LECTURES ON ALGEBRO-GEOMETRIC CHERN-WEIL

AND CHEEGER-CHERN-SIMONS THEORY FOR VECTOR BUNDLES

SPENCER BLOCH AND HÉLÈNE ESNAULT

Abstract. An algebraic theory of differential characters [16], [17] is outlined. A Riemann-Roch theorem for regular flat bundles, using a weak form of these classes, is described. Full details of the proof are available in a manuscript on the web [6].

1. Connections and Characteristic Classes

1.1. Introduction. The purpose of these two lectures will be to describe some work on the general topic of algebro-geometric Chern-Weil and Cheeger-Chern-Simons theory for characteristic classes of vector bundles. The trick is to cling to algebraic methods (K-theory, the Zariski topology, algebraic differential forms,...) so the resulting theory gives information about algebraic cycles, while at the same time drawing inspiration from the beautiful calculus of differential forms and connections lying behind the differential geometry.

We start with an algebraic variety X which will always be taken smooth and quasi-projective over an algebraically closed field k of characteristic 0. Let E be a vector bundle on X. We suppose given an algebraic connection on E, that is a k-linear map

∇ : E → E ⊗ Ω_{X/k}^{1} (1.1.1)

satisfying the Leibniz rule

∇(f · e) = f∇(e) + e ⊗ df (1.1.2)

for f a function and e a section of E. Note that in the C^{∞} category a partition of unity argument can be used to give any bundle E a connection. Algebraically, if e.g. X is projective, most bundles E will not admit a connection and our ship would seem to have sunk before leaving the harbor. In fact, we may dodge that bullet as follows. Proposition 1.1.1. [1] There exists a class

At(E) ∈ H^{1}(X, rmHom(E, E) ⊗ Ω_{X}^{1})

Date: June 3, 1999.

1991 Mathematics Subject Classification. Primary .

2 SPENCER BLOCH AND HÉLÈNE ESNAULT

the vanishing of which is necessary and sufficient for E to admit a connection.

Proof. As the difference of two connections is O_{X}-linear (see (1.1.1) and (1.1.2)), connections on E form a torsor under rmHom(E, E ⊗ Ω_{X}^{1}). For more detail, one can argue as follows. Let I ⊂ O_{X×X} be the ideal of the diagonal. Recall

Ω_{X}^{1} ≅ I/I^{2}; dx ↦ x ⊗ 1 − 1 ⊗ x mod I^{2}

Write P := O_{X×X}/I^{2}, so we have an exact sequence

0 → Ω_{X}^{1} → P → O_{X} → 0 (1.1.3)

Note P has two O_{X}-module structures pulling back via the two projections X × X → X. Tensor (1.1.3) with E via the left O_{X}-structure:

0 → E ⊗_{O_{X}} Ω_{X}^{1} → E ⊗_{O_{X}} P →^{←^{s}} E → 0 (1.1.4)

and view this as a sequence of O_{X}-modules via the right O_{X}-module structure on P. The middle term E ⊗_{O_{X}} P is the vector bundle of 1-jets of sections of E. The evident splitting s(e) = e ⊗ 1 is not an O_{X} splitting. In fact, it is straightforward to check that there is a 1 − 1 correspondence [11]

{O_{X}-linear splittings t of (1.1.4)} ↔ {connections ∇ : E → E ⊗ Ω_{X}^{1}} (1.1.5)

given by t ↦ ∇ := s − t. The proposition follows by considering the sequence obtained by rmHom(E, (1.1.4))

0 → rmHom(E, E ⊗ Ω_{X}^{1}) → rmHom(E, E ⊗ P) → Hom(E, E) → 0 (1.1.6)

and defining conn(E) = ∂(Id_{E}) ∈ H^{1}(X, Hom(E, E ⊗ Ω_{X}^{1})).

Corollary 1.1.2. If X is affine, any vector bundle E admits a connection.

Proposition 1.1.3 (Jouanolou). Let X be quasi-projective. There exists π : X̃ → X with the following properties:

1. X̃ is an affine variety.

2. Locally over X, X̃ ≅ X × 𝔸^{N} for some N.

Proof. Embed X ↪ Y with Y projective. Blowing up Y − X, we may assume Y − X is a Cartier divisor. Since the complement of a Cartier divisor in an affine scheme is affine, we reduce in this way to the case X projective. By pulling back, it suffices to do the case X = ℙ^{N}. Let X′ be the dual projective space, and let I ⊂ X′ × X be the incidence correspondence. Define X̃ := X′ × X − I.

The crucial point is that because of condition 2,

π^{∗} : CH^{∗}(X) ≅ CH^{∗}(X̃),

i.e. the two varieties have the same motive. Thus, we may pull back our bundle to X̃ and assume E admits a connection.

In what follows we will consider two sorts of questions. First, for an arbitrary vector bundle, we develop Chern-Weil and Cheeger-Chern-Simons theory for an arbitrary connection on a pullback to an affine as in proposition 1.1.3. Second, when E comes with an integrable connection, we discuss Riemann-Roch theorems and results about chern classes in the Chow group.

1.2. Differential Characters. We start with the C^{∞} case (compare [33]). Let M be a C^{∞}-manifold and let E be a C^{∞}-vector bundle on M. Write

S_{·}(M, A) := C^{∞} singular chains on M with values in the abelian group A Z_{k}(M, A) ⊂ S_{k}(M, A) := closed chains

S^{·}(M, A) := Hom(S_{·}(M, ℤ), A) (1.2.1)

A^{·}(M) := C^{∞} ℂ-valued forms on M

A^{·}(M)_{0} := {ω ∈ A^{·}(M) | dω = 0 and ∫_{γ} ω ∈ ℤ, ∀γ ∈ Z_{·}(M, ℤ)}

I : A^{·}(M) → S^{·}(M, ℂ) integration map

The group of differential characters of degree k, Ĥ^{k}(M, ℂ/ℤ), is defined by the fibre product

Ĥ^{k}(M, ℂ/ℤ)  → Hom(Z_{k}(M, ℤ), ℂ/ℤ)
↓  ↓ χ ↦ χ ∘ ∂
A^{k+1} → S^{k+1}(M, ℂ) →^{restrict}  S^{k+1}(M, ℂ/ℤ)

(1.2.2)

i.e.

(1.2.3) Ĥ^{k}(M, ℂ/ℤ) =

{(a, ω) ∈ Hom(Z_{k}(M, ℤ), ℂ/ℤ) × A^{k+1} | a ∘ ∂ = restrict ∘ I(ω)}.

Proposition 1.2.1. One has an exact sequence

0 ⟶ H^{k}(M, ℂ/ℤ) ⟶ Ĥ^{k}(M, ℂ/ℤ) ⟶^{(a,ω)↦ω} A_{0}^{k+1} ⟶ 0

Proof. Straightforward.

