GAUSS-MANIN DETERMINANTS FOR RANK 1 IRREGULAR CONNECTIONS ON CURVES

SPENCER BLOCH AND HÉLÈNE ESNAULT

Abstract. Let f : U → Spec (K) be a smooth open curve over a field K ⊃ k, where k is an algebraically closed field of characteristic 0. Let ∇ : L → L ⊗ Ω_{U/k}^{1} be a (possibly irregular) absolutely integrable connection on a line bundle L. A formula is given for the determinant of de Rham cohomology with its Gauß-Manin connection ( det Rf_{∗}(L⊗Ω_{U/K}^{1}), det ∇_{GM}). The formula is expressed as a norm from the curve of a cocycle with values in a complex defining algebraic differential characters [7], and this cocycle is shown to exist for connections of arbitrary rank.

Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

Bertrand Russell

1. Introduction

Let f : U → Spec (K) be a smooth open curve over a field K ⊃ k, where k is an algebraically closed field of characteristic zero. Let ∇ : L → L ⊗ Ω_{U/k}^{1} be a possibly irregular absolutely integrable (or vertical, see definition 2.16) connection on a line bundle L. The Riemann-Roch problem in this context is to describe characteristic classes for the relative de Rham cohomology Rf_{∗}(L ⊗ Ω_{U/K}^{∗}) as a (virtual) vector space over K with an integrable connection, in terms of data on U. The 0-th characteristic class, the Euler characteristic dim R^{0} − dim R^{1}, is well-known to be given by

(1.1)  2 − 2g − n − ∑_{i} max(0, m_{i} − 1)

where g is the genus of the complete curve C, n is the number of missing points, and m_{i} is the order of the polar part of the connection at the ith missing point. The purpose of this article is to give a formula for the first characteristic class, which is the determinant of the Gauß-Manin connection on the relative de Rham cohomology of the line bundle,

Date: August 19, 1999, December 9, 2000.

1991 Mathematics Subject Classification. Primary 14C40 19E20 14C99.

(1.2)  ( det(Rf_{∗}(L ⊗ Ω_{U/K}^{∗})), det ∇_{GM}).

When U = C, so the connection has no poles, the formula given in [2] is

(1.3)  ( det(Rf_{∗}(L ⊗ Ω_{C/K}^{∗})), ∇_{GM}) = −f_{∗}((L, ∇) · c_{1}(Ω_{C/K}^{1})).

Concretely, if one has c_{i} ∈ C(K) with ∑ c_{i} a 0-cycle in the linear series representing Ω_{C/K}^{1}, then the determinant is given by restricting L with its connection to each c_{i} and then tensoring the resulting lines with connection together.

When the connection ∇ has at worst regular singular points at the points in D := C − U there is an analogous formula using linear series given by divisors of rational sections s of Ω_{C/K}^{1}(D) satisfying the rigidity condition res_{D}(s) = 1. Indeed, these formulas are valid also for higher rank connections. One takes the determinant at zeroes and poles of s.

In the case of irregular singular points, a similar formula is possible, but the rigidification taken must depend on the polar part of the connection. Let (L, ∇) be an extension of (L, ∇) to C, D = ∑_{i} m_{i}D_{i} be a divisor with multiplicities m_{i} ≥ 1 supported in C − U such that the relative connection

(1.4)  ∇_{/K} : L → L ⊗ Ω_{C/K}^{1}(D)

yields a complex quasiisomorphic to j_{∗}L → j_{∗}L ⊗ Ω_{(C−D)/K}^{1} and has poles at all points D_{i}. Then ∇_{/K} does not factor through

(1.5)  ∇_{/K} : L → L ⊗ Ω_{C/K}^{1}(D − D_{i})

for any i. Writing D also for the artinian subscheme of C determined by D, this implies that ∇_{/K} induces a function linear isomorphism

(1.6)  ∇|_{D} : L|_{D} →^{≅} L ⊗ Ω_{C/K}^{1}(D)|_{D}

Because these maps are function linear, we may cancel the lines L|_{D} and deduce canonical elements triv_{∇} ∈ Ω_{C/K}^{1}(D)|_{D}. We view triv_{∇} as a trivialization of Ω_{C/K}^{1}(D) along D. It is known ([6], Appendix B) that the coboundary of triv_{∇} in H^{1}(C, ω_{C/K}) ≅ K is given by the degree of L. Our main result is:

Theorem 1.1. Let notation be as above. Assume all D_{i} are defined over K and some m_{i} ≥ 2. Because we are only concerned with the cohomology over X − D, we can take L of degree 0 so that the element triv_{∇} can be lifted to H^{0}(C, Ω_{C/K}^{1}(D)). Let s be any such lifting, and write (s) for the divisor of s as a section of Ω_{C/K}^{1}(D). Note the support of (s) is disjoint from D. Then

(1.7) det(Rf_{∗}(L ⊗ Ω_{U/K}^{∗})), det(∇_{GM}) ≅

− f_{∗}(L · (s)) + τ(L) ∈ Ω_{K}^{1}/d log K^{∗}.

Here τ(L) is a 2-torsion term which can be written

τ(L) = ∑_{i} \frac{m_{i}}{2} d log(g_{i,0}) ∈ \frac{1}{2}d log(K^{×})/d log(K^{×})

where the connection ∇_{/K} = (g_{i,0} + g_{i,1}z_{i} + …)dz_{i}/z_{i}^{m_{i}} for a local coordinate z_{i} at D_{i} ∈ D.

Note that Ω_{K}^{1}/d log K^{×} is the group of isomorphism classes of rank 1 connections on Spec (K). Our assumption that points of D are defined over K is made to avoid complications involving generalized jacobians in §2. We remark, of course, that part of our task will be to give a precise definition of the right hand side of the formula of the theorem. It will appear as a product followed by a trace, and this definition does not depend on the particular choice of L above. In particular, this gives a formulation if we don’t assume that L is of degree 0, and also if we don’t assume that m_{i} ≥ 2 for at least one i, that is if ∇ has regular singular points (see theorem 4.6). The precise general formulation of our theorem is in 4.8. In the case that (s) is a sum of K-points c_{i}, one may simply take the tensor product of the lines with connection L|_{c_{i}}. The right hand side of the formula depends only on the equivalence class of (s) in a generalized Picard (or divisor class) group of line bundles with trivializations along D. Thus, by analogy with (1.3), it is natural to write formula (1.7) in the form

(1.8) ((det(Rf_{∗}(L ⊗ Ω_{U/K}^{∗})), det(∇_{GM})) ≅

f_{∗}(L · c_{1}(Ω_{C/K}^{1}(D), triv_{∇})))^{−1} + τ(L).

The classical Riemann-Roch pattern begins to break down in that the characteristic class c_{1}(Ω_{C/K}^{1}(D), triv_{∇}) depends on more than just the geometry of f : U → Spec (K). This reflects the fact that the de Rham cohomology of an irregular connection depends on more than topology.

There is an analogy here with the case of ℓ-adic sheaves. If E is an unramified ℓ-adic sheaf on a complete curve C over a finite field 𝔽_{q}, then the global epsilon factor is given by

(−F_{q}| det(H_{ét}^{∗}(C\overline{𝔽}, E))) = det(E)^{−1} · c_{1}(Ω_{C/𝔽_{q}}^{1}).

The basic result in the ramified ℓ-adic case ([8]) is that the global epsilon factor can be written as a product of local terms corresponding to points on the curve where the sheaf ramifies or where a chosen meromorphic 1-form has zeroes or poles. We suspect formula (1.8) is analogous to a classical formula for Gauß sums

g(c, ψ) = ∑_{a∈(O/𝔣)^{×}} c(a)ψ(a)

where 𝔣 ⊂ O is an ideal in the ring of integers in a local field, c (resp. ψ) is a character of (O/𝔣)^{×} (resp. (O/𝔣)^{+}), and both c and ψ have conductor 𝔣. If the residue field of O has q elements with q odd, one finds

g(ϵ, ψ) = \begin{cases}q^{n}c(x) & 𝔣 = 𝔪^{2n}, n ≥ 1 \\ q^{n}c(x)σ & 𝔣 = 𝔪^{2n+1}, n ≥ 1. \end{cases}

In this formula x ∈ O/𝔣 is a suitable point, σ = ζσ_{0} with ζ^{q} = 1 and σ_{0}^{2} = (\frac{−1}{𝔽_{q}}). (Here σ_{0} is a quadratic Gauß sum.)

Our proof follows the main idea of Deligne [3]. For computing the ϵ-factor associated to a rank one Galois representation on a curve, he expresses the determinant of the cohomology as the cohomology on a symmetric product of (C − D) and reduces the computation to the geometry of the generalized jacobian. In the geometric situation one is further able to express the determinant Gauß-Manin connection as the connection arising by restricting a certain translation-invariant connection to one specific K-point of the generalized jacobian. The essential point seems to be that the de Rham cohomology of a connection of the form d + ω on a trivial bundle is somehow concentrated at the points where ω = 0.

In section 4 we reinterpret the Riemann-Roch formula in terms of a pairing (4)

∪ : ℍ^{1}(C, O_{C}^{∗} → O_{D}^{∗}) × ℍ^{1}(C, O_{C}^{∗} → Ω_{C}^{1}⟨D⟩(D′))

→ ℍ^{2}(C, K_{2} → Ω_{C}^{2}) (1.9)

and a trace map (4.7)

Tr : ℍ^{2}(C, K_{2} → Ω_{C}^{2}) → Ω_{K}^{1}/d log K^{∗}.

In section 5 we give an analogous “non-commutative” product formula in the higher rank case which we conjecture calculates the determinant connection in the generic situation when the connection defines local isomorphisms E|_{D} ≅ E|_{D} ⊗ ω_{D/K} (see (5.3)) and the poles of the absolute connection behave well (see (5.1)). We verify the formula has the appropriate invariance properties. We also show that there is a more general higher rank product of which it is a special case. Finally, in section 6 we give a general formula which calculates the group of isomorphism classes of irregular, integrable, rank 1 connections in higher dimensions on a smooth projective variety.

We apologize for not expressing our results in the modern language of D-modules, but in fact for the study of Gauß-Manin determinants there is little gain in passing from connections to D-modules. Also, rigidity for connections means that the Gauß-Manin determinant connection is determined by its value at the generic point on the base, so we may work with curves over a function field Spec (K).

It is our pleasure to acknowledge the intellectual debt we owe in this work to P. Deligne. We are also grateful to the Humboldt foundation for financing which enabled us to work together.

2. Connections and forms on Generalized Jacobians

Throughout this paper C will be a smooth projective curve over a field K containing an algebraically closed subfield k of characteristic 0, and D = ∑ m_{i}c_{i} is a divisor on C, with c_{i} ∈ C(K). We write G = J_{D} for the generalized Jacobian parametrizing isomorphism classes of degree 0 line bundles on C with trivialization along D. Fixing a Krational point

c_{0} ∈ (C − D)(K),

there is a cycle map i : C −D → J_{D} associating to a closed point x ∈ C with [K(x) : K] = n the class of the line bundle O(x − nc_{0}) together with the trivialization b|_{D} ∘ (a|_{D})^{−1}, where

O_{C} ↩^{a} O_{C}(−nc_{0}) ↪^{b} O_{C}(x − nc_{0})

are the natural maps.

The aim of this section is to describe invariant line bundles with connection on J_{D}, comparing them via the cycle map i to line bundles with connection on (C − D) with a certain irregularity behavior along D.

When the line bundle in question is the trivial bundle, this amounts to studying invariant (absolute) differential forms on the generalized jacobian, so we should start with that. Before doing so, however, it is necessary to understand global functions on the generalized jacobian. We write

(2.1)  G ↠ G_{0} ↠ J

where J is the usual Jacobian of C, and G_{0} is a semi-abelian variety. We have extensions

0 → T → G_{0} → J → 0 (2.2) 0 → 𝕍 → G → G_{0} → 0 (2.3)

Here 𝕍 is a vector group (isomorphic to Spec (Sym(V^{∗})) for some vector space V) and T is a torus, i.e. TK̄ ≅ 𝔾_{m}^{r}.

Lemma 2.1. The semi-abelian variety G_{0} admits a universal vectorial extension

(2.4)  0 → 𝕎 → G → G_{0} → 0.

In fact, this extension is given by the pullback to G_{0} of the universal vectorial extension over J. In particular, 𝕎 = Γ(J^{∨}, Ω_{J^{∨}/K}^{1}) ⊗ 𝔾_{a}.

Proof. It will suffice to show the pullback vectorial extension is universal. Since Ext^{1}(T, 𝔾_{a}) = (0) = Hom(T, 𝔾_{a}), any extension of G_{0} by a vector group 𝕎 is pulled back from a unique extension of J by 𝕎. This extension of J is a pushout from the universal vectorial extension, so the same holds for the pullbacks to G_{0}. □

Lemma 2.2. Let π : G_{0} → J be an extension of J by T as above. There exists, possibly after a finite field extension, a quotient torus T ↠ S and a diagram

(2.5)  \begin{matrix}T & ↪ & G_{0} \\ surj. ↓ & ↙ a \\ S \end{matrix}

such that

(2.6)  H^{i}(G_{0}, O_{G_{0}}) ≅ H^{i}(J, O_{J}) ⊗_{K} H^{0}(S, O_{S})

Proof. There is a boundary map

(2.7)  ∂ : HomK̄(T, 𝔾_{m}) → Ext1 ; K̄(J, 𝔾_{m})

Define N := ker(∂) ⊂ M := HomK̄(T, 𝔾_{m}). Let S = Hom(N, 𝔾_{m}) be the torus with character group N. For m ∈ M let L(m) be the line bundle on JK̄ corresponding under the map (2.7). As an O_{JK̄}-algebra

(2.8)  π_{∗}O_{G_{0,K̄}} ≅ ⊕_{m∈M}L(m) ≅ H^{0}(O_{S}) ⊗ ( ⊕_{m∈M/N} L(m))

The map a in the diagram (2.5) comes from the above inclusion

H^{0}(O_{S}) ⊗ L(0) ⊂ π_{∗}O_{G_{0}}.

For m ∈ M/N, (as is well known, cf. [9] III 16), L(m) has trivial cohomology in all degrees unless m = 0. The proposition follows by taking cohomology of (2.8). □

Lemma 2.3. Let notation be as above. Let

(2.9) 0 ⟶ H^{0}(J^{∨}, Ω_{J^{∨}/K}^{1}) ⊗ 𝔾_{a} ⟶ G ⟶^{p} G_{0} ⟶ 0

be the universal vectorial extension. Then

(2.10)  H^{0}(G, O_{G}) ≅ H^{0}(G_{0}, O_{G_{0}})

Proof. The O_{G_{0}}-algebra p_{∗}O_{G} is filtered, with

(2.11)  gr_{i}p_{∗}O_{G} = fil_{i}/fil_{i−1} ≅ Sym^{i}(H^{0}(J^{∨}, Ω_{J^{∨}/K}^{1})^{∗}) ⊗ O_{G_{0}}

With respect to the exact sequences

(2.12)  0 ⟶ fil_{i−1} ⟶ fil_{i} ⟶ gr_{i} ⟶ 0

it suffices to show the boundary map

(2.13) b : Sym^{i}(H^{0}(J^{∨}, Ω_{J^{∨}/K}^{1})^{∗}) ⊗ H^{0}(G_{0}, O_{G_{0}}) → H^{1}(G_{0}, fil_{i−1})

is injective. Composing on the right with the evident map, it suffices to show the maps

(2.14) Sym^{i}(H^{0}(J^{∨}, Ω_{J^{∨}/K}^{1})^{∗}) ⊗ H^{0}(G_{0}, O_{G_{0}})

→ H^{1}(G_{0}, O_{G_{0}}) ⊗ Sym^{i−1}(H^{0}(J^{∨}, Ω_{J^{∨}/K}^{1})^{∗})

are injective. But

(2.15) H^{0}(J^{∨}, Ω_{J^{∨}/K}^{1})^{∗} ⊗ H^{0}(G_{0}, O_{G_{0}}) ≅ H^{1}(J, O_{J}) ⊗ H^{0}(G_{0}, O_{G_{0}})

≅ H^{1}(G_{0}, O_{G_{0}})

and the map in (2.14) is the map x^{i} ↦ x ⊗ x^{i−1}, which is injective. □

Lemma 2.4. Let G = J_{D} be a generalized jacobian as above. Then there exists a commutative affine algebraic group 𝔾 over K and a map ψ : G → 𝔾 such that

(2.16)  ψ^{∗} : H^{0}(O_{𝔾}) ≅ H^{0}(O_{G}).

Proof. Take 𝔾 = Spec (H^{0}(G, O_{G})). □

Lemma 2.5. Let A be the coordinate ring of a commutative affine algebraic group H over a field K of characteristic 0. Corresponding to the simplicial algebraic group BH, one has a complex

(2.17) A ⟶^{μ^{∗}−p_{1}^{∗}−p_{2}^{∗}} A ⊗_{K} A ⟶^{p_{23}^{∗}−μ_{12}⊗p_{3}^{∗}+p_{1}^{∗}⊗μ_{23}^{∗}−p_{12}^{∗}} A ⊗ A ⊗ A

This complex is exact at the middle term.

Proof. By proposition 4 on p. 168 of [10], the cohomology in the middle is a subgroup of the group of extensions Ext(H, 𝔾_{a}). (Note, A = Map(H, 𝔾_{a}).) By the classification of commutative algebraic groups in characteristic 0, this ext group vanishes (cf [10], pp. 170-172). □

We write Ω_{G}^{1} (resp. Ω_{G/K}^{1}) for the sheaf of 1-forms relative to k (resp. K). Now we would like to define invariant bundles, connections, differential forms, cohomology classes of O_{G}.

