RELATIVE ALGEBRAIC DIFFERENTIAL CHARACTERS

SPENCER BLOCH AND HÉLÈNE ESNAULT

Abstract. Let f : X → S be a smooth morphism in characteristic zero, and let (E, ∇_{X/S}) be a regular relative connection. We define a cohomology of relative differential characters on X which receives classes of (E, ∇_{X/S}). This is a version in family of the corresponding constructions performed in [9]. It says in particular that the partial vanishing of the trace of the iterated Atiyah classes can be made canonical. When applied to a family of curves and c_{2}, the construction yields a connection on f_{∗}c_{2}(E) ∈ Pic(S). Such a connection has been constructed analytically by A. Beilinson ([5]). We also relate our construction to the trace complex as defined in [3].

1. Introduction

Let f : X → S be a smooth family of curves aver a smooth base S over a field k of characteristic zero, and ∇_{X/S} : E → Ω_{X/S}^{1} ⊗ E be a relative connection. Then the Atiyah class

At(E) ∈ H^{1}(X, Ω_{X}^{1} ⊗ End(E))

lies in the image of H^{1}(X, f^{∗}Ω_{S}^{1} ⊗End(E)), and therefore higher Chern classes in H^{i}(X, Ω_{X}^{i}) die in H^{i}(X, Ω_{X}^{i}/f^{∗}(Ω_{S}^{i})). In this situation, when k is the field of complex numbers, A. Beilinson ([5]) uses Deligne-Beilinson analytic cohomology to show that the images f_{∗}c_{2}(E) and f_{∗}(c_{1}(E)^{2}), as analytic bundles, are endowed with canonical holomorphic connections. We show in this note that a careful analysis of splitting principle developed in [9] leads to the existence of functorial classes

c_{2}((E, ∇_{X/S})) and c_{1}((E, ∇_{X/S}))^{2} in the group ℍ^{2}(X, K_{2} ⟶^{d log} Ω_{S}^{1} ⊗

Ω_{X/S}^{1}). In other words, the vanishing of the class in H^{2}(X, Ω_{X}^{2}/f^{∗}(Ω_{S}^{2})) is made canonical. The trace of those classes define then isomorphism classes of line bundles with an algebraic connection in ℍ^{1}(S, O_{X}^{∗} ⟶^{d log} Ω_{S}^{1}). We show how this example is a particular case of a more general

Date: December 1, 1999.

1991 Mathematics Subject Classification. Primary 14F40 19E20 .

2 SPENCER BLOCH AND HÉLÈNE ESNAULT

theory of relative algebraic differential characters (see precise formulation in in section 2, and in subsequent sections for the construction), which allows higher dimensional smooth morphisms and higher classes as well.

In fact, A. Beilinson does not consider just vector bundles and Chern classes, but rather G-principal bundles under a reductive group G and arbitrary characteristic classes of weight 2. In this note we give two constructions of classes. The first is based on a splitting principle and is valid for vector bundles. The second uses the Weil-algebra homomorphism, as constructed in [4], to define a universal class. The latter should apply to more general G-bundles.

In a final note, we relate our construction to the trace complex as defined in [3].

It is a pleasure to work out A. Beilinson’s idea in a more algebraic language, hoping that he’ll like this framework. It is also a pleasure to thank him for his generosity.

2. Relative Cohomology

Let f : X → S be a smooth morphism between smooth varieties over a field k of characteristic 0. Then, for i ≥ 0, one defines the subcomplex of the de Rham complex

F^{i} = f^{∗}Ω_{S}^{i−•} ∧ Ω_{X}^{2•}[−•] ⟶^{ι}Ω_{X}^{≥i}[i], (2.1)

where f^{∗}Ω_{S}^{i−p} ∧ Ω_{X}^{2p} = Ω_{X}^{i+p} for i − p < 0, and F^{0} = Ω_{X}^{•} for i = 0. (Note F^{i} starts with f^{∗}Ω_{S}^{i} in degree 0.) The basic object of study is the complex

(2.2) AD_{X/S}^{i} := cone(K_{i} ⊕ F^{i} ⟶^{d log ⊕−ι} Ω_{X}^{≥i}[i])[−1],

where the Zariski sheaf K_{i} is defined to be the image of Milnor K sheaf K_{i}^{M} in its value at the generic point i_{k(X)∗}K_{i}^{M}(k(X)), or equivalently, the kernel of the residue map

i_{k(X)∗}K_{i}^{M}(k(X)) → ⊕_{x∈X^{(1)}}i_{x∗}K_{i−1}^{M}(k(x)).

One introduces the following

Definition 2.1. The group of relative algebraic differential characters in degree i is the group

AD^{i}(X/S) = ℍ^{i}(X, AD_{X/S}^{i}). (2.3)

We first define a product structure, following [2], the way we did in [9], which is compatible to the natural product

Ω_{X}^{≥i}[i] × Ω_{X}^{≥j}[j] → Ω_{X}^{≥(i+j)}[i + j], (2.4) the induced product

F^{i} × F^{j} → F^{(i+j)}, (2.5)

on the sub-complexes, and the K-product

K_{i} × K_{j} → K_{(i+j)}. (2.6)

Definition 2.2. Let α ∈ ℝ. We define

AD_{X/S}^{i} × AD_{X/S}^{j} → AD_{X/S}^{i+j}

by

x ∪_{α} y = {x, y}  x ∈ K_{i}, y ∈ K_{j}
= 0  x ∈ K_{i}, y ∈ F^{i}
= (1 − α)d log x ∧ y  x ∈ K_{i}, y ∈ Ω_{X}^{≥j}[j]
= 0  x ∈ F^{i}, y ∈ K_{j}
= x ∧ y  x ∈ F^{i}, y ∈ F^{j}
= (−1)^{degx}αx ∧ y  x ∈ F^{i}, y ∈ Ω_{X}^{≥j}[j]
= αx ∧ d log y  x ∈ Ω_{X}^{≥i}[i], y ∈ K_{j}
= (1 − α)x ∧ y  x ∈ Ω_{X}^{≥i}[i], y ∈ F^{j}
= 0  x ∈ Ω_{X}^{≥i}[i], y ∈ Ω_{X}^{≥j}[j]

Proposition 2.3. a) For every α ∈ ℝ, the formulae of definition 2.2 define a product, compatible with the products (2.4), (2.5), (2.6), with the graded commutativity rule

x ∪_{α} y = (−1)^{degx·degy}y ∪_{(1−α)} x.

b) For all α, β ∈ ℝ, the products ∪_{α} and ∪_{β} are homotopic, so, in particular, the product on AD^{i}(X/S) is commutative.

Proof. One has to show

δ(x ∪_{α} y) = δ(x) ∪_{α} y + (−1)^{degx}x ∪_{α} δ(y). (2.7)

Even if the verification is a bit lengthy, we insert it here, for sake of completeness. We have a complex X, sum of two complexes X = X_{1} ⊕ X_{2}, with maps u_{i} : X_{i} → Y. We denote by φ = u_{1} − u_{2} : X → Y. Then we define Cone(φ) = X[1] ⊕ Y, with differential δ(x, y) = (−d(x), φ(x) + d(y)), and δ[−1] on Cone(φ)[−1] = X ⊕ Y [−1] given by δ[−1] = −δ, that is concretely δ[−1](x, y) = (d(x), −φ(x) − d(y)).

Now take local sections a_{i} ∈ K_{i}, f_{i} ∈ F^{i}, ω_{i} ∈ Ω^{≥i}[i] and similarly for i replaced by j. So here, u_{1} is d log whereas u_{2} is the natural

4 SPENCER BLOCH AND HÉLÈNE ESNAULT

embedding. In order to simplify the notation, we omit the α from the notations (but not from the computations).

One has

δ(a_{i} ∪ a_{j}) = −u_{1}(a_{i} ∪ a_{j}) = −d log a_{i} ∧ d log a_{j} (2.8)

δ(a_{i}) ∪ a_{j} = −αd log a_{i} ∧ d log a_{j}
a_{i} ∪ δ(a_{j}) = (1 − α)d log a_{i} ∧ (−d log a_{j})

thus (2.7) is satisfied.

One has

a_{i} ∪ f_{j} = 0 (2.9)

δ(a_{i}) ∪ f_{j} = −(1 − α)d log a_{i} ∧ f_{j}
δ(f_{j}) = (df_{j}, u_{2}(f_{j}) = f_{j})
a_{i} ∪ df_{j} = 0
a_{i} ∪ u_{2}(f_{j}) = (1 − α)d log a_{i} ∧ f_{j}

thus (2.7) is satisfied.

