ(1)_{X} → 𝒪_{X})

to the representative Čech cocycle to reconstruct a 𝒪_{X}^{×}-2-gerbe G according to refs. [7, 10, 12]. Recall that this is accomplished by gluing the local stacks Gerbes(𝒪_{U_{i}}^{×}) using h_{ijkl}. Secondly, we define a hermitian structure as follows. Assign to any object P_{i} over U_{i} of the so-determined 2-gerbe G the trivial \underline{ℰ}_{U_{i},+}^{0}-gerbe \underline{herm}(P_{i}) = Tors(\underline{ℰ}_{U_{i},+}^{0}). For a triple of such on U_{ijk} we use ρ_{ijk} ∈ \underline{ℰ}_{U_{i},+}^{0}|_{U_{ijk}} as an automorphism of an object r_{i} in \underline{herm}(P_{i}).

Checking that this structure satisfies the properties in definition 5.5.1 and it defines a 2-gerbe with hermitian structure whose class is the one we started with is modeled after the pattern of refs. [7] and [8] and it will be left to the reader.

As mentioned before, a connectivity on a 𝒪_{X}^{×}-2-gerbe is in practice the assignment of compatible connective structures on the local gerbes of morphisms. We have the following definition (see also [9, sect. 7], for the first part):

Definition 5.5.4. Let G be a 𝒪_{X}^{×}-2-gerbe on X.

1. A type (1, 0) concept of connectivity on G is the assignment of a F^{1}\underline{A}_{U}^{1}-gerbe \underline{Co}(P) to each object P in G_{U}. This assignment will have to satisfy properties analogous to those of definition 5.5.1. Of course, in the last condition, the map (5.17) will have to be replaced by a ↦ d log a .

2. A type (1, 0) concept of connectivity is compatible with a hermitian structure if for each object P of G_{U}

there is an equivalence of gerbes

\underline{herm}(P) ⟶ \underline{Co}(P)

satisfying the obvious compatibility conditions with the operations of G_{U} and the restrictions.

The proof of the following theorem can be patterned after an appropriate generalization of the proof of Theorem 5.3.3, so we shall omit it.

Theorem 5.5.5. Let G be a 𝒪_{X}^{×}-2-gerbe with hermitian structure and let D(1)_{h.h.}^{•} be the complex given by (3.7) for l = 1. Equivalence classes of type (1, 0) connectivities on G compatible with the given hermitian structure are classified by the group

H^{4}(X, D(1)_{h.h.}^{•}) .

Furthermore, the equivalence class is unique.

5.6 The symbol (L, L′ ]_{h.h.}

We have seen that given two line bundles L and L′ over X their cup product (L, L′]_{h.h.} defines a class in H_{𝒟_{h.h.}}^{4} (X, 1). According to Theorem 5.5.3 it corresponds to an equivalence class of 2-gerbes with hermitian structure. Using the obvious maps of complexes D(1)_{h.h.}^{•} → ℤ(1)_{𝒟}^{•} and ℤ(2)_{𝒟}^{•} → ℤ(1)_{𝒟}^{•} , the geometric 2-gerbe G that underlies (L, L′]_{h.h.} is the same one as for the standard symbol (L, L′] constructed by Brylinski and McLaughlin.

Recall (see ref. [12] for more details) that objects of G underlying (L, L′] over U ⊂ X are the non-vanishing sections s of L|_{U}, denoted (s, L]. Given another non vanishing section s′ ∈ L|_{U} we have s′ = sg for an invertible function g over U. Then the category of morphisms from (s′, L] to (s, L] is the gerbe (g, L] defined in section 5.4. For a third non vanishing section s′′ of L over U, with s′′ = s′ g′, the morphism composition functor is given by the equivalence

(g, L′] ⊗ (g′, L] ⟶ (gg′, L]

where on the left hand side we have the contracted product of two (abelian) gerbes. To be precise, it turns out that G is an appropriate “2-stackification” of the 2-pre-stack defined here.

A calculation in ref. [12] shows that with respect to the trivializations {g_{ij}} and {g′_{ij}} of L and L′, respectively, the class of G is represented by the cocycle g′_{kl}^{−c_{ijk}} ∈ 𝒪_{X}^{×}(U_{ijkl}) , where the cocycle c_{ijk} represents c_{1}(L).