Definition 2.6. (1) A rank one bundle L ∈ H^{1}(G, O_{G}^{∗}) is called invariant if μ^{∗}L = p_{1}^{∗}L ⊗ p_{2}^{∗}L ∈ H^{1}(G × G, O_{G×G}^{∗}), where μ : G × G → G is the multiplication and p_{i} : G × G → G are the projections.

(2) A global 1-form η ∈ Γ(G, Ω_{G}^{1}) is called invariant if η(0) = 0 ∈ Ω_{K}^{1} and μ^{∗}η = p_{1}^{∗}η + p_{2}^{∗}η ∈ Γ(G × G, Ω_{G×G}^{1}).

(3) A rank one bundle with a connection (L, ∇) ∈ ℍ^{1}(G, O_{G}^{∗} → Ω_{G}^{1}) is called invariant if (L, ∇)|{0} = 0 ∈ ℍ^{1}(Spec K, O_{Spec K}^{∗} → Ω_{Spec K}^{1}) = Ω_{K}^{1}/d log K^{∗} and μ^{∗}(L, ∇) = p_{1}^{∗}(L, ∇) ⊗ p_{2}^{∗}(L, ∇) ∈ ℍ^{1}(G × G, O_{G×G}^{∗} → Ω_{G×G}^{1}).

(4) A class s ∈ H^{i}(G, O_{G}) is called invariant if s|{0} = 0 and μ^{∗}s = p_{1}^{∗}s + p_{2}^{∗}s in H^{i}(G × G, O_{G×G}).

We denote by H^{1}(G, O_{G}^{∗})^{inv}, Γ(G, Ω_{G}^{1})^{inv}, ℍ^{1}(G, O_{G}^{∗} → Ω_{G}^{1})^{inv}, and H^{i}(G, O_{G})^{inv} the corresponding groups of invariant bundles, forms, connections and classes. One defines similarly the groups of relative invariant forms H^{0}(G, Ω_{G/K}^{1})^{inv} and relative invariant connections ℍ^{1}(G, O_{G}^{∗} → Ω_{G/K}^{1})^{inv} without condition on the restriction to the zero section, and observe that the natural map Ω_{G}^{1} → Ω_{G/K}^{1} takes global invariant groups to relative invariant groups.

Remark 2.7. In the above definitions, we could have defined a weaker notion of invariance by allowing constant elements. We adopt here the rigidification at the origin, keeping in mind that without this condition, the corresponding groups obtained are a direct sum of the ones obtained with the rigidification and the value of the group on the zero section. Notice, for example, that with our definition, nonzero constant functions are not invariant! Of course, for relative objects, there is no distinction.

Lemma 2.8. Let G = J_{D} be a generalized Jacobian as above. Let τ ∈ Γ(G, Ω_{G/K}^{1})^{inv} be an invariant relative 1-form on G, and assume τ lifts to an absolute global form. Then τ lifts to an invariant absolute form on G.

Proof. Let η ∈ Γ(G, Ω_{G}^{1}) be an absolute lifting. Replacing η with η − η(0) we may assume η(0) ∈ Ω_{K}^{1} vanishes. Then

(2.18)  (μ^{∗} − p_{1}^{∗} − p_{2}^{∗})(η) ∈ H^{0}(O_{G×_{K}G}) ⊗ Ω_{K}^{1}

vanishes in H^{0}(O_{G×_{K}G×_{K}G}) ⊗ Ω_{K}^{1}. Let ψ : G → 𝔾 be as in lemma 2.4, so ψ^{∗} : A := H^{0}(O_{𝔾}) ≅ H^{0}(O_{G}). The previous lemma implies there exists σ ∈ H^{0}(O_{G}) ⊗ Ω_{K}^{1} with (μ^{∗} − p_{1}^{∗} − p_{2}^{∗})(σ) = (μ^{∗} − p_{1}^{∗} − p_{2}^{∗})(η). Then η − σ is the desired invariant absolute form. □

We next need to relate connections on the curve with invariant connections on the generalized Jacobian. Here G = J_{D} with D = ∑ m_{i}c_{i}. Also,

(2.19)  D := ∑ c_{i}; D′ := D − D.

We assume D ≠ ∅.

First, rather briefly, we consider the question of what poles invariant forms on G have when pulled back to C − D via the jacobian map C−D → G (defined once we have a basepoint in C−D). For simplicity, we continue to assume the c_{i} are defined over K. Consider the diagram:

(2.20)  \begin{matrix}G ↠ G_{0} & ↠ & J \\ ↑ i & & ↑ i′ \\ C − D & ↪^{j} & C \end{matrix}

We want to compute the pullbacks i^{∗}H^{0}(Ω_{G}^{1})^{inv} (resp. i^{∗}H^{0}(Ω_{G/K}^{1})^{inv}) in H^{0}(C, Ω_{C}^{1}(∗D)) (resp. in H^{0}(C, Ω_{C/K}^{1}(∗D))) .

Pulling back G via i′ we get a torseur

(2.21)  i′^{∗}G →^{p} C

under the group O_{D}^{∗}/𝔾_{m} = ∏ O_{m_{ℓ}c_{ℓ}}^{∗}/𝔾_{m}. Fix ℓ and let R = k[[t_{ℓ}]] ⊂ M = k((t_{ℓ})) where t_{ℓ} is a formal parameter at c_{ℓ}. Fix a splitting of the torseur

(2.22)  G_{R} ≅ O_{D}^{∗}/𝔾_{m} × Spec (R).

Let c_{0} ∈ (C − D)(K) be the base point used to define the map i : C −D → G. Let L on C ×C be a line bundle with L|_{{c}×C} ≅ O(c−c_{0}). Note one has

(2.23)  O_{C×C} ← p_{2}^{∗}O(−c_{0}) → L

and these maps are isomorphisms on

(2.24)  Spec (M) × Spec (O_{C,D}) ⊂ Spec (M) × C

Corresponding to L|_{Spec (M)×C} and the above trivialization, one gets a map u : Spec (M) → G. With respect to the above splitting, we view u as an element (2.25)

∏_{i}(u_{i0} + u_{i1}t_{i} + … + u_{i,m_{i}−1}t_{i}^{m_{i}−1}) ∈ ∏_{i}(O_{m_{i}c_{i}} ⊗_{K} M)^{∗} mod M^{∗}

As described in [10] VII 4, 21, the local shape of u around c_{ℓ} is given by taking the rational function (s_{ℓ} − t_{ℓ})^{−1}, where the local coordinates around (c_{ℓ}, c_{ℓ}) in C ×C is (s_{ℓ}, t_{ℓ}), and considering it as a unit in O_{m_{ℓ}c_{ℓ}} ⊗ M ≅ M[t_{ℓ}]/ < t_{ℓ}^{m_{ℓ}} >. (We change notation so s_{ℓ} is the local parameter in R ⊂ M.) Since u is well defined and non-vanishing in c_{i} for i ≠ ℓ, we have (2.24) that

(2.26)  u_{ij} ∈ R, u_{i0} ∈ R^{∗} if i ≠ ℓ, ord(u_{ℓ0}) = 1

The pullbacks to Spec (M) of the invariant relative differential forms on G are given by the pullback of invariant relative forms on J together with the coefficients of powers of the T_{i} mod T_{i}^{m_{i}} in the formal expression

(2.27) ∑_{i}(u_{i0} + u_{i1}T_{i} + … + u_{i,m_{i}−1}T_{i}^{m_{i}−1})^{−1}×

(du_{i0} + du_{i1}T_{i} + … + du_{i,m_{i}−1}T_{i}^{m_{i}−1})

= ∑_{i} τ_{i0} + … + τ_{i,m_{i}−1}T_{i}^{m_{i}−1}.

Then (2.26) implies that

τ_{ij} ∈ Ω_{R}^{1} for i ≠ ℓ

τ_{ℓj} ∈ Ω_{R}^{1}⟨D⟩(jD_{ℓ}) − Ω_{R}^{1}⟨D⟩((j − 1)D_{ℓ}). (2.28)

Here we denote by Ω_{C}^{1}⟨D⟩ the sheaf of absolute differential forms of degree 1 with logarithmic poles along D. (See formula 3.59 for a more precise computation).

These are not all the absolutely invariant forms, however. One also has forms pulled back from J, but these are regular along D. Finally, from lemma 2.8 one has an exact sequence

(2.29) 0 → H^{0}(O_{G})^{inv} ⊗ Ω_{K}^{1} → H^{0}(G, Ω_{G}^{1})^{inv} → H^{0}(G, Ω_{G/K}^{1})^{inv}

→ H^{1}(O_{G})^{inv} ⊗ Ω_{K}^{1}

It shows one must consider invariant forms in H^{0}(O_{G})^{inv} ⊗Ω_{K}^{1}. We will see in the proof of proposition 2.13 below that these map to Ω_{G}^{1}(D′). In sum, the above discussion shows that the maps in the following proposition are defined.

Proposition 2.9. Pullback gives isomorphisms

H^{0}(Ω_{G/K}^{1})^{inv} ⟶^{≅} H^{0}(C, Ω_{C/K}^{1}(D)) (2.30)

H^{0}(Ω_{G}^{1})^{inv} ⟶^{≅} H^{0}(C, Ω_{C}^{1}⟨D⟩(D′)) (2.31)

Proof. Pullback on invariant relative forms is injective, because G is generated by the image of C − D. It follows by dimension count that the first arrow (2.30) above is an isomorphism. For the absolute forms we may consider the diagram

(2.32)

0 → H^{0}(O_{G})^{inv} ⊗ Ω_{K}^{1} →  H^{0}(Ω_{G}^{1})^{inv} → H^{0}(Ω_{G/K}^{1})^{inv} → H^{1}(O_{G})^{inv} ⊗ Ω_{K}^{1}
↓≅ ↓ ↓≅ ↓≅
0 → H^{0}(O_{C}(D′)) ⊗ Ω_{K}^{1} → H^{0}(Ω_{C}^{1}⟨D⟩(D′))→ H^{0}(Ω_{C/K}^{1}(D)) → H^{1}(O_{C}(D′)) ⊗ Ω_{K}^{1}

The left and right hand vertical arrows are shown to be isomorphisms in the proof of proposition 2.13. Hence the isomorphism on invariant relative forms implies the isomorphism (2.31) on invariant absolute forms. □

We now consider invariant connections on line bundles on G.

Lemma 2.10. Assume the toric subquotient T of G has trivial Picard group (e.g. T split). Then the map ℍ^{1}(G, O_{G}^{∗} → Ω_{G/K}^{1})^{inv} → H^{1}(O_{G}^{∗})^{inv} is surjective.

Proof. This follows because

(2.33)  ℍ^{1}(J, O_{J}^{∗} → Ω_{J/K}^{1})^{inv} ↠ H^{1}(O_{J}^{∗})^{inv} ↠ H^{1}(O_{G}^{∗})^{inv}.

The second arrow is surjective because we have a diagram

(2.34) \begin{aligned}0 & ⟶ & H^{1}(O_{J}^{∗})^{inv} & ⟶ & H^{1}(O_{J}^{∗}) & ⟶ & H_{DR}^{2}(J/K) \\ & & ↓ & & ↓^{surj.} & & ↓^{inj.} \\ 0 & ⟶ & H^{1}(O_{G}^{∗})^{inv} & ⟶_{a} & H^{1}(O_{G}^{∗}) & ⟶_{b} & H_{DR}^{2}(G/K) \end{aligned}

The bottom row is not a priori exact, but b∘a = 0 (because (Nδ)^{∗} acts by N^{2} on H_{DR}^{2}(G/K).) The middle vertical arrow is onto e.g. because the Picard group of the generic fibre of G → J is zero. Indeed, G → J is rationally split, and the kernel has trivial Picard group by hypothesis. (Since the function field of the generic fibre equals the function field of G, any divisor on G can be moved by rational equivalence to avoid the generic fibre, i.e. to be a pullback from the base.)

Finally, the right hand vertical arrow is injective because, after making a base change K ⊂ ℂ, one can think of G and J as quotients of vector spaces by lattices, and the map on lattices is surjective ⊗ℚ. □

Lemma 2.11. Let a ∈ ℍ^{1}(G, O_{G}^{∗} → Ω_{G/K}^{1})^{inv} be an invariant connection on a line bundle on the generalized jacobian G. Suppose a lifts to an absolute connection b′ ∈ ℍ^{1}(G, O_{G}^{∗} → Ω_{G}^{1}). Then a lifts to an invariant absolute connection b ∈ ℍ^{1}(G, O_{G}^{∗} → Ω_{G}^{1})^{inv}.

Proof. One has as in (2.17)

(2.35) (μ^{∗} − p_{1}^{∗} − p_{2}^{∗})(b′) ∈ Im H^{0}(O_{G×G}) ⊗ Ω_{K}^{1}

⊂ ℍ^{1}(G × G, O_{G×G}^{∗} → Ω_{G×G}^{1})

Now

Im H^{0}(O_{G×G}) ⊗ Ω_{K}^{1} =
H^{0}(O_{G×G}) ⊗ Ω_{K}^{1}/d log Ker {H^{0}(O_{G×G}^{∗}) → H^{0}(Ω_{G×G/K}^{1})} =
H^{0}(O_{G×G}) ⊗ (Ω_{K}^{1}/d log K^{∗}).

Exactness of the sequence in (2.17) implies that there exists an element x ∈ H^{0}(G, O ⊗ Ω_{K}^{1}) with (μ^{∗} − p_{1}^{∗} − p_{2}^{∗})(x) = (μ^{∗} − p_{1}^{∗} − p_{2}^{∗})(b′). Take b = b′ − x. □

Proposition 2.12. One has an exact sequence

(2.36) 0 → H^{0}(O_{G})^{inv} ⊗ Ω_{K}^{1} → ℍ^{1}(G, O_{G}^{∗} → Ω_{G}^{1})^{inv}

→ ℍ^{1}(G, O_{G}^{∗} → Ω_{G/K}^{1})^{inv} → H^{1}(O_{G})^{inv} ⊗ Ω_{K}^{1}

Proof. This is immediate from the lemma. □

Recall our notation. C is a smooth, projective, geometrically connected curve over K. G = J_{D} with D = ∑ m_{i}c_{i}, D = ∑ c_{i}, D′ = D−D.

Proposition 2.13. There exists a diagram of exact sequences, with vertical arrows isomorphisms:

(2.37)

0 →  H^{0}(O_{G})^{inv} ⊗ (Ω_{K}^{1}/d log K^{∗})  →  ℍ^{1}(G, O_{G}^{∗} → Ω_{G}^{1})^{inv}
↓ e  ↓
0 → (H^{0}(O_{C}(D′))/H^{0}(O_{C})) ⊗ (Ω_{K}^{1}/d log K^{∗}) → \frac{ℍ^{1}(C,j_{∗}(O_{C−D}^{∗})→Ω_{C}^{1} ⟨D⟩(D′))}{(Ω_{K}^{1} /K^{×})}

→  ℍ^{1}(G, O_{G}^{∗} → Ω_{G/K}^{1})^{inv}  →  H^{1}(O_{G})^{inv} ⊗ Ω_{K}^{1}
↓  ↓ h
→  ℍ^{1}(C, j_{∗}(O_{C−D}^{∗}) → Ω_{C/K}^{1}(D))  →  H^{1}(O_{C}(D′)) ⊗ Ω_{K}^{1}

Proof. The first step is to compute H^{i}(O_{G})^{inv} for i = 0, 1. Let W be a finite dimensional K-vector space, and suppose G = J_{D} is a vectorial extension

(2.38)  0 → W ⊗ 𝔾_{a} → G →^{p} G_{0} → 0

We know by lemma 2.1 that this sequence pulls back from an extension of J by W ⊗ 𝔾_{a}. Let

(2.39)  0 → O_{J} → fil_{1} → W^{∗} ⊗ O_{J} → 0

be the exact sequence of functions of filtration degree ≤ 1 as in lemma 2.3, and let ∂ : W^{∗} → H^{1}(O_{J}) be the boundary map in cohomology.

Lemma 2.14. We have

H^{0}(O_{G})^{inv} ≅ ker(∂ : W^{∗} → H^{1}(O_{J})) (2.40)

H^{1}(O_{G})^{inv} ≅ H^{1}(O_{J})/∂(W^{∗}). (2.41)

proof of lemma. One has a filtration fil_{·}p_{∗}O_{G} with fil_{0} = O_{G_{0}} and gr_{r} = Sym^{r}(W^{∗}) ⊗ O_{G_{0}}. The corresponding spectral sequence looks like (2.42) E_{1}^{pq} = H^{p+q}(G_{0}, gr_{−p}) = H^{p+q}(G_{0}, Sym^{−p}(W^{∗}) ⊗ O_{G_{0}}) ⇒ H^{p+q}(O_{G}).

Let

(2.43)  0 → H_{0} → G_{0} → S → 0

be as in lemma 2.2, so S is the maximal quotient torus of G_{0}.