One has

δ(a_{i} ∪ ω_{j}) = δ((1 − α)d log a_{i} ∧ ω_{j})) (2.10)

= −(1 − α)d(d log a_{i} ∧ ω_{j}) = (1 − α)d log a_{1} ∧ dω_{j}
δ(a_{i}) ∪ ω_{j} = 0
a_{i} ∪ δ(ω_{j}) = (a_{i}, −dω_{j}) = (1 − α)d log a_{i} ∧ (−dω_{j})

thus (2.7) is satisfied.

One has

f_{i} ∪ a_{j} = 0 (2.11)

δ(f_{i}) = (df_{i}, u_{2}(f_{i}))
df_{i} ∪ a_{j} = 0
u_{2}(f_{i}) ∪ a_{j} = αf_{i} ∪ d log a_{j}

f_{i} ∪ δ(a_{j}) = f_{i} ∪ (−d log a_{j}) = (−1)^{deg(f_{i})}αf_{i} ∧ (−d log a_{j})

thus (2.7) is satisfied.

One has

δ(f_{i} ∪ f_{j}) = δ(f_{i} ∧ f_{j}) = (2.12)

(df_{i} ∧ f_{j} + (−1)^{deg(f_{i})}f_{i} ∧ df_{j}, u_{2}(f_{i} ∧ f_{j})) ∈ (F^{i+j+1}, Ω_{X}^{≥(i+j)}) δ(f_{i}) ∪ f_{j} = df_{i} ∪ f_{j} + u_{2}(f_{i}) ∪ f_{j} = (df_{i} ∧ f_{j}, (1 − α)f_{i} ∧ f_{j})

f_{i} ∪ δ(f_{j}) = f_{i} ∪ df_{j} + f_{i} ∪ u_{2}(f_{j}) = (f_{i} ∧ df_{j}, (−1)^{deg(f_{i})}αf_{i} ∧ f_{j})

thus (2.7) is satisfied.

One has

δ(f_{i} ∪ ω_{j}) = δ((−1)^{deg(f_{i})}αf_{i} ∧ ω_{j}) = −d((−1)^{deg(f_{i})}αf_{i} ∧ ω_{j}) (2.13)

δ(f_{i}) ∪ ω_{j} = df_{i} ∪ ω_{j} + u_{2}(f_{i}) ∪ ω_{j} = (−1)^{deg(df_{i})}αdf_{i} ∧ ω_{j} + 0
f_{i} ∪ δ(ω_{j}) = f_{i} ∪ dω_{j} = (−1)^{deg(f_{i})}αf_{i} ∧ (−dω_{j})

thus (2.7) is satisfied.

One has

δ(ω_{i} ∪ a_{j}) = δ(αω_{i} ∧ d log a_{j}) = −d(αω_{i} ∧ d log a_{j}) (2.14)

δ(ω_{i}) ∪ a_{j} = −d(ω_{i}) ∪ a_{j} = −αd(ω_{i}) ∪ d log a_{j}
ω_{i} ∪ δ(a_{j}) = ω_{i} ∪ (−d log a_{j}) = 0

thus (2.7) is satisfied.

One has

δ(ω_{i} ∪ f_{j}) = δ((1 − α)ω_{i} ∧ f_{j})) = −d((1 − α)ω_{i} ∧ f_{j})) (2.15)

δ(ω_{i}) ∪ f_{j} = −dω_{i} ∪ f_{j} = −(1 − α)dω_{i} ∧ f_{j}
ω_{i} ∪ δ(f_{j}) = ω_{i} ∪ (df_{j}, u_{2}(f_{j})) = (1 − α)ω_{i} ∧ df_{j}.

The degree of ω_{i} in Cone(φ)[−1] equals the degree of ω_{i} in the complex Ω_{X}^{≥i} plus 1. Thus (2.7) is fulfilled.

One has

ω_{i} ∪ ω_{j} = 0 = δ(ω_{i}) ∪ ω_{j} = ω_{i} ∪ δ(ω_{j}) = 0

thus (2.7) is satisfied.

The homotopy is defined as follows

h(x ⊗ y) = (−1)^{m}(α − β)x ∧ y x ∈ (Ω_{X}^{≥i}[i])^{m−1}, y ∈ Ω_{X}^{≥j}[j]
= 0  otherwise .

3. Splitting Principle

Let f : X → S be a smooth proper morphism between smooth varieties, and let ∇_{X/S} : E → Ω_{X/S}^{1} ⊗ E be a relative connection on a vectorbundle of rank r. Let π : ℙ := ℙ(E) → X be the projective bundle associated to E. The relative connection ∇_{X/S} defines a splitting

τ : Ω_{ℙ/S}^{1} → π^{∗}Ω_{X/S}^{1} (3.1)

of the exact sequence

0 → π^{∗}Ω_{X/S}^{1} ⟶^{ι_{/S}} Ω_{ℙ/S}^{1} ⟶^{p_{/S}} Ω_{ℙ/X}^{1} → 0, (3.2)

6 SPENCER BLOCH AND HÉLÈNE ESNAULT

such that τ ∘ π^{∗}∇_{X/S} stabilizes the tautological sequence

0 → Ω_{ℙ/X}^{1}(1) → π^{∗}E → O_{ℙ}(1) → 0 (3.3)

(see [8], [6], [9]). Let i and p be defined by the exact sequence

0 → π^{∗}Ω_{X}^{1} ⟶^{i}Ω_{ℙ}^{1} ⟶^{p}Ω_{ℙ/X}^{1} → 0, (3.4)

and q : Ω_{ℙ}^{1} → Ω_{ℙ/S}^{1} be the projection.

Definition 3.1. We define the sheaf

Ω := Ker(Ω_{ℙ}^{1} ⟶^{τ∘q} π^{∗}Ω_{X/S}^{1}) (3.5)

From the definition, one has an exact sequence

0 → π^{∗}f^{∗}Ω_{S}^{1} → Ω → Ω_{ℙ/X}^{1} → 0 (3.6)

Lemma 3.2. The extension class

[Ω] ∈ H^{1}(ℙ, T_{ℙ/X} ⊗ π^{∗}f^{∗}Ω_{S}^{1}) = H^{1}(X, End^{0}(E) ⊗ Ω_{S}^{1})

given by (3.6) is the trace free part of the lifting of the Atiyah class of E in H^{1}(X, End^{0}(E) ⊗ Ω_{X}^{1}) defined by the choice of ∇_{X/S}.

Proof. The Atiyah sequence can be written

0 → End(E) → F → T_{X}^{1} → 0. (3.7)

Here F is interpreted as infinitesimal symmetries of the bundle E. A connection ∇_{X/S} relative to S gives rise to an action of T_{X/S}^{1}, i.e. a lifting ρ : T_{X/S}^{1} → F of the inclusion T_{X/S}^{1} ⊂ T_{X}^{1}.

The corresponding sequence for infinitesimal symmetries of the projective bundle ℙ = ℙ(E) looks like

0 → End^{0}(E) → π_{∗}T_{ℙ}^{1} → T_{X}^{1} → 0

Since symmetries of the vector bundle give rise to symmetries of the projective bundle, we get a diagram (End^{0}(E) = End(E)/k · id)

T_{X/S}^{1}
↓
0 ⟶ End(E) ⟶ F ⟶ T_{X}^{1} ⟶ 0
↓  ↓  ∥
0 ⟶ End^{0}(E) ⟶ π_{∗}T_{ℙ}^{1} ⟶ T_{X}^{1} ⟶ 0

In particular, the bottom line represents the pushout of the Aityah extension to the tracefree endomorphisms. On the other hand the composite of the vertical arrows gives rise by adjunction to a map π^{∗}T_{X/S}^{1} → T_{ℙ}^{1} which is dual to τ in (3.1). It follows that the reduction of structure of the tracefree Atiyah class defined by the connection ∇_{X/S} is given by the bottom line in the diagram

0  0
↓  ↓
T_{X/S}^{1}  T_{X/S}^{1}
↓^{ρ′}  ↓
0 ⟶ End^{0}(E) ⟶ π_{∗}T_{ℙ}^{1} ⟶ T_{X}^{1} ⟶ 0
∥  ↓  ↓
0 ⟶ End^{0}(E) ⟶ π_{∗}Ω^{∨} ⟶ f^{∗}T_{S}^{1} ⟶ 0
↓  ↓
0  0

Given the relation between the map labelled ρ′ above and the map τ in (3.1), it is straightforward to check that this bottom line is obtained from (3.6) by dualizing and pushing forward.