We can define a hermitian structure on G as follows. To an object (s, L′] of G_{U} we assign

(5.20)  (s, L′] ⇝ \underline{herm}((s, L′]) = trivial \underline{ℰ}_{U,+}^{0}-gerbe.

Furthermore, as remarked above we have \underline{Hom}_{U}((s′, L′], (s, L′]) ≅ (g, L′] . Thus we set

(5.21)  \underline{Hom}_{U}(\underline{herm}((s′, L′]), \underline{herm}((s, L′])) = (g, L′]_{h.h.} ,

where on the right hand side we use the hermitian structure on the gerbe (g, L′] as defined in section 5.4. On the left hand side of (5.21) we have the equivalences of the two \underline{ℰ}_{U,+}^{0}-gerbes.

The proof of the following proposition is a straightforward generalization of the one for proposition 5.4.1.

Proposition 5.6.1. The class of the 𝒪_{X}^{×}-2-gerbe G underlying the symbol (L, L′] with hermitian structure defined by eqs. (5.20) and (5.21) is given by the product (L, L′]_{h.h.} in the group H^{4}(X, ℤ(1)_{X} → 𝒪_{X} → \underline{ℰ}_{X}^{0}) ≅ H_{𝒟_{h.h.}}^{4} (X, 1) .

5.7 Comparisons and relations with other definitions

Recall from refs. [10, 12], that analytic connective structures on gerbes with band 𝒪_{X}^{×} are classified by the group H_{𝒟}^{3} (X, ℤ(2)). Similarly, for 2-gerbes with the same band, the relevant group is H_{𝒟}^{4} (X, ℤ(2)). In the previous sections we have introduced hermitian structures1 and type-(1, 0) connective structures on gerbes and 2-gerbes with band 𝒪_{X}^{×}. We define the concept of compatibility analogously to the case of line bundles in sect. 4.4 as follows.

Let 𝒢 be a 𝒪_{X}^{×}-gerbe on X. Let \underline{Co}(·)^{an} be a (holomorphic) connective structure on 𝒢 in the sense of refs. [10, 12], and let \underline{Co}(·)^{h} be a connective structure on the same gerbe in the sense of sect. 5.3.

The relevant group classifying 𝒢 equipped with both types of connections is therefore H^{3}(X, Γ̃(2)^{•}), where the complex Γ̃(2)^{•} has been introduced in sect. 4.4.

Definition 5.7.1. We say that \underline{Co}(·)^{an} and \underline{Co}(·)^{h} are compatible if for any object P of 𝒢_{U}, U ⊂ X, there is an isomorphism of torsors \underline{Co}(P)^{an} ≅ \underline{Co}(P)^{h} (after lambda-extension of \underline{Co}(P)^{an} from \underline{Ω}_{U}^{1} to \underline{A}_{U}^{1,0}.)

Similarly, if G is a 𝒪_{X}^{×}-2-gerbe on X, carrying both types of connective structures, its class is an element of the group H^{4}(X, Γ̃(2)^{•}). We can also repeat the above definition, taking care that now for any object of G over U ⊂ X, \underline{Co}(P)^{an} ≅ \underline{Co}(P)^{h} must be an equivalence of gerbes.

The next lemma immediately follows from the definitions.

Lemma 5.7.2. Let Γ(2)^{•} be the complex defined in sect. 4.4.

1. Classes of 𝒪_{X}^{×}-gerbes with compatible connective structures in the sense of definition 5.7.1 are classified by the elements of the group H^{3}(X, Γ(2)^{•}).

2. Similarly, classes of 𝒪_{X}^{×}-2-gerbes with compatible connective structures are classified by H^{4}(X, Γ(2)^{•}).

5.7.1 Compatibility and flatness conditions

While these definitions seem to follow the pattern of line bundles analyzed in sect. 4.4, there in an important difference, namely gerbes (or 2-gerbes) satisfying the compatibility condition of definition 5.7.1 are not necessarily flat! Moreover, in the present framework the compatibility condition is less special than it was seen in the case of line bundles. This can be seen by way of the following cohomological argument.

