A FORMULA FOR GAUSS-MANIN DETERMINANTS

SPENCER BLOCH AND HÉLÈNE ESNAULT

Abstract. We give an explicit formula for the determinant of the Gauß-Manin connection for an irregular connection on a Zariski open set of the projective line ℙ_{K}^{1} over a function field K over a field k of characteristic zero.

I hasten to write down in verse what I saw then, For the scene lost to sight can’t be revived again.

Su Shi

1. Introduction

Let K be a function field over a field k of characteristic 0, and let j : U ⊂ ℙ_{K}^{1} be a Zariski open set of the projective line. We consider a flat connection (E, ∇) on U. The de Rham cohomology groups H_{DR}^{i}(U/K, ∇_{/K}) carry a K/k connection, the Gauß-Manin connection, and taking the alternate tensor of the determinant connections

⊗(det H_{DR}^{i}(U/K, ∇_{/K}), Gauß − Manin)^{(−1)^{i}},

one defines the Gauß-Manin determinant connection, denoted by

det H_{DR}(U/K, ∇_{/K}).

This invariant is living in the group of isomorphism classes of K-lines endowed with a connection, which is the abelian group

Ω_{K}^{1}/d log K^{×}.

The aim of this article is to give an explicit formula for it (see theorem 1.3 for a vague formulation, and 2.8 for a precise one) under a genericity assumption on (E, ∇). Special examples are contained in [4].

We comment briefly on the meaning and interest in such a formula. There is a deep analogy between connections on curves over function

Date: April 12, 2000.

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

This work has been partly supported by the NSF grant DMS-9423007-A1, the DFG Forschergruppe ”Arithmetik und Geometrie”, the Humboldt Foundation and the University Paris VII.

2 SPENCER BLOCH AND HÉLÈNE ESNAULT

fields in characteristic 0 and ℓ-adic sheaves E_{ℓ} on curves U over finite fields 𝔽_{q}. Irregular singular points for the connection correspond to wild ramification at ∞ for the ℓ-adic sheaf, and the Gauß-Manin determinant connection corresponds to the global epsilon factor

ϵ(E_{ℓ}) := det(−f | det RΓ(U, E_{ℓ})).

Our hope is that the local formula we obtain for higher rank irregular connections will suggest local formulas for ϵ-factors extending the abelian Tate theory.

The Gauß-Manin construction is fairly standard and we do not recall it in detail. By way of example, we cite two classical formulas (Gauß hypergeometric and Bessel functions, respectively) :

\frac{Γ(b)Γ(c − b)}{Γ(c)}F (a, b; c; z) = ∫_{0}^{1} u^{b−1}(1 − u)^{c−b−1}(1 − uz)^{−a}du

J_{n}(z) = \frac{1}{2πi} ∫_{S_{0}} u^{−n} exp \frac{z}{2}(u + \frac{1}{u})\frac{du}{u} (S_{0} = circle about 0).

In both cases, the integrand is a product of a solution of a rather simple degree 1 differential equation in u, the solution being either

u^{b−1}(1 − u)^{c−b−1}(1 − uz)^{−a} or u^{−n} exp \frac{z}{2}(u + \frac{1}{u}),

with an algebraic 1-form (du or \frac{du}{u}). The integral is taken over a chain in the u-plane. The resulting functions F (a, b; c; z) and J_{n}(z) satisfy Gauß-Manin equations, which are much more interesting degree 2 equations in z.

It is not our purpose to go further into the classical theory, but, to understand the role of the determinant, we remark that in each of the above cases, there is a second path and a second algebraic 1-form such that the two integrals, say f_{1}(z) and f_{2}(z), satisfy the same second order equation. The Wronskian determinant

|f_{1} f_{2}_{\frac{df_{1}}{dz} \frac{df_{2}}{dz}} |

satisfies the degree 1 equation given by the determinant of Gauß-Manin. It seems to us to be possible using the theory of Stokes structures to formulate a theory of period integrals for irregular connections in such a way that the Gauß-Manin determinant connection has as solution the determinant of the period matrix. We hope to return to this in a future paper.

Let X be a complete, smooth curve over K. For purposes of this article, we define the group of relative algebraic differential characters

AD^{2}(X/K) := ℍ^{2}(X, K_{2} ⟶^{d log} Ω_{X}^{2}/(O_{X} ⊗ Ω_{K}^{2})) (1.1)

(The notation here differs from [7], as one has factored out 2-forms coming from the base and in particular truncated the differential forms of degree 3). The transfer

f_{∗} : AD^{2}(X/K) → AD^{1}(K) = Ω_{K}^{1}/d log K^{×} (1.2)

maps the group of relative algebraic differential characters of degree 2 on X to the group of algebraic characters of degree 1 on K, which is the group of connections on K. Indeed this is an isomorphism (lemma 2.7). But to write connections on K as coming from differential characters on X allows to single out two types of classes, global decomposable classes and local classes, which we discuss in the sequel.

Let D = ∑ m_{i}x_{i} be an effective divisor on X, and let D = ∑ x_{i} be the corresponding reduced divisor. We define a sheaf of meromorphic 1-forms

Ω_{X}^{1}(D − D) ⊂ Ω_{X}^{1}{D} ⊂ Ω_{X}^{1}(D) (1.3)

as follows. If z is a local parameter at a point x of multiplicity m in D, a 1-form is a section of Ω_{X}^{1}{D} if it can be written in local coordinates in the form

\frac{fdz}{z^{m}} + \frac{η}{z^{m−1}} (1.4)

where f ∈ O_{X,x} and η ∈ O_{X,x} ⊗ Ω_{K}^{1} are regular at x. We define Ω_{X}^{p}{D} = Ω_{X}^{1}{D} ∧ Ω_{X}^{p−1} ⊂ Ω_{X}^{p}(D). There is an exact sequence

0 → O_{X}(D − D) ⊗ Ω_{K}^{1} → Ω_{X}^{1}{D} → ω_{X/K}(D) → 0 (1.5)

where we write ω_{X/K} for the sheaf of relative 1-forms.

The graded algebra ⊕_{n} ∧^{n} (Ω_{X}^{1}{D}) is closed under exterior d. Further, writing I = O(−D) ⊂ O_{X} for the ideal sheaf, we have

d log((1 + I)^{×}) ∧ Ω_{X}^{p}{D} ⊂ Ω_{X}^{p+1}. (1.6)

Let E be a vector bundle on X, and let

∇ : E → E ⊗ Ω_{X}^{1}{D} (1.7)

be an absolute connection (i.e. parameters from K are also differentiated).

Definition 1.1. The connection ∇ is vertical, if the curvature

∇^{2} : E → E(∗D) ⊗ Ω_{K}^{2}.

4 SPENCER BLOCH AND HÉLÈNE ESNAULT

We assume our connection ∇ is vertical. (Of course the most important case is that of a flat connection ∇^{2} = 0.) We write ∇_{/K} : E → ω_{X/K}(D) for the corresponding relative connection. Viewing D as a nilpotent subscheme of X, it is easy to check that ∇ induces a function linear “polar part” map

(1.8) ∇_{D} : E|_{D} → E ⊗ (Ω_{X}^{1}{D}/Ω_{X}^{1}) (Absolute)

∇_{D/K} : E|_{D} → E ⊗ ω_{D/K} = E ⊗ (ω_{X/K}(D)/ω_{X/K}) (Relative)

Definition 1.2. The connection (1.7) is said to be admissible if the relative polar parts map ∇_{D/K} : E|_{D} → E ⊗ ω_{D/K} is an isomorphism in a singular point of multiplicity ≥ 2, and if in a singular point of multiplicity 1, Deligne’s condition [6] that the eigenvalues of the residue do not belong to {0, 1, 2 … } is fulfilled.