The equation (2.6) identifies H^{i}(G_{0}, O_{G_{0}}) with H^{i}(J ×S, O_{J×S}), and the invariance condition might be looked at on J × S. Let us write H^{0}(S, O_{S}) = K ⊕ V, f ↦ f(0) ⊕ (f − f(0)), where V consists of the regular functions which vanish at {0} ∈ S. Then H^{i}(G_{0}, O_{G_{0}})^{inv} = H^{i}(J, O_{J})^{inv} ⊕ (H^{i}(J, O_{J}) ⊗ V)^{inv}. Thus if a class F = ∑ φ_{f} ⊗ f ∈ H^{i}(O_{J}) ⊗ V is invariant, where φ_{f} ∈ H^{i}(J, O_{J}) and the f ∈ V are linearly independent over K, then

(μ^{∗} − p_{1}^{∗} − p_{2}^{∗})(F)|(J × J × {0} × S) =
∑(μ^{∗} − p_{2}^{∗})(φ_{f}) ⊗ f = 0

thus μ^{∗}φ_{f} = p_{i}^{∗}φ_{f} and μ^{∗}φ_{f}|{0} × J = φ_{f} = φ_{f}|{0}. So for i ≥ 1, this implies that φ_{f} = 0 and for i = 0 this implies that F ∈ H^{0}(O_{S})^{inv} = 0.

In short: (2.44)

H^{∗}(O_{G_{0}})^{inv} ≅ (H^{∗}(O_{J}) ⊗ H^{0}(O_{S}))^{inv} ≅ H^{∗}(O_{H_{0}})^{inv} ≅ H^{∗}(O_{J})^{inv}

Thus, it suffices to prove the lemma with G_{0} replaced by H_{0}, so we may assume the quotient torus S = (0). Since in this case H^{i}(O_{G_{0}}) ≅ ∧^{i}H^{1}(O_{J}), one sees that pullback under the multiplication by N map, Nδ : G → G acts on E_{1}^{pq} by multiplication by N^{q}. It follows that the spectral sequence (2.42) degenerates at E_{2}. In particular the eigenspace where Nδ^{∗} acts by multiplication by N on H^{1}(O_{G}) is

(2.45)  H^{1}(O_{J})/∂(W^{∗}) ≅ E_{2}^{0,1} ≅ E_{∞}^{0,1} ↪ H^{1}(O_{G}).

Note that as a quotient of H^{1}(O_{J}) the space E_{∞}^{0,1} is clearly invariant. Conversely, let Δ : G → G × G be the diagonal. Since μ ∘ Δ = 2δ it follows that for a ∈ H^{1}(O_{G})^{inv} we have

(2.46)  (2δ)^{∗}(a) = Δ^{∗}μ^{∗}(a) = Δ^{∗}(p_{1}^{∗}(a) + p_{2}^{∗}(a)) = 2a

so necessarily a ∈ E_{∞}^{0,1}, proving (2.41). A similar argument on E_{∞}^{−1,1} proves (2.40).

We remark here that H^{0}(J, fil_{1}) is in a natural way a subspace of the regular functions on G, and (2.40) takes the quotient of this by H^{0}(J, O_{J}) = K. This is because we have forced the rigidification condition in the definition 2.6. □

We return to the proof of proposition 2.13. The exact sequence

(2.47)  0 → O_{C} → O_{C}(D′) → O_{C}(D′)/O_{C} → 0

defines a map

(2.48)  ψ : W^{∗} := H^{0}(O_{C}(D′)/O_{C}) → H^{1}(O_{C}).

By lemma 2.1, as a group extension of G_{0}, the group G is defined by a unique map from H^{0}(C, Ω_{C/K}^{1}) to a vector space. We claim that this map is the dual of ψ. To see this, one identifies J and J^{∨}. Then it is well know that the universal vectorextension on J^{∨} is

0 → H^{0}(C, Ω_{C/K}^{1}) → ℍ^{1}(C, O_{C}^{∗} → Ω_{C/K}^{1}) → Pic^{0}(C) → 0

inducing the universal vectorextension

0 → H^{0}(C, Ω_{C/K}^{1}) → ℍ^{1}(C, O_{C,D}^{∗} → Ω_{C/K}^{1}) → Pic^{0}(C, D) → 0

on

Pic^{0}(C, D) := Ker(H^{1}(C, O_{C,D}^{∗}) → H^{1}(C, Ω_{C/K}^{1}))

where

O_{C,Z}^{∗} := Ker(O_{C}^{∗} → O_{Z}^{∗})

for any subscheme Z ⊂ C. The map of complexes

a : {O_{C,D}^{∗} → 0} → {O_{C,D}^{∗} → Ω_{C/K}^{1}|_{D′}}

induces an isomorphism on ℍ^{1}. Indeed, a sends the exact sequence

0 → H^{0}(C, (1 + O_{D′}(−D))) → H^{1}(C, O_{C,D}^{∗}) → H^{1}(C, O_{C,D}^{∗}) → 0

to the exact sequence

0 → H^{0}(C, Ω_{C/K}^{1}|_{D′}) → ℍ^{1}(C, O_{C,D}^{∗} → Ω_{C/K}^{1}|_{D′}) → H^{1}(C, O_{C,D}^{∗}) → 0,

so one has just to see that

d log : H^{0}(C, (1 + O_{D′}(−D))) → H^{0}(C, Ω_{C/K}^{1}|_{D′})

is an isomorphism. But H^{0}(C, O_{D′}(−D)) ≅ H^{0}(C, (1 + O_{D′}(−D))) via the exponential map and the quasiisomorphism [4]

{O_{C}(−D) → Ω_{C/K}^{1}(−D′)} → {O_{C}(−D) → Ω_{C/K}^{1}}

allows to conclude.

Define a bundle E on C by pullback

(2.49) \begin{aligned}0 & ⟶ & O_{C} & ⟶ & E & ⟶ & W^{∗} ⊗ O_{C} & ⟶ & 0 \\ & & ∥ & & ↓ & & ↓ \\ 0 & ⟶ & O_{C} & ⟶ & O(D′) & ⟶ & O_{C}(D′)/O_{C} & ⟶ & 0 \end{aligned}

Because of the isomorphism H^{1}(O_{J}) ≅ H^{1}(O_{C}), the top row of the diagram (2.49) pulls back uniquely from an extension of W^{∗} ⊗ O_{J} by O_{J}. There is a unique vectorial extension

(2.50)  0 → W ⊗ 𝔾_{a} → H →^{r} J → 0

such that the above extension of vector bundles coincides with

(2.51)  0 → O_{J} → fil_{1}r_{∗}O_{H} → W^{∗} ⊗ O_{J} → 0

From this we get a diagram (defining t and u. Here i : C ↪ J)

(2.52) \begin{aligned}0 & ⟶ & O_{J} & ⟶ & fil_{1}r_{∗}O_{H} & ⟶ & W^{∗} ⊗ O_{J} & ⟶ & 0 \\ & & ↓ & & ↓^{t} & & ↓^{u} \\ 0 & ⟶ & O_{C} & ⟶ & O_{C}(D′) & ⟶ & O_{C}(D′)/O_{C} & ⟶ & 0 \end{aligned}

We get a diagram with exact rows

(2.53)

0  → H^{0}(O_{J}) → H^{0}(fil_{1}r_{∗}O_{H}) → H^{0}(W^{∗} ⊗ O_{J}) →^{∂} H^{1}(O_{J}) →
↓≅ ↓ t ↓≅ ↓≅
0  → H^{0}(O_{C}) → H^{0}(O_{C}(D′)) → H^{0}(O_{C}(D′)/O_{C}) → H^{1}(O_{C}) →
H^{1}(O_{H})^{inv} →  0
↓ v
H^{1}(O_{C}(D′)) →  0

The diagram (2.53) gives isomorphisms

(2.54) e : H^{0}(O_{G})^{inv} ≅ ker(∂) ≅ Ker(H^{0}(O_{C}(D′)/O_{C}) → H^{1}(O_{C}))

h : H^{1}(O_{G})^{inv} ≅ H^{1}(O_{C}(D′)). (2.55)

These are two of the desired arrows for the diagram in the proposition.

Lemma 2.15. The natural map on relative connections

(2.56) ℍ^{1}(G, O_{G}^{∗} → Ω_{G/K}^{1})^{inv} → ℍ^{1}(C, j_{∗}(O_{C−D}^{∗}) → Ω_{C/K}^{1}(D))

is an isomorphism.

proof of lemma. We note the following facts:

(1) H^{1}(G, Ω_{G/K}^{1})^{inv} = (H^{0}(G, Ω_{G/K}^{1})^{inv} ⊗ H^{1}(O_{G}))^{inv} = (0).

(2) H^{1}(O_{G}^{∗})^{inv} ≅ H^{1}(O_{C−D}^{∗}). Indeed, as remarked in the proof of lemma 2.10 one has J(K) ↠ H^{1}(O_{G}^{∗})^{inv}. One checks that the kernel is generated by divisors of degree 0 supported on D.

(3) H^{0}(O_{G}^{∗})/consts. ≅ H^{0}(O_{G}^{∗})^{inv} ≅ H^{0}(O_{C−D}^{∗})/consts..

(4)

(H^{0}(Ω_{G/K}^{1})/d log(H^{0}(O_{G}^{∗})))^{inv} = H^{0}(Ω_{G/K}^{1})^{inv}/d log(H^{0}(O_{G}^{∗})^{inv})

(This is seen by noting H^{0}(O_{G}^{∗})^{inv} = Hom(G, 𝔾_{m}), so one has a homomorphism ψ : G → 𝔾_{m}^{r} such that ψ^{∗} is an isomorphism on global units modulo constants. The assertion then reduces to the case G = 𝔾_{m}^{r}, which is easy.)

We build a diagram

(2.57)

0 → \frac{H^{0}(Ω_{G/K}^{1} )^{inv}}{H^{0}(O_{G}^{∗})^{inv}} →  ℍ^{1}(O_{G}^{∗} → Ω_{G/K}^{1})^{inv}  →  H^{1}(O_{G}^{∗})^{inv}  → 0
↓  ↓  ↓
0 → \frac{H^{0}(Ω_{C/K}^{1} (D))}{H^{0}(O_{C−D}^{∗})} → ℍ^{1}(C, j_{∗}(O_{C−D}^{∗}) → Ω_{C/K}^{1}(D)) → H^{1}(C − D, O^{∗}) → 0

Since the left and right hand arrows are isomorphisms, it follows that the central arrow is as well, proving the lemma. □

The assertions of the proposition follow easily from the lemma. □

Finally, we need an analogous result for integrable connections. More precisely, we consider a slightly weaker condition.

Definition 2.16. Let X be a variety over K. A connection ∇ : E → E⊗Ω_{X}^{1}⊗K(X) (so possibly with poles) is said to have vertical curvature if the curvature

(2.58)  ∇^{2} : E → E ⊗ Ω_{X}^{2} ⊗ K(X)

has values in the subsheaf E ⊗ Ω_{K}^{2} ⊗ K(X) ⊂ E ⊗ Ω_{X}^{2} ⊗ K(X). The group of line bundles with vertical curvature will be denoted

ℍ^{1}(X, O_{X}^{∗} → Ω_{X}^{1})^{vert}

and similarly for invariant line bundles with vertical curvature

ℍ^{1}(X, O_{G}^{∗} → Ω_{G}^{1})^{inv,vert} .

Proposition 2.17. With notation as above, we have isomorphisms

(2.59) ℍ^{1}(G, O_{G}^{∗} → Ω_{G/K}^{1})^{inv,vert} = ℍ^{1}(C, j_{∗}O_{C−D}^{∗} → Ω_{C/K}^{1}(D))^{vert}

(2.60) ℍ^{1}(G, O_{G}^{∗} → Ω_{G}^{1})^{inv,vert} ≅ \frac{ℍ^{1}(C, j_{∗}O_{C−D}^{∗} → Ω_{C}^{1}⟨D⟩(D′))^{vert}}{Ω_{K}^{1}/K^{×}}

Proof. For example, in the absolute case, the curvature of a line bundle with invariant absolute connection on G is a section η ∈ H^{0}(Ω_{G}^{2}/Ω_{K}^{2} ⊗ O_{G}) satisfying μ^{∗}(η) = p_{1}^{∗}(η)+p_{2}^{∗}(η). It is easy to see that such a section lies in the subsheaf Ω_{G/K}^{1} ⊗ Ω_{K}^{1}. The isomorphism (2.60) follows from proposition 2.13 and the fact that pullback to C of invariant forms is injective by (2.30). The case of relative forms is similar and is left for the reader. □

3. The Geometric Setup

We continue to work with a curve C/K and a line bundle L on C of degree 0. Let ∇_{/K} : L → L ⊗ Ω_{C/K}^{1}(D), where D = ∑ m_{i}c_{i}. As in the previous section, write D = ∑ c_{i} and D′ = D − D.

Lemma 3.1. Assume ∇_{/K}|C − D lifts to an absolute integrable connection ∇′ : L_{C−D} → L_{C−D} ⊗ Ω_{C−D/k}^{1}. Then ∇′ extends to an absolute integrable connection

(3.1)  ∇ : L → L ⊗ Ω_{C/k}^{1}⟨D⟩(D′).

The notation ⟨D⟩ refers to log poles at D as in (1.4).

Proof. Let e be a basis for L at c a point with multiplicity m ≥ 1 in D, and let x be a local parameter at c on C. Write (3.2)

∇′(e) = A(x)dx + ∑_{i} B_{i}(x)dτ_{i}; dτ_{i} basis in Ω_{K}^{1}, x^{m}A(x) ∈ O_{C,c}.

We must show x^{m−1}B_{i}(x) is regular at c. But integrability of ∇′ implies that ∂A/∂τ_{i} = ∂B_{i}/∂x, from which the assertion is clear. □

We know from proposition 2.17 that the restriction to C − D of an integrable absolute connection of the form (3.1) pulls back from a unique invariant integrable absolute connection L → L ⊗ Ω_{G}^{1} on G = J_{D}. More precisely, we fix a basepoint c_{0} ∈ (C − D)(K) and normalize our connection (3.1) to be trivial at the basepoint by tensoring with a pullback from Spec (K).

We consider now the basic geometric picture of Deligne [3]

(3.3)  π : Sym^{N}(C − D) → G_{N}; N = 2g − 2 + ∑_{i} m_{i}

where G_{N} is the J_{D}-torseur of degree N line bundles trivialized along D and π(∑ z_{i}) = O(∑ z_{i}) with trivialization given by restricting to D the canonical (upto scalar in K) map O_{C} → O_{C}(∑ z_{i}). Note that N = deg(Ω_{C/K}^{1}(D)) and dim G = g −1+∑ m_{i} = N −g +1. (Recall we assume D ≠ ∅.) We identify G_{N} ≅ G by sending the point [O(Nc_{0})] ↦ 0, and we write L for the resulting line bundle with connection on G_{N}. The basic remark of Deligne is

Proposition 3.2. Assume

H_{DR}^{0}(C − D, (L, ∇)) = H_{DR}^{2}(C − D, (L, ∇)) = 0.

Then

(3.4) det(H_{DR}^{∗}(C − D, (L, ∇)) = det(H_{DR}^{1}((C − D, (L, ∇))))

≅ H^{N}(Sym^{N}(C − D), (π^{∗}(L, ∇)))

as a line with connection on K.

Proof. Our hypotheses imply H_{DR}^{1}(C − D, (L, ∇)) has dimension N. Consider the diagram

(3.5)  \begin{aligned}(C − D)^{N} & →^{p} & Sym^{N}(C − D) \\ & ↘ q & ↓ π \\ & & G_{N} \end{aligned}

We have q^{∗}(L) ≅ L ⊠ ⋯ ⊠ L (exterior tensor product on (C − D)^{N}. The Künneth formula gives

(3.6) H^{N}((C − D)^{N}, (L ⊠ ⋯ ⊠ L, ∇)) ≅ H_{DR}^{1}(C − D, (L, ∇))^{⊗N}.

There is an action of the symmetric group S_{N} on the pair

((C − D)^{N}, L ⊠ ⋯ ⊠ L).

The resulting action on (H_{DR}^{1})^{⊗N} is alternating because of the odd degree cohomology, so the invariants are precisely det H_{DR}^{1}. There is an evident map

(3.7) p^{∗} : H_{DR}^{N}(Sym^{N}(C − D), π^{∗}L) → H_{DR}^{N}((C − D)^{N}, L ⊠ ⋯ ⊠ L)^{S_{N}}

= det H_{DR}^{1}(C − D, L)

To show this map is an isomorphism, it suffices to remark that one has a trace map

(3.8) p_{∗} : H_{DR}^{N}((C − D)^{N}, L ⊠ ⋯ ⊠ L) → H_{DR}^{N}(Sym^{N}(C − D), π^{∗}L)

Because L ⊠ ⋯ ⊠ L = p^{∗}π^{∗}L, the existence of such a trace follows from the projection formula and the trace in de Rham cohomology with constant coefficients. □

Now one uses the geometry of the map π and (3.4) to compute the determinant.

Lemma 3.3. Let X be a smooth variety over a field of characteristic 0. Let A ⊂ X be a smooth subvariety of codimension p. Let (E, ∇) be an integrable connection on X. Then

(3.9)  ℍ_{A}^{n}(X, E ⊗ Ω_{X}^{∗}) ≅ H^{n−2p}(A, E ⊗ Ω_{A}^{∗}).

Proof. Write \underline{\underline{H}}_{A}^{r}(F) for the Zariski sheaf associated to the presheaf U ↦ H_{A}^{r}(U, F) for any Zariski sheaf F on X. For F locally free, \underline{\underline{H}}_{A}^{r}(F) = (0) for r ≠ p by purity. Duality theory gives (here X ⊃ A_{α} ⊃ A runs through nilpotent thickenings)

(3.10) E ⊗ Ω_{A}^{m} → Ext^{p}(O_{A}, E ⊗ Ω_{X}^{m+p}) → \underrightarrow{lim}_{α} Ext^{p}(O_{A_{α}}, E ⊗ Ω_{X}^{m+p})

≅ \underline{\underline{H}}_{A}^{p}(E ⊗ Ω_{X}^{m+p}).