We now construct a τ version of the AD complex. Certainly, one has

d(∧^{i}Ω) ⊂ ∧^{i−1}Ω ∧ Ω_{ℙ}^{2}. (3.8)

Definition 3.3. Formally replacing f^{∗}Ω_{S}^{i} by ∧^{i}Ω and Ω_{X}^{≥i} by Ω_{ℙ}^{≥i} in the definition 2.1, we define the subcomplex

F_{τ}^{i} = ∧^{i−•}Ω ∧ Ω_{ℙ}^{2•}[−•] ⟶^{ι}Ω_{ℙ}^{≥i}[i] (3.9)

of the de Rham complex. Note F_{τ}^{0} = Ω_{ℙ}^{•} and, in degree 0, (F_{τ}^{i})^{0} = ∧^{i}Ω.

This allows one to define

Definition 3.4.

AD_{τ}^{i} = cone(K_{i} ⊕ F_{τ}^{i} ⟶^{d log ⊕−ι} Ω_{ℙ}^{≥i}[i])[−1],

and

AD_{τ}^{i}(ℙ) = ℍ^{i}(ℙ, AD_{τ}^{i}).

Then one defines a product formally as in definitiom 2.2, replacing AD_{X/S}^{i} by AD_{τ}^{i}. To be precise:

Definition 3.5. Let α ∈ ℝ. We define

AD_{τ}^{i} × AD_{τ}^{j} → AD_{τ}^{i+j}

8 SPENCER BLOCH AND HÉLÈNE ESNAULT

by

x ∪_{α} y = {x, y}  x ∈ K_{i}, y ∈ K_{j}
= 0  x ∈ K_{i}, y ∈ F^{i}
= (1 − α)d log x ∧ y  x ∈ K_{i}, y ∈ Ω_{ℙ}^{≥j}[j]
= 0  x ∈ F_{τ}^{i}, y ∈ K_{j}
= x ∧ y  x ∈ F_{τ}^{i}, y ∈ F_{τ}^{j}
= (−1)^{degx}αx ∧ y  x ∈ F_{τ}^{i}, y ∈ Ω_{ℙ}^{≥j}[j]
= αx ∧ d log y  x ∈ Ω_{ℙ}^{≥i}[i], y ∈ K_{j}
= (1 − α)x ∧ y  x ∈ Ω_{ℙ}^{≥i}[i], y ∈ F_{τ}^{j}
= 0  x ∈ Ω_{ℙ}^{≥i}[i], y ∈ Ω_{ℙ}^{≥j}[j]

Of course, proposition 2.3 holds true as well, replacing AD(X/S) by AD_{τ}(ℙ).

Definition 3.6. We denote by ξ the class of the induced partial connection

(O(1)_{ℙ}, τ ∘ ∇_{X/S}) ∈

AD_{τ}^{1}(ℙ) = ℍ^{1}(ℙ, K_{1} ⟶^{τ∘q∘d log} π^{∗}Ω_{X/S}^{1}),

and by [ξ] its image in ℍ^{1}(ℙ, F_{τ}^{1}) induced by the connecting morphism of the exact sequence

0 → F_{τ}^{1}[−1] → (K_{1} ⟶^{d log} Ω_{ℙ}^{1} ⟶^{d}Ω_{ℙ}^{2} → …) → AD_{τ}^{1} → 0.

Theorem 3.7. The product

AD_{τ}^{i}(ℙ) ∪ ξ^{∪j} ∈ AD_{τ}^{i+j}(ℙ)

induces an isomorphism

AD_{τ}^{r}(ℙ) = AD^{r}(X/S) ⊕ AD^{r−1}(X/S) ∪ ξ ⊕ … ⊕ AD^{1}(X/S) ∪ ξ^{∪(r−1)}.

Proof. From the exact sequence

… → ℍ^{n−1}(ℙ, Ω_{ℙ}^{≥i}[i]) → ℍ^{n}(ℙ, AD_{τ}^{i}) → H^{n}(ℙ, K_{i}) ⊕ ℍ^{n}(ℙ, F_{τ}^{i}) → …, (3.10)

the splitting principle on H^{n}(ℙ, K_{i}) and ℍ^{n−1}(ℙ, Ω_{ℙ}^{≥i}[i]), and the compatiblity of the products, one just has to see that

ℍ^{ℓ}(ℙ, F_{τ}^{i}) = ℍ^{ℓ}(X, F^{i}) ⊕ ℍ^{ℓ−1}(X, F^{i−1}) ∪ [ξ] ⊕ … H^{0}(X, F^{i−ℓ}) ∪ [ξ]^{∪ℓ} (3.11)

for ℓ = i, i − 1.

To this end, we will define a quasi-isomorphism

⨁_{j} F^{n−j}[−j] → ℝπ_{∗}(F_{τ}^{n}). (3.12)

Definition 3.8. a) We write the term in degree q in F_{τ}^{n} as ∧^{n−q}Ω ·

Ω_{ℙ}^{2q} ⊂ Ω_{ℙ}^{n+q}.

b) The exact sequence (3.6) defines a decreasing filtration on the exterior powers of Ω, with fil^{j} ∧^{p}Ω generated by wedges with at least j entries in π^{∗}f^{∗}Ω_{S}^{1}.

Lemma 3.9. fil^{p−j}Ω_{X}^{p+q−j} →^{≅} R^{j}π_{∗}(∧^{p}Ω · Ω_{ℙ}^{q}).

proof of lemma. By projecting [ξ] we find a canonical element θ ∈ H^{0}(X, R^{1}π_{∗}(Ω)). Taking powers gives θ^{j} ∈ H^{0}(X, R^{j}π_{∗}(∧^{j}Ω). Since π^{∗}f^{∗}Ω_{S}^{p} ⊂ ∧^{p}Ω, multiplication by θ^{j} gives a map as in the statement of the lemma. To see this map is an isomorphism, we argue by induction on q. Consider the diagram

R^{j−1}π_{∗}(∧^{p}Ω) ⊗ Ω_{X/S}^{q} ⟶^{a} R^{j}π_{∗}(∧^{p+1}Ω · Ω_{ℙ}^{q−1}) ⟶ R^{j}π_{∗}(∧^{p}Ω · Ω_{ℙ}^{q})) ⟶^{c} R^{j}π_{∗}(∧^{p}Ω) ⊗ Ω_{X/S}^{q}
≅↑_{b} _{∪θ^{j}} ↑ ≅↑_{d}
0  ⟶ fil^{p+1−j}Ω_{X}^{p+q−j} ⟶ fil^{p−j}Ω_{X}^{p+q−j} ⟶ f^{∗}Ω_{S}^{p−j} ⊗ Ω_{X/S}^{q} ⟶ 0

where the top row comes from the short exact sequence of sheaves

0 → ∧^{p+1}Ω · Ω_{ℙ}^{q−1} → ∧^{p}Ω · Ω_{ℙ}^{q} → ∧^{p}Ω ⊗ Ω_{X/S}^{q} → 0.

Suppose for a moment we know the lemma is true when q = 0. It follows that the maps d in the above diagram are isomorphisms in all degrees. It follows that the maps c are surjective and the maps a are zero. By induction the maps b are isomorphisms for all j, so it follows that the middle vertical arrow is an isomorphism as desired.

It remains to prove the lemma when q = 0. One proves by descending induction on j that

R^{k}π_{∗}(fil^{j} ∧^{p} Ω) ≅ \begin{cases}0 & k > p − j \\ f^{∗}Ω_{S}^{p−k} & k ≤ p − j. \end{cases}

We have fil^{p} ∧^{p} Ω = f^{∗}Ω_{S}^{p}, so the assertion is clear when j = p. The induction step comes from consideration of the long exact sequence

R^{k−1}π_{∗}(Ω_{ℙ/X}^{p−j}) ⊗ f^{∗}Ω_{S}^{j} → R^{k}π_{∗}(fil^{j+1} ∧^{p} Ω) → R^{k}π_{∗}(fil^{j} ∧^{p} Ω) →

→ R^{k}π_{∗}(Ω_{ℙ/X}^{p−j}) ⊗ f^{∗}Ω_{S}^{j} → R^{k+1}π_{∗}(fil^{j+1} ∧^{p} Ω),

noting that R^{a}π_{∗}Ω_{ℙ/X}^{b} ≅ O_{X} when a = b and is zero otherwise. Details are omitted.