Notice that the notion of admissibility depends on the extension of E to all of X. Although the definition makes reference only to the relative connection, admissibility depends also on the absolute connection, which is required to take values in Ω_{X}^{1}{D} ⊂ Ω_{X}^{1}(D). It is a local formal property. The motivation for this definition comes from a struture theorem ([8], [12], [9], compare also with [10], p.124) asserting that locally formally, after ramification of the curve, a flat connection (E, ∇) becomes a direct sum of summands L ⊗ Λ, where Λ is a higher rank flat connection with logarithmic singularities, and L is either a rank 1 trivial connection or a rank 1 flat connection with multiplicity ≥ 2 (strictly speaking, this is proven in [9] only over ℂ, outside of a Baire set). Since rank 1 vertical connections are admissible ([3], lemma 3.1), we see that up to ramification, any flat connection is locally of sum of admissible and logarithmic connections.

For a rank 1 vertical connection we may view

∇_{D/K} : E|_{D} ≅ E|_{D} ⊗ ω_{X/K}(D)|_{D} (1.9)

as defining a trivialization of ω_{X/K}(D)|_{D}, i.e. a class (ω_{X/K}(D), ∇_{D/K}) ∈ Pic(X, D). There is a cohomological pairing (j : X − D ↪ X)

{ , } : Pic(X, D) ⊗ ℍ^{1}(X, j_{∗}O_{X−D}^{×} → Ω_{X}^{1}{D}) (1.10)

→ AD^{2}(X) = ℍ^{2}(X, K_{2} ⟶^{d log} Ω_{X}^{2})

and in the rank 1 case (see main theorem of [3])

det H_{DR}(U/K, ∇_{/K}) = −f_{∗}{(ω_{X/K}(D), ∇_{D/K}), (E, ∇)} (1.11)

∈ Ω_{K}^{1}/d log K^{×} ⊗ ℚ.

In higher rank, however, we have examples of connections (E, ∇) with (det(E), det(∇)) trivial but non-trivial Gauß-Manin determinant connection (see [4], remark 3.3, equation 3.27, and remark 2.9).

Still, the choice of a meromorphic section s ∈ ω_{X/K}(D) ⊗ K(X) which generates the sheaf at the points of D defines a rigidification c_{1}(ω_{X/K}(D), s) ∈ Pic(X, D), and allows to define a class

{c_{1}(ω_{X/K}(D), s), det(E, ∇)} ∈ AD^{2}(X).

We refer to it as the global factor (see (2.5)).

In each singularity, we define local factors (see proposition 2.4) which play the rôle of the local epsilon factors defined to express the global epsilon of an ℓ-adic sheaf. If A_{i} = g_{i} \frac{dz}{z^{m}} + \frac{η_{i}}{z^{m−1}} is the local equation of ∇ in a local basis around the point a_{i} ∈ D, then one defines Trdg_{i}g_{i}^{−1}A_{i} ∈ AD^{2}(X/K). We also define a 2-torsion local factor \frac{1}{2}d log(det g_{i}(a_{i}))) ∈ AD^{2}(X/K) (see definition 2.5).

The main theorem of this article says

Theorem 1.3. Let (E, ∇) be an admissible connection on ℙ_{K}^{1} having at least one point of multiplicity ≥ 2. Then

det H_{DR}(U, ∇_{/K}) = −f_{∗}{c_{1}(ω_{X/K}(D), s), det(E, ∇)} +

∑_{i} res_{a_{i}} Tr(dg_{i}g_{i}^{−1}A_{i}) + \frac{1}{2} ∑_{i} m_{i}d log(det(g_{i}(a_{i})))

∈ Ω_{K}^{1}/d log(K^{×}).

(See theorem 2.8 for a slightly more precise formulation).

The cases rank (E) = 1, resp. ∇ with logarithmic poles, were considered in the earlier articles [3], resp. [2], except that for the rank 1 case, our results did not include torsion. In those two cases, there is a well defined class γ(E, ∇) ∈ AD^{2}(X) such that f_{∗}γ(E, ∇) = det H_{DR}(U/K, ∇_{/K}). In the higher rank, non-logarithmic case considered here, one still has the global factor in AD^{2}(X), but the local factors are well defined in AD^{2}(X/K) only (see proposition 2.4).

It would be of great interest to find some variant of this formula which applied to epsilon factors for ℓ-adic sheaves.

We now discuss the proof of the main theorem. Let K be a field in characteristic 0. A classical theorem by Euler ([11], III, 6, lemma 2) asserts that if g ∈ K[t] is a polynomial, and h ∈ L = K[u]/gK[u], then the trace of multiplication by h, viewed as a K-linear map from L to itself, is computed by

−res_{u=∞}dgg^{−1}h.

We define a generalization of this in the non-commutative situation as follows. Let V be a finite dimensional K-vectorspace, and g =

6 SPENCER BLOCH AND HÉLÈNE ESNAULT

∑_{i=0} g_{i}u^{i} ∈ End(V)[u] be a polynomial with coefficients in the endomorphisms of E, such that the leading coefficient g_{m} ∈ Aut(V) is invertible. The invertibility of g_{m} allows us to write an element of W = V [u]/gV [u] as the class of an element ∑_{r=0}^{m−1} v_{i}u^{i}, where v_{i} ∈ V, yielding a splitting σ : W → V [u] of the natural projection p : V [u] → W. For any h ∈ End(V)[u], ϕ(h) := p ∘ h ∘ σ : W → W will have a trace. Proposition 5.1 says

Tr_{V [u]/gV [u]}(ϕ(h)) = −Tr_{V} res_{u=∞}(dgg^{−1}h).

The second point is to relate the trace of this linear operator with the trace of a differential operator. The crucial case to understand is that of a connection on a trivial bundle E = V ⊗_{K} O_{ℙ^{1}} on ℙ_{K}^{1}. Such a connection is given by a matrix which has the shape

∑_{i=1}^{N} ∑_{r=1}^{m_{i}} \frac{g_{r}^{(i)}d(t − a_{i})}{(t − a_{i})^{r}} + η = g + η, (1.12)

where g_{r}^{(i)} ∈ End(V) and η ∈ End(V) ⊗ Ω_{K}^{1} ⊗ O_{ℙ_{K}^{1}}(∗D). We write g also for the corresponding matrix of relative forms, so ∇_{/K} = d+g. We fix a certain finite-dimensional vector subspace σ : H ↪ H^{0}(ℙ_{K}^{1}, V ⊗_{K} ω(∗D)) such that composition with the natural projections give isomorphisms

H →^{σ} H^{0}(ℙ_{K}^{1}, V ⊗_{K} ω(∗D)) →^{p_{∇}} H_{DR}^{1}(U/K, ∇_{/K}) (1.13)

= H^{0}(ℙ_{K}^{1}, V ⊗_{K} ω(∗D))/Im∇_{/K}

H →^{σ} H^{0}(ℙ_{K}^{1}, V ⊗_{K} ω(∗D)) →^{p_{g}} H^{0}(ℙ_{K}^{1}, V ⊗_{K} ω(∗D))/Im g

The operators

η_{∇} := (p_{∇} ∘ σ)^{−1}p_{∇} ∘ η ∘ σ : H → H ⊗ Ω_{K}^{1}
η_{γ} := (p_{g} ∘ σ)^{−1}p_{g} ∘ η ∘ σ : H → H ⊗ Ω_{K}^{1}

will be referred to as the (Gauß-Manin) de Rham operator and Higgs operator respectively. The traces of these operators play a central role in the Gauß-Manin determinant, and the remarkable fact is that for an admissible, vertical connection on a trivial bundle on ℙ_{K}^{1} one finds

(1.14) Tr(η_{∇} − η_{γ}) ≡ \frac{1}{2} ∑_{i} m_{i}d log(det(g_{m_{i}}^{(i)})) mod d log(K^{×}).