Now let ∇ be a C^{∞}-connection on E, and let N = rank(E). Let

F = ∇^{2} ∈ Hom(E, E ⊗ A^{2}(M)) (1.2.4)

4 SPENCER BLOCH AND HÉLÈNE ESNAULT

be the curvature of ∇. Let P ∈ Sym^{p}(gl_{N}^{∗}) be a polynomial which is invariant under the adjoint action. The central point of Chern-Weil theory is that P (F) ∈ A^{2p}(M) is closed and represents the characteristic de Rham class in cohomology associated to E and P. Write D for a diagonal matrix with diagonal entries x_{1}, …, x_{N}. If P (D) is the p-th elementary symmetric function (resp. P (D) = ∑_{i} \frac{x_{i}^{p}}{p!} ), P (F) represents the p-th chern class c_{p}(E) (resp. the component of the chern character ch(E) in cohomological degree 2p). Cheeger-Chern-Simons theory refines this:

Proposition 1.2.2. With notation as above, assume further that the characteristic class defined by P is integral. Then there exist canonically defined classes ĉ(P, E, ∇) ∈ Ĥ^{2p−1}(M, ℂ/ℤ) lifting P (F) ∈ A^{2p}(M)_{0}. These classes are functorial for pullbacks and additive for exact sequences.

Proof. Omitted.

Now we return to the case X, E algebraic and ∇ an algebraic connection. Suppose for a moment E is a line bundle. Define

(1.2.5) AD^{1}(X) := ℍ^{1}(X_{zar}, O_{X}^{∗} →^{d log} Ω_{X}^{1} →^{d} Ω_{X}^{2} → …)

ÂD^{1}(X) := ℍ^{1}(X_{zar}, O_{X}^{∗} →^{d log} Ω_{X}^{1})

(Here O_{X}^{∗} is the Zariski sheaf of invertible functions, and AD stands for “algebraic differential character”.)

Proposition 1.2.3. 1. ÂD^{1}(X) is isomorphic to the group Pic^{∇}(X) of isomorphism classes of line bundles with connections on X.

2. AD^{1}(X) is isomorphic to the group Pic^{0,∇}(X) of isomorphism classes of line bundles with integrable connections on X.

3. Let Γ(X, Ω_{X}^{2})_{0} denote the group of global closed algebraic 2-forms ω on X such that [ω] ∈ H_{DR}^{2}(X) is the class of an algebraic divisor. Then there is an exact sequence

0 → AD^{1}(X) → ÂD^{1}(X) → Γ(X, Ω_{X}^{2})_{0} → 0.

4. Assume k = ℂ and X is projective. Then

AD^{1}(X) ≅ H^{1}(X_{rman}, ℂ/ℤ2πi).

Proof. Let (L, ∇) be a line bundle with connection on X. With respect to an open cover {U_{i}} trivializing L, we have transition functions

a_{ij} ∈ Γ(U_{i} ∩ U_{j}, O^{∗})

and 1-forms ω_{i} = ∇(1_{i}) ∈ Γ(U_{i}, Ω_{X}^{1}). The data {a_{ij}, ω_{i}} represent a class in ÂD^{1}(X). The assertions of the proposition are straightforward from this.

The idea to define the higher AD-groups [15], [17] is to work with the higher Milnor K-sheaves

(1.2.6) K_{n} := Image((O_{X}^{∗})^{⊗n} → K_{n}^{Milnor}(k(X)))

These sheaves are less abstract than the Quillen K-sheaves. In [30] they are shown to satisfy many of the same properties. In particular, one has a Gersten resolution, and

H^{n}(X, K_{n}) ≅ CH^{n}(X). (1.2.7)

There is a d log map

d log : K_{n} → Ω_{X}^{n}; d log\{x_{1}, …, x_{n}\} = \frac{dx_{1}}{x_{1}} ∧ ⋯ ∧ \frac{dx_{n}}{x_{n}} . (1.2.8)

We define

AD^{n}(X) := ℍ^{n}(X, K_{n} →^{d log} Ω_{X}^{n} → Ω^{n+1} → …) (1.2.9)

ÂD^{n}(X) := ℍ^{n}(X, K_{n} →^{d log} Ω_{X}^{n} → Ω_{X}^{n+1} → … → Ω_{X}^{2n−1})

Write

(1.2.10) Γ(X, Ω_{X}^{2n})_{0} =

ker (Γ(X, Ω_{X,closed}^{2n}) → ℍ^{n}(X, Ω^{n} → Ω^{n+1} → …)/d log H^{n}(X, K_{n}))

= {ω ∈ Γ(X, Ω_{X,closed}^{2n}) | [ω] ∈ ℍ^{2n}(X, Ω_{X}^{≥n})
is the class of an algebraic cycle}

Proposition 1.2.4. 1. There is an exact sequence

0 → AD^{n}(X) → ÂD^{n}(X) → Γ(X, Ω_{X}^{2n})_{0} → 0

2. Let (E, ∇) be a rank N algebraic bundle with connection on X, and let P be an invariant polynomial on gl_{N} as above. Then one has defined functorial characteristic classes

ĉ(P, E, ∇) ∈ ÂD^{n}(X) ↦ P [F] ∈ Γ(X, Ω_{X}^{2n})_{0}.

3. When k = ℂ, there is a canonical map of exact sequences from the above sequence to the classical differential character sequence from proposition 1.2.1. This map carries the algebraic classes from 2 to the classes defined in proposition 1.2.2.

6 SPENCER BLOCH AND HÉLÈNE ESNAULT

Proof. For full details the reader is referred to [17]. Let us sketch one of the constructions of the characteristic classes which uses a “Weil algebra” construction due to Beilinson and Kazhdan [2]. First, we push out the sequence (1.1.6) along the trace on the left to get an exact sequence (defining Ω_{X,E}^{1})

0 → Ω_{X}^{1} → Ω_{X,E}^{1} → End(E) → 0 (1.2.11)

We consider the composed differential O_{X} ⟶^{δ} Ω_{X,E}^{1} and build a differential graded algebra Ω_{X,E}^{∗} subject only to the relation that δ(f) = df for f ∈ O_{X}. Even the algebra structure here is tricky; Ω_{X,E}^{∗} is not simply an exterior algebra on Ω_{X,E}^{1}. One has as graded algebras

⋀ Ω_{X,E}^{1} ⊗ Sym(End(E)) ≅ Ω_{X,E}^{∗} (1.2.12)

where End(E) has graded degree 2. (To see this, think about the case X=point, Ω_{X}^{1} = (0). The universal DGA structure in this case makes

Ω_{X,E}^{∗} = ext. algebra on complex in deg. [1, 2] End(E) →^{=} End(E).)

One defines a Hodge filtration

F^{p}Ω_{X,E}^{∗} := Image( ⊕_{a+b≥p} ∧^{a}Ω_{X,E}^{1} ⊗ Sym^{b}(End(E))) → Ω_{X,E}^{∗}. (1.2.13)

With respect to the Hodge filtrations, the natural map

(Ω_{X}^{∗}, Ω^{≥p}) → (Ω_{X,E}^{∗}, F^{p}Ω_{X,E}^{∗}) (1.2.14)

is a filtered quasi-isomorphism.

A connection ∇ on E is equivalent to a splitting of (1.2.11) and hence also (1.2.14):

\begin{aligned}O_{X} & → & Ω_{X,E}^{1} & → & ∧^{2}Ω_{X,E}^{1} ⊕ End(E) & → & … \\ ∥ & & ↓ ∇ & & ↓ ↙ ∇^{2} \\ O_{X} & → & Ω_{X}^{1} & → & Ω_{X}^{2} & … \end{aligned} (1.2.15)

The curvature ∇^{2} is the obstruction to compatibility of this splitting with the Hodge filtration.