The complex Γ(2)^{•} introduced in sect. 4.4 is easily seen to be a quotient of the complex D(2)_{h.h.}^{•}:

D(2)_{h.h.}^{•} ⟶ Γ(2)^{•} ⟶ 0 .

The kernel is complicated, but up to quasi-isomorphism, it can be reduced (by direct computation) to the one-element complex \underline{ℰ}_{X}^{2}(1) ∩ \underline{A}_{X}^{1,1}[−4] so that we have the triangle:

\underline{ℰ}_{X}^{2}(1) ∩ \underline{A}_{X}^{1,1}[−4] ⟶ D(2)_{h.h.}^{•} ⟶ Γ(2)^{•} ⟶^{+1}

Focusing our attention to degree 3 and 4, we get the sequence:

0 → H^{2}(X, ℝ(2)/ℤ(2)) → H^{3}(X, Γ(2)^{•}) → E^{2}(X)(1) ∩ A^{1,1}(X) → H_{𝒟_{h.h.}}^{4} (X, 2) → H^{4}(X, Γ(2)^{•}) → 0 ,

where we have used lemma 3.2.4. Moreover, the exact sequence from the proof of lemma 4.4.3 relating Γ̃(2)^{•} to Γ(2)^{•} yields the following completion of (4.15):

0 → H^{1}(X, ℝ(2)/ℤ(2)) → Pic(X, ∇, h) → E^{1}(X)(1) → H^{3}(X, Γ(2)^{•}) → H^{3}(X, Γ̃(2)^{•}) → 0

and

H^{4}(X, Γ(2)^{•}) ⟶^{≅} H^{4}(X, Γ̃(2)^{•}) ,

where we have used that \underline{ℰ}_{X}^{1}(1) is soft. In summary we have:

Proposition 5.7.3.

1. The class of a 𝒪_{X}^{×}-gerbe supporting both types of connective structures can be lifted to a class of compatible connective structures on a (possibly equivalent) gerbe.

2. A 𝒪_{X}^{×}-gerbe with compatible connective structures is flat if the (trivial) (1, 1)-curving is zero (cf. sect. 5.3, remarks 5.3.6 and 5.3.7.)

3. A 𝒪_{X}^{×}-2-gerbe supporting both types of connective structures is equivalent to a 2-gerbe with compatible connective structures. Its class can be lifted to H_{𝒟_{h.h.}}^{4} (X, 2)

5.7.2 Comparing (f, L] and (L, L′] with their hermitian variants

The higher symbols (f, L] and (f, L]_{h.h.} have the same underlying gerbe, and similarly (L, L′] and (L, L′]_{h.h.} determine the same 2-gerbe. Let us denote them, respectively, by {f, L} and {L, L′}. By construction, they determine classes in H^{3}(X, Γ̃(2)^{•}) and H^{4}(X, Γ̃(2)^{•}), respectively. The proposition specializes to this case as follows:

Corollary 5.7.4. The connective structures \underline{Co}(·)^{an} and \underline{Co}(·)^{h} on \{f, L\} are compatible (up to \underline{ℰ}_{U}^{1}-torsor automorphism).

The analytic and hermitian connective structures on the 2-gerbe {L, L′} are compatible.

Proof. The statement follows at once from the calculations preceding the proposition.

Remark 5.7.5. As an alternative proof of the corollary, note that a calculation analogous to that of the proof of proposition 4.4.4 from the cocycle representations (5.1) and (5.3), yields the 1-cocycle r_{2}(f, g_{ij}) with values in \underline{ℰ}_{X}^{1}(1), where g_{ij} are the transition functions of L. This cocycle represents the zero class (softness of \underline{ℰ}_{X}^{1}(1)), therefore r_{2}(f, g_{ij}) = η_{j} − η_{i} , and this choice is determined up to a global section of \underline{ℰ}_{X}^{1}(1).

Similarly, in the case of {L, L′} we get the 2-cocycle r_{2}(g_{ij}, g′_{jk}) which again represents the zero class.