(Here the g_{m_{i}}^{(i)} are as in (1.12).) This result is theorem 3.6. It is reminiscent of Hitchin’s comparison of de Rham and Higgs twisted cohomologies on projective manifolds, and of Kontsevich’s theorem comparing de Rham and Higgs cohomology of df, where f is a regular function on a manifold. Algebraically, if

(\frac{g_{m}}{z^{m}} + \frac{g_{m−1}}{z^{m−1}} + …)dz + \frac{η_{m−1}}{z^{m−1}} + \frac{η_{m−2}}{z^{m−2}} + …,

represents the polar part of our admissible, vertical connection at a point z = 0, the essential result (proposition 3.8) is that

Tr ∑_{s=0}^{m−1} g_{m}^{−1}[g_{m−s}, η_{s}]

is identically vanishing. We must confess that, even after performing the computation, we don’t really understand its meaning.

Acknowledgements: It is a pleasure to thank A. Beilinson, C. Sabbah and T. Saito for interesting discussions related to the topics discussed in this article.

2. Admissible Connections

Let K be a function field over a field k of characteristic 0, f : X → Spec K be a smooth projective curve, j : U ⊂ X a non-trivial Zariski open set such that the closed points of D = X \ U are K rational points, and (E, ∇) a global connection of rank r on X which is regular on U. Barring express mention to the contrary, we shall always assume ∇ to be vertical (definition 1.1). We shall need a small generalization of the notion of admissibility introduced in definition 1.2.

Definition 2.1. The connection (E, ∇) is pseudo-logarithmic at x ∈ X \ U if the local equation of ∇ in some basis of E has the shape

A = g \frac{dz}{z} + \frac{η}{z} ,

where z is a local parameter around x, η ∈ M(r × r, Ω_{K}^{1} ⊗ O_{X}), g ∈ GL(r, O_{x}). The connection (E, ∇) is pseudo-admissible if it is admissible in singularities with multiplicities m ≥ 2 and pseudo-logarithmic in points with multiplicities m = 1.

We will use a very special simple shape of pseudo-logarithmic singularities, which we single out in the following definition.

Definition 2.2. A special pseudo-logarithmic point of a connection (E, ∇) is pseudo-logarithmic, and there is local basis

(e_{ν}) = ((e_{1}, …, e_{s}), (e_{s+1}, …, e_{r}))

with respect to which the block-matrix of the connection has the shape

( \begin{matrix}A + m\frac{dz}{z} & zB \\ \frac{C}{z} & D + n\frac{dz}{z} \end{matrix} ),

where the connection matrix

( \begin{matrix}A & B \\ C & D \end{matrix} )

has no poles and m, n ∈ k.

A connection is special pseudo-logarithmic if it is admissible, and special pseudo-logarithmic in pseudo-logarithmic points.

Working with pseudo-admissible connections will enable us to reduce the Gauß-Manin determinant computation for a general (E, ∇) on ℙ^{1} to the case where E ≅ O_{ℙ^{1}}^{⊕r}. We use the following:

Theorem 2.3. [Compare with [2], lemma 4.2 and reduction 4.1] Let (E, ∇) be an admissible connection on ℙ_{K}^{1} having a singularity of multiplicity ≥ 2. Then there are finitely many points p_{i} ∈ U(K), such that if λ : V = U \ {p_{i}} → ℙ_{K}^{1} denotes the open embedding, then (E|_{V} , ∇|_{V}) extends to a special pseudo-admissible connection (⊕_{1}^{r}O_{ℙ_{K}^{1}}, ∇) on ℙ_{K}^{1} such that

(⊕_{1}^{r}O_{ℙ_{K}^{1}}) ⟶^{∇_{/K}} ω(D + ∑_{i} p_{i}) ⊗ (⊕_{1}^{r}O_{ℙ_{K}^{1}})

→ (λ_{∗}E_{V} ⟶^{∇_{/K}} λ_{∗}(ω ⊗ E_{V}))

is a quasiisomorphism.

Proof. Without loss of generality, we may assume that ∞ is a smooth point of the connection. Let x be a point of multiplicity ≥ 2, and let z be a local coordinate at x. Note the effect of twisting (i.e. replacing E by E(Nx)) is to replace the local connection matrix A at x with respect to a basis e_{i} with A − N \frac{dz}{z}I for the basis \frac{e_{i}}{z^{N}}. In particular, x will remain an admissible singularity for the new connection. After such a twist, we may assume E = ⊕_{i=1}^{r}O(n_{i}), with 0 ≤ n_{1} ≤ n_{2} ≤ . . .. We argue by induction on n_{r} − n_{1}. If n_{r} − n_{1} = 0, we replace E by E(−n_{1}x) and argue as above.

Assume n_{r} − n_{1} > 0. Let E′ = ⊕_{i=1}^{r−1}O(n_{i}) ⊕ O(n_{r} − 1), and embed E′ in E via O((n_{r} − 1)∞) → O(n_{r}∞). If z is a local parameter at ∞, and e_{ν} is a local basis of O(n_{ν}) at ∞, then ((e_{ν}, μ ≤ r − 1), ze_{r}) is a local basis of E′, and if

( \begin{matrix}A & B \\ C & D \end{matrix} ) (2.1)

is the local block matrix of the connection ∇ in the basis ((e_{1}, …, e_{r−1}), e_{r}), then

( \begin{matrix}A & zB \\ \frac{C}{z} & D + \frac{dz}{z} \end{matrix} ) (2.2)

is the local block matrix of the connection in the basis ((e_{1}, …, e_{r−1}), ze_{r}). If C has a local expansion C = C_{0} + C_{1}z + …, then the polar part of this connection is

( 0 0_{\frac{C_{0}}{z} \frac{dz}{z}} ). (2.3)

Thus replacing now E′ by E′′ = E′(2∞) ≅ ⊕_{i=1}^{r−1}O(n_{i} + 2) ⊕ O(n_{r} + 1), the local equation of the connection at ∞ becomes

( \begin{matrix}A − 2\frac{dz}{z} & 0 \\ \frac{C}{z} & D − \frac{dz}{z} \end{matrix} ), (2.4)

and therefore, is pseudo-admissible. On the other hand, n_{r} − n_{1} has decreased. We conclude by induction.

Next we describe the class γ(E, ∇) ∈ AD^{2}(X/K) := ℍ^{2}(X, K_{2} ⟶^{d log} Ω_{X}^{2}/O_{X} ⊗Ω_{K}^{2}) from theorem 2.8. If we choose a meromorphic section s of ω_{X/K}(D) which generates this sheaf in a neighborhood of D, we may view s as defining a trivialization of ω_{X/K}(D)|_{D}, i.e. a class (ω(D), s) ∈ Pic(X, D). As in (1.10), we may consider the product

{(ω(D), s), (det(E), det(∇))} ∈ AD^{2}(X). (2.5)

We refer to this class as the global factor.