Let N = rank(E). Invariant polynomials in Sym^{p}(End(E)) define a subcomplex with trivial differential

S := (Sym(k_{X}^{⊕N}[−2]))^{S_{N}} → Ω_{X,E}^{∗} (1.2.16)

Note S^{2p} (viewed as a complex placed in degree 2p) is generated by invariant polynomials of degree p and maps to F^{p}Ω_{X,E}^{∗} (viewed as a complex beginning in degree p). A universal argument on BGL_{N} shows that characteristic classes for E associated to invariant polynomials of degree p can be defined in

ℍ^{2p}(X, Cone(K_{p}[−p] ⊕ S^{2p} → F^{p}Ω_{X,E}^{∗})[−1]) (1.2.17)

Finally, the splitting (1.2.15) induced by a connection carries S^{2p} → F^{2p}Ω_{X}^{∗}, so we get characteristic classes

(1.2.18) ĉ(P, E, ∇) ∈ ℍ^{2p}(X, Cone(K_{p}[−p] ⊕ F^{2p}Ω_{X}^{∗} → F^{p}Ω_{X}^{∗})[−1])

≅ ÂD^{p}(X).

If ∇ is integrable, the map on S^{2p} is zero and the classes fall in

AD^{p}(X) = ℍ^{2p}(X, Cone(K_{p}[−p] → F^{p}Ω_{X}^{∗})[−1]). (1.2.19)

1.3. Cheeger-Chern-Simons Classes. This is all a bit abstract, but there are much more down to earth variants on the differential character construction which can be studied locally. Examining the complexes (1.2.9) defining the ÂD groups, one sees there are vertical maps and commutative diagrams:

0 ⟶  AD^{1}(X)  ⟶  ÂD^{1}(X)  ⟶ Γ(X, Ω_{X}^{2})_{0} ⟶ 0
↓≅  ↓≅  ↓≅
0 ⟶ Γ(X, Ω_{X,cl}^{1}/d log(O_{X}^{∗})) ⟶ Γ(X, Ω_{X}^{1}/d log(O_{X}^{∗})) ⟶_{d} Γ(X, Ω_{X}^{2})_{0} ⟶ 0

(1.3.1)

We write H^{n} := Ω_{X,cl}^{n}/dΩ_{X}^{n−1}. For p > 1, the diagram corresponding to (1.3.1) becomes

0 ⟶ AD^{p}(X) ⟶  ÂD^{p}(X)  ⟶ Γ(X, Ω_{X}^{2p})_{0} ⟶ 0
↓  ↓  ↓
0 ⟶ Γ(X, H^{2p−1}) ⟶ Γ(X, Ω_{X}^{2p−1}/dΩ_{X}^{2p−2}) ⟶_{d} Γ(X, Ω_{X,cl}^{2p})

(1.3.2)

Note when p > 1 the vertical arrows are no longer isomorphisms. We will write

(1.3.3) H_{CS}^{2p}(X) := \begin{cases}Γ(X, Ω_{X}^{1}/d log O_{X}^{∗}) & p = 1 \\ Γ(X, Ω_{X}^{2p−1}/dΩ_{X}^{2p−2}) & p ≥ 2. \end{cases}

For an invariant polynomial P of degree p, there are classes

w(P, E, ∇) ∈ H_{CS}^{2p}(X) (1.3.4)

deduced from the ÂD classes via the vertical maps in (1.3.1) and (1.3.2).

Proposition 1.3.1. Let η = Spec (k(X)) →^{j} X be the generic point of X. Then the pullback maps j^{∗} : H_{CS}^{2p}(X) → H_{CS}^{2p}(η) are injective.

Proof. When p = 1 this amounts to the assertion that a logarithmic 1-form \frac{df}{f} is regular along a divisor D if and only if D is neither a zero nor a pole of f. For p > 1 it follows from the Gersten style resolution [8]

0 → H^{n} → j_{∗}(H_{DR}^{n}(η)) → … (1.3.5)

where H^{n} = Ω_{cl}^{n}/dΩ^{n−1}.

Thus, as sections of a sheaf, the w(P, E, ∇) may be calculated locally. Further, by the above proposition, they are determined by their value at the generic point. One of the main results in [5] is that these classes can be calculated locally using the connection matrix as in Chern-Simons theory.

Proposition 1.3.2. Let U ⊂ X be non-empty and open such that E|U ≅ O_{U}^{⊕N}. Let A be the corresponding connection matrix of 1-forms on U. For an invariant polynomial P of degree p ≥ 2 define

T P (A) := p ∫_{0}^{1} P (A ∧ F (tA)^{p−1})dt; F (tA) := tdA − t^{2}A ∧ A.

then

dT P (A) = P (F (A)),

and

T P (A)|_{U} represents w(P, E|_{U}, ∇) ∈ Γ(U, Ω^{2p−1}/dΩ^{2p−2}) = H_{CS}^{2p}(U).

Proof. The first assertion is the basic result of Chern-Simons theory. The reader is referred to theorem (4.0.1) in [5] for the proof of the second assertion.

Example 1.3.3. Suppose E is a rank 2 bundle with structure group SL_{2}, and ∇ is a connection which is written on some open U via its connection matrix

(\begin{matrix}α & β \\ γ & −α \end{matrix})

after the chhoice of a basis. Then for the invariant polynomial P (M) = tr(M^{2}) the Chern-Simons class w(P, E, ∇) is represented by

2αdα + βdγ + γdβ − 4αβγ ∈ Γ(U, Ω^{3}).

If ∇ is flat, that is ∇^{2} = 0, then

dα = βγ, dγ = 2γα, dβ = 2αβ

and thus w(P, E, ∇) is represented by

2αdα ∈ Γ(U, H^{3}) ⊂ H_{CS}^{4}(U).

Important question: Do there exist examples with X projective where this class is not trivial, i.e. where αdα is not an exact form in Ω_{k(X)}^{3}?

Example 1.3.4. Let π : Y → X be a finite étale morphism, and define E = π_{∗}O_{Y} with the natural (Gauß-Manin) connection ∇. Then the class associated to P (M) = tr(M) is calculated locally as follows. We may suppose X = Spec (R), Y = Spec (R[t]/(f(t)), where f is a monic polynomial. Write disc(f) for the discriminant of f. Then

w(trace, π_{∗}O_{Y} , ∇) = [\frac{1}{2}d log(disc(f))] ∈ Γ(Spec (R), Ω_{R,cl}^{1}/d log(O^{∗})).

This class is trivial if and only if the discriminant is a square in R. Said another way, if we factor f = ∏(t − r_{i}), then the discriminant is ∏_{i≠j}(r_{i} − r_{j}) and the class is represented by d log(∏_{i<j}(r_{i} − r_{j})).

Finally, we want to explain the relation between these classes and the Griffiths group of algebraic cycles. For simplicity we will stick to the case of codimension 2. Analogous results are available in higher codimensions but the appropriate equivalence relation on cycles is defined via a spectral sequence and the business becomes more technical.