We want to show that this map is an isomorphism, compatible with the connection, thus yielding a quasiisomorphism of complexes

(3.11)  E ⊗ Ω_{A}^{∗} → \underline{\underline{H}}_{A}^{p}(E ⊗ Ω_{X}^{∗}).

The problem is local, so we can assume

A ⊂ A_{1} ⊂ … ⊂ A_{p} = X

with A_{i} smooth of codimension p−i in X. Now \underline{\underline{H}}_{A}^{p}(E⊗Ω_{X}^{∗}) represents R\underline{\underline{Γ}}_{A}(E ⊗ Ω_{X}^{∗})[p] in the derived category of Zariski sheaves on A, and in the derived category we may write

(3.12) R\underline{\underline{Γ}}_{A}(E ⊗ Ω_{X}^{∗})[p] = R\underline{\underline{Γ}}_{A}[1] ∘ R\underline{\underline{Γ}}_{A_{1}}[1] ∘ … ∘ R\underline{\underline{Γ}}_{A_{p−1}}(E ⊗ Ω_{X}^{∗})[1]

In this way, we reduce to verifying (3.9) in the case p = 1. So, suppose A : t = 0 in X = Spec (R). We have

(3.13)  \underline{\underline{H}}_{A}^{1}(E ⊗ Ω_{X}^{∗}) ≅ E_{R[t^{−1}]} ⊗ Ω_{R[t^{−1}]}^{∗}/E_{R} ⊗ Ω_{R}^{∗}

as H_{DR}^{1}(X, E) ⊂ H_{DR}^{1}(X − A, E).

By [4], since E has no singularity along t = 0, one has

(3.14) (E_{R} ⊗ Ω_{R}^{∗}(log(t = 0)), ∇) →^{q.iso.} (E_{R[t^{−1}]} ⊗ Ω_{R[t^{−1}]}^{∗}, ∇|_{SpecR[t^{−1}]})

Thus res : \underline{\underline{H}}_{A}^{1}(E ⊗ Ω_{X}^{∗}, ∇)) → E_{R/tR} ⊗ (Ω_{R/tR}^{∗}[−1], ∇|_{Spec(R/tR)}) is an isomorphism. □

Lemma 3.4. Let p : G_{N} → J_{N} be the projection to the corresponding torseur over the absolute jacobian. Write [Ω_{C/K}^{1}(D)] ∈ J_{N} for the point corresponding to the canonical bundle twisted by O(D). Let a ∈ G_{N}. We have

(3.15)  π^{−1}(a) = \begin{cases}𝔸^{g−1} & p(a) ≠ [Ω_{C/K}^{1}(D)] \\ 𝔸^{g} & p(a) = [Ω_{C/K}^{1}(D)]; ∂(a) = 0 \\ ∅ & p(a) = [Ω_{C/K}^{1}(D)]; ∂(a) ≠ 0 \end{cases}

where by definition 𝔸^{g−1} = ∅ is g = 0. Note that if p(a) = [Ω_{C/K}^{1}(D)], then a corresponds to a trivialization O_{D} ≅ Ω_{C/K}^{1}(D)|_{D} defined upto scalar. ∂(a) in the above refers to the evident boundary of this trivialization in H^{1}(Ω_{C/K}^{1}) = K (again upto scale).

Proof. Let p(a) correspond to a line bundle M of degree N, we consider the exact sequence

(3.16)  0 → M(−D) → M → M|_{D} → 0

Suppose first M ≠ Ω_{C/K}^{1}(D). Then H^{1}(M(−D)) = (0), so any trivialization in H^{0}(M|_{D}) lifts to H^{0}(M), and the space of such liftings is a torseur under H^{0}(M(−D)), a vector space of dimension g − 1. (Note this is an affine torseur, not a projective torseur.) If M = Ω_{C/K}^{1}(D), H^{0}(M(−D)) has dimension g, and the image H^{0}(M) → H^{0}(M|_{D}) has codimension 1. □

Remark 3.5. If we choose local parameters t_{i} at c_{i} ∈ D, then

H^{0}(Ω_{C/K}^{1}(D)|_{D})

can be identified with the space of polar parts of 1-forms with poles along D, and the map ∂ is given by the residue

(3.17)  ∂(∑_{i} ∑_{j=0}^{m_{i}−1} u_{ij}dt_{i}/t_{i}^{m_{i}−j}) = ∑_{i} u_{i,m_{i}−1}

Note the (open) condition for an element in H^{0}(Ω_{C/K}^{1}(D)|_{D}) to be a trivialization is simply

(3.18)  ∏_{i} u_{i0} ≠ 0.

Because B ⊂ G_{N}, we must factor out by the action of 𝔾_{m}, which we can normalize away by setting u_{10} = 1. Thus we have

(3.19) B := π(Sym^{N}(C − D)) ∩ p^{−1}[Ω_{C/K}^{1}(D)] =

{ ∑_{i} ∑_{j=0}^{m_{i}−1} u_{ij}dt_{i}/t_{i}^{m_{i}−j} | ∑_{i} u_{im_{i}−1} = 0; ∏_{i} u_{i0} ≠ 0; u_{10} = 1}.

Define A := π^{−1}(B) ⊂ Sym^{N}(C − D) =: X. Using the localization sequence and lemma 3.4, we get an exact sequence (3.20)

H_{DR}^{N−2g}(A/K, π^{∗}L) ⟶ H_{DR}^{N}(X/K, π^{∗}L) ⟶ H_{DR}^{N}(X − A/K, π^{∗}L)
↑≅  ↑≅
H_{DR}^{N−2g}(B/K, L|_{B})  H_{DR}^{N}(G_{N} − p^{−1}[Ω_{C/K}^{1}(D)]/K, L)

where p : G_{N} → J_{N} with J = J(C) the absolute Jacobian. The fact that the vertical arrows in this diagram are isomorphisms follows because the maps are maps of affine space bundles and the line bundles with connection are pulled back from the base.

To simplify the presentation, we will assume that m_{1} ≥ 2. Another proof of our formula for the de Rham determinant in the case D = D = ∑ c_{i}, (i.e. for regular singular points) will be given in theorem 4.6.

Lemma 3.6. Assume the line bundle L on C has degree 0, and that D is minimal, i.e. ∇ : L|_{D} ≅ L ⊗ Ω_{C/K}^{1}(D)|_{D}. We continue to assume also that D = ∑ m_{i}c_{i} with m_{1} ≥ 2. Then

(3.21)  H_{DR}^{∗}(Sym^{N}(C − D) − A/K, π^{∗}L) = (0).

Proof. The isomorphism on the right in (3.20) implies we must show H_{DR}^{∗}(G_{N} − p^{−1}[Ω_{C/K}^{1}(D)]/K, L) = (0). The assumption m_{1} ≥ 2 means we have a 𝔾_{a} action by translation on G_{N} − p^{−1}[Ω_{C/K}^{1}(D)]/K, L, and minimality of D implies that the connection is nontrivial on the fibres. The fibration is Zariski-locally trivial, so the Leray spectral sequence for de Rham cohomology reduces us to showing H_{DR}^{∗}(𝔾_{a,S}/S, (O, Ξ)) = (0) where Ξ is an everywhere non-zero, translation invariant, relative 1form on O_{𝔾_{a,S}}. In other words, for s ∈ O_{S}^{×} we must show

(3.22)  O_{S}[t] ⟶^{d+sdt} O_{S}[t]dt

has trivial cohomology. This is straightforward. □

Lemma 3.7. Assume H_{DR}^{0}(C − D, L) = (0). Then

(3.23)  H_{DR}^{m−2}(B/K, L|_{B}) ≅ H_{DR}^{N}(Sym^{N}(C − D)/K, π^{∗}L) ≅ K

as a line with a connection over K.

Proof. Note m − 2 = N − 2g. Extending the top sequence in (3.20) one step to the left and using the previous lemma gives the left isomorphism. We have already seen the isomorphism on the right. □

Our task now is to calculate H_{DR}^{m−2}(B/K, L|_{B}) with its connection. We assume that L ∈ Pic^{0}(J) as in lemma 3.6. Then L carries a relative invariant connection d_{/K} on J, and ∇_{/K} = d_{/K} + Ξ for some invariant form Ξ ∈ H^{0}(G, Ω_{G/K}^{1})^{inv}. Changing the choice of d_{/K} changes Ξ to Ξ + p^{∗}(α), where α ∈ H^{0}(J, Ω_{J/K}^{1})^{inv} and p : G_{N} → J_{N} is the torseur under the affine group G := ker(G → J). In particular, Ξ|p^{−1}[Ω_{C/K}^{1}(D)] does not depend on the choice of d_{/K}. As p^{−1}(Ω_{G/K}^{1}(D)) is isomorphic to G, and we see that

(3.24)  (L, ∇_{/K})|_{B} = (O_{B}, d + Ξ|_{B}).

We say that Ξ|_{B} vanishes at a point b ∈ B if Ξ(b) = 0 in 𝔪_{b}/𝔪_{b}^{2}.

Lemma 3.8. Let b ∈ B ⊂ G_{N} correspond to the trivialization on Ω_{C/K}^{1}(D)|_{D} given by ∇|_{D} : L|_{D} → L ⊗ Ω_{C/K}^{1}(D)|_{D}. Then Ξ|_{B} vanishes at b. Ξ|_{B} does not vanish at any other point of B.

Proof. Let us write η := i^{∗}Ξ|C − D, where i : C − D → G is the cycle map. Then by definition, the trivialization of Ω_{C/K}^{1}(D)|_{D} associated to i^{∗}(L, ∇) depends only on η|D, or equivalently only on Ξ|p^{−1}[Ω_{C/K}^{1}(D)]. We have

(3.25)  π^{∗}Ξ = ∑_{i=1}^{N} η_{i}

where η_{i} is the pullback of η via the i-th projection (C −D)^{N} → C −D.

Suppose for a moment that the divisor of η (viewed as a section of Ω_{C/K}^{1}(D)) is reduced, (η) = ∑ e_{i}; e_{i} ∈ (C − D)(K̄). Let e := (e_{1}, ⋯, e_{N}) ∈ Sym^{N}(C − D) be the point corresponding to η. We have e ∈ A = π^{−1}(B) ⊂ Sym^{N}(C − D) and b = π(e). Since A → B is a projective bundle, there is a surjection on tangent spaces

(3.26)

T_{A}(e) ↪ T_{Sym^{N} (C−D)}(e)
surjective ↓  ↓ π_{∗}
T_{B}(b) ↪  T_{G_{N}} (b)

Since T_{Sym^{N} (C−D)}(e) is spanned by expressions ∑ τ_{i}|_{e_{i}}, to show Ξ|_{B} vanishes at b, it suffices to show

(3.27)  < Ξ, π_{∗}(∑ τ_{i}|_{e_{i}}) >= 0.

This expression equals

(3.28)  < π^{∗}(Ξ), ∑ τ_{i}|_{e_{i}} >= ∑ < η, τ_{i}|_{e_{i}} > .

Each term on the right vanishes because η(e_{i}) = 0 in 𝔪_{e_{i}}/𝔪_{e_{i}}^{2}. The general case ((η) not necessarily reduced) follows from this by a specialization argument.

We postpone until lemma 3.10 the proof that Ξ|_{B} doesn’t vanish at any other point of B. □

By assumption we start with an absolute, invariant, integrable connection on L of degree 0 on J. Restricting to B, we get an absolute closed invariant 1-form Ψ, whose corresponding relative form is Ξ. Recall (3.19) we have coordinates u_{ij} on B with u_{10} = 1, ∏ u_{i0} ≠ 0, and u_{1,m_{1}−1} = − ∑_{i≥2} u_{i,m_{i}−1}.

Lemma 3.9. Under the assumption that

{i^{∗}∇_{/K} : i^{∗}L → i^{∗}L⊗Ω_{C/K}^{1}(D)} → {j_{∗}i^{∗}L|(C−D) → j_{∗}i^{∗}L⊗Ω_{(C−D)/K}^{1}}

is a quasiisomorphism, and i^{∗}∇_{/K} has poles along all points of D, we can arrange that a K-basis for

H_{DR}^{m−2}(B/K, L|_{B}) = H_{DR}^{m−2}(B/K, (O_{B}, d + Ξ))

is given by the closed form

(3.29)  θ := ∏_{\begin{matrix}(i,j) \\ m_{i}≥2 \end{matrix}} du_{ij} ∧ ∏_{\begin{matrix}i \\ m_{i}=1 \end{matrix}} \frac{du_{i0}}{u_{i0}}

Proof. Recall (2.27) the relative invariant forms on G := ker(G → J) are the τ_{ij} defined by the expression

(3.30)  ∑_{j=0}^{m_{i}−1} τ_{ij}T_{i}^{j} = ( ∑_{j} u_{ij}T_{i}^{j})^{−1} ∑_{j} du_{ij}T_{i}^{j}.

Write

(3.31)  Ξ = ∑_{i,j} λ_{ij}τ_{ij}; λ_{i,m_{i}−1} ≠ 0

where the nonvanishing condition comes from the requirement that the form restricted to C − D gives a trivialization along D (see (2)). If we write ( mod T_{i}^{m_{i}}) (3.32)

(u_{i0} + u_{i1}T_{i} + … + u_{i,m_{i}−1}T_{i}^{m_{i}−1})^{−1} = ν_{i0} + ν_{i1}T_{i} + … + ν_{i,m_{i}−1}T_{i}^{m_{i}−1}

we get the table

(3.33)  τ_{i0} = ν_{i0}du_{i0}
τ_{i1} = ν_{i0}du_{i1} + ν_{i1}du_{i0}
⋮
τ_{i,m_{i}−1} = ν_{i0}du_{i,m_{i}−1} + … + ν_{i,m_{i}−1}du_{i0}

Note if we give u_{ij}, du_{ij}, ν_{ij} all weight j, then τ_{ij} will be homogeneous of weight j. Comparing (3.33) and (3.31), it follows that if we expand Ξ|_{B} in terms of the du_{ij}, omitting du_{10} and du_{1,m_{1}−1}, we find for suitable α_{ij} ≠ 0

(3.34) Ξ|_{B} = ∑_{i} g_{ij}du_{ij} = ∑_{i,j} [(α_{i,m_{i}−1}u_{i0}^{−1} − α_{1,m_{1}−1})du_{i,m_{i}−1}+

u_{i0}^{−1} ∑_{p=0}^{m_{i}−2} (α_{ip} \frac{u_{i,m_{i}−1−p}}{u_{i0}} + ∑ terms at least quadratic in \frac{u_{ik}}{u_{i0}} )du_{ip}]

Looking at the weights, we see that for p ≤ m_{i} − 2

(3.35) g_{ip} = nonzero multiple of \frac{u_{i,m_{i}−p−1}}{u_{i0}} +

terms only involving u_{ij}, j < m_{i} − p − 1

while

(3.36)  g_{i,m_{i}−1} = α_{i,m_{i}−1}u_{i0}^{−1} − α_{1,m_{1}−1}

with neither α coefficient 0.

Now generators of H_{DR}^{m−2}(B, (O, d + Ξ)) are of the form Mθ where M is a monomial in the u_{ij}, u_{i0}^{−1} and θ is as in (3.29). Relations are

(3.37)  (\frac{∂}{∂u_{ij}} + g_{ij})(M)θ = 0

Because of (3.35) one can use these relations to eliminate u_{ij}, j > 0 from M by downward induction on j, starting from u_{i,m_{i}−1}. We are left with the case M = u_{2,0}^{n_{2}} ⋯ u_{r,0}^{n_{r}} with n_{i} ∈ ℤ. In this case we can apply (3.36). If m_{i} ≥ 2, we get the relation

(3.38)  u_{2,0}^{n_{2}} ⋯ u_{r,0}^{n_{r}}θ ≡ \frac{α_{i,m_{i}−1}}{α_{1,m_{1}−1}} u_{2,0}^{n_{2}} ⋯ u_{i,0}^{n_{i}−1} ⋯ u_{r,0}^{n_{r}}θ

Using this, we can get n_{i} = 0. If m_{i} = 1 and i ≥ 2 the relation becomes

(3.39)  u_{2,0}^{n_{2}} ⋯ u_{r,0}^{n_{r}}θ ≡ \frac{n_{i} + α_{i,0}}{α_{1,m_{1}−1}} u_{2,0}^{n_{2}} ⋯ u_{i,0}^{n_{i}−1} ⋯ u_{r,0}^{n_{r}}θ

If α_{i,0} is not a positive integer, we can arrange n_{i} = 0.

On the other hand, we claim that if m_{i} = 1, then α_{i0} is the residue of i^{∗}∇ along c_{i}. Indeed (2.27) shows that in this case, τ_{i0} = d log u_{i0}, and u_{i0} is then just the local parameter in the point c_{i} (see (2.26)). Thus the quasiisomorphism

(3.40) {i^{∗}∇_{/K} : i^{∗}L → i^{∗}L ⊗ Ω_{C/K}^{1}(D)} → {j_{∗}i^{∗}L|(C − D)

→ j_{∗}i^{∗}L ⊗ Ω_{(C−D)/K}^{1}} forces a_{i} not to lie in ℕ − {0}. □

The following was left open in the proof of lemma 3.8:

Lemma 3.10. Let Ξ = ∑_{i,j} λ_{ij}τ_{ij} be as in lemma 3.8. Then Ξ vanishes at a unique point b ∈ B.