Fix a basis e_{i} for E in a neighborhood of D. the choice of e_{i} determines a local connection matrix A, so ∇ = d + A. Let O_{X,D} denote the semi-local ring of functions regular at all points of D. The choice of s determines g ∈ GL_{r}(O_{X,D}) such that the relative connection ∇_{/K} = d+gs. Note the hypothesis of pseudo-admissibility insures that g is invertible.

The basic local invariant we consider is

Tr(dgg^{−1}A) ∈ H^{0}(X, Ω_{X}^{2}{D}/(Ω_{X}^{2} + O_{X}(D − D) ⊗ Ω_{K}^{2})). (2.6)

The boundary map from the exact sequence

(2.7) 0 → Ω_{X}^{2}/O_{X} ⊗ Ω_{K}^{2} → Ω_{X}^{2}{D}/(O_{X}(D − D) ⊗ Ω_{K}^{2})

→ Ω_{X}^{2}{D}/(Ω_{X}^{2} + O_{X}(D − D) ⊗ Ω_{K}^{2}) → 0

together with the evident map H^{1}(X, Ω_{X}^{2}/O_{X} ⊗ Ω_{K}^{2}) → AD^{2}(X/K) enables us to define an element, which we denote by abuse of notation

Tr(dgg^{−1}A) ∈ AD^{2}(X/K). (2.8)

Proposition 2.4. Let (E, ∇) be a pseudo-admissible vertical connection on X. The element Tr(dgg^{−1}A) (2.8) is independent of the choice of local bases around D. The element

(2.9) −{(ω(D), s), (det(E), det(∇))} + Tr(dgg^{−1}A) ∈ AD^{2}(X/K)

is independent both of the choice of local bases and the trivializing meromorphic section s of ω(D).

Proof. We show first that Tr(dgg^{−1}A) is independent of the local bases. We work locally around a point x which is a singular point of the connection with multiplicity m. We assume first that s = \frac{dz}{z^{m}} for a local coordinate, and that the connection matrix is

A = \frac{gdz}{z^{m}} + \frac{η}{z^{m}} .

with g ∈ GL_{r}(O) and η ∈ M_{r}(O ⊗ Ω_{K}^{1}). (This includes both the admissible and pseudolog cases.) We take a gauge transformation of the form A ↦ ϕAϕ^{−1} + dϕϕ^{−1} with ϕ ∈ GL_{r}(O). This amounts to

g ↦ ϕgϕ^{−1} + z^{m} \frac{dϕ}{dz}ϕ^{−1}; η ↦ ϕηϕ^{−1} + z^{m}d_{K}ϕϕ^{−1}.

Here d = d_{z} + d_{K}. We claim

Tr(dgg^{−1}A) ∈ Ω_{X}^{2}(D)/(O(D) ⊗ Ω_{K}^{2} + Ω_{X}^{2})

A FORMULA FOR GAUSS-MANIN DETERMINANTS 11

is invariant. We compute (writing T (A) := Tr(dgg^{−1}A), and computing modulo Ω_{X}^{2}) (2.10) T (ϕAϕ^{−1} + dϕϕ^{−1}) =

Tr(d(ϕgϕ^{−1} + z^{m} \frac{dϕ}{dz}ϕ^{−1})(ϕgϕ^{−1} + z^{m} \frac{dϕ}{dz}ϕ^{−1})^{−1}(ϕAϕ^{−1} + dϕϕ^{−1})) ≡

Tr(d(ϕgϕ^{−1} + z^{m} \frac{dϕ}{dz}ϕ^{−1})ϕg^{−1}ϕ^{−1}ϕAϕ^{−1}) ≡

Tr((dϕϕ^{−1} + ϕdgg^{−1}ϕ^{−1} − ϕgϕ^{−1}dϕϕ^{−1}ϕg^{−1}ϕ^{−1}

+ mz^{m−1} \frac{dϕ}{dz}g^{−1}ϕ^{−1}dz)ϕAϕ^{−1}) ≡

Tr(ϕ^{−1}dϕA + dgg^{−1}A − gϕ^{−1}dϕg^{−1}A + mz^{m−1}ϕ^{−1} \frac{dϕ}{dz}g^{−1}dz ∧ A) ≡

Tr(dgg^{−1}A + ϕ^{−1}dϕ(A − g^{−1}Ag) + mz^{m−1}ϕ^{−1} \frac{dϕ}{dz}g^{−1}dz ∧ A) ≡

Tr(dgg^{−1}A + ϕ^{−1} \frac{dϕ}{dz}g^{−1}dz(gA − Ag + mz^{m−1}A))

(Note that gA − Ag has entries in Ω_{K}^{1} ⊗ K(X), justifying replacing dϕ by \frac{dϕ}{dz}dz.) To show invariance, it will suffice to show

dz ∧ (gA − Ag + mz^{m−1}A) ≡ 0 mod Ω_{X}^{2}.

This expression can be written

(∗) = \frac{dz}{z^{m}} ∧ [g, η] + m\frac{dz}{z} ∧ η.

Verticality gives

dg ∧ \frac{dz}{z^{m}} + mη ∧ \frac{dz}{z^{m+1}} + \frac{d_{z}η}{z^{m}} = \frac{dz}{z^{2m}} [g, η]

Multiplying through by z^{m}, the expression (∗) above becomes

(∗) ≡ m\frac{dz}{z} ∧ η + mη ∧ \frac{dz}{z} ≡ 0 mod Ω_{X}^{2}.

It remains to show independence of s. Let s′ = fs where f is meromorphic on X and invertible on D. Consider a diagram

H^{0}(O_{D}^{×})  ⟶^{∂} H^{1}((1 + I_{D})^{×}) = Pic(X, D) ⟶ H^{1}(O_{X}^{×}) = Pic(X)
×  ×  ×
Ω_{X}^{1}{D}/(Ω_{X}^{1} + d log j_{∗}O_{X−D}^{×}) ⟵ ℍ^{1}(j_{∗}O_{X−D}^{×} → Ω_{X}^{1}{D}) ⟵ ℍ^{1}(O_{X}^{×} → Ω_{X}^{1})
↓  ↓  ↓
Ω_{X}^{2}{D}/(Ω_{X}^{2} + d log j_{∗}K_{2,O_{X−D}} ) ⟶  AD^{2}(X/K)  AD^{2}(X/K)

(2.11)

12 SPENCER BLOCH AND HÉLÈNE ESNAULT

The classes of the rigidified bundles (ω(D), s) and (ω(D), fs) differ by ∂(f̃), where f̃ ∈ H^{0}(O_{D}) is the image of f. It follows that (with notation as above)

(2.12) \{(ω(D), fs)(ω(D), s)^{−1}, (det(E), det(∇))\} = Tr(\frac{df̃}{f̃}A)

Replacing s with fs in (2.9), this gives the desired invariance.

To complete the construction of the class γ(E, ∇) ∈ AD^{2}(X/K) we need one more invariant. Let x ∈ D and assume m ≥ 2, where m is the multiplicity of x in D. Let z be a local coordinate at x. Write the local relative connection ∇_{/K} = d + g \frac{dz}{z^{m}} = \frac{g_{m}dz}{z^{m}} + \frac{g_{m−1}dz}{z^{m−1}} + … with g_{m} ∈ GL_{r}(K).

Definition 2.5. If the multiplicity is ≥ 2, the invariant

τ_{x}(E, ∇) := \frac{m}{2}d log(det(g_{m})) ∈ \frac{1}{2}d log(K^{×})/d log(K^{×})

is associated to the rank 1 quadratic form

det g_{m}^{m} ∈ K^{×}/(K^{×})^{2} ≅ H^{1}(K, ℤ/2ℤ).