6 Concluding remarks

In this paper we have put forward a definition for the concept of hermitian structure, and associated compatible connective structure for gerbes and 2-gerbes with band 𝒪_{X}^{×}. We have presented classification results in terms of low degree hermitian holomorphic Deligne cohomology groups. Notable examples are provided by higher versions of the classical notion of tame symbol associated to two invertible functions. Indeed, our second main result that there exists a modified version of the cup product in low degree Deligne cohomology taking values in the first hermitian holomorphic Deligne complex, naturally provides the symbols (f, L] and (L, L′] with hermitian structures according to our definition.

Two questions naturally arise. Since (f, L] and (L, L′] also carry an analytic connective structure, we may ask to what degree the latter and the hermitian one are compatible. Remark 5.3.5 prompts a second obvious question regarding the relation between our classification theorems 5.3.3 and 5.5.5 and others’, notably Brylinski’s ([9, Proposition 6.9 (1)]).

We have analyzed the compatibility in cohomological terms, first for line bundles (in the sense of 𝒪_{X}^{×}-torsors) and then for gerbes and 2-gerbes with band-𝒪_{X}^{×}, with somewhat surprising results. Whereas the compatibility may be regarded as exceptional for a line bundle—and it implies its flatness—it is not so for gerbes (or 2-gerbes). Thus flatness is not a necessary condition. In the specific case of the tame symbols and their generalizations, we have found that while the compatibility of (f, g] and (f, g]_{h.h.} (that is, their respective connections) may in general be obstructed, (f, L] and (f, L]_{h.h.} can always be made compatible, and (L, L′] and (L, L′]_{h.h.} are automatically so.

As for the relation with other notions of “hermitian gerbe” with “hermitian connective structure” (or 2gerbe) there appear to be subtle differences in the definitions which we can trace to what aspect of line bundles with connection we decide to generalize. Our approach has been to copy the concept of metrized analytic (or algebraic) line bundle familiar from Arakelov geometry (cf. ref. [21]). On the other hand, one could describe a metrized 𝒪_{X}^{×}-line bundle by means of the 𝕋-reduction of its associated smooth line bundle plus a unitary connection. Whereas these two approaches are equivalent in the case of line bundles, they seem to diverge as soon as we move on to gerbes. (And possibly matters worsen in the case of 2-gerbes.) This may also serve to explain the lack of uniqueness found by Hitchin’s student D. Chatterjee in his thesis. Although that school’s approach to gerbes lacks the categorical input (in fact for them a gerbe is just the “torsor cocycle” in the sense of [7]) the definition of hermitian gerbe is along Brylinski’s lines.

Another difference is the following. Our cohomological characterization via the group H_{𝒟_{h.h.}}^{k} (X, 1) ≅ H^{k}(X, D(1)_{h.h.}^{•}), k = 3, 4, involves forms of degree two, which points to a natural notion of curving naturally associated with the structures we have defined (cf. remarks 5.3.6 and 5.3.7). This is obviously absent in the truncated group in remark 5.3.5. The cohomological analysis of sect. 5.7, where the group H_{𝒟_{h.h.}}^{4} (X, 2) appears, suggests that curvings can be a very nuanced structure, however dealing with them in detail falls outside the scope of the present work.

We hope to further elucidate matters in the future in another publication.

A Remarks on Hodge-Tate structures

The relation between the “imaginary part” map made in sect. 4.3 together with the product ℤ(1)_{𝒟}^{•} ⊗ ℤ(1)_{𝒟}^{•} → 2π\sqrt{−1}⊗D(1)_{h.h.}^{•}, and the cup product ℤ(1)_{𝒟}^{•} ⊗ℤ(1)_{𝒟}^{•} → ℤ(2)_{𝒟}^{•} giving rise to the tame symbol becomes more transparent from the point of view of Hodge-Tate structures.

A.1 A Mixed Hodge Structure

Let us briefly recall the following well known MHS on ℂ^{3}, see [13, 4]. Consider, as before,

(A.1)  M^{(2)} = (\begin{matrix}1 \\ x & 1 \\ z & y & 1 \end{matrix})

with complex entries x, y, z. Consider also its canonical version

(A.2)  A^{(2)} = (\begin{matrix}1 \\ x & 2π\sqrt{−1} \\ z & 2π\sqrt{−1} y & (2π\sqrt{−1})^{2} \end{matrix}) .