Let X be smooth and projective over ℂ. For an abelian group A, write H^{n}(A) for the Zariski sheaf associated to the presheaf U ↦ H^{n}(U_{an}, A). Thus, the H^{n} above is H^{n}(ℂ). One has a spectral sequence of Leray type associated to the “continuous map” X_{an} → X_{zar}

E_{2}^{p,q} = H^{p}(X_{zar}, H^{q}(A)) ⇒ H^{p+q}(X_{an}, A) (1.3.6)

The analysis of this spectral sequence carried out in [8] shows that one gets an exact sequence (ℤ(2) := ℤ · (2πi)^{2}.)

(1.3.7) H^{3}(X_{an}, ℤ(2)) →^{a} Γ(X, H^{3}(ℤ(2))) →^{d_{2}} H^{2}(X_{zar}, H^{2}(ℤ(2)))

→^{b} H^{4}(X_{an}, ℤ(2))

Further, H^{2}(X_{zar}, H^{2}(ℤ)) is identified with the group of codimension 2 algebraic cycles modulo algebraic equivalence, so Griff^{2}(X) := ker(b) is the Griffiths group of codimension 2 algebraic cycles homologous to zero modulo algebraic equivalence. Write H̃^{3}(X, ℤ(2)) := image(a), so we get an exact sequence

0 → H̃^{3}(X, ℤ(2)) → Γ(X, H^{3}(ℤ(2))) → Griff^{2}(X) → 0. (1.3.8)

this can be interpreted as an exact sequence of (possibly infinite dimensional) mixed Hodge structures, where the Griffiths group is given

10 SPENCER BLOCH AND HÉLÈNE ESNAULT

a trivial mixed Hodge structure (direct limit of Hodge structures of the form ⊕ ℤ(0) + torsion).

Now let (E, ∇) be a vector bundle of rank ≥ 2 with a flat connection on X. We write

w_{p}(E, ∇) = w(P, E, ∇)

where P is the invariant polynomial associated to the p-th elementary symmetric function. Thus w_{2} is associated to

P (M) = \frac{1}{2}(−tr(M^{2}) + tr(M))^{2})

Let z = c_{2}(E) be the second Chern class, which we interpret as an element in Griff^{2}(X) ⊗ ℚ. Tensoring (1.3.8) with ℚ and pulling back to the fibre over ℚ · z yields an exact sequence of finite dimensional ℚ-mixed Hodge structures, which we write

0 → H̃^{3}(X_{an}, ℚ(2)) → E_{z} → ℚ(0) → 0 (1.3.9)

The Hodge structure on the left is pure of weight −1, so the sequence splits canonically when tensored with ℝ. That is, there exists a unique w ∈ F^{0}E_{z} ⊗ ℝ lifting z.

Proposition 1.3.5. Under the inclusion E_{z} ⊗ ℝ ⊂ Γ(X, H^{3}(ℂ)), the above element w lifting z is identified with w_{2}(E, ∇). In particular, w_{2}(E, ∇) = 0 ⟺ c_{2}(E) = 0 in Griff^{2}(X) ⊗ ℚ.

Proof. This is theorem (5.6.2) in [5].

2. Riemann-Roch

2.1. Introduction. In this lecture we will explain a sort of Riemann-Roch theorem for bundles with flat connections having regular singular points. A preprint with details of these results is on the algebraic geometry server [6]. Such results extend to the irregular rank one case [7], but we don’t address this subject in those notes. We just say that then, the top Chern class of differential forms, which in our theory plays the role of the Todd class, then depends on the connection.

A bundle with a flat connection is an example of a holonomic Dmodule, and very probably the correct Riemann-Roch theorem should be formulated in that context (possibly including also the hypothesis of regular singular points.) One has various Riemann-Roch theorems for D-modules ([27], [23]) but in the context of flat connections these give little more than the Euler characteristic of the underlying coherent sheaf. On a projective variety they depend only on the rank of the local system. A model for a more precise theorem has been proven in the analytic category by Bismut and Lott [3]. A Riemann-Roch theorem with values in the AD-classes explained in the previous lecture could be expected to yield an algebraic proof of the Bismut-Lott result. For technical reasons, however, the algebraic result we want to explain is currently limited to the Chern-Simons classes in H_{CS}^{∗}(X).

Definition 2.1.1. Let f : X → S be a flat morphism of smooth varieties. Let Y ⊂ X and T ⊂ S be normal crossings divisors, and assume that set-theoretically f^{−1}(T) ⊂ Y. The data {f : X → S, Y ⊂ X, T ⊂ S} is said to be a relative normal crossings if

(2.1.1) Ω_{X/S}^{1}(log(Y)) := Ω_{X}^{1}(log Y)/f^{∗}Ω_{S}^{1}(log T)

is locally free of rank d = dim(X/S).

Let Z_{i} denote the components of Y not lying in f^{−1}(T). One has residue maps

res_{Z_{i}} : Ω_{X/S}^{1}(log(Y)) → O_{Z_{i}} (2.1.2)

which may be viewed as “partial trivializations” of Ω_{X/S}^{1}(log(Y)). Using ideas of T. Saito [32] one can define relative top chern classes

(2.1.3) c_{d}(Ω_{X/S}^{1}(log(Y)), {res_{Z_{i}}}) ∈ CH^{d}(X, Z_{•}) :=

ℍ^{d}(X, K_{d,X} → K_{d,Z^{(1)}} → K_{d,Z^{(2)}} → …)

Here as before K_{d} denotes the Milnor K-sheaf, and Z^{(i)} is the i-fold intersections in Z = ⋃ Z_{i}.

12 SPENCER BLOCH AND HÉLÈNE ESNAULT

We continue to assume {f : X → S, Y ⊂ X, T ⊂ S} is a relative normal crossing. Let E be a locally free sheaf on X, and let

∇ : E → E ⊗ Ω_{X}^{1}(log(Y))

be a flat connection with logarithmic poles along Y. As in the first lecture, such a bundle with connection supports CS-classes with log poles. For the Riemann-Roch theorem we will need only the “Newton classes”. Let P be the invariant polynomial of degree p which maps the diagonal matrix with x_{i} entries on the diagonal to ∑ x_{i}^{p}. One defines

(2.1.4) Nw_{p}(E, ∇) = w(P, E, ∇) ∈ H_{CS}^{2p}(X(log Y))

:= \begin{cases}Γ(X, Ω_{X}^{1}(log Y)/d log(j_{∗}O_{X−Y}^{∗})) & p = 1 \\ Γ(X, Ω_{X}^{2p−1}(log Y)/dΩ_{X}^{2p−1}(log Y)) & p ≥ 2 \end{cases}

Assume further that f : X → S is projective. The partial trivialization built into the class (2.1.3) compensates for the log poles in (2.1.4), so one has a well-defined product-pushforward

(2.1.5) f_{∗}(Nw_{p}(E, ∇) · c_{d}(Ω_{X/S}^{1}(log Y), res_{Z_{i}})) ∈ H_{CS}^{2p}(S(log T)).