In a point of multiplicity 1, we set τ_{x}(E, ∇) = 1.

A change of local gauge replaces g with hgh^{−1} + z^{m} \frac{dh}{dz}h^{−1} with h ∈ GL_{r}(K[[z]]). It follows that det(g_{m}) is invariant under gauge transformation of m ≥ 2. On the other hand, replacing z with z′ = uz leads to g′_{m} = u^{m−1}g_{m}, whence det(g′_{m}) = u^{r(m−1)} det(g_{m}) and

\frac{m}{2}d log(det(g′_{m})) = \frac{m}{2}d log(det(g_{m})) + \frac{rm(m − 1)}{2}d log(u),

so the definition is independent of the choice of z.

For x ∈ X(K), we define a map

ρ_{x} : K^{×}/K^{×2} → AD^{2}(X/K) (2.13)

as follows. One has a map of exact sequence of complexes

\begin{matrix}K_{2X} & ⟶ & j_{x∗}K_{2,X−\{x\}} & ⟶ & i_{x∗}K^{×} \\ ↓ & & ↓ & & ↓^{a} \\ Ω_{X}^{2}/O_{X} ⊗ Ω_{K}^{2} & ⟶ & Ω_{X}^{2}\{x\}/O_{X} ⊗ Ω_{K}^{2} & ⟶ & Ω_{X}^{2}\{x\}/(O_{X} ⊗ Ω_{K}^{2} + Ω_{X}^{2}). \end{matrix} (2.14)

Write z for a local coordinate and let a be as in (2.14). The mapping

K^{×} ⊗ ℚ/ℤ → coker(a); κ ⊗ \frac{1}{n} ↦ \frac{1}{n} \frac{dz}{z} ∧ \frac{dκ}{κ} (2.15)

is well-defined. We compose this with the boundary map from (2.14) to define ρ_{x}.

Definition 2.6. Let (E, ∇) be a pseudo-admissible connection as above. Then

(2.16) γ(E, ∇) = −{(ω(D), s), (det(E), det(∇))}

+ Tr(dgg^{−1}A) + ∑_{x∈D,m_{x}≥2} ρ_{x}(τ_{x}(E, ∇)) ∈ AD^{2}(X/K).

We continue to assume f : X → Spec (K) is a smooth, projective curve. The transfer map f_{∗} : AD^{2}(X/K) → Ω_{K}^{1}/d log(K^{×}) is defined as follows. We remark that H^{2}(X, K_{2}) = (0) and Ω_{X}^{2}/O_{X} ⊗ Ω_{K}^{2} ≅ ω_{X/K} ⊗ Ω_{K}^{1}, and we define f_{∗} from the diagram

H^{1}(X, K_{2}) ⟶ H^{1}(X, ω_{X/K}) ⊗ Ω_{K}^{1} ⟶ AD^{2}(X/K) ⟶ 0
↓^{Tr}  ↓≅  ↓^{f_{∗}}
K^{×}  ⟶  Ω_{K}^{1}  ⟶ Ω_{K}^{1}/d log(K^{×}) ⟶ 0

(2.17)

The check that with γ(E, ∇) defined as in (2.16), f_{∗}γ(E, ∇) has the form as in theorem 1.3 is straightforward and will be omitted. Since the trace map Tr : H^{1}(X, K_{2}) → K^{×}, which is simply defined on the generators ⊕_{x}λ_{x} ∈ ⊕_{x∈X^{(1)}}K(x)^{×} by Tr_{[K(x):K]}d log λ_{x}, is surjective, we obtain the

Lemma 2.7. The transfer map

f_{∗} : AD^{2}(X/K) → AD^{1}(K) = Ω_{K}^{1}/d log K^{×}

is an isomorphism.

Now we are in the position to give a slightly more precise formulation of our main theorem.

Theorem 2.8. Let (E, ∇) be a special pseudo-admissible connection on ℙ_{K}^{1}, smooth over ∅ ≠ U ⊂ ℙ_{K}^{1}, such that the singularities ℙ_{K}^{1}\U = D of ∇ consist of K-rational points. Then, with the notation of definition 2.6, one has

det H_{DR}(U/K, ∇_{/K}) = f_{∗}γ(E, ∇) ∈ AD^{1}(K) = Ω_{K}^{1}/d log K^{×}.

Finally, we take a moment to point out some simple consequences of theorem 2.8.

Remark 2.9. (i). Let (E, ∇) be an admissible connection on ℙ_{K}^{1}, and let (E^{∨}, ∇^{∨}) be the dual connection. then

det H_{DR}(E, ∇) ≅ det H_{DR}(E^{∨}, ∇^{∨})^{(−1)}. (2.18)

Indeed, replacing E with E^{∨} replaces g with −^{t}g and A with −^{t}A. Verticality and admissibility imply that [g, A] has no pole, so

Tr(d(−^{t}g)(−^{t}g)^{−1}(−^{t}A)) = −Tr(^{t}(g^{−1}dg)^{t}A) = −Tr(g^{−1}dgA)

= −Tr(dgAg^{−1}) = −Tr(dgg^{−1}A).

(ii). In [4], remark 3.3, equation 3.27, we give an example of an admissible connection on ℙ_{K}^{1} with trivial determinant, for which the determinant of the Gauß-Manin connection is not torsion. More generally, given the main theorem of [3], if (L, ∇_{L}) and (M, ∇_{M}) are two connections, say on ℙ_{K}^{1}, with the same singularities D, which are generic enough so that the singularites of (L ⊗ M, ∇_{L} ⊗ ∇_{M}) are exactly D as well, then det A, with

A = ( \begin{matrix}L ⊗ M & 0 & 0 \\ 0 & L^{−1} & 0 \\ 0 & 0 & M^{−1} \end{matrix} ) (2.19)

is trivial, while the main theorem of [3] says that the Gauß-Manin determinant is computed by

c_{1}(ω(D), Γ_{L} + Γ_{M}) · c_{1}(∇_{L} + ∇_{M}) (2.20)

−c_{1}(ω(D), Γ_{L}) · c_{1}(∇_{L}) − c_{1}(ω(D), Γ_{M}) · c_{1}(∇_{M}),

where Γ_{L} and Γ_{M} are the principal parts of ∇_{L} and ∇_{M}. For generic L and M, this won’t vanish.

3. Higgs and de Rham Traces

In this section we introduce the concepts of Higgs and de Rham operators associated to a vertical pseudo-admissible connection on the trivial bundle on ℙ_{K}^{1}, and analyse the difference between the traces of those operators.

It will be convenient to write E = V ⊗ O with V = Γ(ℙ^{1}, E). Let (e_{μ}), μ = 1, …, r be a basis of V. The connection ∇ will be a vertical pseudo-admissible connection on E with poles on D = ∑_{i=1}^{N}(a_{i}) where for simplicity we take a_{i} ∈ ℙ^{1}(K). We assume ∞ ∉ D. The relative connection is given by

∇_{/K}e_{μ} = ∑_{i=1}^{N} ∑_{r=1}^{m_{i}} \frac{g_{r}^{(i)}(e_{μ})dt}{(t − a_{i})^{r}}. (3.1)

Here the g_{r}^{(i)} are matrices with entries in K. The admissibility condition implies that g_{m_{i}}^{(i)} is invertible over O_{ℙ_{K}^{1},a_{i}}. Regularity of the connection

A FORMULA FOR GAUSS-MANIN DETERMINANTS 15

at infinity means

∑_{i=1}^{N} g_{1}^{(i)} = 0. (3.2)

Definition 3.1. Let M_{i} = m_{i} − 1 if m_{i} ≥ 2, and M_{i} = m_{i} = 1 else. The absolute connection has the following equation

∇e_{μ} = ∑_{i=1}^{N} ∑_{r=1}^{m_{i}} \frac{g_{r}^{(i)}(e_{μ})d(t − a_{i})}{(t − a_{i})^{r}} + ∑_{i=1}^{N} ∑_{r=1}^{M_{i}} \frac{η_{r}^{(i)}(e_{μ})}{(t − a_{i})^{r}} + η_{0}(e_{μ}), (3.3)

where the η are matrices with entries in Ω_{K}^{1}. Definition 3.2. We define γ_{K} by ∇_{/K} = d + γ_{K}, thus concretely

γ_{K} = ∑_{i=1}^{N} ∑_{r=1}^{m_{i}} \frac{g_{r}^{(i)}dt}{(t − a_{i})^{r}} .