10 SPENCER BLOCH AND HÉLÈNE ESNAULT

To finish the proof of theorem (3.7), it suffices to note that the maps in the lemma are compatible with the de Rham differentials, so we get quasi-isomorphisms in the derived category, which gives (3.12). The desired decomposition (3.11) follows by taking cohomology on X.

Definition 3.10. The decomposition of theorem 3.7 allows us to defines classes c_{i}(E, ∇_{X/S}) ∈ AD^{i}(X/S) in the usual way as the coefficients of the equation

(ξ)^{∪r} = ∑_{i=0}^{r−1}(−1)^{(i−1)}c_{r−i}(E, ∇_{X/S}) ∪ (ξ)^{∪i}.

Theorem 3.11. Let

0 → (E′′, ∇′′_{X/S}) → (E, ∇_{X/S}) → (E′, ∇′_{X/S}) → 0

be an exact sequence of relative connections. Then one has a Whitney product formula

c_{n}(E, ∇_{X/S}) = ∑_{i=0}^{n} c_{i}(E′, ∇′_{X/S})) ∪ c_{n−i}((E′′, ∇′′_{X/S}) (3.13)

Proof. Let ℙ′ = ℙ(E′) ↪^{i} ℙ = ℙ(E), and let U := ℙ − ℙ′ ↪^{j} ℙ. Let π′ : ℙ′ → X and p : U → ℙ′′ = ℙ(E′′) be the natural maps.

Lemma 3.12. Define AD_{τ}(U) by restricting the cone of complexes of sheaves used to define AD_{τ}(ℙ) to U. One has a pullback map

p^{∗} : AD_{τ′′}(ℙ′′) → AD_{τ}(U),

and p^{∗}(ξ′′) = j^{∗}(ξ). Finally, in degree r′′ = rank(E′′) we have

ker (j^{∗} : AD_{τ}^{r′′}(ℙ) → AD_{τ}^{r′′}(U)) ≅ ℤ.

proof of lemma. Compatibility of the connections on E and E′′ leads to a commutative diagram of sheaves on U

0  0
↓  ↓
0 ⟶ p^{∗}Ω_{τ′′} ⟶ p^{∗}Ω_{ℙ′′}^{1} ⟶ p^{∗}π′′^{∗}Ω_{X/S}^{1} ⟶ 0
↓^{a}  ↓a (∗)  ∥
0 ⟶ Ω_{τ}|_{U} ⟶ Ω_{U}^{1} ⟶ p^{∗}π′′^{∗}Ω_{X/S}^{1} ⟶ 0
↓  ↓
Ω_{U/ℙ′′}^{1}  Ω_{U/ℙ′′}^{1}
↓  ↓
0  0

Here of course we use the notation Ω_{τ}, Ω_{τ′′} etc to indicate in which projective bundle we consider the construction 3.5 of Ω. Exterior powers of the maps labelled a enable one to define

p^{∗} : AD_{τ′′}(ℙ′′) → AD_{τ}(U).

Commutativity of the square labelled (∗) above implies, using definition (3.6) that p^{∗}(ξ′′) = j^{∗}(ξ).

Finally, regarding the kernel of j^{∗} in degree r′′, we show here that it is a subgroup of ℤ. Once we describe the Gysin homomorphism below, it will be clear this kernel is nonzero. Since ℙ′ ⊂ ℙ has codimension r′′, purity results for local cohomology imply

ℍ_{ℙ′}^{r′′}(ℙ, AD^{r′′}) ↪ ker(ℍ_{ℙ′}^{r′′}(ℙ, K_{r′′} ⊕ ∧^{r′′}Ω_{τ}) → H_{ℙ′}^{r′′}(ℙ, Ω_{ℙ}^{r′′})).

Further we have

H_{ℙ′}^{r′′}(ℙ, K_{r′′}) ≅ ℤ; H_{ℙ′}^{r′′}(ℙ, ∧^{r′′}Ω_{τ}) ↪ H_{ℙ′}^{r′′}(ℙ, Ω_{ℙ}^{r′′}).

It follows that the local cohomology of AD_{τ}, which maps onto the kernel of j^{∗}, is a subgroup of ℤ as claimed.

Lemma 3.13. Let i : ℙ′ ↪ ℙ be the inclusion. There is defined a Gysin map

i_{∗} : AD_{τ′}^{k}(ℙ′) → AD_{τ}^{k+r′′}(ℙ).

The image i_{∗}(AD_{τ′}^{0}(ℙ′)) = ℤ and 1 maps to the cycle class of ℙ^{r′} in CH^{r′′}(ℙ). There is a projection formula

i_{∗}(x · i^{∗}y) = i_{∗}(x)y.

Proof. We have a commutative diagram

0  0
↓  ↓
N_{ℙ′/ℙ}^{∨}  N_{ℙ′/ℙ}^{∨}
↓  ↓
0 ⟶ i^{∗}Ω_{τ} ⟶ i^{∗}Ω_{ℙ}^{1} ⟶ π′^{∗}Ω_{X/S}^{1} ⟶ 0
↓  ↓  ∥
0 ⟶ Ω_{τ′} ⟶ Ω_{ℙ′}^{1} ⟶ π′^{∗}Ω_{X/S}^{1} ⟶ 0
↓  ↓
0  0

The left column leads to maps

det(N_{ℙ′/ℙ}^{∨}) ⊗ ∧^{p}Ω_{τ′} → i^{∗} ∧^{p+r′′} Ω_{τ}.

On the other hand, Grothendieck duality theory, [12], gives

det(N_{ℙ′/ℙ}) ⊗ i^{∗} ∧^{p+r′′} Ω_{τ} ≅ Ext_{O_{ℙ}}^{r′′}(O_{ℙ′}, ∧^{p+r′′}Ω_{τ}) → H_{ℙ′}^{r′′}(ℙ, ∧^{p+r′′}Ω_{τ}),

where H_{ℙ′} is the Zariski local cohomology sheaf. Composing these arrows, we get

∧^{p}Ω_{τ′} → H_{ℙ′}^{r′′}(ℙ, ∧^{p+r′′}Ω_{τ}).

An elaboration on this construction, using the middle column of the previous commutative diagram and the analogous map on K-sheaves

K_{p,ℙ′} → H_{ℙ′}^{r′′}(K_{p+r′′,ℙ})

gives a map of complexes

AD_{ℙ′}^{p} → H_{ℙ′}^{r′′}(AD^{p+r′′}).

Finally, using purity, we find

AD_{τ′}^{p}(ℙ′) = ℍ^{p}(ℙ′, AD_{τ′}^{p}) → H^{p}(ℙ′, H_{ℙ′}^{r′′}(AD_{τ}^{p+r′′})) →

ℍ_{ℙ′}^{p+r′′}(ℙ, AD_{τ}^{p+r′′}) → ℍ^{p+r′′}(ℙ, AD_{τ}^{p+r′′}) = AD_{τ}^{p+r′′}(ℙ).

In the special case p = 0, we find

ℤ = AD^{0}(ℙ′) → ℍ_{ℙ′}^{r′′}(ℙ, AD_{τ}^{r′′}) → AD_{τ}^{r′′}(ℙ).

Composing with the map AD_{τ}^{r′′}(ℙ) → CH^{r′′}(ℙ), we see that the above map is injective, and in fact is an isomorphism, taking 1 to the class of ℙ′.

Finally, the projection formula is a straightforward consequence of the multiplication on the complex AD_{τ}, which makes AD_{τ′}(ℙ′) an AD_{τ}(ℙ)-module.

The proof of the Whitney formula (3.13) is now straightforward. Write

F′(ξ) = ∑(−1)^{r′−i}c_{r′−i}(E′, ∇′)ξ^{i}, F′′(ξ) = ∑(−1)^{r′′−i}c_{r′′−i}(E′′, ∇′′)ξ^{i}

for the polynomials associated to E′ and E′′. We know that

j^{∗}F′′(ξ) = F′′(ξ′′) = 0,

so necessarily F′′(ξ) = i_{∗}(1). Also, F′(ξ′) = 0, so

0 = i_{∗}(F′(ξ′)) = i_{∗}(1)F′(ξ) = F′′(ξ)F′(ξ).

Since F (ξ) = ∑(−1)^{r′+r′′−i}c_{r′+r′′−i}(E, ∇)ξ^{i} is the unique monic polynomial in ξ of degree r′ + r′′ vanishing in AD_{τ}(ℙ), we conclude F = F′F′′.