We can use the classical Gauß-Manin construction to push forward E as follows (here again it would be much more convenient to work with D-modules). Consider the diagram of complexes (with differentials defined using the flat connection ∇)

(2.1.6) 0 → E ⊗ Ω_{X/S}^{∗}(log Y)[−1] ⊗ Ω_{S}^{1}(log T) → E ⊗ Ω_{X}^{∗}(log Y) → E ⊗ Ω_{X/S}^{∗}(log Y) → 0

The boundary map puts S-connections on the cohomology sheaves

(2.1.7) ∇_{GM} : ℝ^{i}f_{∗DR}(E) := ℝ^{i}f_{∗}(E ⊗ Ω_{X/S}^{∗}(log Y))

→ ℝ^{i}f_{∗DR}(E) ⊗_{O_{S}} Ω_{S}^{1}(log T).

these connections are again integrable, and one can try to relate their CS-classes to the CS-classes of E on X. (Warning: because of log poles, the cohomology sheaves are not necessarily locally free on S.)

One further notation: write Ẽ := E−rk(E)O_{X}. One extends relative de Rham cohomology and characteristic classes to such virtual bundles with connection by additivity.

Theorem 2.1.2 (Riemann-Roch for flat bundles). With notation as above (d = dim(X/S)) we have

w_{p}(ℝ^{∗}f_{∗DR}(Ẽ), ∇_{GM}) = (−1)^{d}f_{∗}(c_{d}(Ω_{X/S}^{1}(log(Y)), {res_{Z_{i}}})·w_{p}(E, ∇)).

Remark 2.1.3. 1. Notice this Riemann-Roch has a very simple shape. w_{p}(ℝ^{∗}f_{∗DR}(Ẽ), ∇_{GM}) is determined by w_{p}(E, ∇). This should be contrasted with Riemann-Roch for coherent sheaves, where the inhomogeneous nature of the Todd class Td(T_{X/S}) results in a mixing of degress of characteristic classes. Essentially, for flat bundles one can think one is taking the chern character of E⊗Ω_{X/S}^{∗}. The homogeneous result then results from the identity

(2.1.8) (−1)^{dim(X/S)}c_{d}(Ω_{X/S}^{1}) = Td(T_{X/S}) · ∑_{i}(−1)^{i}ch(Ω_{X/S}^{i}).

2. When p = 1, this result fixes the isomorphism class of the local system determinant of cohomology

(det(ℝ^{∗}f_{∗DR}(Ẽ)), ∇_{GM})

For p ≥ 2 the CS-classes are rather mysterious. Using non-flat connections (see 4.), one can give examples where they are non-zero, but the situation is far from being understood.

3. This Riemann-Roch theorem only gives information for virtual bundles of rank 0. It would be interesting to have a “Noether’s theorem” calculating the Nw_{p}(ℝ^{∗}f_{∗DR}(O_{X}). Suppose, e.g., S is a curve, and the morphism f degenerates over a divisor T ⊂ S. The class Nw_{1} represents the determinant det of de Rham cohomology, which is a rank 1 local system on S − T. One can show (e.g. by looking at the representation on cohomology with ℤ-coefficients) that det^{2} is trivial, so

det := det ℝf_{∗}(ℂ) ∈ H^{1}(S − T, ℤ/2ℤ) →^{∂} H^{0}(T, ℤ/2ℤ). (2.1.9)

One question would be for t ∈ T to compute the t-component of ∂(det) in ℤ/2ℤ in terms of characteristic classes supported on the singular fibre f^{−1}(t).

4. The Riemann-Roch theorem proved in [6] is somewhat more precise in that the curvature ∇^{2} is assumed to be basic, i.e. in End(E)⊗f^{∗}Ω_{S}^{2}, rather than actually zero.

2.2. Sketch of Proof. We want to give some indication of the basic ideas of the proof of theorem 2.1.2.

2.2.1. Step 1. Since the CS-classes are determined at the generic point, one reduces to the case S = Spec (F) is the spectrum of a function field. By a Lefschetz pencil argument, one reduces further to the case X = ℙ_{F}^{1} , E = ⊕O_{ℙ^{1}}(m_{i}), m_{i} ≥ 0. The Nw_{p} take values in a ℚ-vector space (except for Nw_{1}, which requires a special argument) so we can replace S by a finite cover and assume the connection has logarithmic poles along a divisor D = {a_{1}, …, a_{δ}} ⊂ ℙ^{1}(F). One shows further

14 SPENCER BLOCH AND HÉLÈNE ESNAULT

that one is free to augment D by adding points where the connection does not have log poles. In particular, we may assume ∞ ∈ D but ∇ does not have a pole at ∞.

2.2.2. Step 2. We may assume E = O_{ℙ^{1}}^{⊕n}. In the original version of the paper, this reduction contained a serious error, which was pointed out to us by O. Gabber. Thus, for the sake of credibility, I will give the argument here in full. Identify E = ⊕O(m_{i} · ∞), and define

E′ = ⨁_{m_{i}=0} O_{P} ⊕ ⨁_{m_{i}>0} O((m_{i} − 1) · ∞) ⊂ E

Lemma 2.2.1. With notation as above, E′ ⊂ E is stable under ∇.

proof of lemma. The assertion is invariant under an extension of F, so we may assume D = {a_{1}, …, a_{δ}, ∞} with all a_{ν} ∈ P (F). Let 1_{j}, 1 ≤ j ≤ r be the evident basis of E on 𝔸^{1} = P − {∞}, and let z be the standard parameter on P. An element γ ∈ Γ(P, O(n)⊗Ω_{ℙ_{F}^{1}}^{1}(log D)) for n ≥ 0 can be uniquely written in the form

∑_{ν=1}^{δ} A_{ν}d log(z − a_{ν}) + ∑_{i=0}^{n} z^{i}η_{i} + ∑_{j=1}^{n} C_{j}(z − a_{1})^{j}d log(z − a_{1}),

with A_{ν}, C_{j} ∈ F and η_{i} ∈ Ω_{F}^{1} . Since ∇(z^{m_{j}}1_{j}) ∈ Γ(E ⊗ Ω_{ℙ_{F}^{1}}^{1}(log D)), we may write

(2.2.1) ∇(1_{j}) = ∑ 1_{k} ⊗ [ ∑_{i=0}^{m_{k}−m_{j}} η_{j}^{ki}(z − a_{1})^{ℓ} + ∑ A_{j}^{kν}d log(z − a_{ν})

+ ∑_{ℓ=1}^{m_{k}−m_{j}} C_{j}^{kℓ}(z − a_{1})^{ℓ}d log(z − a_{1})]

If m_{j} > m_{k}, the sums over i and ℓ on the right are not there. If m_{j} = m_{k}, the sum over ℓ is absent. With respect to (2.2.1) we have the following facts:

C_{j}^{k,m_{k}−m_{j}} = 0 (2.2.2)

For m_{j} ≥ m_{k},

(2.2.3) ∑_{ν} A_{j}^{kν}d log(z − a_{ν}) ∈ Γ(ℙ^{1}, Ω_{ℙ_{F}^{1}}^{1}(log D)((m_{k} − m_{j}) · ∞)).