Proof. We have seen in the proof of lemma 3.8 that Ξ vanishes at a point in B. We must show it vanishes at at most one point. Let b = (…, b_{ij}, …) ∈ B be a point. Write b = (…, y_{ij}, …) with respect to the coordinates ν_{ij} (3.32). Staring at (3.33), the conditions that Ξ|_{b} = 0 are seen to be (recall ∑_{i} du_{i,m_{i}−1} = 0 = du_{10}) for i ≥ 2

λ_{i,m_{i}−1}y_{i1} + λ_{i,m_{i}−2}y_{10} = 0 (3.41)

λ_{i,m_{i}−1}y_{i2} + λ_{i,m_{i}−2}y_{i1} + λ_{i,m_{i}−3}y_{i0} = 0
⋮ ⋮
λ_{i,m_{i}−1}y_{i,m_{i}−1} + λ_{i,m_{i}−2}y_{i,m_{i}−2} + … + λ_{i,0}y_{i0} = 0

For i = 1 one gets the same list but with the last line (coefficient of du_{i0}) omitted. Finally, using ν_{10} = 1 and du_{1,m_{1}−1} = − ∑_{i≥2} du_{i,m_{i}−1} one gets

(3.42)  λ_{1,m_{1}−1} = y_{i0}λ_{i,m_{i}−1}; 2 ≤ i ≤ r

Since λ_{i,m_{i}−1} ≠ 0, equations (3.41) and (3.42) admit a unique solution for the y_{ij}. Since we know Ξ vanishes at at least one point of B, this point must lie in B. □

Finally, we must calculate the Gauß-Manin connection on

H_{DR}^{m−2}(B, (O_{B}, d + Ξ))

Define Ψ to be an absolute invariant form lifting Ξ. By assumption our connection on C −D comes from an absolute integrable connection which, by proposition 2.17, comes from an absolute integrable connection on G. Restricting this connection to B gives our Ψ.

Lemma 3.11. With notation as above, there exists F ∈ O_{B}, η ∈ Ω_{K}^{1}, and a_{i} ∈ k, i ≥ 2, such that

(3.43)  Ψ = ∑_{i=2}^{r} a_{i} \frac{du_{i0}}{u_{i0}} + dF + η.

If moreover ∇ is integrable, then η is closed.

Proof. Since Ξ is (relatively) closed on B, one can write

(3.44)  Ξ = ∑_{i≥2} a_{i} \frac{du_{i0}}{u_{i0}} + d_{/K}F; a_{i} ∈ K.

Lifting to an absolute form forces

(3.45)  Ψ = ∑_{i≥2} a_{i} \frac{du_{i0}}{u_{i0}} + dF + ∑_{j} f_{j}η_{j}; f_{j} ∈ O_{B}, η_{j} ∈ Ω_{K}^{1}.

Here the η_{j} are linearly independent in Ω_{K}^{1}. Using dΨ = 0 modulo Ω_{K}^{2} ⊗ O_{B} and taking residues along u_{i0} = 0 yields a_{i} ∈ k ⊂ K. Then computing dΨ mod O_{B} ⊗ Ω_{K}^{2} yields

(3.46)  0 = ∑_{j} d_{/K}f_{j} ⊗ η_{j} ∈ Ω_{B/K}^{1} ⊗ Ω_{K}^{1}.

It follows that f_{j} ∈ K, so η := ∑ f_{j}η_{j} ∈ Ω_{K}^{1}. Taking d again shows η is closed if ∇ is integrable. □

We now compute the Gauß-Manin connection. We have the diagram of global sections

(3.47)

Ω_{B/k}^{m−2} ⟶_{onto} Ω_{B/K}^{m−2}
↓^{d+Ψ}
Ω_{B/K}^{m−2} ∧ Ω_{K}^{1} ⟶_{≅} \frac{Ω_{B/k}^{m−1}}{Ω_{B/k}^{m−3}·Ω_{K}^{2}}

The connection is determined by its value on θ (3.29). To calculate, one lifts θ to θ̃ ∈ Ω_{B/k}^{m−2} and then applies d + Ψ. But for θ̃ one can choose the form with the same expression (3.29). This form is closed, so

(3.48)  ∇_{GM}(θ) = Ψ ∧ θ = (d_{K}(F) + η) ∧ θ

Here we write F = ∑_{I} a_{I}u^{I}, a_{i} ∈ K and d_{K}(F) := ∑ da_{I}u^{I}.

Let b ∈ B be the point corresponding to the trivialization of Ω_{C/K}^{1}(D) given by the polar part of the original relative connection. It really lies in B since we have assumed that degL = 0.

Lemma 3.12. With notation as above, the Gauß-Manin connection on the rank 1 K-vector space

H_{DR}^{m−2}(B, (O, d + Ξ))

described by (3.48) is isomorphic to the connection on K given by

1 ↦ Ψ|_{b} + \frac{1}{2}d log(κ)

for a suitable κ ∈ K^{×}.

Proof. We have seen (lemma 3.8) that this point b is determined by the condition that Ξ(b) = 0 ∈ 𝔪_{B,b}/𝔪_{B,b}^{2}. Changing Ψ by a closed form pulled back from K changes the Gauß-Manin connection and the connection at b in the same way, so we can assume η = 0, i.e. Ψ = ∑ a_{i} \frac{du_{i0}}{u_{i0}} + dF. Write

(3.49)  g_{ij} = \begin{cases}\frac{∂F}{∂u_{ij}} & j > 0 \\ \frac{∂F}{∂u_{i0}} + a_{i}/u_{i0} & j = 0. \end{cases}

Write F = ∑_{I} a_{I}u^{I}. Then

(3.50)  Ψ = ∑ g_{ij}du_{ij} + ∑_{I} u^{I}da_{I}; Ψ ∧ θ = ∑_{I} u^{I}da_{I} ∧ θ

We have

(3.51) g_{ij}(b) = 0, j < m_{i} − 1; g_{i,m_{i}−1}(b) = g_{k,m_{k}−1}(b); all i, k.

Since ∑_{i} du_{i,m_{i}−1}|_{B} = 0, we see from (3.51) that

(3.52)  Ψ|_{{b}} = ∑_{I} b^{I}da_{I}

Thus, it will suffice to relate u^{I}θ and b^{I}θ in H_{DR}^{m−2}. Note that each monomial u^{I} involves u_{ij} for only one value of i, and the weight of u^{I} is ≤ m_{i} − 1 (see the discussion after (3.33)).

Suppose first the weight of u^{I} is strictly less than m_{i} − 1. Let j be maximal such that u_{ij} appears in u^{I}. From (3.35) it follows that

(3.53) g_{i,m_{i}−1−j} = α_{i,m_{i}−1−j} \frac{u_{ij}}{u_{i0}^{2}} + terms involving only u_{ik}; k < j.

Here α_{i,m_{i}−1−j} ≠ 0. Define u^{L} = u^{I}u_{i0}^{2}u_{ij}^{−1}. Note the weight of u^{L} is < m_{i} − 1 − j, so in H_{DR}^{m−2} we have (compare (3.37))

(3.54) u^{I}θ = (u^{I} − α_{i,m_{i}−1−j}^{−1}(\frac{∂}{∂u_{i,m_{i}−1−j}} + g_{i,m_{i}−1−j})u^{L})θ =

(u^{I} − α_{i,m_{i}−1−j}^{−1}g_{i,m_{i}−1−j}u^{L})θ = σ_{I}θ

where σ_{I} is a sum of terms of weights < |I| and terms of weight |I| only involving u_{i0}, …, u_{i,j−1}. Note that b^{I} = σ_{I}(b) because g_{i,m_{i}−1−j}(b) = 0. In this way we reduce to the case u^{I} = u_{i0}^{p}. Our assumption on the weight implies m_{i} ≥ 2, so

(3.55)  (\frac{∂}{∂u_{i,m_{i}−1}} + g_{i,m_{i}−1})u_{i0}^{p} = g_{i,m_{i}−1}u_{i0}^{p}.

Together with (3.35) and g_{ij}(b) = 0, this enables us to reduce to p = 0.

Suppose now the weight of I is m_{i} − 1. If m_{i} ≥ 2 we can use the above argument, except in the case u^{I} = u_{ij}u_{i,m_{i}−1−j}u_{i0}^{−2}. Here there are two subcases. If j ≠ m_{i} − 1 − j, the α_{i,m_{i}−1−j} in (3.54) is a_{I} in the expansion F = ∑_{I} a_{I}u^{I}, so

(3.56)  u^{I}da_{I} ∧ θ = (b^{I}da_{I} + \frac{da_{I}}{a_{I}}) ∧ θ.

This completes the proof in this case because the connections b^{I}da_{i} and b^{I}da_{I} + d log(a_{I}) are isomorphic. If, on the other hand, m_{i} is odd and j = \frac{m_{i}−1}{2}, the monomial u^{I} = u_{ij}^{2}u_{i0}^{−2} and dF contains the term 2a_{I}u_{ij}du_{ij}. Thus, from (3.53) we conclude α_{i, \frac{m_{i}−1}{2}} = 2a_{I}. The lefthand identity in (3.54) yields in this case

(3.57)  u^{I}da_{I} ∧ θ = (b^{I}da_{I} + \frac{1}{2}d log a_{I}) ∧ θ.

In the statement of the lemma, we take κ to be the product of the corresponding a_{I}.

Suppose finally m_{i} = 1. In this case \frac{∂F}{∂u_{i0}} = 0, so the corresponding a_{I} = 0 and by (3.52) this term contributes nothing to Ψ|_{{b}}. Similarly, by (3.50) there is no contribution to Ψ ∧ θ.

□

We give two interpretations of the 2-torsion term \frac{1}{2}d log(κ) occurring in the previous lemma.

Definition 3.13. Let σ be a closed 1-form relative to K on

Spec (K[[t_{1}, …, t_{N}]]).

Assume σ(0) = 0 ∈ 𝔪/𝔪^{2}. Write σ = dh with h(0) = 0, so h = h_{2} + h_{3} + … with h_{i} homogeneous of degree i. If h_{2} is nondegenerate, we may define disc(σ) = discriminant(h_{2}) ∈ K^{×}/K^{×2}. This is welldefined independent of the choice of parameters.

Theorem 3.14. The Gauß-Manin connection on H_{DR/K}^{m−2}(B, (O, d+Ξ)) is isomorphic to

(d + Ψ)|_{\{b\}} + \frac{1}{2}d log(disc(Ξ|Ô_{B,b})).

(In particular, the quadratic term in h = ∫ Ξ|Ô_{B,b}is non-degenerate.)

Proof. First we collect some facts about Ξ = ∑_{i,j} λ_{ij}τ_{ij} = ∑ g_{ij}du_{ij}. We have u_{1,0} = 1 and ν_{i,0} = u_{i,0}^{−1}. It follows from (3.33) of the paper that u_{1,m_{1}−1} does not appear in the expression for Ξ and du_{1,m_{1}−1} only appears with constant coefficient. Restricting to B : u_{1,m_{1}−1} = − ∑_{i≥2} u_{i,m_{i}−1} thus has the effect of surpressing the term in du_{1,m_{1}−1} and changing the coefficients g_{i,m_{i}−1} by a constant for i ≥ 2. Expressed in this way, it follows that the coefficient of du_{ij} in Ξ|_{B} involves only monomials in u_{ip} for the same i. Giving u_{ij} and du_{ij} both weight j, the terms in g_{ij}du_{ij} all have weight ≤ m_{i} − 1. It follows from formulas (3.34)-(3.36) that, writing U_{ij} = u_{ij} − b_{ij} so U_{ij}(b) = 0, we may write

Ξ|_{B} = ∑ G_{ij}(U)dU_{ij}.

Here G_{ij}(0) = 0. giving U_{ij} and dU_{ij} weights j, all terms with first index i have weights ≤ m_{i} − 1. All terms of the form

U_{ij}dU_{i,m_{i}−1−j}, 0 ≤ j ≤ m_{i} − 1, i ≥ 2 (resp. i = 1, 1 ≤ j ≤ m_{1} − 2)

occur with nonzero coefficient. Notice that replacing u_{ij} with U_{ij} + b_{ij} introduces monomials of lower degree, but these have weight < m_{i} − 1.

It follows that disc(Ξ|Ô_{B,b}) is the determinant of a matrix

M = (\begin{matrix}M_{1} & 0 & … & … & 0 \\ 0 & M_{2} & 0 & … & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & … & … & M_{r} \end{matrix})

where M_{i} is symmetric, m_{i} × m_{i} (resp. (m_{1} − 2) × (m_{1} − 2)), and has the shape

(\begin{matrix}… & … & … & … & • \\ … & … & … & • & 0 \\ … & … & • & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ • & 0 & … & … & 0 \end{matrix})

with the entries • non-zero.

Mod squares, det(M_{i}) is 1 if m_{i} is even, and is given by

\frac{1}{2} · coefficient of(U_{i, \frac{m_{i}−1}{2}} dU_{i, \frac{m_{i}−1}{2}} )

if m_{i} is odd. Writing

(3.58)  Ψ|_{B} = ∑ a_{i} \frac{du_{i0}}{u_{i0}} + dF

as just above (3.49) with F = ∑ a_{I}u^{I} we find

\frac{1}{2}d log disc(Ξ|Ô_{B,b}) = \frac{1}{2} ∑ \frac{da_{I}}{a_{I}} = \frac{1}{2}d log(κ).

(The sum on the right is over all I such that u^{I} = (u_{i, \frac{m_{i}−1}{2}} )^{2}.) □

Another interpretation of the 2-torsion is the following. As in (2.25) for s_{i} a local parameter at c_{i} ∈ D and t_{i} another copy of s_{i} (so s_{i} − t_{i} is a local defining equation for the diagonal in (C × C), the pullback of u_{ip} to K((s_{i})) is the coefficient s_{i}^{−(p+1)} of t_{i}^{p} in (s_{i} − t_{i})^{−1}. It follows from (3.30) that

(3.59) τ_{ij} pulls back to \frac{−ds_{i}}{s_{i}^{j+1}} .

Write the polar part of the connection at s_{i} = 0 in the form (g_{0} + g_{1}s_{i} + …)\frac{ds_{i}}{s_{i}^{m_{i}}}. Since Ξ = ∑_{i,j} λ_{ij}τ_{ij} pulls back to this connection form, we get g_{0} = −λ_{i,m_{i}−1}. On the other hand, again from (3.30) the coefficient of u_{i,0}^{−2}u_{i, \frac{m_{i}−1}{2}} du_{i, \frac{m_{i}−1}{2}} in Ξ is −λ_{i,m_{i}−1} if m_{i} is odd. This coefficient is the contribution to disc(Ξ|_{O_{B,b}}) from the point c_{i} ∈ D, so we conclude

Theorem 3.15. Write the relative connection at c_{i} ∈ D in the form (g_{i,0}+g_{i,1}s_{i}+…)\frac{ds_{i}}{s_{i}^{m_{i}}}. Then the Gauß-Manin connection on H_{DR/K}^{m−2}(B, (O, d+ Ξ)) is isomorphic to

(d + Ψ)|_{\{b\}} + ∑_{i} \frac{m_{i}}{2} d log(g_{i,0}(0)).

Definition 3.16. With notation as above, write

τ(L) = ∑_{i} \frac{m_{i}}{2} d log(g_{i,0}(0)).

To summarize, we have proven

Theorem 3.17. Let C/K be a complete smooth curve of genus g over a field K ⊃ k. Let ∇_{/K} : L → L ⊗ Ω_{C/K}^{1}(D) be a connection, such that

(L → L ⊗ Ω_{C/K}^{1}(D)) → (j_{∗}L|(C − D) → j_{∗}L ⊗ Ω_{C/K}^{1}|(C − D))

is a quasiisomorphism. This implies that the divisor D is minimal such that ∇|_{C−D} extends with values in Ω_{C/K}^{1}(D) (see section 4, (4.2)). We also assume that L has degree 0, and that the connection on L|_{C−D} lifts to an integrable, absolute (i.e. /k) connection ˜∇. Then

(3.60)  ∇|_{D} : L|_{D} → L ⊗ Ω_{C/K}^{1}(D)|_{D}

is an O_{D}-linear isomorphism and determines a trivialization of

Ω_{C/K}^{1}(D)|_{D}

Write J_{D} for the generalized jacobian and J_{D,N} for the torseur of divisors of degree N := 2g−2+deg D. The above trivialization corresponds to a K-point b ∈ J_{D,N}. Write π_{N} : (C − D)^{N} → J_{D,N} for the natural map, and let (L_{N}, ˜∇_{N}) be the evident bundle and absolute connection on (C −D)^{N}. Then there exists a unique invariant, absolute connection (L, Φ) on J_{D,N} such that π_{N}^{∗}(L, Φ) = (L_{N}, ˜∇_{N}). Moreover, we have

(3.61) ( det(H_{DR}^{∗}(C − D, (L, ∇)), ∇_{GM})^{−1} ≅ (L, Φ)|_{{b}} + τ(L)

where τ(L) is as in definition 3.16.

Remark 3.18. 2-torsion also occurs in the determinant of de Rham cohomology for the trivial connection [2]. By virtue of the following lemma this can only happen when the variety has even dimension.

Lemma 3.19. Let X/K be a smooth projective variety of odd dimension n = 2m + 1 over a function field in characteristic 0. Then the Gauß-Manin determinant

det(H_{DR}(X/K), d)

is trivial in Ω_{K}^{1}/d log K^{∗}.

Proof. The strong Lefschetz theorem identifies the determinant connections on H_{DR}^{p} and H_{DR}^{2n−p} so we need only consider the connection on det H_{DR}^{n}. As well known, the Poincaré duality morphism

φ : H_{DR}^{n}(X/K) ⊗ H_{DR}^{n}(X/K) → H_{DR}^{2n}(X/K) = K

is compatible with the Gauß-Manin connection, which is trivial on H_{DR}^{2n}(X/K) = K. On the other hand, it is alternating, thus its determinant

det(φ) : det(H_{DR}^{n}(X/K)) ⊗ det(H_{DR}^{n}(X/K)) → K

fulfills

det(φ)(e ⊗ e) = p^{2} · 1

where p ∈ K^{∗} is the Pfaffian of the determinant of φ, written in the basis e. Thus if ∇(e) = α ⊗ e, one has

det(φ)(∇(e ⊗ e)) = 2αp^{2} · 1 = 2pd(p) · 1.