The MHS ℳ_{2} corresponding to M^{(2)}, or more precisely A^{(2)}, comprises the following data. The integer lattice is the ℤ span of the columns of A^{(2)}, and similarly for ℚ and ℝ. Let v_{0}, v_{1}, v_{2} denote the columns of A^{(2)} starting from the left. The weight spaces are W_{−2k}ℳ^{(2)} = span⟨v_{k}, …, v_{2}⟩ (over the appropriate ring), and the Hodge filtration is given by F^{−k}ℳ^{(2)}(ℂ) = ℂ⟨e_{0}, …, e_{k}⟩ , where the e_{i}’s are the standard basis vectors in ℂ^{2}. The graded quotients Gr_{−2k}^{W} ℳ^{(2)} are the Tate structures ℤ(0), ℤ(1), and ℤ(2). A change of the generators v_{i} preserving the structure clearly amounts to a change of A^{(2)} by right multiplication by a lower unipotent matrix over ℤ (or ℚ or ℝ). This is the same as changing M^{(2)} by a matrix in H_{ℤ} (or the appropriate ring thereof) as in sect. 4.2. ^{3}

The real structure underlying ℳ^{(2)} is linked to the hermitian structure on the bundle H_{ℂ}/H_{ℤ} as presented in sect. 4.3.1. In [4] the image of A^{(2)} in GL_{2}(ℂ)/ GL_{2}(ℝ) is obtained by computing the matrix

B =^{def} AĀ^{−1} ( \begin{matrix}1 \\ & −1 \\ & & 1 \end{matrix} ) ,

(we have dropped the superscript (2) for ease of notation). The logarithm is:

\frac{1}{2} log B = (\begin{matrix}1 \\ π_{0}(x) & 1 \\ π_{1}(z) − π_{1}(x)π_{0}(y) & π_{0}(y) & 1 \end{matrix}) .

We immediately recognize the expression of the hermitian form as given in sect. 4.3.1.

A.2 The big period

In ref. [17] Goncharov defines a tensor

P (ℳ) ∈ ℂ ⊗_{ℚ} ℂ

associated to a MHS (technically, a framed one) ℳ. For the MHS defined by the period matrix (A.1) it is computed as follows. Let f_{0}, f_{1}, f_{2} be the dual basis to v_{0}, v_{1}, v_{2}. Then, according to ref. [17],

P (ℳ^{(2)}) = ∑_{k}⟨f_{2}, M^{(2)}v_{k}⟩ ⊗_{ℚ} ⟨f_{k}, M^{(2)}^{−1}v_{0}⟩ .

Performing the calculation we find:

\begin{aligned}P (ℳ^{(2)}) & = \frac{z}{(2π\sqrt{−1})^{2}} ⊗ 1 − 1 ⊗ \frac{z}{(2π\sqrt{−1})^{2}} \\ & + 1 ⊗ \frac{xy}{(2π\sqrt{−1})^{2}} − \frac{y}{2π\sqrt{−1}} ⊗ \frac{x}{2π\sqrt{−1}} \end{aligned} ; (A.3)

Clearly, P (ℳ^{(2)}) is invariant under the action (4.5) (over ℚ). Moreover, P (ℳ^{(2)}) belongs to the kernel ℐ of the multiplication map ℂ ⊗_{ℚ} ℂ → ℂ . As a consequence, we have:

Proposition A.2.1. The “connection form” (4.6) and the (logarithm of the) hermitian fiber metric on the Heisenberg bundle correspond to the images of P (ℳ^{(2)}) under the two projections

ℐ ⟶ ℐ /ℐ^{2} = Ω_{ℂ/ℚ}^{1}

and

ℐ ⊂ ℂ ⊗_{ℚ} ℂ ⟶ ℝ(1) ,

respectively.