We define η by

η = ∑_{i=1}^{N} ∑_{r=1}^{M_{i}} \frac{η_{r}^{(i)}}{(t − a_{i})^{r}} + η_{0},

and γ by

γ = ∑_{i=1}^{N} ∑_{r=1}^{m_{i}} \frac{g_{r}^{(i)}d(t − a_{i})}{(t − a_{i})^{r}},

so that ∇ = d + γ + η.

We have natural identifications

Γ(E(∗D)) = V [\frac{1}{t − a_{1}} , …, \frac{1}{t − a_{N}}] (3.4)

Γ(E ⊗ ω(∗D)) ⊊ V [\frac{1}{t − a_{1}} , …, \frac{1}{t − a_{N}}]dt (3.5)

where the strict inclusion in (3.5) comes from the requirement of no poles at infinity.

The following lemma will be useful in the sequel. Lemma 3.3. For integers r, s ≥ 1 one has a formal identity

\frac{1}{(t − a)^{r}(t − b)^{s}} = ∑_{p=1}^{r} \frac{A_{p}(a, b)}{(t − a)^{p}} + ∑_{q=1}^{s} \frac{B_{q}(a, b)}{(t − b)^{q}}.

One has

A_{r} = (a − b)^{−s}; B_{s} = (−1)^{r}(a − b)^{−r} = (b − a)^{−r}.

16 SPENCER BLOCH AND HÉLÈNE ESNAULT

The partial fraction expansion of \frac{1}{(t−b)^{s}(t−a)} begins

\frac{1}{(t − b)^{s}(t − a)} = \frac{(a − b)^{−s}}{t − a} − \frac{(a − b)^{−s}}{t − b} + …

In particular, we have

(3.6) \frac{1}{(t − b)^{s}} (\frac{d(a − b) ∧ dt}{t − a} − \frac{d(a_{N} − b) ∧ dt}{t − a_{N}})

= (a − b)^{−s}d(a − b) ∧ dt(\frac{1}{t − a} − \frac{1}{t − a_{N}} )

+ ((a_{N} − b)^{−s}d(a_{N} − b) ∧ dt − (a − b)^{−s}d(a − b) ∧ dt)

× (\frac{1}{t − b} − \frac{1}{t − a_{N}} ) + terms involving \frac{1}{(t − b)^{r}} for r ≥ 2.

Proof. We have

(3.7) \frac{1}{(t − b)^{s}(t − a)^{r}}

= \frac{1}{(s − 1)!(r − 1)!}(\frac{d}{db})^{s−1}(\frac{d}{da})^{r−1}(\frac{(a − b)^{−1}}{t − a} − \frac{(a − b)^{−1}}{t − b} ).

The formulas in the lemma follow easily from this. Definition 3.4. We denote by H ↪^{σ} Γ(E ⊗ ω(∗D)) be the K-vector subspace with basis

\frac{e_{μ}dt}{(t − a_{i})^{r}} , \begin{cases}2 ≤ r ≤ m_{i} & 1 ≤ i ≤ N − 1 \\ 2 ≤ r ≤ m_{N} − 1 & i = N \end{cases} , (3.8)

e_{μ}dt(\frac{1}{t − a_{i}} − \frac{1}{t − a_{N}} ), i < N.

There are two splittings of σ, π_{γ} : Γ(E ⊗ ω(∗D)) → Γ(E ⊗ ω(∗D))/γ_{K}Γ(E(∗D)) ≅ H π_{∇} : Γ(E ⊗ ω(∗D)) → Γ(E ⊗ ω(∗D))/∇_{/K}Γ(E(∗D)) ≅ H

There is a multiplication map

η : Γ(E ⊗ ω(∗D)) → Γ(E ⊗ ω(∗D)) ⊗ Ω_{K}^{1}.

We define now two K-linear operators. Definition 3.5. The composite map

η_{γ} := π_{γ} ∘ η ∘ σ : H → H ⊗ Ω_{K}^{1}

(resp.

η_{∇} := π_{∇} ∘ η ∘ σ : H → H ⊗ Ω_{K}^{1})

will be called the Higgs (resp. de Rham) operator. Similarly, if h is one of the terms \frac{η_{r}^{(i)}}{(t−a_{i})^{r}} appearing in the definition of η, we denote by h_{γ} and h_{∇} the corresponding Higgs and de Rham operators.

The rest of this section is devoted to the comparison of the trace of those two K-linear operators. We will show

Theorem 3.6. Let (E, ∇) be a pseudo-admissible connection on the trivial bundle E ≅ ⊕_{1}^{r}O_{ℙ_{K}^{1}} on ℙ_{K}^{1} having at least one singularity of order ≥ 2. Then

Tr(η_{γ} − η_{∇}) ≡ \frac{1}{2} ∑_{m_{i}≥2} m_{i}d log(det(g_{m_{i}}^{(i)})) mod d log K^{×}.

We assume henceforth that m_{N} ≥ 2.

Suppose first a_{i} is a pseudo-logarithmic point for the connection. We write h = \frac{η^{(i)}}{t−a_{i}}, and we compute Tr(h_{∇}) − Tr(h_{γ}).

The notation x = y + (H) will mean x and y differ by an element in H. The pattern is then we take x in the basis of H. We write

h(x) = γ_{K}y + (H) = ∇_{/K}y′ + (H). (3.9)

Then

z := h_{∇}(x) − h_{γ}(x) = −dy′ + γ_{K}(y − y′). (3.10)

Of course, if h(x) ∈ H then y = y′ = z = 0. Also, we are only interested in the trace, so if the expansion of z in the basis of H does not involve x, we can ignore it. Suppose e.g. x = \frac{e_{μ}dt}{(t−a_{j})^{r}} , j ≠ i.

Then h(x) = \frac{(∗)dt}{(t−a_{i})(t−a_{j})^{r}} ∈ H (the condition to lie in H amounts to a bound on the pole order together with no pole of the differential form at infinity.) Thus such elements x contribute 0. Similarly, if j ≠ i then

\frac{(∗)dt}{(t − a_{i})(t − a_{j})} − \frac{(∗)dt}{(t − a_{i})(t − a_{N})} ∈ H (3.11)

so

x = e_{μ}dt(\frac{1}{t − a_{j}} − \frac{1}{t − a_{N}} ) (3.12)

contributes 0.