Thus α = d log p and the determinant of the Gauß-Manin connection is trivial. □

4. Product and Trace

In this section, we introduce a product which is reminiscent of Deligne’s product explained in [5].

We keep the notations of sections 2 and 3 for C/K, (L, ∇), j : U = C − D → C, D = ∑ m_{i}c_{i}, and D′ = D − D. Further,

∇ : L → L ⊗ Ω_{U}^{1} (4.1)

is an absolute connection with vertical curvature ∇^{2}(L) ⊂ L ⊗ Ω_{K}^{2} ⊗ K(X). Let ∇_{/K} : L → L ⊗ Ω_{C/K}^{1}(D) be an extension of (L, ∇_{/K}) such that

{L → L ⊗ Ω_{C/K}^{1}(D)} → {j_{∗}L → j_{∗}L ⊗ Ω_{U/K}^{1}} (4.2)

is a quasiisomorphism. We assume ∇_{/K} has a pole at every c_{i} ∈ D. Note this implies that ∇_{/K} does not factor through Ω_{C/K}^{1}(D − c_{i}) for any i. Indeed, by assumption, the complex

(4.3)  j_{∗}L/L → (j_{∗}L/L) ⊗ Ω_{C/K}^{1}(D)

is acyclic. Take e a local basis of L at c_{i} and z a local parameter, and suppose the connection can be written locally as ∇_{/K}e = a(z)dz/z^{m−1}e with m = m_{i}. Then ∇_{/K}(z^{−1}e) = (a(z)dz/z^{m} − z^{−2}dz)e. The assumption that ∇_{/K} does have a pole at c_{i} implies that m ≥ 2, so z^{−1}e would represent a nontrivial element in H^{0} of the complex (4.3), a contradiction.

By lemma 3.1 we know that the verticality condition implies that the absolute connection extends as

∇ : L → L ⊗ Ω_{C}^{1}⟨D⟩(D′). (4.4)

From now on, we fix such a (L, ∇).

As we have seen, the map L|_{D} → L ⊗ Ω_{C/K}^{1}(D)|_{D} is function linear. Since the connection does not factor through lower order poles, this gives a trivialization (denoted triv(∇)) of Ω_{C/K}^{1}(D)|_{D}. We have

(c_{1}(Ω_{C/K}^{1}(D)), triv (∇)) ∈ ℍ^{1}(C, O_{C}^{∗} → O_{D}^{∗}) (4.5)

(L, ∇) ∈ ℍ^{1}(C, O_{C}^{∗} → Ω_{C}^{1}⟨D⟩(D′)).

The aim of this section is to define a product

∪ : ℍ^{1}(C, O_{C}^{∗} → O_{D}^{∗}) × ℍ^{1}(C, O_{C}^{∗} → Ω_{C}^{1}⟨D⟩(D′))

→ ℍ^{2}(C, K_{2} → Ω_{C}^{2}) (4.6)

Here K_{2} is the Milnor sheaf associated to K_{2}, and the map K_{2} → Ω_{C}^{2} is the d log map \{a, b\} ↦ \frac{da}{a} ∧ \frac{db}{b}. For a more detailed study of characteristic classes for connections defined in the hypercohomology of such complexes, the reader is referred to [7]. In addition, we will define a trace

Tr : ℍ^{2}(C, K_{2} → Ω_{C}^{2}) → Ω_{K}^{1}/d log K^{∗} (4.7)

We write

(4.8)  A · B := Tr(A ∪ B)

so for example

(4.9) (c_{1}(Ω_{C/K}^{1}(D)), triv (∇)) · (L, ∇) :=

Tr ((c_{1}(Ω_{C/K}^{1}(D)), triv (∇)) ∪ (L, ∇))

Let O_{C,D}^{∗} = Ker (O_{C}^{∗} → O_{D}^{∗}). Then

Lemma 4.1. d log O_{C,D}^{∗} ∧ Ω_{C}^{1}⟨D⟩(D′) ⊂ Ω_{C}^{2}

Proof. Since O_{C,D}^{∗} ⊂ 1 + I_{D}, where I_{D} is the ideal sheaf of D,

d log O_{C,D}^{∗} ⊂ O_{C}dI_{D} ⊂ Ω_{C}^{1}(∗D).

Also one has dI_{D} ⊂ I_{D} ⊗_{O_{C}} Ω_{C}^{1}⟨D⟩. Thus

d log O_{C,D}^{∗} ∧ Ω_{C}^{1}⟨D⟩(D′) ⊂ I_{D} ⊗_{O_{C}} Ω_{C}^{2}⟨D⟩ ⊂ Ω_{C}^{2}.
□

We define ∪ by

O_{C,D}^{∗} ∪ O_{C}^{∗} → K_{2} (4.10)

(λ, c) ↦ {λ, c}
O_{C,D}^{∗} ∪ Ω_{C}^{1}⟨D⟩(D′) → Ω_{C}^{2}
(λ, ω) ↦ d log λ ∧ ω.

Concretely, we can write the product in terms of Cech cocyles. Here C^{i} refers to Cech cochains, δ is the Cech coboundary, and d is a boundary in the complex:

(λ_{ij}, μ_{i}) ∈ (C^{1}(O_{C}^{∗}) × C^{0}(O_{D}^{∗}))_{d−δ}
(c_{ij}, ω_{i}) ∈ (C^{1}(O_{C}^{∗}) × C^{0}(Ω_{C}^{1}⟨D⟩(D′))_{d−δ}

one has

(λ, μ) ∪ (c, ω) = ({λ_{ij}, c_{jk}}, d log λ_{ij} ∧ ω_{j}, −d log μ̃_{i} ∧ ω_{i}) (4.11)

∈ (C^{2}(K_{2}) × C^{1}(Ω_{C}^{2}⟨D⟩(D′)) × C^{0}(Ω_{C}^{2}⟨D⟩(D′)/Ω_{C}^{2}))_{d+δ}

where μ̃_{i} ∈ C^{0}(O_{C}^{∗}) is a local lifting of μ_{i}. Note we have replaced the complex K_{2} → Ω_{C}^{2} with the quasiisomorphic complex

K_{2} → Ω_{C}^{2}⟨D⟩(D′) → Ω_{C}^{2}⟨D⟩(D′)/Ω_{C}^{2}.

Proposition 4.2. The product ∪ extends to

∪ : ℍ^{1}(C, O_{C}^{∗} → O_{D}^{∗}) × ℍ^{1}(C, j_{∗}O_{U}^{∗} → Ω_{C}^{1}⟨D⟩(D′))
→ ℍ^{2}(C, K_{2} → Ω_{C}^{2}).

Proof. The map

ℍ^{1}(C, O_{C}^{∗} → Ω_{C}^{1}⟨D⟩(D′)) → ℍ^{1}(C, j_{∗}O_{U}^{∗} → Ω_{C}^{1}⟨D⟩(D′))

is surjective, and its kernel is the ℤ-module generated by (O(D_{i}), d_{i}) where d_{i} is the connection with logarithmic poles along D_{i} with residue -1. Let z_{1} be a local coordinate around c_{1}. Let U_{i} be a Cech covering of C, with c_{1} ∈ U_{1} ⊂ V_{1}, and c_{1} ∉ U_{i}, i ≠ 1. Assume c_{1} is the only zero or pole of z_{1} on U_{1}. Let

(λ, μ) ∈ ℍ^{1}(C, O_{C}^{∗} → O_{D}^{∗})

be a Cech representative of a class in ℍ^{1}(C, O_{C}^{∗} → O_{D}^{∗}). Then (c_{ij}, ω_{i}) with c_{1j} = z_{1}^{−1}, c_{ij} = 1 for i ≠ 1, ω_{1} = −d log z_{1}, ω_{i} = 0 for i ≠ 1 is a Cech representative of (O(D_{1}), d_{1}). Thus considering Z ∈ C^{0}(O_{C}[z_{1}^{−1}]^{∗}) with Z_{1} = z_{1} and Z_{i} = 1 for i ≠ 1, the cocyle of (4.7) is just the coboundary

(d − δ)({λ_{ij}, Z_{j}}, d log μ̃_{i} ∧ d log Z_{i}) ∈ (d − δ)(C^{1}(K_{2}) × C^{0}(Ω_{C}^{2}⟨D⟩(D′)).

(Note Z_{j} is invertible on U_{ij} for i ≠ j so the K_{2}-cochain is defined.) □

Now we define the trace. We have (with standard K-theoretic notation, [1])

H^{2}(C, K_{2}) = 0 (4.12)

Nm : H^{1}(C, K_{2}) = {⊕_{x∈C^{(1)}}κ(x)^{∗}}/Tame(K_{2}(K(C))) → K^{∗}

∑_{x} φ_{x} ↦ Π_{x}Nm (φ_{x})

and of course H^{1}(C, Ω_{C}^{2}) = Ω_{K}^{1} ⊗ H^{1}(C, Ω_{C/K}^{1}) = Ω_{K}^{1}. This defines

Tr : ℍ^{2}(C, K_{2} → Ω_{C}^{2}) = H^{1}(C, Ω_{C}^{2})/H^{1}(C, K_{2}) → Ω_{K}^{1}/d log K^{∗}. (4.13)

Lemma 4.3. The trace

Tr : ℍ^{2}(C, K_{2} → Ω_{C}^{2}) = ℍ^{2}(C, K_{2} → Ω_{C}^{2}⟨D⟩(D′) → Ω_{C}^{2}⟨D⟩(D′)/Ω_{C}^{2})

→ Ω_{K}^{1}/d log K^{∗}

factors through

ℍ^{2}(C, K_{2} → Ω_{K}^{1} ⊗ Ω_{C/K}^{1}(D) → Ω_{K}^{1} ⊗ (Ω_{C/K}^{1}(D)/Ω_{C/K}^{1})) (4.14)

≅ Ω_{K}^{1}/d log K^{∗}
≅ Ω_{K}^{1} ⊗_{K} H^{0}(D, ω_{D/K})/ℍ^{1}(C, K_{2} → Ω_{K}^{1} ⊗ Ω_{C/K}^{1}(D))

where ω_{D/K} is the relative dualizing sheaf of the scheme D, containing K ≅ ω_{D/K}.

Proof. Note that

Ω_{K}^{1} ⊗_{K} Ω_{C/K}^{1}(D) ≅ Ω_{C}^{2}⟨D⟩(D′)/(Ω_{K}^{2} ⊗ O_{C}(D′))

so the complex in (4.14) is indeed a quotient. From the diagram

K_{2}  K_{2}
↓  ↓
0 ⟶ Ω_{K}^{1} ⊗ Ω_{C/K}^{1} ⟶ Ω_{K}^{1} ⊗ Ω_{C/K}^{1}(D) ⟶ Ω_{K}^{1} ⊗ (Ω_{C/K}^{1}(D)/Ω_{C/K}^{1}) ⟶ 0

one deduces that the left hand side of (4.14) is isomorphic to

ℍ^{2}(C, K_{2} → Ω_{K}^{1} ⊗ Ω_{C/K}^{1}) ≅ coker(H^{1}(C, K_{2}) → Ω_{K}^{1} ⊗ H^{1}(C, Ω_{C/K}^{1})).

The right hand side here is identified under the norm with Ω_{K}^{1}/d log K^{∗}, which proves the second equality. The third one comes from the map

Ω_{K}^{1} ⊗ ω_{D/K}[−2] → {K_{2} → Ω_{K}^{1} ⊗ Ω_{C/K}^{1}(D) → Ω_{K}^{1} ⊗ ω_{D/K}}

and the vanishing of ℍ^{2}(K_{2}Ω_{K}^{1} ⊗ Ω_{C}^{1}(D)). Note that this cumbersome way of writing this cohomology allows to write local contribution of a class in this cohomology group. □

The first main result of this section is the following

Theorem 4.4. Let (L, ∇) and (L′, ∇′) be two extensions of the vertical connection (L, ∇) on U as above satisfying the quasiisomorphism condition (4.2). Then, with notation as in (4.10),

((c_{1}(Ω_{C/K}^{1}), triv ∇) · (L, ∇)) = ((c_{1}(Ω_{C/K}^{1}), triv ∇′) · (L′, ∇′)).

Proof. The quasiisomorphism condition is local about each point of D, so we may assume our line bundles are L(νc) ⊂ L for some ν < 0 and c ∈ D.

Choose local coordinates z_{i} near c_{i} and a Cech covering U_{i} of C such that c_{i} ∈ U_{i}, z_{i} ∈ O^{∗}(U_{i} − c_{i}), c_{i} ∉ U_{j} for i ≠ j. Let us denote by

(c_{1}(Ω_{C/K}^{1}), z_{i}) ∈ ℍ^{1}(C, O_{C}^{∗} → O_{D}^{∗})

the class defined by the local trivialization

\frac{dz_{i}}{z_{i}^{m_{i}}} : O_{m_{i}c_{i}} → Ω_{C/K}^{1}(D) ⊗ O_{m_{i}c_{i}}.

Let (L, ∇) = (c_{ij}, ω_{i}). Then ω_{i} = a_{i} \frac{dz_{i}}{z_{i}^{m_{i}}} + _{\frac{b_{i}}{z_{i}^{m_{i}−1}}} , with a_{i} ∈ O_{C} such

that a_{i}|_{m_{i}c_{i}} ∈ O_{m_{i}c_{i}}^{∗} and b_{i} ∈ Ω_{K}^{1} ⊗ O_{C}. Suppose c = c_{i}. We drop the index i for convenience. One has

((c_{1}(Ω_{C/K}^{1}), triv ∇) · (L, ∇)) = (4.15)

(c_{1}(Ω_{C/K}^{1}), z) · (L, ∇) + (0, 0, −d log(a) ∧ (a\frac{dz}{z^{m}} + \frac{b}{z^{m−1}} )).

where the last term is a cocycle as in (4.12) or the quotient complex (4.17). For (L(νc), ∇(νc)) one replaces a by a − νz^{m−1}, leaving b and m unchanged.

By proposition 4.2, one has

(c_{1}(Ω_{C/K}^{1}), z) · (L, ∇) = (c_{1}(Ω_{C/K}^{1}), z) · (L(νc), ∇(νc))

Thus

((c_{1}(Ω_{C/K}^{1}), triv ∇) · (L, ∇)) − ((c_{1}(Ω_{C/K}^{1}), triv ∇′) · (L′, ∇′))

= (0, 0, (d(a − νz^{m−1}) − d(a)) ∧ \frac{dz}{z^{m}} + d log(\frac{a − νz^{m−1}}{a}) ∧ \frac{b}{z^{m−1}} )

= (0, 0, \frac{ν(m − 1)d log z ∧ \frac{b}{a}}{(1 − ν \frac{z^{m−1}}{a})} ).

The nontrivial part in the last expression is computed in

H^{0}(Ω_{C}^{2}⟨c⟩/Ω_{C}^{2}) ≅ Ω_{c}^{1}.

Computing using the residue at c we find the above difference is

(0, 0, ν(m − 1)\frac{b(c)}{a(c)})

The verticality condition for the curvature reads

da ∧ \frac{dz}{z^{m}} + \frac{db}{z^{m−1}} − (m − 1)\frac{dz}{z^{m}} ∧ b = 0 ∈ Ω_{K}^{1} ⊗ Ω_{C/K}^{1}(D).

In particular, (da − (m − 1)b)_{|c} = 0. The difference of the two products is therefore (0, 0, d log a^{ν}), which vanishes in Ω_{K}^{1}/d log K^{∗}. □

Remark 4.5. A version of the formula (4.15) in higher rank plays a central role in section 5.

Suppose now

m_{i} = 1 for all i. (4.16)

In this case, the class (c_{1}(Ω_{C/K}^{1}(D), z_{i}) ∈ ℍ^{1}(C, O_{C}^{∗} → O_{D}^{∗}) as defined in the proof of theorem 4.4 does not in fact depend on the choice of the local coordinate z_{i}. Indeed, the trivialization

O_{D} → (Ω_{C/K}^{1}(D)/Ω_{C/K}^{1} = O_{D}), 1 ↦ \frac{dz_{i}}{z_{i}}

is just the canonical identification given by the residue along c_{i}. In other words, the class (c_{1}(Ω_{C/K}^{1}(D), z_{i}) is what is denoted by (c_{1}(Ω_{C/K}^{1}(D), res_{D}) in [2], and appears on the right hand side of the Riemann-Roch formula. The second main result of this section is

Theorem 4.6. Let (L, ∇) be as above, with m_{i} = 1 for all i. Then

det (H_{DR}^{∗}(U, L)), Gauß - Manin connection) =

− c_{1}(Ω_{C/K}^{1}(D), triv∇) · (L, ∇).

Proof. Given the main result of [2], and lemma 3.19, the theorem is of course equivalent to

(4.17) c_{1}(Ω_{C/K}^{1}(D), res_{D}) · (L, ∇) = c_{1}(Ω_{C/K}^{1}(D), triv ∇) · (L, ∇).

Keeping the same notations as in the proof of theorem 4.4, one has

c_{1}(Ω_{C/K}^{1}(D), res_{D}) − c_{1}(Ω_{C/K}^{1}(D), triv ∇) = (0, a_{i})

and thus

(c_{1}(Ω_{C/K}^{1}(D), res_{D}) − c_{1}(Ω_{C/K}^{1}(D), triv ∇)) · (L, ∇) =
(0, 0, −d(a_{i}) ∧ d log z_{i} − d log a_{i} ∧ b_{i}) =
(0, 0, −d(a_{i}) ∧ d log z_{i}).