∑ A_{j}^{kν}d log(z − a_{ν}) ∈ Γ(ℙ^{1}, Ω_{ℙ_{F}^{1} /F}^{1}(log D)((m_{k} − m_{j} − 1) · ∞)). (2.2.4)

To check (2.2.2) we may suppose m_{k} > m_{j}. The composition

(2.2.5) O(m_{j}) ↪ E →^{∇} E ⊗ Ω_{ℙ_{F}^{1}}^{1}(log D) → E ⊗ Ω_{ℙ_{F}^{1} /F}^{1}(log D)

→^{res_{∞}} E|_{∞} ↠ O(m_{k})|_{∞} = O(m_{k} · ∞)/O((m_{k} − 1) · ∞)

maps

(z − a_{1})^{m_{j}}1_{j} ↦ C_{j}^{k,m_{k}−m_{j}}(z − a_{1})^{m_{k}} (mod O((m_{k} − 1) · ∞))

By assumption, the connection has zero residue at infinity, so this is zero. The inclusion (2.2.3) follows because

∇((z − a_{1})^{m_{j}}1_{j}) =

∑(z − a_{1})^{m_{k}}1_{k} ⊗ [δ_{m_{j},m_{k}} · η_{j}^{k0} + (z − a_{1})^{m_{j}−m_{k}} ∑ A_{j}^{kν}d log(z − a_{ν})]

+ m_{j}(z − a_{1})^{m_{j}}1_{j} ⊗ d log(z − a_{1})

is assumed to extend across infinity. finally, (2.2.4) holds because of the vanishing of the residue (2.2.5). In the case m_{j} ≥ m_{k} the residue map is

1_{j} ↦ ((z − a_{1})^{m_{j}−m_{k}} ∑ A_{j}^{kν}d log(z − a_{ν}))|_{∞}.

Now view (2.2.1) as defining the connection matrix B = (b_{j}^{k}) for ∇ on P − {∞}. The above assertions can be summarized as follows. For m_{k} > m_{j}

b_{j}^{k} ∈ Γ(ℙ^{1}, f^{∗}Ω_{F}^{1} ((m_{k} − m_{j}) · ∞) + Ω_{ℙ_{F}^{1}}^{1}(log D)((m_{k} − m_{j} − 1) · ∞))

and for η ∈ Ω_{F}^{1} and m_{k} ≤ m_{j},

b_{j}^{k} ∈ Γ(ℙ^{1}, Ω_{ℙ_{F}^{1}}^{1}(log D)((m_{k} − m_{j}) · ∞))
b_{j}^{k} ∈ Γ(ℙ^{1}, Ω_{ℙ_{F}^{1} /F}^{1}(log D)((m_{k} − m_{j} − 1) · ∞))
η ∧ b_{j}^{k} ∈ Γ(ℙ^{1}, Ω_{F}^{1} ⊗ Ω_{ℙ_{F}^{1} /F}^{1}(log D)((m_{k} − m_{j} − 1) · ∞)).

It follows that whenever m_{i} ≤ m_{k} we have

b_{i}^{j}b_{j}^{k} ∈ Γ(ℙ^{1}, Ω_{F}^{1} ⊗ Ω_{ℙ_{F}^{1} /F}^{1}(log D)(m_{k} − m_{i} − 1) · ∞)).

Vanishing of C_{i}^{k,m_{k}−m_{i}} and the trivial curvature condition

db_{i}^{k} = ∑_{j} b_{i}^{j}b_{j}^{k}

implies

db_{i}^{k} = (m_{k} − m_{i})η_{i}^{k,m_{k}−m_{i}}(z − a_{1})^{m_{k}−m_{i}}d log(z − a_{1}) + ϵ

for ϵ ∈ Γ(ℙ^{1}, Ω_{ℙ_{F}^{1}}^{2}(log(D))((m_{k} − m_{i} − 1)∞))). It follows for m_{k} > m_{i}

that η_{i}^{k,m_{k}−m_{i}} = 0 as claimed.

It follows now from (2.2.1) that ∇ stabilizes E′ ⊂ E, proving the lemma.

The lemma implies reduction to the case E = ⊕O_{ℙ^{1}} by induction on max{m_{i}}.

2.2.3. Step 3. We continue to assume E = ⊕O_{P} with P := ℙ_{F}^{1} and connection given by

∇(1_{j}) = ∑_{kν} A_{j}^{kν}1_{k} ⊗ d log(z − a_{ν}) + ∑_{k} 1_{k} ⊗ η_{j}^{k}. (2.2.6)

we define an F -linear splitting σ of the natural reduction from absolute to relative E-valued 1-forms

(2.2.7) σ : Γ(P, Ω_{P/F}^{1} (log D)) → Γ(P, Ω_{P/k}^{1}(log D)); σ(1_{k} ⊗ (z − a_{1})^{ν}dz) = 1_{k} ⊗ (z − a_{1})^{ν}d(z − a_{1}),

σ(1_{k} ⊗ d log(z − a_{ν})) = 1_{k} ⊗ d log(z − a_{ν})

Now consider the diagram

Γ(E)  =  Γ(E)
↓ ∇_{1}  ↓ ∇_{P/F}
Γ(E) ⊗ Ω_{F}^{1}  → Γ(E ⊗ Ω_{P}^{1} (log D)) →^{←^{σ}} Γ(E ⊗ Ω_{P/F}^{1} (log D))
↓ ∇_{P/F} ⊗ 1  ↓ ∇_{2}
Γ(E ⊗ Ω_{P/F}^{1} (log D)) ⊗ Ω_{F}^{1} = Γ(E ⊗ Ω_{P}^{2} (log D)/Ω_{F}^{2} )

(2.2.8)

Here ∇_{1} and ∇_{2} are the absolute connection maps. Define

Φ := ∇_{1} − σ∇_{P/F}; Ψ = −∇_{2}σ. (2.2.9)

The diagram

Γ(E)  ⟶^{Φ}  Γ(E) ⊗ Ω_{F}^{1}
∇_{P/F} ↓  ↓^{∇_{P/F} ⊗1}
Γ(E ⊗ Ω_{P/F}^{1} (log D)) ⟶^{Ψ} Γ(E ⊗ Ω_{P/F}^{1} (log D)) ⊗ Ω_{F}^{1}

(2.2.10)

represents (∑_{i=0}^{i=2}(−1)^{i}(R^{i}f_{∗}(E⊗Ω_{P/F}^{∗} , ∇_{P/F}), GM^{i}(∇)) in the Grothendieck group K(F) of F -vector spaces with connection. We see from (2.2.6) that

Φ(1_{j}) = (∇ − σ∇_{P/F})(1_{j}) = ∑_{k} 1_{k} ⊗ η_{j}^{k}. (2.2.11)

Also

(2.2.12) Ψ(1_{j} ⊗ d log(z − a_{ν}) = −∇(1_{j}) ∧ d log(z − a_{ν}) =

= ( − ∑_{kτ} A_{j}^{kτ}1_{k} ⊗ d log(z − a_{τ}) − 1_{k} ⊗ η_{j}^{k}) ∧ d log(z − a_{ν}) =

= − ∑_{kτ} A_{j}^{kτ}1_{k} ⊗ (d log(z − a_{τ}) − d log(z − a_{ν})) ⊗ d log(a_{ν} − a_{τ}) +

+ ∑_{k} 1_{k} ⊗ d log(z − a_{ν}) ⊗ η_{j}^{k}.