It remains to consider x = e_{μ}dt(\frac{1}{t−a_{i}} − \frac{1}{t−a_{N}} ). We have

h(x) = \frac{η_{1}^{(i)}(e_{μ})dt}{(t − a_{i})^{2}} + (H). (3.13)

18 SPENCER BLOCH AND HÉLÈNE ESNAULT

So we can take

y = \frac{(g_{1}^{(i)})^{−1}η_{1}^{(i)}(e_{μ})dt}{t − a_{i}}; y′ = \frac{(g_{1}^{(i)} − I)^{−1}η_{1}^{(i)}(e_{μ})dt}{t − a_{i}}. (3.14)

Write γ_{K} = \frac{g_{1}^{(i)}dt}{t−a_{i}} + γ′_{K}. Then z = γ′_{K}(y − y′). (3.15) Since we are interested in the trace, we need only consider the coefficient of e_{μ}dt(\frac{1}{t−a_{i}} − \frac{1}{t−a_{N}} ) in the expansion of z in the basis of H. This coefficient is the coefficient of e_{μ} in

γ′_{K}|_{t=a_{i}}(y − y′) = ∑_{\begin{matrix}j≠i \\ r \end{matrix}}(a_{i} − a_{j})^{−r}g_{r}^{(j)}((g_{1}^{(i)} − I)^{−1} − (g_{1}^{(i)})^{−1})η_{1}^{(i)}(e_{μ}). (3.16)

Summing over μ yields finally

Tr(h_{∇} − h_{γ}) = Tr( ∑_{\begin{matrix}j≠i \\ r \end{matrix}}(a_{i} − a_{j})^{−r}g_{r}^{(j)}((g_{1}^{(i)} − I)^{−1} − (g_{1}^{(i)})^{−1})η_{1}^{(i)}). (3.17)

Next we consider i with m_{i} ≥ 2. Take first

x = \frac{e_{μ}dt}{(t − a_{i})^{r}}; \begin{cases}2 ≤ r ≤ m_{i} & 1 ≤ i ≤ N − 1 \\ 2 ≤ r ≤ m_{N} − 1 & i = N \end{cases} .

Since we have already handled the h = \frac{η^{(i)}}{t−a_{i}} in pseudo-logarithmic points, we introduce the notation

(3.18) η′ = η − ∑_{pseudo−log} \frac{η^{(i)}}{t − a_{i}} ,

γ′_{K} = γ_{K} − ∑_{pseudo−log} \frac{g_{i}^{(i)}}{(t − a_{i})}dt.

Take y_{0} = (g_{m_{i}}^{(i)})^{−1}η_{m_{i}−1}^{(i)}(e_{μ})/(t − a_{i})^{r−1}. (3.19) It follows from lemma 3.3 that η′x = γ′_{K}y_{0} + lower order terms. Here “lower order terms” means terms with denominators (t − a_{j})^{s} where s ≤ m_{j}, j ≠ i and s ≤ m_{i} + r − 2 for j = i. Now continue in this way,

A FORMULA FOR GAUSS-MANIN DETERMINANTS 19

replacing y_{0} by

y = y_{0} + ∑_{s=0}^{r−2} \frac{v_{s}}{(t − a_{i})^{s}} . (3.20)

We may write

η′(e_{μ})dt/(t − a_{i})^{r} = γ′_{K}y + ∑_{j=1}^{N−1} ∑_{u=1}^{m_{j}} \frac{w_{j,u}dt}{(t − a_{j})^{u}} + ∑_{u=1}^{m_{N} −1} \frac{w_{N,u}dt}{(t − a_{N})^{u}} . (3.21)

From equations (3.19) and (3.20) we may also write

(3.22) \frac{η′(e_{μ})dt}{(t − a_{i})^{r}} = (d + γ′_{K})y + ∑_{j=1}^{N−1} ∑_{u=1}^{m_{j}} \frac{w_{j,u}dt}{(t − a_{j})^{u}} + ∑_{u=1}^{m_{N} −1} \frac{w_{N,u}dt}{(t − a_{N})^{u}}

+ (r − 1)(g_{m_{i}}^{(i)})^{−1}η_{m_{i}−1}^{(i)}(e_{μ})dt/(t − a)^{r} + ∑_{s=1}^{r−2} \frac{sv_{s}dt}{(t − a_{i})^{s+1}} .

For r = 1 we may write for suitable v, w_{j,u} ∈ Γ(E)

η′(e_{μ})dt(\frac{1}{t − a_{i}} − \frac{1}{t − a_{N}}) = γ′_{K}v + ∑_{j=1}^{N−1} ∑_{u=1}^{m_{j}} \frac{w_{j,u}dt}{(t − a_{j})^{u}} + ∑_{u=1}^{m_{N} −1} \frac{w_{N,u}dt}{(t − a_{N})^{u}} . (3.23)

Since dv = 0 in (3.23) we conclude from equations (3.17), (3.22) and (3.23) that

(3.24) Tr(η_{∇} − η_{γ}) = ∑_{i=1}^{N−1} ∑_{r=2}^{m_{i}}(r − 1)Tr((g_{m_{i}}^{(i)})^{−1}η_{m_{i}−1}^{(i)})

+ ∑_{r=2}^{m_{N} −1}(r − 1)Tr((g_{m_{N}}^{(N)})^{−1}η_{m_{N} −1}^{(N)})

+ ∑_{\begin{matrix}i \\ m_{i}=1 \end{matrix}} Tr( ∑_{\begin{matrix}j≠i \\ r \end{matrix}}(a_{i} − a_{j})^{−r}g_{r}^{(j)}((g_{1}^{(i)} − I)^{−1} − (g_{1}^{(i)})^{−1})η_{1}^{(i)}).

(Notice that replacing γ_{K} by γ′_{K} in (3.21), (3.22) and (3.23) will not affect the trace calculation.) We now use the verticality condition dA = A ∧ A mod Ω_{K}^{2} ⊗ K(X).

(3.25) 0 = ∑_{i=1}^{N} ∑_{r=1}^{m_{i}} \frac{dg_{r}^{(i)}dt}{(t − a_{i})^{r}} + ∑_{i=1}^{N} ∑_{s=1}^{M_{i}} \frac{sη_{s}^{(i)}dt}{(t − a_{i})^{s+1}}

+ ∑_{i,j=1}^{N} ∑_{r,s=1}^{m_{i},M_{j}}[g_{r}^{(i)}, η_{s}^{(j)}]\frac{dt}{(t − a_{i})^{r}(t − a_{j})^{s}} + ∑_{i=1}^{N} ∑_{r=1}^{m_{i}}[g_{r}^{(i)}, η_{0}]\frac{dt}{(t − a_{i})^{r}} .

Dropping terms with poles at t = a_{i} of degree > M_{i} + 1, we find

(3.26) 0 = ∑_{i=1}^{N} ∑_{r=1}^{m_{i}} \frac{dg_{r}^{(i)}dt}{(t − a_{i})^{r}} + ∑_{i=1}^{N} ∑_{s=1}^{M_{i}} \frac{sη_{s}^{(i)}dt}{(t − a_{i})^{s+1}}

+ ∑_{i,j=1; i≠j}^{N} ∑_{r,s=1}^{m_{i},M_{j}}[g_{r}^{(i)}, η_{s}^{(j)}]\frac{dt}{(t − a_{i})^{r}(t − a_{j})^{s}}

+ ∑_{i=1}^{N} ∑_{r+s≤M_{i}+1}[g_{r}^{(i)}, η_{s}^{(i)}]\frac{dt}{(t − a_{i})^{r+s}} + ∑_{i=1}^{N} ∑_{r=1}^{m_{i}}[g_{r}^{(i)}, η_{0}]\frac{dt}{(t − a_{i})^{r}} .