Proof. The images under the two projections are, respectively, equal to

−d(\frac{z}{(2π\sqrt{−1})^{2}} ) + \frac{x}{2π\sqrt{−1}} d(\frac{y}{2π\sqrt{−1}})

and

\frac{1}{(2π\sqrt{−1})^{2}} (π_{1}(z) − π_{1}(x) π_{0}(y)) .

3These data correspond to the case N = 2 of a MHS on ℂ^{N} defined for any integer N, cf. [4]

A.3 The extension class

The big period can be obtained as a symmetrization of an extension class of MHS. Indeed, the weight −2 subspace W_{−2}ℳ^{(2)} ≅ ℳ^{(1)} ⊗ 2π\sqrt{−1} ≡ ℳ^{(1)}(1) is itself a MHS (twisted by 2π\sqrt{−1}) defined by

(A.4)  A^{(1)} = \binom{1}{y 2π\sqrt{−1}} .

(The data are as for ℳ^{(2)}, replacing 2 by 1.) We thus have an extension of MHS:

(A.5)  0 ⟶ ℳ^{(1)}(1) ⟶ ℳ^{(2)} ⟶ ℤ(0) ⟶ 0 .

Following the procedure explained in ref. [6], it is seen that the class of the extension (A.5) belongs to

ℳ_{ℂ}^{(1)}(1)/ℳ_{ℚ}^{(1)}(1) ,

and it is given by the vector

(A.6)  e = −\frac{x}{2π\sqrt{−1}} v_{1} − \frac{z − xy}{(2π\sqrt{−1})^{2}} v_{2}

taken modulo ℳ_{ℚ}^{(1)}. This computation can be refined by noticing ([6]) that ℳ^{(1)} is itself an extension,

0 ⟶ ℤ(1) ⟶ ℳ^{(1)} ⟶ ℤ(0) ⟶ 0

mapping (over ℚ) to the “universal extension” ℋ ^{(1)}:

(A.7)  0 ⟶ ℚ(1) ⟶ ℂ ⟶ ℂ^{×} ⊗ ℚ ⟶ 0

obtained by tensoring the standard exponential sequence by ℚ. Over the complex numbers, we have

0 ⟶ ℂ ⟶ ℂ ⊗_{ℚ} ℂ ⟶ ℂ^{×} ⊗_{ℤ} ℂ ≅ ℂ/ℚ(1) ⊗_{ℚ} ℂ ⟶ 0 ,

Here we have ℋ_{ℚ}^{(1)} = ℂ and ℋ_{ℂ}^{(1)} = ℂ ⊗_{ℚ} ℂ . According to the same principle the class of the extension (A.7) lives in

(A.8)  ℋ_{ℂ}^{(1)}/ℋ_{ℚ}^{(1)} ≅ ℂ ⊗_{ℚ} ℂ/ℂ ≅ ℂ ⊗_{ℤ} ℂ^{×} .

The image of (A.6) in ℂ ⊗_{ℚ} ℂ is given by

(A.9)  ẽ = −y ⊗ x − 2π\sqrt{−1} ⊗ \frac{z − xy}{2π\sqrt{−1}} .

Taking (A.9) modulo ℋ_{ℚ}^{(1)} ≅ ℂ we finally have

(A.10)  (Id ⊗ exp)(ẽ) = y ⊗ e^{−x} + 2π\sqrt{−1} ⊗ e^{−(z−xy)/2π\sqrt{−1}} .

This is the (image of) the class of the extension (A.5) as computed in ref. [6]. It is easily seen that the element (A.10) is invariant under the transformations (4.5).

Lemma A.3.1. There is a unique well defined lift of the class (A.10) to F^{0}ℋ_{ℂ}^{(1)} = ker(m: ℂ ⊗_{ℚ} ℂ → ℂ) . This can be obtained by adding to (A.9) a (necessarily unique, see ref. [6]) element from ℋ_{ℚ}^{(1)} ≅ ℂ to (A.9). The lift is

2π\sqrt{−1} ⊗ 2π\sqrt{−1} · P (ℳ^{(2)}) .

Proof. We can identify ℋ_{ℚ}^{(1)} ≅ ℂ inside ℋ_{ℂ}^{(1)} via a ↦ a ⊗ 2π\sqrt{−1}. Thus add any such element to ẽ and consider the image under the multiplication map:

m(ẽ + a ⊗ 2π\sqrt{−1}) = −z + 2π\sqrt{−1}a .