Corollary 3.14. Let Y ⊂ X be a fiber of f. The classes c_{i}(E, ∇_{X/S}) ∈ AD^{i}(X/S) specialize to the classes c_{i}(E|_{Y} , ∇_{X/S}|_{Y}) ∈ AD^{i}(Y) defined in [9].

Let d be the relative dimension of the morphism f. The existence for n − a < d of a trace or transfer map

ℝf_{∗}(Ω_{X}^{n}/f^{∗}Ω_{S}^{a} · Ω_{X}^{n−a}) → Ω_{S}^{n−d}[−d]

leads to a transfer map

f_{∗} : AD^{n}(X/S) → ℍ^{n−d}(S, K_{n−d} ⟶^{d log} Ω_{S}^{n−d} ⟶^{d}… → Ω_{S}^{n−[ \frac{d}{2} ]−1}) (3.14)

defined for example by taking the Gersten-Quillen resolution of the K sheaves, and the Cousin resolution of the coherent sheaves of forms. In particular

Corollary 3.15.

f_{∗}(c_{d+1}(E, ∇_{X/S})) ∈ ℍ^{1}(S, K_{1} ⟶^{d log} Ω_{S}^{1} ⟶^{d}… → Ω_{S}^{[ \frac{d+1}{2} ]})

is the isomorphism class of a rank one line bundle with a connection, which is flat for d ≥ 3.

14 SPENCER BLOCH AND HÉLÈNE ESNAULT

4. Universal construction via the Weil algebra

In this section we give another construction of the relative classes, using unpublished work of A. Beilinson and D. Kazhdan [4]. These authors define a filtered differential graded algebra Ω_{X,E}^{•} ⊃ F^{n}Ω_{X,E}^{•}, such that (Ω_{X}^{•}, Ω_{X}^{≥n}) ⟶^{≅} (Ω_{X,E}^{•}, F^{n}Ω_{X,E}^{•}), together with a Weil homo-

morphism S^{n}(G^{∗})^{G} ⟶^{w^{n}(E)} F^{n}Ω_{X,E}^{•}[2n]. For us, the algebraic group is G = GL(r), and S^{n}(G)^{G} are the G-invariant polynomials on the Lie algebra G over k.

We begin by recalling the Beilinson-Kazhdan construction. Let p : 𝔼 → X be the GL(r)-torsor corresponding to a vector bundle E on X. Define

Ω_{X,E}^{1} := (p_{∗}Ω_{𝔼}^{1})^{GL(r)}.

The Atiyah sequence can be reinterpreted as the exact sequence of GL(r)-invariant relative differentials pushed down to X

0 → Ω_{X}^{1} → Ω_{X,E}^{1} ⟶^{π} End(E) → 0. (4.1)

Given an exact sequenc of vector bundles 0 → A → B →^{π} C → 0, the generalized Koszul sequence yields for any n

(4.2) 0 → ∧^{n}A → ∧^{n}B ⟶^{δ} ∧^{n−1}B ⊗ C ⟶^{δ}

∧^{n−2} B ⊗ Sym^{2}(C) → … → Sym^{n}(C) → 0

with

δ(b_{1}∧…∧b_{p}⊗c_{1}·. . .·c_{n−p}) = ∑_{i}(−1)^{i−1}b_{i}∧… b̂_{i} …∧b_{p}⊗π(b_{i})·c_{1}·. . .·c_{n−p}

Combining (4.1) and (4.2) we define a complex Ω_{X,E}^{•} to be the simple complex associated to the first quadrant double complex:

0  ⟶ ⋯; ↑ ; 0  ⟶ Sym^{2}(End(E)) ⟶ ⋯; ↑  ↑ ; 0 ⟶ End(E) ⟶^{d′} End(E) ⊗ Ω_{X,E}^{1} ⟶ ⋯; ↑  _{d′′=π}↑  ↑_{d′′} ; O_{X} ⟶^{d′} Ω_{X,E}^{1} ⟶^{d′}  ∧^{2}Ω_{X,E}^{1}  ⟶ ⋯ (4.3)

Grading by total degree gives

Ω_{X,E}^{n} = ⨁_{\begin{matrix}a+b=n \\ a≥b \end{matrix}} ∧^{a−b}Ω_{X,E}^{1} ⊗ Sym^{b}(End(E)) := ⨁ Ω_{X,E}^{a,b} (4.4)

The n-th column in (4.3) is a resolution of Ω_{X}^{n}, so defining

F^{p}Ω_{X,E}^{n} := ⨁_{\begin{matrix}a+b=n \\ a≥p \end{matrix}} Ω_{X,E}^{a,b} (4.5)

we get a filtered quasi-isomorphism

(Ω_{X}^{•}, Ω_{X}^{≥p}) →^{≅} (Ω_{X,E}^{•}, F^{p}Ω_{X,E}^{•}). (4.6)

In addition, End(E) is a twisting of the Lie algebra 𝔤 of GL(r), so there is a map

w^{n}(E) : Sym^{n}(𝔤^{∨})^{GL(r)} → Sym^{n}(End(E)) → F^{n}Ω_{X,E}^{•}[2n]. (4.7)

Assume now we are in a relative situation, with f : X → S and (E, ∇_{X/S}) as in section 2.

Lemma 4.1. The connection ∇_{X/S} determines a descent of the Atiyah extension (4.1) to an extension

0 → f^{∗}(Ω_{S}^{1}) → Ω_{∇}^{1} → End(E) → 0. (4.8)

I.e. the Atiyah extension comes from (4.8) by pushout f^{∗}Ω_{S}^{1} → Ω_{X}^{1}.

Proof. A connection defines an infinitesimal action of vector fields over X on E, i.e. a splitting of the Atiyah sequence (3.7). Thus, a relative connection leads to a diagram

\begin{aligned}0 & ⟶ & Ω_{X}^{1} & ⟶ & Ω_{X,E}^{1} & ⟶ & End(E) & ⟶ & 0 \\ & & ↓ & & ↓^{h} & & ∥ \\ 0 & ⟶ & Ω_{X/S}^{1} & ⟶ & Ω_{X/S,E}^{1} & ⟶^{⟵^{σ}} & End(E) & ⟶ & 0 \end{aligned} (4.9)

The descent comes by defining

Ω_{∇}^{1} := {z ∈ Ω_{X,E}^{1} | h(z) ∈ im(σ)}. (4.10)

Recall the complex F^{n} defined in (2.1). Our next objective is to mimick the above Weil construction, replacing (4.1) by (4.8). The construction leads to complexes F_{∇}^{n} and quasi-isomorphisms F^{n} ↪ F_{∇}^{n} analogous to (4.6). Note by (4.10) that Ω_{∇}^{1} ⊂ Ω_{X,E}^{1}, so one may define ∧^{a}Ω_{∇}^{1} · ∧^{b}Ω_{X,E}^{1} ⊂ ∧^{a+b}Ω_{X,E}^{1}.

Lemma 4.2. Recall (F^{n})^{j} = f^{∗}Ω_{S}^{n−j} · Ω_{X}^{2j}. There is a Koszul type resolution

(4.11) 0 → (F^{n})^{j} → ∧^{n−j}Ω_{∇}^{1} · ∧^{2j}Ω_{X,E}^{1} →

( ∧^{n−j−1} Ω_{∇}^{1} · ∧^{2j}Ω_{X,E}^{1}) ⊗ End(E) → …
→ ( ∧^{n−j−p} Ω_{∇}^{1} · ∧^{2j}Ω_{X,E}^{1}) ⊗ Sym^{p}(End(E)) → …
→ Sym^{n+j}(End(E)) → 0

Proof. We start with the resolution

(4.12) 0 → Ω_{X}^{q} → ∧^{q}Ω_{X,E}^{1} → ∧^{q−1}Ω_{X,E}^{1} ⊗ End(E) → … → Sym^{q}(End(E)) → 0

which we filter

fil^{i}(Ω_{X}^{q}) = f^{∗}Ω_{S}^{i} · Ω_{X}^{q−i}; (4.13)

fil^{i}( ∧^{q−k} Ω_{X,E}^{1} ⊗ Sym^{k}(End(E))) =

= \begin{cases}( ∧^{i−k} Ω_{∇}^{1} · ∧^{q−i}Ω_{X,E}^{1}) ⊗ Sym^{k}(End(E)) & i ≥ k \\ 0 & i < k \end{cases}

This filtration is compatible with the differential, and (using Ω_{X,E}^{1}/Ω_{∇}^{1} ≅ Ω_{X/S}^{1}), we find

(4.14) gr^{i}( ∧^{q−k} Ω_{X,E}^{1} ⊗ Sym^{k}(End(E)))

= \begin{cases}∧^{i−k}Ω_{∇}^{1} ⊗ Ω_{X/S}^{q−i} ⊗ Sym^{k}(End(E)) & i ≥ k \\ 0 & i < k \end{cases}

The complex gr^{i} is obtained by tensoring the Koszul resolution

0 → f^{∗}Ω_{S}^{i} → ∧^{i}Ω_{∇}^{1} → ∧^{i−1}Ω_{∇}^{1} ⊗ End(E) → …

with Ω_{X/S}^{q} and is thus exact. It follows that fil^{i} is a resolution of f^{∗}Ω_{S}^{i} · Ω_{X}^{q−i}. Taking i = n − j; q = n + j, we obtain the desired resolution of (F^{n})^{j}.