Define

B_{jν}^{kτ} = \begin{cases}−A_{j}^{kτ}d log(a_{ν} − a_{τ}) & τ ≠ ν \\ η_{j}^{k} + ∑_{θ≠ν} A_{j}^{kθ}d log(a_{ν} − a_{θ}) & τ = ν \end{cases} . (2.2.13)

Then

Ψ(1_{j} ⊗ d log(z − a_{ν})) = ∑_{kτ} 1_{k} ⊗ d log(z − a_{τ}) ∧ B_{jν}^{kτ}. (2.2.14)

The left hand side of the Riemann-Roch formula for E (in this case there is no need to subtract off O^{⊕rk(E)}) is given by

(2.2.15) Nw_{n}((∑_{i=0}^{i=2}(−1)^{i}(R^{i}f_{∗}(E ⊗ Ω_{P/F}^{∗} , ∇_{P/F}), GM^{i}(∇)))

= Nw_{n}(Φ) − Nw_{n}(Ψ).

2.2.4. Step 4. We next make some observations about Nw_{n}(Ψ). Define B_{ν}^{τ} (resp. B) to be the N × N matrix (resp. δ × δ matrix of N × N matrices)

B_{ν}^{τ} := (B_{jν}^{kτ})_{1≤j,k≤N} (resp. B = (B_{ν}^{τ})_{1≤ν,τ≤n}). (2.2.16)

Lemma 2.2.2. Let M(B) = B^{r_{1}}(dB)^{r_{2}} ⋯ B^{r_{2s−1}}(dB)^{r_{2s}} be some (noncommuting) monomial in B and dB. Then

Tr (M(B)) = ∑_{τ=1}^{n} Tr (M(B_{τ}^{τ})).

Proof. Write as above

M(B)_{ν}^{τ} := (M(B)_{jν}^{kτ})_{1≤j,k≤N} (resp. M(B) = (M(B)_{ν}^{τ})_{1≤ν,τ≤n}).

Then Trace(M(B)) = ∑_{τ} Trace(M(B)_{τ}^{τ}). Now

M(B)_{τ}^{τ} = ∑_{τ_{1},...,τ_{r_{2s}−1}} B_{τ}^{τ_{1}}B_{τ_{1}}^{τ_{2}} ⋯ B_{τ_{r_{1}−1}}^{τ_{r_{1}}}dB_{τ_{r_{1}}}^{τ_{r_{1}+1}} ⋯ dB_{τ_{r_{2s}−1}}^{τ}.

18 SPENCER BLOCH AND HÉLÈNE ESNAULT

For ν ≠ τ we can write B_{ν}^{τ} = C_{ν}^{τ}d log(a_{ν} − a_{τ}). Possibly introducing some signs, the d log terms can be pulled to the right. Suppose, among {τ_{1}, τ_{2}, …, τ_{r_{2s}−1}} we have τ_{j_{1}}, …, τ_{j_{a}} ≠ τ and all the other τ_{k} = τ. Then that particular summand on the right multiplies

d log(a_{τ} − a_{τ_{j_{1}}}) ∧ ⋯ ∧ d log(a_{τ_{ja}} − a_{τ}) = 0.

(Note x_{1} + ⋯ + x_{a+1} = 0 ⇒ dx_{1} ∧ … ∧ dx_{a+1} = 0.) Thus, one one term on the right is non-zero, and

M(B)_{τ}^{τ} = M(B_{τ}^{τ}),

proving the lemma.

Since Nw_{n}(Ψ) is a sum of terms Tr (M(B)) as in the lemma, we conclude

Nw_{n}(Ψ) = ∑_{τ=1}^{δ} Nw_{n}(Ψ_{τ}), (2.2.17)

where Ψ_{τ} is the connection on F^{⊕N} given (with notation as above) by

1_{j} ↦ ∑_{k=1}^{N} 1_{k} ⊗ (η_{j}^{k} + ∑_{θ≠τ} A_{j}^{kθ}d log(a_{τ} − a_{θ}))

The connection matrix for Ψ_{τ} is thus

Φ + ∑_{θ≠τ} A^{θ}d log(a_{τ} − a_{θ}) (2.2.18)

where Φ = (η_{j}^{k}) and A^{θ} = (A_{j}^{kθ}).

2.2.5. Step 5. We now consider the right hand side of the Riemann-Roch formula, which in our case takes the form

−Nw_{n}(E, ∇) · c_{1}(Ω_{P/F}^{1} (log D), res_{D})

Since Ω_{P/F}^{1} (log D) has rank 1, the relative chern class can be computed in a standard way to be the divisor of any meromorphic section ω of the bundle such that ω is regular along D and res_{D}(ω) = 1. We shall assume that 0 ∉ D. (This is easy to arrange by applying an automorphism to P.) We take for our meromorphic section

ω := ( ∑_{τ=1}^{δ} \frac{1}{z − a_{τ}} − \frac{δ + 1}{z})dz. (2.2.19)

ALGEBRAIC CHARACTERISTIC CLASSES 19

Clearing denominators

ω = \frac{F (z)}{z ∏(z − a_{τ})}dz; F (z) = ∑_{τ=1}^{δ} a_{τ} ∏_{θ≠τ}(z − a_{θ}) − ∏_{τ=1}^{δ}(z − a_{τ}). (2.2.20)

Writing formally F = ∏_{i=1}^{δ}(z − β_{i}), we get

c_{1}(Ω_{P/F}^{1} (log D), res) = (ω) = ∑(β_{i}) − (0).

2.2.6. Step 6. We shall need to compute

− ∑_{i=1}^{δ} Nw_{n}(E, ∇)|_{z=β_{i}} + Nw_{n}(E, ∇)|_{z=0} (2.2.21)

and compare the answer to Nw_{p}(Φ) − Nw_{p}(Ψ) (cf. (2.2.11), (2.2.13), (2.2.15), (2.2.17), (2.2.18)). Proposition 2.2.3. Suppose given {j_{1}, …, j_{r}} ⊂ {1, …, δ}. Then

d log(z − a_{j_{1}}) ∧ ⋯ ∧ d log(z − a_{j_{r}})|(ω) :=

∑_{i=1}^{δ} d log(β_{i} − a_{j_{1}}) ∧ ⋯ ∧ d log(β_{i} − a_{j_{r}}) − d log(a_{j_{1}}) ∧ ⋯ ∧ d log(a_{j_{r}}) =

= ∑_{t≠j_{1},...,j_{r}} d log(a_{t} − a_{j_{1}}) ∧ ⋯ ∧ d log(a_{t} − a_{j_{r}}).

Proof. To simplify notation we consider the case j_{k} = k, 1 ≤ k ≤ r. Lemma 2.2.4.

∑_{i=1}^{δ} d log(β_{i} − a_{1}) ∧ ⋯ ∧ d log(β_{i} − a_{r}) =

= ∑_{j=1}^{r}(−1)^{j−1}d log(F (a_{j})) ∧ d log(a_{j} − a_{1}) ∧ ⋯ ∧ { \atop d log(a_{j} − a_{j})} ∧ ⋯

⋯ ∧ d log(a_{j} − a_{r}).

proof of lemma 2.2.4. We can argue universally. Consider the rings

A = ℚ[z_{1}, …, z_{δ}, t_{1} …, t_{r}] ⊂ B = A[x]/(x^{δ} + ∑_{i=1}^{δ} z_{i}x^{δ−i}) ≅

≅ ℚ[z_{1}, …, z_{δ−1}, t_{1} …, t_{r}, X].