This lies in Ω_{K}^{1}⊗ω_{D/K} = Ω_{K}^{1} and by lemma 4.3, its trace factors through Ω_{K}^{1} ⊗ H^{1}(C, Ω_{C/K}^{1}). But the image of γ = ∑_{i} a_{i}d log z_{i} ∈ ω_{D/K} in H^{1}(C, Ω_{C/K}^{1}) is the relative Atiyah class at_{/K}(L) ([6], appendix B), thus the image of dγ = ∑_{i} d(a_{i})∧d log z_{i} ∈ Ω_{K}^{1} ⊗ω_{D/K} in Ω_{K}^{1} ⊗H^{1}(C, Ω_{C/K}^{1}) is d(at(L)), where at(L) ∈ H^{1}(C, Ω_{C}^{1}) is the absolute Atiyah class of L. Indeed, d : H^{1}(C, Ω_{C}^{1}) → Ω_{K}^{1} ⊗ H^{1}(C, Ω_{C/K}^{1}) factors through H^{1}(C, Ω_{C/K}^{1}) by Hodge theory. On the other hand, if c_{ij} ∈ C^{1}(O_{C}^{∗}) is a cocyle for L, then d log c_{ij} ∈ C^{1}(Ω_{C}^{1}) is a cocyle for at(L), and consequently, d(at(L)) = 0. □

We want to explain briefly a fundamental compatibility satisfied by the pairing (4.8). Let b = ∑ b_{i} be a 0-cycle on C with support disjoint from D, and let ∇ : L → L⊗Ω_{C}^{1} < D > (D′) be an absolute, integrable connection. We can interpret ∇|_{b_{i}} ∈ Ω_{K(b_{i})}^{1}/d log K(b_{i})^{∗}.

Proposition 4.7. With notation as above, let [b] ∈ ℍ^{1}(C, O_{C}^{∗} → O_{D}^{∗}) be the class of the 0-cycle b. Then

[b] · (L, ∇) = ∑_{i} Tr_{K(b_{i})/K}(∇|_{b_{i}}) ∈ Ω_{K}^{1}/d log K^{∗}.

Let (L_{0}, ∇_{0}) be the invariant connection on J_{𝔻} which pulls back to (L, ∇) via the cycle map i : (C − D) → J_{D} (proposition 2.17). Then

[b] · (L, ∇) = Tr_{i_{0}(b)/}(L_{0}, ∇_{0})|_{i_{0}(b)}.

Proof. One reduces easily to the case b is a single K-point. Let U_{2} be a Zariski-open set containing D and b. Shrink U_{2} if necessary so there exists z ∈ H^{0}(O_{U_{2}}) with z|_{D} = 1 and (z) = b. Let U_{1} = C − {b} − D so C = U_{1} ∪ U_{2}. Shrinking the U_{i} if necessary, we can assume L|_{U_{i}} ≅ O_{U_{i}}, so (L, ∇) is represented by some cocycle (μ_{12}, ω_{1}, ω_{2}). Then

∇|_{b} = ω_{2}|_{b} ∈ Ω_{K}^{1}/d log K^{∗}

On the other hand, by the definition (4.12), [b] · (L, ∇) is represented by the image of the cocycle

d log(z) ∧ ω_{2}|_{U_{12}} ∈ H^{1}(C, Ω_{C}^{2}) → Ω_{K}^{1} ⊗_{K} H^{1}(C, Ω_{C/K}^{1}) ≅ Ω_{K}^{1}.

Write ω_{2} = ω_{2}(b) + zη_{2} with η_{2} regular on U_{2}. Since d log(z) ∧ zη_{2} extends to U_{2}, it is homologous to zero, so

[b] · (L, ∇) = ω_{2}(b)[b] ∈ Ω_{K}^{1} ⊗ ℍ^{1}(C, Ω_{C/K}^{1}) ↦ ω_{2}(b) ∈ Ω_{K}^{1}.

Now, since one obviously has

Tr(∇|_{b_{i}}) = Tr(∇_{0})|_{i_{0}(b_{i})}

and the translation ∇_{0} is invariant, the second equality is a direct interpretation of the first one. □

Now we can formulate and prove a variant of theorem 3.17.

Theorem 4.8. Let (C/K, U/K, (L, ∇), D) be as in (4.1), (4.2), (4.3). Then

det (H_{DR}^{∗}(U, L)) = −(c_{1}(Ω_{C/K}^{1}(D), triv (∇)) · (L, ∇) ∈ Ω_{K}^{1}/d log K^{∗} (4.18)

modulo torsion (see remark 3.18), where ∇_{/K} : L → L ⊗ Ω_{C/K}^{1}(D) is any extension of (L, ∇_{/K}) having poles along all points of D such that

{L → L ⊗ Ω_{C/K}^{1}(D)} → {j_{∗}L → j_{∗}(L ⊗ Ω_{U/K}^{1})}

is a quasiisomorphism.

Proof. If all m_{i} = 1 this is just theorem 4.6. So we assume that m_{1} ≥ 2 in the sequel. Then as in the proof of theorem 4.4, replacing L by L(−c_{1}) changes a_{1} to (a_{1} + z_{1}^{m_{1}−1}) and keeps the rest unchanged. Thus the quotient complex

L/L(−c_{1}) → L/L(−c_{1}) ⊗ Ω_{C/K}^{1}(D)

is O_{c_{1}}-linear and the map is the multiplication by a_{1} ∈ O_{c_{1}}^{∗}. In particular, L(νc_{1}) fulfills (4.2) for all ν ∈ ℤ and taking ν = −deg L, we may assume by theorem 4.4 that deg L = 0. If H_{DR}^{0}(U, L) ≠ 0, then there is a meromorphic section φ of L verifying the flatness condition

dφ + ωφ = 0.

This implies in particular that ω has at most logarithmic poles along D, which contradicts the condition m_{1} ≥ 2. On the other hand, H_{DR}^{2}(U, L) = 0 for dimension reasons, thus we can apply theorem 3.17 together with proposition 4.7 to obtain the result, after we have replaced (L, ∇) by (L, ∇)⊗f^{∗}((L, ∇)|_{c_{0}}^{−1}) to trivialize the connection at c_{0} and applied the projection formula to this tensor connection. □

We finish this section with an example. Let U = 𝔸_{K}^{1} be the affine line over SpecK, with parameter t, and ∇ be a connection on the trivial bundle. Then up to a twist by a form of the base, ∇ has equation A = df, where f = ∑_{i=1}^{m−1} a_{i}t^{i}, a_{i} ∈ K, a_{m−1} ≠ 0. Write df = d_{K}f +f′dt with d_{k}f = ∑_{i=1}^{m−2} da_{i}t^{i} and f′ = ∑_{i=1}^{m−2} ia_{i}t^{i−1}. Let b_{i}, i = 1, …, (m−2) be the zeroes of f′ (defined over K after some finite field extension), and let N_{ℓ}(\underline{b}) = ∑_{i=1}^{m−2} b_{i}^{ℓ} be the Newton classes of the zeroes of f′, which of course are expressable in the a_{i} already on SpecK. Then the main theorem says

det(GM)^{−1} = ∑_{i=1}^{m−2} da_{i}N_{i}(\underline{b}).

5. A Formula in Higher Rank

In this section, we want to define a sort of non-commutative product of a higher rank connection with the Chern class of the dualizing sheaf of C with poles. The notations are as in the whole article: C is a curve defined over a function field K over an algebraically closed field k of characteristic 0, U is an open set such that D = X −U = ∑_{i} c_{i} consists of K−rational points. Let (E, ∇) be a rank r-connection on U with vertical curvature (2.58). Let m_{i} be the multiplicity of the relative connection at the point c_{i}, that is, the minimal multiplicity such ∇ factors

∇_{/K} : E → E ⊗ Ω_{C/K}^{1}(∑_{i} m_{i}c_{i}).

Lemma 3.1 no longer holds true in the higher rank case. We say that the poles of the global connection behave well if

∇ : E → E ⊗ Ω_{C}^{1} < D > (D′) (5.1)

where D = ∑_{i} m_{i}c_{i} and D′ = D − D.

Let s = {s_{i}} be a trivialization of ω_{C/K}(D)|_{D} ≅ ω_{D/K}. That is, s_{i} ∈ ω_{m_{i}c_{i}} and the map 1 ↦ s_{i} is an isomorphism O_{D} ≅ ω_{m_{i}c_{i}}. For example, if z_{i} is a local parameter, one can take s_{i} = \frac{dz_{i}}{z_{i}^{m_{i}}}. We will abuse notation and write s_{i} also for a lifting of the trivialization to a local section of Ω_{C}^{1} < D > (D′). The local matrix of the connection has the shape

A_{i} = g_{i}s_{i} + \frac{η_{i}}{z_{i}^{m_{i}−1}} (5.2)

where g_{i} and η_{i} are r × r matrices with coefficients in O_{C} and Ω_{K}^{1} ⊗ O_{C} respectively. Note that the matrix of functions g_{i} depends only on the lifting of s_{i} to a section of ω_{C/K}(D). (Indeed, the relative connection has matrix gs_{i}.)

We assume

Image(g_{i}) ∈ M(r × r, O_{D})

lies in GL(r, O_{D}) (5.3)

Under this assumption, we define

{c_{1}(ω(D)), ∇} := (5.4)

c_{1}(ω(D), s_{i}) · det(∇) − ∑_{i} res Tr(dg_{i}g_{i}^{−1}A_{i})

∈ Ω_{K}^{1}/d log K^{∗}

Conjecture 5.1. Assuming (5.1) and (5.3), we have

detH_{DR}^{∗}(U, (E, ∇))^{−1} = {c_{1}(ω(D)), ∇} ∈ (Ω_{K}^{1}/d log K^{∗}) ⊗_{ℤ} ℚ.

We discuss the assumption (5.3) (see proposition 5.6) at the end of this section. The assumption (5.1) on the poles behaving well is not very well understood. It reflects a sort of stability in all possible directions for the poles of the global connection.

First, we justify the conjecture by establishing some rather surprising invariance properties for {c_{1}(ω(D)), ∇}.

Lemma 5.2. Fix an index i and write the connection matrix locally in the form

A = g \frac{dz}{z^{m}} + \frac{η}{z^{m−1}}

where g is an invertible matrix of functions and η is a matrix with entries from Ω_{K}^{1} ⊗ O_{C}. Then

(1) res Tr(dgg^{−1}A) = res Tr(dgg^{−1} \frac{η}{z^{m−1}} ).

(2) [η, g]z^{1−m} has no pole at the point z = 0.

Proof. The assumption that the curvature is vertical implies

(5.5) dA = dg \frac{dz}{z^{m}} +d(\frac{η}{z^{m−1}} ) ≡ A^{2} = [η, g]\frac{dz}{z^{2m−1}} mod Ω_{K}^{2} ⊗O_{C}[z^{−1}]

Multiplying through by z^{m} and contracting against \frac{∂}{∂z} we deduce 2.

For 1, we must show res Tr(dg \frac{dz}{z^{m}}) = 0. From 5.5, using Tr[g, η] = 0 we reduce to showing res Trd(\frac{η}{z^{m−1}} ) = 0. Since η has entries Ω_{K}^{1}, one has

res Trd(\frac{η}{z^{m−1}} ) = res Trd_{C/K}(\frac{η}{z^{m−1}} ).

And the residue of an exact form is vanishing. □

Lemma 5.3. {c_{1}(ω(D)), ∇} is independent of the choice of the trivializations s_{i}.

Proof. First we show independence of the choice of lifting of s. As remarked above, g is determined by the local lifting of s to ω(D), so {c_{1}(ω(D)), ∇} depends only on that choice. If s and s′ are two such local liftings, with gs = g′s′, we have

dgg^{−1} = dg′g′^{−1} + d log(\frac{s′}{s})I = dg′g′^{−1} + z^{m}h

for some h ∈ M_{r}(O_{C}). It follows immediately that

res Tr(dgg^{−1}A) = res Tr(dg′g′^{−1}A)

as desired.

Next we show independence of the choice of trivializations themselves. Let f be a rational function on C whose divisor (f) is disjoint from the singular locus of ∇. It will suffice to show that s and fs as trivializations give rise to the same invariant, i.e.

(5.6) c_{1}(ω(D), s) · det(∇) − ∑_{i} res Tr(dg_{i}g_{i}^{−1}A_{i}) =

c_{1}(ω(D), fs) · det(∇) − ∑_{i} res Tr((dg_{i}g_{i}^{−1} − dff^{−1}I)A_{i})

Recall we can calculate c_{1}(ω(D), s) · det(∇) by choosing δ a divisor in the linear series ω(D) compatible with the rigidification s and then restricting ∇|_{δ} and taking the norm to Spec (K). Associated to the trivialization fs we may take the divisor δ + (f). It follows that

c_{1}(ω(D), fs) · det(∇) − c_{1}(ω(D), s) · det(∇) = Norm det ∇|_{(f)}

(To get this relation, one could have taken the formula (4.15) as well). On the other hand, since the formula depends only on the local behavior of f near D, by suitably choosing f, we may assume ∇ is defined by A in a neighborhood of (f) and that f ≡ 1 modulo some large power of the maximal ideal at the finite set of points where the connection is not given by A. We can interpret Tr(dff^{−1}A) ∈ Ω_{K}^{1} ⊗ ω_{k(C)}, so the sum of the residues over all closed points of C will vanish in Ω_{K}^{1}. Thus

∑_{i} res Tr(dff^{−1}A_{i}) = − ∑_{(f)} res Tr(dff^{−1}A) = Tr(A|_{(f)})

Since the connection matrix for the determinant connection is the trace of the connection matrix, the contributions to (5.6) cancel.

□

Now consider what happens to the expression

(5.7)  {c_{1}(ω(D)), ∇}

under a change of coordinates given by a matrix h of functions. We have

(5.8)  A ↦ A′ := hAh^{−1} + dhh^{−1} = g′ \frac{dz}{z^{m}} + \frac{η′}{z^{m−1}}

Note that h is regular, so dhh^{−1} does not contribute to the polar part of the connection, i.e.

(5.9)  g′ = hgh^{−1} + z^{m}a; η′ = hηh^{−1} + z^{m−1}b

with a and b regular.

We compute

(5.10) dg′g′^{−1} = d(hgh^{−1})hg^{−1}h^{−1} + ez^{m} + fz^{m−1}dz =

= dhh^{−1} + hdgg^{−1}h^{−1} − hgh^{−1}dhg^{−1}h^{−1} + ez^{m} + fz^{m−1}dz with e and f regular. Thus

(5.11) res Tr(dg′g′^{−1}A′) = res Tr(dg′g′^{−1}hAh^{−1}) =

res Tr(h^{−1}dhA) + res Tr(dgg^{−1}A) − res Tr(h^{−1}dhg^{−1}Ag).

Note

(5.12)  A − g^{−1}Ag = z^{1−m}(η − g^{−1}ηg) = z^{1−m}g^{−1}(gη − ηg)

From lemma 5.2,2, this expression is regular. Plugging into (5.11), we conclude.

Lemma 5.4. {c_{1}(ω(D)), ∇} is independent of the choice of basis for the bundle E.

Remarks 5.5. (1) The definition of {c_{1}(ω(D)), ∇} was inspired by the calculations in section 4 (cf. formula (4.15)) for a rank one connection. The formula

detH_{DR}^{∗}(U, (E, ∇))^{−1} = {c_{1}(ω(D)), ∇} (5.13)

when applied to the rank 1 case, gives back the main theorem of this article.

(2) When m_{i} = 1 for all c_{i}, that is when ∇ has regular singularities, then the argument from theorem 4.6 (slighlty modified in the higher rank case) gives

{c_{1}(ω(D)), ∇} = c_{1}(ω(D), res_{D}) · det(∇) (5.14)

where, as in theorem 4.6, res_{D} refers to the natural trivialization coming from the residue.

(3) Finally, twisting ∇ by f^{∗}α, where α ∈ Ω_{K}^{1} comes from the base, changes the right hand side of the formula by

(2g − 2 + n − ∑_{i} m_{i})rα = −χ(H_{DR}^{∗}(U, (E, ∇_{/K})))α,

as it should. Here r = rankE.

Proposition 5.6. Let D ⊂ C be a divisor on a smooth curve, and let f ∈ O_{C,D} be a local defining equation for D. Let

∇ : E → E ⊗ ω(D)

be a connection, and write

g = ∇|_{D} : E/E(−D) → (E(D)/E) ⊗ ω

Let j : E − D ↪ E be the inclusion and consider the connection

j_{∗}j^{∗}∇ : j_{∗}j^{∗}E → j_{∗}j^{∗}E ⊗ ω.

There is a natural map

ι : {E → E ⊗ ω} → {j_{∗}j^{∗}∇ : j_{∗}j^{∗}E → j_{∗}j^{∗}E ⊗ ω}.

The map ι is a quasi-isomorphism if and only if for any n ≥ 0 the natural map

g − n · id ⊗ \frac{df}{f} : E(nD)/E((n − 1)D) → (E((n + 1)D)/E(nD)) ⊗ ω

given by f^{−n}e ↦ f^{−n}g(e) − nf^{−n−1}e ⊗ df is a quasi-isomorphism.

In particular, if, every point of D has multiplicity ≥ 2, then g is an isomorphism if and only if ι is a quasi-isomorphism.

Proof. The usual exact sequence reduces us to showing the condition is equivalent to

j_{∗}j^{∗}E/E →^{≅} (j_{∗}j^{∗}E/E) ⊗ ω(D).