Definition 4.3. The complex

(F_{∇}^{i})^{•} = ⨁_{\begin{matrix}a+b=• \\ a≥b \end{matrix}}(F_{∇}^{i})^{a,b} ⊂ ⨁_{\begin{matrix}a+b=• \\ a≥b \end{matrix}} Ω_{X,E}^{a,b} (4.15)

is defined by

(F_{∇}^{i})^{a,b} = \begin{cases}( ∧^{2i−a−b} Ω_{∇}^{1} · ∧^{2a−2i}Ω_{X,E}^{1}) ⊗ Sym^{b}(End(E)) & 2i ≥ a + b; a ≥ i \\ 0 & i > a \\ Ω_{X,E}^{a,b} & a ≥ i; 2i ≤ a + b \end{cases} . (4.16)

The differentials are induced by the differentials on Ω_{X,E}^{•}.

Remark 4.4. The content of lemma (4.2) is that one has a resolution

0 → F^{i}[−i] → (F_{∇}^{i})^{i,0} → (F_{∇}^{i})^{i,1} → …

with differential d′′.

Remark 4.5. The Beilinson-Kazhdan-Weil homomorphism w_{n}(E), (4.7), takes values in (F_{∇}^{n})^{n,n} = Ω_{X,E}^{n,n} = Sym^{n}(End(E)) ⊂ F^{n}Ω_{X,E}^{2n}.

In particular, the complex

(4.17) cone(K_{n} ⊕ S^{n}(G^{∗})^{G}[−n] ⟶^{d log ⊕w_{n}(E)} F^{n}Ω_{X,E}^{•}[n])[−1]

used in [9] to define the absolute differential characters is endowed with a map

(4.18) cone(K_{n} ⊕ S^{n}(G^{∗})^{G}[−n] ⟶^{d log ⊕w_{n}(E)} F^{n}Ω_{X,E}^{•}[n])[−1]

⟶^{ι_{∇}=(1⊕w_{n},1)} cone(K_{n} ⊕ F_{∇}^{n}[n] ⟶^{d log ⊕w_{n}(E)} F^{n}Ω_{X,E}^{•}[n])[−1]

where the latter complex is quasi-isomorphic to AD_{X/S}^{n}.

Let X = ∪_{i}X_{i} be a Zariski covering trivializing E and let [E] : X_{•} → BG_{•} be the induced map to the simplicial classifying scheme BG_{•}. We denote by E_{un} the universal bundle on BG_{•}. Since

ℍ^{n}(BG_{•}, cone(K_{n} ⊕ S^{n}(G^{∗})^{G}[−n] ⟶^{d log ⊕w_{n}(E_{un})} F^{n}Ω_{BG_{•},E_{un}}^{•}[n])[−1]) (4.19)
= H^{n}(BG_{•}, K_{n})

(see [9]), one has a well defined universal class

(4.20) c_{n,an} ∈

ℍ^{n}(BG_{•}, cone(K_{n}⊕S^{n}(G^{∗})^{G}[−n] ⟶^{d log ⊕w_{n}(E_{un})} F^{n}Ω_{BG_{•},E_{un}}^{•}[n])[−1]).

One way to define c_{n,an} is again via the splitting principle on BG_{•}. Via the functoriality map

[E]^{∗} : cone(K_{n} ⊕ S^{n}(G^{∗})^{G}[−n] ⟶^{d log ⊕w_{n}(E_{un})} F^{n}Ω_{BG_{•},E_{un}}^{•}[n])[−1]

→ R[E]_{∗}cone(K_{n} ⊕ S^{n}(G^{∗})^{G}[−n] ⟶^{d log ⊕w_{n}(E)} F^{n}Ω_{X_{•},E}^{•}[n])[−1]

this defines a class

(4.21) [E]^{∗}(c_{n,un}) ∈

ℍ^{n}(X_{•}, cone(K_{n} ⊕ S^{n}(G^{∗})^{G}[−n] ⟶^{d log ⊕w_{n}(E)} F^{n}Ω_{X_{•},E}^{•}[n])[−1])

and via ι_{∇}, one obtains a class

Definition 4.6.

c′_{n}(E, ∇_{X/S}) := ι_{∇}([E]^{∗}(c_{n,un})) ∈ AD^{n}(X/S).

It remains to compare those classes to the classes c_{n}(E, ∇_{X/S}) constructed in the previous section with the splitting principle. As in [9], theorem 3.26, one has to verify the Whitney formula (4). One proceeds as in (4) of loc. cit.

Proposition 4.7. One has

c_{n}(E, ∇_{X/S}) = c′_{n}(E, ∇_{X/S}) ∈ AD^{i}(X/S)

5. The image of c_{2} in a family of curves

We now specialize the discussion to the case where f : X → S is a smooth family of curves. Let (E, ∇_{X/S}) be a relative connection, to which one has assigned classes

c_{2}(E, ∇_{X/S}), c_{1}(E, ∇_{X/S})^{2} ∈ AD^{2}(X/S). (5.1)

One considers the trace map

f_{∗} : AD^{2}(X/S) → AD^{1}(S/k) (5.2)

introduced in corollary 3.15. Since AD^{1}(S/k) is the group of isomorphism classes of line bundles with connection, we have constructed in this way a connection on the line bundles

f_{∗}c_{2}(E) and f_{∗}(c_{1}(E)^{2}).

We want to identify the curvature of the connection.

We consider the exact sequence of complexes on X

0 → (f^{∗}Ω_{S}^{2} → f^{∗}Ω_{S}^{2} ⊗ Ω_{X/S}^{1})[−1] (5.3)

→ (K_{2} ⟶^{d log} Ω_{X}^{2} → Ω_{X}^{3}/f^{∗}Ω_{S}^{3}) → AD_{X/S}^{1} → 0.

(Recall that f has dimension 1, thus Ω_{X}^{3}/f^{∗}Ω_{S}^{3} = f^{∗}Ω_{S}^{2} ⊗ Ω_{X/S}^{1}.) Then given a class γ ∈ AD^{2}(X/S), the connecting morphism δ of 5.3 defines a class

δ(γ) ∈ ℍ^{2}(X, f^{∗}Ω_{S}^{2} → f^{∗}Ω_{S}^{2} ⊗ Ω_{X/S}^{1}) →^{f_{∗}} (5.4)

H^{0}(S, Ω_{S}^{2} ⊗ R^{2}f_{∗}(Ω_{X/S}^{•})) = H^{0}(S, Ω_{S}^{2}).

Proposition 5.1. Let f_{∗}(γ) = (L, ∇) ∈ AD^{1}(S/k), with curvature ∇^{2} ∈ H^{0}(S, Ω_{S}^{2}). Then one has

∇^{2} = f_{∗}δ(γ).

Proof. Briefly, one has diagrams of complexes of sheaves on X

K_{2,X}  K_{2,X}
↓  ↓
f^{∗}Ω_{S}^{2}  ⟶ Ω_{X}^{2}  ⟶ Ω_{S}^{1} ⊗ Ω_{X/S}^{1}
↓  ↓
f^{∗}Ω_{S}^{2} ⊗ Ω_{X/S}^{1} ⟶^{≅} Ω_{X}^{3}/f^{∗}Ω_{S}^{3},

(5.5)

and on S

O_{S}^{×}  O_{S}^{×} ; ↓  ↓ ; Ω_{S}^{1}  Ω_{S}^{1} ; ↓ ; Ω_{S}^{2}  Ω_{S}^{2}. (5.6)

Using Gersten resolutions and Cousin complexes, one defines a trace map

ℝf_{∗}(diagram (5.5))[1] → diagram (5.6),

which yields a commutative diagram

AD^{2}(X/S) ⟶^{δ} ℍ^{2}(X, f^{∗}Ω_{S}^{2} → f^{∗}Ω_{S}^{2} ⊗ Ω_{X/S}^{1})
↓^{f_{∗}}  ↓^{f_{∗}}
AD^{1}(S) ⟶^{d}  Γ(S, Ω_{S}^{2}).