20 SPENCER BLOCH AND HÉLÈNE ESNAULT

Note that both A and B are polynomial rings over ℚ. Let L ⊂ M be their quotient fields, and consider the symbol

S = {X − t_{1}, …, X − t_{r}} ∈ K_{r}(M).

We will compute the norm, N(S) ∈ K_{r}(L). Note the tame symbol is given by

tame(S) = ∑_{k=1}^{r}(−1)^{k−1}\{t_{k} − t_{1}, …, { \atop t_{k} − t_{k}}, …, t_{k} − t_{r}\}|_{X=t_{k}}

Let π : Spec (B) → Spec (A). We have a map on divisors

π_{∗}(X − t_{i} = 0) = (F (t_{i}) = 0)

with degree 1, so

tame(N(S)) = N(tame(S)) =

= ∑_{k=1}^{r}(−1)^{k−1}\{t_{k} − t_{1}, …, { \atop t_{k} − t_{k}}, …, t_{k} − t_{r}\}|_{F (t_{k})=0} =

= tame( ∑_{k=1}^{r}(−1)^{k−1}\{F (t_{k}), t_{k} − t_{1}, …, { \atop t_{k} − t_{k}}, …, t_{k} − t_{r}\}).

(The last equality holds because F (t_{j})/F (t_{k}) = 1 on the divisor t_{j} = t_{k}.). Since L is purely transcendental over ℚ, this determines N(S) upto constant symbols, which can be ignored because we want to apply d log. Specializing the z_{i} to the coefficients of our F and the t_{i} ↦ a_{i} and applying d log, we deduce the lemma.

Lemma 2.2.5.

d log(b_{1}) ∧ ⋯ ∧ d log(b_{r}) =

= ∑_{k=1}^{δ}(−1)^{k−1}d log(b_{k}) ∧ d log(b_{k} − b_{1}) ∧ ⋯ ∧ { \atop d log(b_{k} − b_{k})} ∧ ⋯

⋯ ∧ d log(b_{k} − b_{r}).

proof of lemma 2.2.5. As above, we argue universally and prove the corresponding identity for symbols. For this it suffices to compare the images under the tame symbol. At the divisor b_{j} − b_{k} = 0 for j < k we need

0 = (−1)^{j+k}\{b_{k}, b_{k} − b_{1}, …, { \atop b_{k} − b_{j}}, …, { \atop b_{k} − b_{k}}, …, b_{k} − b_{r}\}|_{b_{k}=b_{j}} +

+ (−1)^{j+k−1}\{b_{j}, b_{j} − b_{1}, …, { \atop b_{j} − b_{k}}, …, { \atop b_{j} − b_{j}}, …, b_{j} − b_{r}\}|_{b_{k}=b_{j}},

which is clear. Finally at the divisor b_{k} = 0 we need

(−1)^{k−1}{−b_{1}, …, ˆ−b_{k}, …, −b_{r}} = (−1)^{k−1}{b_{1}, …, b̂_{k}, …, b_{r}} + ϵ

where ϵ dies under d log. Again this is clear.

Returning to the proof of proposition 2.2.3, we apply lemmas 2.2.4 and 2.2.5 (with b_{j} = a_{j}) to conclude

d log(z − a_{j_{1}}) ∧ ⋯ ∧ d log(z − a_{j_{r}})|_{(ω)} =

= ∑_{s=1}^{r}(−1)^{s−1}d log ( ∏_{k∉{j_{1},...,j_{r}}}(a_{j_{s}} − a_{k})) ∧ d log(a_{j_{s}} − a_{j_{1}}) ∧ ⋯

⋯ ∧ { \atop d log(a_{j_{s}} − a_{j_{s}})} ∧ ⋯ ∧ d log(a_{j_{s}} − a_{j_{r}})

= ∑_{\begin{matrix}k≠j_{1},...,j_{r} \\ s=1 \end{matrix}}^{s=r}(−1)^{s−1}d log(a_{j_{s}}−a_{k})∧d log(a_{j_{s}}−a_{j_{1}})∧⋯∧{ \atop d log(a_{j_{s}} − a_{j_{s}})}∧⋯

⋯ ∧ d log(a_{j_{s}} − a_{j_{r}}).

Finally we apply lemma 2.2.4 again to this last expression, taking b_{s} = a_{j_{s}} − a_{k}, to get the assertion of the proposition:

d log(z − a_{j_{1}}) ∧ ⋯ ∧ d log(z − a_{j_{r}})|_{(ω)} =

= ∑_{k≠j_{1},...,j_{r}} d log(a_{j_{1}} − a_{k}) ∧ ⋯ ∧ d log(a_{j_{r}} − a_{k}).

Proposition 2.2.6. With notation as above, the Riemann-Roch formula holds for (E, ∇).

Proof. The computation mentioned in (2.2.21) can be done as follows. Let ρ_{ν} be closed 1-forms. For J = {j_{1} < … < j_{r}} ⊂ {1, …, δ} define ρ_{J} = ρ_{j_{1}} ∧ ⋯ ∧ ρ_{j_{r}}. Write

Nw_{n}(O_{P}^{⊕N}, ∑_{ν=1}^{δ} A^{ν}ρ_{ν} + Φ) = ∑_{J⊂{1,...,δ}} P_{J}(A^{ν}, dA^{ν}, Φ, dΦ)ρ_{J} + Nw_{n}(O_{P}^{⊕N}, Φ) (2.2.22)

Here A^{ν} (resp. Φ) are matrices with coefficients in F (resp. Ω_{F}^{1} ), and the P_{J} are independent of the ρ_{j}. Then, using proposition 2.2.3, we get

(2.2.23) − Nw_{n}(∑_{ν=1}^{δ} A^{ν}ρ_{ν} + Φ)|_{(ω)} =

= − ∑_{\begin{matrix}J⊂\{1,...,δ\} \\ r=|J|≥1 \end{matrix}} P_{J}(A^{ν}, dA^{ν}, Φ, dΦ) ∑_{k∉J} d log(a_{j_{1}}−a_{k})∧⋯∧d log(a_{j_{r}}−a_{k})+

+ (1 − δ)Nw_{n}(F^{⊕N}, Φ).

On the other hand, if we fix τ ≤ δ and take ρ_{ν} = d log(a_{τ} − a_{ν}) for ν ≠ τ and ρ_{τ} = 0 we find

(2.2.24) Nw_{n}(ℝf_{∗}(E ⊗ Ω_{P/F}^{∗} (log D))) = Nw_{n}(F^{N}, Φ) − ∑_{τ=1}^{δ} Nw_{n}(F^{N}, Ψ_{τ}) =

= − ∑_{τ=1}^{δ} ∑_{\begin{matrix}J⊂\{1,...,δ\} \\ τ∉J \end{matrix}} P_{J}(A^{ν}, dA^{ν}, Φ, dΦ)d log(a_{j_{1}}−a_{τ})∧⋯∧d log(a_{j_{r}}−a_{τ})+

+ (1 − δ)Nw_{n}(Φ).

The right hand sides of (2.2.23) and (2.2.24) coincide, proving the proposition.

This completes the sketch of the proof of Riemann-Roch for bundles with regular connections. For more details, the reader is referred to [6].

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

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

FB 6, Mathematik, Universität Essen, 45 117 Essen, Germany

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