It is equal to zero iff a = z/2π\sqrt{−1}, hence

˜ẽ = ẽ + \frac{z}{2π\sqrt{−1}} ⊗ 2π\sqrt{−1}

= −y ⊗ x + 2π\sqrt{−1} ⊗ \frac{xy}{2π\sqrt{−1}} + \frac{z}{2π\sqrt{−1}} ⊗ 2π\sqrt{−1} − 2π\sqrt{−1} ⊗ \frac{z}{2π\sqrt{−1}}

is the required element.

References

[1] Ettore Aldrovandi, On hermitian-holomorphic classes related to uniformization, the dilogarithm and the liouville action, Communications in Mathematical Physics 251 (2004), 27–64, arXiv:math.CV/0211055.

[2] Alexander A. Beĭlinson, Higher regulators and values of L-functions, Current problems in mathematics, Vol. 24, Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 181–238.

[3] , Notes on absolute Hodge cohomology, Applications of algebraic K-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), Amer. Math. Soc., Providence, RI, 1986, pp. 35–68.

[4] Alexander A. Beĭlinson and Pierre Deligne, Interprétation motivique de la conjecture de Zagier reliant polylogarithmes et régulateurs, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 97–121.

[5] Spencer Bloch, The dilogarithm and extensions of Lie algebras, Algebraic K-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980), Springer, Berlin, 1981, pp. 1–23.

[6] , Function theory of polylogarithms, Structural properties of polylogarithms, Math. Surveys Monogr., vol. 37, Amer. Math. Soc., Providence, RI, 1991, pp. 275–285.

[7] Lawrence Breen, On the classification of 2-gerbes and 2-stacks, Astérisque (1994), no. 225, 160.

[8] Jean-Luc Brylinski, Loop spaces, characteristic classes and geometric quantization, Birkhäuser Boston Inc., Boston, MA, 1993.

[9] , Geometric construction of Quillen line bundles, Advances in geometry, Birkhäuser Boston, Boston, MA, 1999, pp. 107–146.

[10] Jean-Luc Brylinski and Dennis A. McLaughlin, The geometry of degree-four characteristic classes and of line bundles on loop spaces. I, Duke Math. J. 75 (1994), no. 3, 603–638.

[11] , Holomorphic quantization and unitary representations of the Teichmüller group, Lie theory and geometry, Progr. Math., vol. 123, Birkhäuser Boston, Boston, MA, 1994, pp. 21–64.

[12] , The geometry of degree-4 characteristic classes and of line bundles on loop spaces. II, Duke Math. J. 83 (1996), no. 1, 105–139.

[13] Pierre Deligne, Le symbole modéré, Inst. Hautes Études Sci. Publ. Math. (1991), no. 73, 147–181.

[14] Hélène Esnault, Characteristic classes of flat bundles, Topology 27 (1988), no. 3, 323–352.

[15] Hélène Esnault and Eckart Viehweg, Deligne-Beĭlinson cohomology, Beĭlinson’s conjectures on special values of L-functions, Academic Press, Boston, MA, 1988, pp. 43–91.

[16] Jean Giraud, Cohomologie non abélienne, Springer-Verlag, Berlin, 1971, Die Grundlehren der mathematischen Wissenschaften, Band 179.

[17] Alexander B. Goncharov, Volumes of hyperbolic manifolds and mixed Tate motives, J. Amer. Math. Soc. 12 (1999), no. 2, 569–618, arXiv:alg-geom/9601021.

[18] , Explicit regulator maps on polylogarithmic motivic complexes, Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), Int. Press Lect. Ser., vol. 3, Int. Press, Somerville, MA, 2002, pp. 245–276.

[19] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978, Pure and Applied Mathematics.

[20] Richard M. Hain, Classical polylogarithms, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 3–42.

[21] Serge Lang, Introduction to Arakelov theory, Springer-Verlag, New York, 1988.

[22] Dinakar Ramakrishnan, A regulator for curves via the Heisenberg group, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 191–195.