5.1. Beilinson’s gerbe construction. Now let us explain the construction (see [5]), which is the whole motivation for this note: A. Beilinson constructs such a connection on f_{∗}c_{2}(E) (and more generally on the image of classes of G-bundles of weight 2) in the analytic Deligne cohomology. We want to compare our construction to the one in [5]. We should emphasize that this comparison lacks precision in two points. First, we give our own interpretation of Beilinson’s construction in terms of a specific cocyle, and then we do not give the details of how this precise cocyle yields the same class as ours.

Beilinson uses gerbes as follows. Consider the Atiyah torsor of E. Given a local trivialization of E on X = ∪_{i}X_{i}, this defines transition functions g_{ij} ∈ C^{1}(Aut(E_{X})), and dg_{ij}g_{ij}^{−1} ∈ C^{1}(Ω_{X}^{1} ⊗ End(E_{X})) is a cocyle for the Atiyah torsor. Then the Ω_{X}^{2}-gerbe c_{1}^{2} −2c_{2} is represented by the cocyle Tr(dg_{ij}g_{ij}^{−1}dg_{jk}g_{jk}^{−1}) ∈ C^{2}(Ω_{X}^{2}) whereas the Ω_{X}^{2}-gerbe c_{1}^{2} is represented by the cocyle (Trdg_{ij}g_{ij}^{−1})^{2} ∈ Ω_{X_{ijk}}^{2}. Let us consider the Cech resolution (j_{I} : X_{I} ↪ X)

Ω_{X}^{2} → ∏_{i} j_{i∗}Ω_{X_{i}}^{2} ⟶^{d_{0}} ∏_{i<j} j_{ij∗}Ω_{X_{ij}}^{2} ⟶^{d_{1}} ∏_{i<j<k} j_{ijk∗}Ω_{X_{ijk}}^{2} ⟶^{d_{2}} … (5.7)

Then a cocyle c ∈ H^{0}(X, ∏_{i<j<k} j_{ijk∗}Ω_{X_{ijk}}^{2}) defines a gerbe, with objects a on U ⊂ X given by a ∈ d_{1}^{−1}(c|_{U}) and morphisms

Hom(a, b) = d_{0}^{−1}(b − a).

We consider the image gerbe c′ ∈ H^{0}(X, ∏_{i<j<k} j_{ijk∗}(f^{∗}Ω_{S}^{1}⊗Ω_{X/S}^{1})|_{X_{ijk}}). In particular, for X_{i} the complement of étale multisections on a sufficiently small affine open sets S_{α} ⊂ S, the residue (k_{I} : S_{I} ↪ S)

res(c′) ∈ H^{0}(S, ∏_{α<β} k_{αβ∗}Ω_{S_{αβ}}^{1})

represents the image torsor

(5.8) 0 → Ω_{S}^{1} → d_{0}^{−1}(res(c′)) ∈ ∏_{α} k_{α∗}Ω_{S_{α}}^{1} → res(c′) → 0

where

Ω_{S}^{1} → ∏_{α} k_{α∗}Ω_{S_{α}}^{1} ⟶^{d_{0}} ∏_{α<β} k_{αβ∗}Ω_{S_{αβ}}^{1} ⟶^{d_{1}} … (5.9)

is the Cech resolution of Ω_{S}^{1}.

Assume now that c′ is a coboundary, and choose

γ ∈ H^{0}(X, ∏_{i<j} j_{ij∗}(f^{∗}Ω_{S}^{1} ⊗ Ω_{X/S}^{1})|_{X_{ij}}) (5.10)

with d_{1}(γ) = c. So γ is a trivialization of the gerbe c′ and res(γ) ∈ d_{0}^{−1}(res(c′)) is a trivialization of the torsor res(c′). In particular, for c equal, as above, c_{1}^{2} − 2c_{2} or c_{1}^{2}, the torsor res(c′) is the Atiyah torsor of the line bundle f_{∗}c ∈ Pic(S). Thus the choice of γ defines a splitting res(γ) of the Atiyah torsor, or equivelently, it defines a connection on the line bundle f_{∗}(c).

It remains to see why the choice of a relative connection ∇_{X/S} : E → Ω_{X/S}^{1} ⊗ E defines a γ. The gauge transformation equation reads

dg_{ij}g_{ij}^{−1} = A_{i} − g_{ij}A_{j}g_{ij}^{−1} + ρ_{ij} (5.11)

where A_{i} ∈ H^{0}(X_{i}, Ω_{X}^{1} ⊗End(E)) is a lifting to global forms of the local relative forms of the connection, and ρ_{ij} ∈ H^{0}(X_{ij}, f^{∗}(Ω_{S}^{1}) ⊗ Ω_{X/S}^{1} ⊗ End(E)) is a cocyle. This defines in natural way

Tr(dg_{ij}g_{ij}^{−1}dg_{jk}g_{jk}^{−1}) = δ(η_{ij}) (5.12)

Tr(dg_{ij}g_{ij}^{−1})Tr(dg_{jk}g_{jk}^{−1}) = δ(ξ_{ij}) (5.13)

with

(5.14) η_{ij} = Tr(A ∪ δA + [A, ρ])_{ij}

= Tr(A_{i}δ(A)_{ij} − ρ_{ij}A_{j} + A_{i}ρ_{ij}) ∈ H^{0}(X_{ij}, f^{∗}Ω_{S}^{1} ⊗ Ω_{X/S}^{1})

ξ_{ij} = (a_{i}δ(a)_{ij} − κ_{ij}a_{j} + a_{i}κ_{ij}) ∈ H^{0}(X_{ij}, f^{∗}Ω_{S}^{1} ⊗ Ω_{X/S}^{1})

for a = Tr(A), κ = Tr(ρ).

We claim now that this construction leads to the same class as ours. It is obvious for f_{∗}(c_{1}^{2}) because of the product defined in section 3. As for f_{∗}(c_{1}^{2} − 2c_{2}), we can exhibit the AD class as follows. The relative AD-complex in this case is

K_{2} → Ω_{X}^{2}/f^{∗}Ω_{S}^{2}.

To write down a cocycle, we fix a flag in E, so the transition matrices g_{ij} are upper triangular:

g_{ij} = (\begin{matrix}ℓ_{ij}^{(1)} & … \\ 0 & ℓ_{ij}^{(2)} & … \\ ⋮ & ⋮ \\ 0 & 0 & … ℓ_{ij}^{(n)} \end{matrix}) ,

and we take a_{ijk} := ∑_{p}{ℓ_{ij}^{(p)}, ℓ_{jk}^{(p)}} ∈ K_{2}(U_{ijk}). Clearly,

d log(a_{ijk}) = Tr(dg_{ij}g_{ij}^{−1}dg_{jk}g_{jk}^{−1}).

The claim is now that the AD-class is represented by the 2-hypercocycle

(a_{ijk}, η_{ij}) ∈ (C^{2}(K_{2}) × C^{1}(f^{∗}Ω_{S}^{1} ⊗ Ω_{X/S}^{1})) (5.15)

where η_{ij} is the same cochain (5.14) which trivializes the gerbe c′ in Beilinson’s construction.

To see this, let us introduce the complete flag variety q : Q → X, such that q^{∗}E carries a tautological complete flag. The construction of Ω (see 3.5) carries over to the complete flag variety, and the connection q^{∗}∇ on q^{∗}E induces a Ω_{Q}^{1}/Ω connection, which then stabilizes the flag. The cohomology group in which the class considered is living is AD_{τ}^{2}(Q), and since Ω_{Q}^{3}/Ω · Ω_{Q}^{2} = q^{∗}Ω_{X/S}^{3} = 0 for a family of curves, one has

AD_{τ}^{2}(Q) = ℍ^{2}(Q, K_{2} ⟶^{d log} Ω_{Q}^{2}/ ∧^{2} Ω).