Writing as usual D = D′ + D where D is the reduced divisor with support equal to the support of D, we have a commutative square

E/E(−D)  ⟶^{g−nid⊗ \frac{df}{f}}  (E(D′)/E(−D)) ⊗ ω(D)
≅↓^{“·f^{−n}”}  ≅↓^{“·f^{−n}”}
E(nD)/E((n − 1)D) ⟶^{g} (E(nD + D′)/E((n − 1)D + D′))⊗ω(D)

The map ι is a quasi-isomorphism if and only if for all n ≥ 0 the map g on the bottom line of the above square is an isomorphism, and this will hold only if the top line is.

In particular, since \frac{df}{f} has poles of order 1, if all multiplicities are ≥ 2, then ι is a quasiisomorphism if and only if g is an isomorphism. □

We close those remarks by a numerical computation for E = ⊕_{1}^{r}O on U = 𝔸_{K}^{1}, with parameter t . There is only one singular point at ∞. Let us write

A = ∑_{i=0}^{m−1} B_{i}t^{i} + ∑_{i=0}^{m−2} C_{i}t^{i}dt

where the B_{i} and the C_{i} are matrices with coefficients in Ω_{K}^{1} respectively K. The assumption 5.3 means that C_{m−2} ∈ GL(r, K). We consider the cases m = 2, 3. For m = 2 both sides of the formula are 0, and for m = 3 they are equal to

Tr(B_{0} − B_{1}C_{0}C_{1}^{−1} + B_{2}C_{0}C_{1}^{−1}C_{0}C_{1}^{−1}) (5.15)

Note that −C_{0}C_{1}^{−1} is the zero of the “polynomial” C_{0} + C_{1}t, where t is a “variable” of matrices, and thus the formula could also be written as

TrA|_{zero(C_{0}+C_{1}t)} (5.16)

if it made sense. For higher m, the right hand side should be a sort of restriction of ∇ to the “Newton” classes of ∑_{i=0}^{m−2} C_{i}t^{i}.

In the remaining part of this section, we show that the product {c_{1}(ω(D)), ∇} is a particular case of a more general product between higher rank connections and a larger class of trivializations along D.

We consider the tuples {E, L, ∇, D, g}, where

∇ : E → E ⊗ Ω_{C}^{1} < D > (D′) (5.17)

is a connection on a rank r vector bundle E. We denote by ∇|_{D} the O_{C}-linear map

∇ : E → E ⊗ (Ω_{C}^{1} < D > (D′)/Ω_{C}^{1}). (5.18)

Further, L is a rank 1 bundle, g is a trivialization

g : E|_{D} ≅ E ⊗ L|_{D}, (5.19)

which fulfills

0 = [g, ∇|_{D}] : E|_{D} → E ⊗ L|_{D} (5.20)

By lemma 5.2, if L = ω_{C}(D) and g is the trivialization E|_{D} → E ⊗ (ω_{C}(D)/ω_{C}) arising from the principal part of ∇_{C/K}, then the condition 5.20 is fulfilled.

Let us introduce the cocyles of the tuple {E, L, ∇, D, g}, as in section 4. If E has cocycle c_{ij} ∈ GL(r, O_{C}), L has cocycle λ_{ij} ∈ O_{C}^{∗}, g has cocyle μ_{i} ∈ GL(r, O_{D}), and ∇ has cocyle ω_{i} ∈ M(r×r, Ω_{C}^{1} < D > (D′)) then one has

dc_{ij}c_{ij}^{−1} = ω_{i} − c_{ij}ω_{j}c_{ij}^{−1} (5.21)

μ_{i} = c_{ij}μ_{j}c_{ij}^{−1}λ_{ij}. (5.22)

The commutativity condition 5.20 then reads

[μ_{i}, ω_{i}] = 0. (5.23)

Let μ̃_{i} ∈ GL(r, O_{C}) be a local lifting ot μ_{i} ∈ GL(r, O_{D}). Then one has

Theorem 5.7. The cochain

(5.24)  {{λ_{ij}, det(c_{jk})}, d log λ_{ij} ∧ Tr(ω_{j}), Tr(−dμ̃_{i}μ̃_{i}^{−1} ∧ ω_{i})}

is a cocyle in

(5.25)  K_{2} → Ω_{X}^{2}⟨D⟩(D′) → (Ω_{X}^{2}⟨D⟩(D′)/Ω_{X}^{2})

and defines a cohomology class {c_{1}(L), g, ∇} in

(5.26)  ℍ^{2}(C, K_{2} → Ω_{X}^{2}⟨D⟩(D′) → (Ω_{X}^{2}⟨D⟩(D′)/Ω_{X}^{2}))

Proof. First

(5.27) δ(d log(λ) ∧ Tr(ω))(ijk) = d log(λ_{ij}) ∧ Tr(ω_{j}) + d log(λ_{jk}) ∧ Tr(ω_{k}) − d log(λ_{ik}) ∧ Tr(ω_{k}) =

d log(λ_{ik}) ∧ Tr(ω_{j} − ω_{k}) − d log(λ_{jk}) ∧ Tr(−ω_{k} + ω_{j}) = d log(λ_{ij}) ∧ Tr(ω_{j} − ω_{k}) = d log(λ_{ij}) ∧ d log(det(c_{jk})).

Next computing mod the ideal of D and so ignoring the tilde,

(5.28) δ(Tr(−dμ̃_{i}μ̃_{i}^{−1} ∧ ω_{i}))(ij) = Tr(−dμ̃_{j}μ̃_{j}^{−1} ∧ ω_{j} + dμ̃_{i}μ̃_{i}^{−1} ∧ ω_{i}) =

Tr((dλ_{ij}λ_{ij}^{−1} · I + c_{ij}^{−1}dc_{ij} − c_{ij}^{−1}dμ_{i}μ_{i}^{−1}c_{ij} − c_{ij}^{−1}μ_{i}dc_{ij}c_{ij}^{−1}μ_{i}^{−1}c_{ij}) ∧ ω_{j}+

+ dμ_{i}μ_{i}^{−1} ∧ ω_{i}) = Tr((dλ_{ij}λ_{ij}^{−1} · I + c_{ij}^{−1}dc_{ij} − c_{ij}^{−1}dμ_{i}μ_{i}^{−1}c_{ij}
− c_{ij}^{−1}μ_{i}dc_{ij}c_{ij}^{−1}μ_{i}^{−1}c_{ij}) ∧ (c_{ij}^{−1}ω_{i}c_{ij} − c_{ij}^{−1}dc_{ij}) + dμ_{i}μ_{i}^{−1} ∧ ω_{i})

By our commutation assumption 5.20

(5.29) Tr(c_{ij}^{−1}dc_{ij} ∧ c_{ij}^{−1}ω_{i}c_{ij}) = Tr(c_{ij}^{−1}μ_{i}dc_{ij}c_{ij}^{−1}μ_{i}^{−1}c_{ij} ∧ c_{ij}^{−1}ω_{i}c_{ij})

Also terms with no poles (i.e. terms not involving ω) can be ignored. We get

(5.30) δ(Tr(−dμ̃_{i}μ̃_{i}^{−1} ∧ ω_{i}))(ij) = dλ_{ij}λ_{ij}^{−1} ∧ Tr(ω_{i}) = dλ_{ij}λ_{ij}^{−1} ∧ Tr(ω_{j})

which is what we want. Note the right hand equality holds because we are computing mod forms regular along D, and Tr(ω_{i}) ≡ Tr(ω_{j}) mod regular forms by the cocycle condition 5.21.

It remains to show that our cocyle 5.24 does not depend on the choice of the cocycles {c, ω, λ, μ}.

If λ_{ij} is replaced by λ_{ij}δ(ν)_{ij}, then the new 5.24 differs from the old one by the cochain

δ({ν_{i}, det(c_{ij})}, d log ν_{i} ∧ Tr(ω_{i})).

If c_{ij} is replaced by γ_{i}c_{ij}γ_{j}^{−1}, then ω_{i} is replaced by dγ_{i}γ_{i}^{−1} + γ_{i}ω_{i}γ_{i}^{−1}, and μ_{i} is replaced by γ_{i}μ_{i}γ_{i}^{−1}. The commutativity relation implies dγ_{i}μ_{i}γ_{i}^{−1} − γ_{i}μ_{i}γ_{i}^{−1}dγ_{i}γ_{i}^{−1} = 0. Then the new 5.24 differs from the old one by

{{λ_{ij}, det(δ(γ))_{jk}}, dλ_{ij} ∧ Trd log(γ_{i}),

res Tr(γ_{i}^{−1}dγ_{i}ω_{i}dμ_{i}μ_{i}^{−1}γ_{i}^{−1}dγ_{i} − μ_{i}γ_{i}^{−1}dγ_{i}μ_{i}^{−1}γ_{i}^{−1}dγ_{i} − μ_{i}γ_{i}^{−1}dγ_{i}μ_{i}^{−1}ω_{i})} (5.31)

Using the commutativity relation 5.20 as well, we see that this is this expression is the cochain

δ({λ_{ij}, detγ_{j}}, 0, 0). (5.32)

□

Now we consider the image under the map f : C → Spec (K):

(5.33)  f_{∗}({c_{1}(L), g, ∇}) ∈ Ω_{K}^{1}/d log K^{∗}

of {c_{1}(L), g, ∇}. We want to study closedness for forms in the image of this map.

Lemma 5.8. Let {a, b, c} be a cocycle as in (5.24) representing a class in ℍ^{2}(C, K_{2} → Ω_{C}^{2}), with db = 0. Then df_{∗}{a, b, c} = res_{D}(dc) ∈ Ω_{K}^{2}.

Proof. Another representative of {a, b, c} in

ℍ^{1}(C, Ω_{C}^{2} < D > (D′) → Ω_{C}^{2} < D > (D′)/Ω_{C}^{2})/d log(H^{1}(C, K_{2}))

is of the shape {0, b + d log α, c}, thus its derivative in

ℍ^{1}(C, Ω_{C}^{3} < D > (D′) → Ω_{C}^{3} < D > (D′)/Ω_{C}^{3})

is of the shape {0, 0, dc} since db = 0. Then one applies the commutativity of the diagramm

(5.34)

ℍ^{1}(C, Ω_{C}^{2}⟨D⟩(D′) → Ω_{C}^{2}⟨D⟩(D′)/Ω_{C}^{2}) ⟶^{d} ℍ^{1}(C, Ω_{C}^{3}⟨D⟩(D′) → Ω_{C}^{3}⟨D⟩(D′)/Ω_{C}^{3})
↓  ↓
Ω_{K}^{1}  ⟶^{d}  Ω_{K}^{2}
□

Theorem 5.9. If ∇^{2} = 0, that is if ∇ is flat, then f_{∗}({c_{1}(L), g, ∇}) ∈ Ω_{K,clsd}^{1}/d log K^{∗}, that is the image is flat as well. In particular, this is true for {c_{1}(ω(D)), ∇}, as predicted by conjecture 5.1.

Proof. Since, under the integrabiltiy assumption, one has in particular d(d log λ_{ij} ∧ Trω_{j}) = 0, one can apply lemma 5.8. One has to compute

γ = −res Tr d(dμμ^{−1}ω)

(5.35) = −res Tr(dμμ^{−1})^{2}ω + res Tr dμμ^{−1}dω

48 SPENCER BLOCH AND HÉLÈNE ESNAULT

We omit the indices since we compute only with one index. Note in the calculations which follow μ is regular and ω has poles along D. We write a ≡ b to indicate that the polar parts of a and b coincide.

The condition 5.20 implies dμω + μdω ≡ dωμ − ωdμ (5.36) thus dμμ^{−1}ω + μdωμ^{−1} ≡ dω − ωdμμ^{−1} (5.37) which implies (dμμ^{−1})^{2}ω + dμdωμ^{−1} ≡ dμμ^{−1}dω − dμμ^{−1}ωdμμ^{−1} (5.38) So taking the trace, one obtains 2Tr((dμμ^{−1})^{2}ω) ≡ Tr((dμμ^{−1} − μ^{−1}dμ)dω) (5.39) Now, under the integrability assumption dω = ωω (5.40) (using (5.23)) one obtains 2Tr((dμμ^{−1})^{2}ω) = 0 (5.41) Now we consider the other term γ′ = Tr(dμμ^{−1}(ω)^{2}) (5.42) Choosing a local basis of the bundle E, we write μ as a matrix M, and ω as matrix Ω. Then the condition 5.20 reads MΩ = ΩM (5.43) and one has γ′ = Tr(dMM^{−1}Ω^{2}) (5.44) The condition 5.43 implies Ω^{2} = M^{−1}Ω^{2}M (5.45) thus γ′ = Tr(dMΩ^{2}M^{−1}) = Tr(M^{−1}dMΩ^{2}) (5.46) On the other hand, differentiating the condition 5.43, one obtains

M^{−1}dMΩ^{2} + dΩΩ =

M^{−1}dΩMΩ − M^{−1}ΩdMΩ =

M^{−1}dΩΩM − ΩM^{−1}dMΩ (5.47) So taking the trace, one obtains γ′ = Tr(M^{−1}dMΩ^{2}) = −Tr(M^{−1}dMΩ^{2}) = −γ′ (5.48)

Thus γ′ = 0.

□

6. Rank 1 Irregular Connections in Arbitrary Dimension on Projective Manifolds

Let X be a smooth, projective variety in characteristic 0, and let D ↪ X be a normal crossings divisor. Given m_{i} ≥ 0, define C(X, D, \underline{m}) to be the group of isomorphism classes of line bundles L on X together with an integrable connection

∇ : L → L ⊗ Ω_{X}^{1} < D > (D)

where D = ∑_{i} m_{i}D_{i}. Define for m_{i} ≥ 1

Irreg(X, D, \underline{m}) := C(X, D, \underline{m})/C(X, D, \underline{0}).

Theorem 6.1. With notation as above, there is a canonical isomorphism

Irreg(X, D, \underline{m}) ≅ Γ(X, O_{X}(D)/O_{X}),

where O_{X}(D)/O_{X} is the sheaf of principal parts of degree ≤ m_{i} along D_{i}.

Lemma 6.2. Exterior differentiation induces an isomorphism

d : O_{X}(D)/O_{X} →^{≅} Ω_{X}^{1} < D > (D)_{clsd}/Ω_{X}^{1} < D >_{clsd}

proof of lemma. We first check injectivity. Let x = 0 be a local defining equation for D. Suppose for some n with 1 ≤ n ≤ m we had

d(\frac{a}{x^{n}}) = \frac{1}{x^{n}} (da − na\frac{dx}{x}) ∈ Ω_{X}^{1} < D >

Multiplying by x^{n} and taking residue along D, it would follow that a|_{D} = 0, i.e. \frac{a}{x^{n}} = \frac{b}{x^{n−1}} .

To show surjectivity, write a local section of Ω^{1} < D > (nD)_{clsd} (here 1 ≤ n ≤ m) in the form

(6.1)  ω = \frac{adx}{x^{n+1}} + \frac{B}{x^{n}}

where B does not involve dx. Replacing ω by ω+d(\frac{a}{nx^{n}}), we can assume

ω = \frac{adx + B}{x^{n}} .

Then

0 = dω = \frac{da ∧ dx + dB − n\frac{dx}{x} ∧ B}{x^{n}} .

Multiplying by x^{n} and taking residue along D, we see B|_{D} = 0. Since B does not involve dx, it follows that B = xC and ω can be written

ω = \frac{adx}{x^{n}} + \frac{C}{x^{n−1}}

Comparing with (6.1), we have lowered the order of pole by 1. This process continues until ω has log poles. □

proof of theorem 6.1. Using the lemma, we get a diagram with exact rows

0 ⟶  O_{X}^{∗}  O_{X}^{∗}  ⟶  0  ⟶ 0
↓  ↓  ↓
0 ⟶ Ω_{X}^{1} < D >_{clsd} ⟶ Ω_{X}^{1} < D > (D)_{clsd} ⟶ O_{X}(D)/O_{X} ⟶ 0

We view this as a diagram of complexes written vertically. Using the standard hypercohomological interpretation of line bundles with connection, this yields an exact sequence

0 → C(X, D, 0) → C(X, D, \underline{m}) → H^{0}(X, O_{X}(D)/O_{X})

→^{δ} ℍ^{2}(X, O_{X}^{∗} → Ω_{X}^{1} < D >_{clsd})

We claim the map δ above is zero. In the derived category, δ factors

O_{X}(D)/O_{X} →^{≅} \frac{Ω_{X}^{1} < D > (D)_{clsd}}{Ω_{X}^{1} < D >_{clsd}} →^{∂} Ω_{X}^{1}(< D >)_{clsd}[1]

→ {O_{X}^{∗} → Ω_{X}^{1}(< D >)_{clsd}}[2].

We have a factorization of ∂:

H^{0}(O_{X}(D)/O_{X}) → H^{1}(O_{X}) →^{d̃} H^{1}(Ω_{X,clsd}^{1}) → H^{1}(Ω_{X}^{1} < D >_{clsd}),

so it suffices to show the map d̃ is zero. By Hodge theory, the composition

H^{1}(O_{X}) →^{d̃} H^{1}(Ω_{X,clsd}^{1}) →^{ι} ℍ^{1}(Ω_{X}^{1} → Ω_{X}^{2})

is zero, and the map ι is injective as the complex {Ω_{X}^{1}/Ω_{X,clsd}^{1} → Ω_{X}^{2}} is quasi-isomorphic to the complex {0 → Ω_{X}^{2}/Ω_{X,exact}^{2}}, and in particular starts in degree 1. □

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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