The equation of the connection becomes

dh_{ij}h_{ij}^{−1} = B_{i} − h_{ij}B_{j}h_{ij}^{−1} + σ_{ij} (5.16)

where h_{ij} is an upper triangular matrix of functions, B_{i} is an upper triangular matrix of forms in Ω_{Q}^{1}, σ_{ij} is an upper triangular matrix of forms in Ω. In particular, since h, B and σ are upper triangular, the rule for the product implies that q^{∗}(c_{1}^{2} − 2c_{2}) is

(b_{ijk}, Tr(B ∪ δ(B) + [B, σ]) ∈ (C^{2}(K_{2}) × C^{1}(Ω_{Q}^{2}/ ∧^{2} Ω)), (5.17)

where b_{ijk} = ∑_{p}{λ_{ij}^{(p)}, λ_{jk}^{(p)}} and where the λ_{ij}^{(p)} are the invertible diagonal entries of h_{ij}.

To finish the argument, one just needs that (5.17) is compatible with gauge transformation.

Remark 5.2. The construction of a connection on f_{∗}c_{2}(E) is used in [11], Theorem IV.3 in Faltings’ construction of Hitchin’s connection, when f is the projection C × M_{∇} → M_{∇} to the moduli (or stack) of connections on a fixed curve C, and E is the universal bundle. In particular, it identifies M_{∇} with the Atiyah torsor of f_{∗}c_{2}(E_{0}), where f_{0} : C × M → M is the projection to the moduli of stable bundles (see Lemma IV.4 of [11]), and E_{0} is the universal bundle on C × M.

5.2. The trace complex. Let f : X → S be a smooth family of curves, and let E be a vector bundle on X with a relative connection ∇_{X/S}. Under what conditions can one construct a connection on

RELATIVE CHARACTERS 23

det Rf_{∗}(E)? Intuitively, the Riemann Roch theorem gives [det Rf_{∗}(E)] ∈ Pic(S) as a linear combination of

f_{∗}c_{2}(E), f_{∗}(c_{1}(E)^{2}), f_{∗}(c_{1}(E) · K_{X/S}), and f_{∗}(K_{X/S}^{2}). We have shown how to put connections on f_{∗}c_{2}(E) and f_{∗}(c_{1}(E)^{2}). Suppose in addition that X = Y × S → S is a product family so K_{X/S}^{2} is trivial, and assume further that we have an absolute connection D on det E (e.g. det E = O_{X}). Intuitively again, D gives a connection on f_{∗}(c_{1}(E) · K_{X/S}). We sketch how in this situation one may use the trace complex of Beilinson and Schechtman [3] (see also [10]) to define a connection on det Rf_{∗}(E).

Associated to the family X/S and the bundle E one has the Atiyah algebra 0 → End(E) → A_{E,X} → T_{X} → 0. (5.18) One has a filtration T_{X/S} ⊂ T_{f} ⊂ T_{X}, where T_{f} consists of vector fields whose image falls in f^{−1}T_{S} ⊂ f^{∗}T_{S}. By pullback one defines subalgebras End(E) ⊂ A_{E,X/S} ⊂ A_{E,f} ⊂ A_{E,X}. (5.19) Pushing out by the trace map yields that corresponding Atiyah algebra for det E

\begin{aligned}0 & ⟶ & End(E) & ⟶ & A_{E,?} & ⟶ & T_{?} & ⟶ & 0 \\ & & ↓^{Tr} & & ↓ & & ∥ \\ 0 & ⟶ & O_{X} & ⟶ & A_{det E,?} & ⟶ & T_{?} & ⟶ & 0. \end{aligned} (5.20)

Thus we get 0 → End^{0}(E) → A_{E,?} → A_{det E,?} → 0. (5.21)

The trace complex trA^{•} = {A^{−2} → A^{−1} → A^{0}} (5.22) fits into an exact sequence of complexes 0 → ω_{X/S}[2] → ^{tr}A^{•} → {A_{E,X/S} → A_{E,f}} → 0 (5.23) One has A^{−2} = O_{X}; A^{0} = A_{E,X/S} (5.24) and A^{−1} is an extension 0 ⟶ ω_{X/S} ⟶ A^{−1} ⟶ A_{E,X/S} ⟶ 0. (5.25)

An absolute connection on det E induces compatible splittings

\begin{aligned}A_{det E,X/S} & ↠^{←} & T_{X/S} \\ ↓ & & ↓ \\ A_{det E,f} & ↠^{←} & T_{f}. \end{aligned} (5.26)

Assuming X = Y ×S (or, more generally, that we are given a connection for the map f : X → S) we get a decomposition

T_{f} = T_{X/S} ⊕ f^{−1}T_{S} (5.27)

Combining (5.26) and (5.27) yields an injective quasiisomorphism of complexes

{0 → f^{−1}T_{S}} ↪ {A_{det E,X/S} → A_{det E,f}}. (5.28)

We can pull back the sequences (5.21) along this map to get a quasiisomorphic subcomplex (defining B_{det E,f})

{End(E)^{0} → B_{det E,f}} ↪ {A_{E,X/S} → A_{E,f}} (5.29)

Finally, we can pull back (5.23) along this map, defining a quasiisomorphic subcomplex ^{tr}B^{•} ⊂ ^{tr}A^{•} with

B^{−2} = A^{−2} = O_{X}; B^{0} = B_{det E,f} (5.30)

0 → ω_{X/S} → B^{−1} → End(E)^{0} → 0 (5.31)

The sequence (5.31) is the trace-free part of the dual of the relative Atiyah sequence, so it is split by the given relative connection on E. there results a map of complexes

trB^{•} → ω^{•}[2] (5.32)

splitting the sequence (5.23) upto quasiisomorphism. Applying R^{0}f_{∗} yields a splitting for the exact sequence

0 → O_{S} ≅ R^{2}f_{∗}ω^{•} → R^{0}f_{∗}^{tr}B^{•} → T_{S} → 0. (5.33)

But, by [3], this is the Atiyah sequence for det Rf_{∗}(E), and a splitting defines a connection on that line bundle. In summary, we have shown

Theorem 5.3. Let f : X → S be a trivial family of curves. Let E be a bundle on X, together with a global connection D : det(E) → Ω_{X}^{1} ⊗ det(E) and a relative one ∇_{X/S} : E → Ω_{X/S}^{1} ⊗ E. Then D and ∇_{X/S} define a subcomplex ^{tr}B^{•} ⊂ ^{tr}A^{•} of the trace complex, quasiisomorphic to it, and a splitting ^{tr}B^{•} → ω[2]. In particular, these data induce a connection on the determinant of Rf_{∗}(E).

We admit to not having worked out the precise relation between this connection and the connections on f_{∗}(c_{2}) and f_{∗}(c_{1}^{2}) described earlier.

References

[1] Atiyah, M.: Complex analytic connections in fiber bundles, Trans. Am. Math. Soc. 85 (1957), 181-207.

[2] Beilinson, A.: Higher regulators and values of L-functions, J. Soviet Math. 30 (1985), 2036-2070.

[3] Beilinson, A.; Schechtman, V.: Determinant Bundle and Virasoro Algebras, Comm. in Math. Physics 118 (1988), 651-701.

[4] Beilinson, A.; Kazhdan, D.: Appendix B, Chern classes, unpublished.

[5] Beilinson, A.: letter to H. Esnault, June 4, 1999.

[6] Bloch, S.; Esnault, H.: Algebraic Chern-Simons theory, Am. J. of Math. 119 (1997), 903-952.

[7] Bloch, S.; Esnault, H.: Gauß-Manin determinant for rank 1 irregular connections on curves, preprint 1999, 47 pages.

[8] Esnault, H.: Characteristic classes of flat bundles II, K-Theory 6 (1992), 45-56.

[9] Esnault, H.: Algebraic Differential Characters (preprint) 1996, 28 pages, appears in the Proceedings of the Conference on Motives at the Landau Institute in Jerusalem, March 1996, Birkhäuser Verlag, in print.

[10] Esnault, H.; Tsai, I-H.: Determinant bundle in a family of curves, after A. Beilinson and V. Schechtman, preprint 1999, 6 peages, appears in Comm. in Math. Phy.

[11] Faltings, G.: Stable G-bundles and projective connections, J. Algebraic Geom. 2 (1993), 507-568.

[12] Grothendieck, A.: Théorèmes de dualité pour les faisceaux algébriques cohérents, Sém. Bourbaki exp. 149, mai 1957.

Dept. of Mathematics, University of Chicago, Chicago, IL 60637, USA

E-mail address: bloch@math.uchicago.edu

Mathematik, Universität Essen, FB6, Mathematik, 45117 Essen, Germany

E-mail address: esnault@uni-essen.de