In an admissible point, we calculate as before, multiplying through by

(g_{m_{i}}^{(i)})^{−1}(t − a_{i})^{m_{i}}/dt

and set t = a_{i} to get

(3.27) 0 = (g_{m_{i}}^{(i)})^{−1}dg_{m_{i}}^{(i)} + (m_{i} − 1)(g_{m_{i}}^{(i)})^{−1}η_{m_{i}−1}^{(i)}

+ ∑_{j≠i} ∑_{s=1}^{M_{j}}(g_{m_{i}}^{(i)})^{−1}[g_{m_{i}}^{(i)}, η_{s}^{(j)}]/(a_{i} − a_{j})^{s} + ∑_{r+s=m_{i}}(g_{m_{i}}^{(i)})^{−1}[g_{r}^{(i)}, η_{s}^{(i)}]

+ (g_{m_{i}}^{(i)})^{−1}[g_{m_{i}}^{(i)}, η_{0}].

Taking traces gives

(3.28) 0 = d log(det(g_{m_{i}}^{(i)})) + (m_{i} − 1)Tr((g_{m_{i}}^{(i)})^{−1}η_{m_{i}−1}^{(i)})

+ ∑_{r+s=m_{i}} Tr((g_{m_{i}}^{(i)})^{−1}[g_{r}^{(i)}, η_{s}^{(i)}]).

On the other hand, in a pseudo-logarithmic point, we multiply through by (t − a_{i})^{2}/dt and then set t = a_{i} getting

η_{1}^{(i)} = [η_{1}^{(i)}, g_{1}^{(i)}]. (3.29)

Now discard those terms, multiply by (t − a_{i})/dt and set t = a_{i}. One gets

0 = dg_{1}^{(i)} + ∑_{j≠i} ∑_{s=1}^{M_{j}}(a_{i} − a_{j})^{−s}([g_{1}^{(i)}, η_{s}^{(j)}] + [g_{s}^{(j)}, η_{1}^{(i)}]) + [g_{1}^{(i)}, η_{0}]. (3.30)

Formula (3.29) gives

η_{1}^{(i)}(g_{1}^{(i)} − I) = g_{1}^{(i)}η_{1}^{(i)}, (3.31) whence, assuming the indicated matrices invertible, one has

η_{1}^{(i)}(g_{1}^{(i)} − I)^{−1} = (g_{1}^{(i)})^{−1}η_{1}^{(i)}. (3.32) Using Tr(a[b, c]) = 0 if [a, b] = 0, multiplying equation (3.30) on the left by (g_{1}^{(i)})^{−1} (resp. by (g_{1}^{(i)} − I)^{−1}) and taking traces yields

Tr( ∑_{j≠i} ∑_{s=1}^{M_{j}}(a_{i} − a_{j})^{−s}(g_{1}^{(i)})^{−1}[g_{s}^{(j)}, η_{1}^{(i)}]) ∈ d log K^{×}, (3.33)

Tr( ∑_{j≠i} ∑_{s=1}^{M_{j}}(a_{i} − a_{j})^{−s}(g_{1}^{(i)} − I)^{−1}[g_{s}^{(j)}, η_{1}^{(i)}]) ∈ d log K^{×}.

We now get

(3.34) ∑_{\begin{matrix}i \\ m_{i}=1 \end{matrix}} Tr( ∑_{\begin{matrix}j≠i \\ r \end{matrix}}(a_{i} − a_{j})^{−r}g_{r}^{(j)}((g_{1}^{(i)} − I)^{−1} − (g_{1}^{(i)})^{−1})η_{1}^{(i)})

= ∑_{\begin{matrix}i \\ m_{i}=1 \end{matrix}} Tr( ∑_{\begin{matrix}j≠i \\ r \end{matrix}}(a_{i} − a_{j})^{−r}((g_{1}^{(i)} − I)^{−1} − (g_{1}^{(i)})^{−1})η_{1}^{(i)}g_{r}^{(j)})

≡ ∑_{\begin{matrix}i \\ m_{i}=1 \end{matrix}} Tr( ∑_{\begin{matrix}j≠i \\ r \end{matrix}}(a_{i} − a_{j})^{−r}(η_{1}^{(i)}(g_{1}^{(i)} − I)^{−1} − (g_{1}^{(i)})^{−1}η_{1}^{(i)})g_{r}^{(j)})

≡ 0 mod (d log K^{×}).

Now one can compare (3.24), (3.28), and (3.34) and deduce:

Proposition 3.7. With notation as in definition 3.5 above, and assuming that a_{N} has multiplicty m_{N} ≥ 2, one has

Tr(η_{∇} −η_{γ}) ≡ − ∑_{i=1}^{N−1} \frac{m_{i}}{2} (d log(det(g_{m_{i}}^{(i)}))+Tr ∑_{r+s=m_{i}}(g_{m_{i}}^{(i)})^{−1}[g_{r}^{(i)}, η_{s}^{(i)}])

− \frac{m_{N} − 2}{2}(d log(det(g_{m_{N}}^{(N)})) + Tr ∑_{r+s=m_{N}}(g_{m_{N}}^{(N)})^{−1}[g_{r}^{(N)}, η_{s}^{(N)}])

mod d log K^{×}.

To complete the proof of theorem 3.6, we must show

Proposition 3.8. Let (E, ∇) be a vertical admissible connection, with local equation

\frac{g_{m}dz}{z^{m}} + \frac{g_{m−1}dz}{z^{m−1}} + … + \frac{η_{m−1}}{z^{m−1}} + \frac{η_{m−2}}{z^{m−2}} + …

in an admissible point. Then

Φ = Tr(g_{m}^{−1} ∑_{s=0}^{m−1}[g_{m−s}, η_{s}]) = 0

Proof. The connection ∇ is vertical. Vanishing for curvature terms involving z^{−p} for p ≥ m + 1 implies

[g_{m}, η_{ℓ}] + [g_{m−1}, η_{ℓ+1}] + … [g_{ℓ+1}, η_{m−1}] = 0 (3.35)

for ℓ = 1, …, m − 1.

The tactic is to eliminate first η_{1} from Φ, then η_{2} etc. One easily verifies matrix relations

[a^{−1}, b] = −a^{−1}[a, b]a^{−1} (3.36)

Tr(g^{−1}[a, b]) = −Tr(a[g^{−1}, b]) = Tr(ag^{−1}[g, b]g^{−1})
Tr(ag^{−1}[b, η]g^{−1}) = Tr(ag^{−1}bηg^{−1}) − Tr(bg^{−1}ag^{−1}η).

In particular,

Tr(ag^{−1}[b, η]g^{−1}) = Tr(ag^{−1}bg^{−1}[g, η]g^{−1}) if ag^{−1}b = bg^{−1}a. (3.37)

Write

Φ = Trg_{m}^{−1}[g_{m−1}, η_{1}] + Tr ∑_{s=2}^{m−1} g_{m}^{−1}[g_{m−s}, η_{s}]. (3.38)

This yields Φ = −Trg_{m}^{−1}g_{m−1}g_{m}^{−1}[g_{m−1}, η_{2}] + Trg_{m}^{−1}[g_{m−2}, η_{2}] (3.39)

−Tr ∑_{s=3}^{m−1} g_{m}^{−1}g_{m−1}g_{m}^{−1}[g_{m−s+1}, η_{s}] + Tr ∑_{s=3}^{m−1} g_{m}^{−1}[g_{m−s}, η_{s}].

Applying again now the relations (3.36) yields Φ = Tr(−g_{m}^{−1}g_{m−1}g_{m}^{−1}g_{m−1}g_{m}^{−1} + g_{m}^{−1}g_{m−2}g_{m}^{−1})[g_{m}, η_{2}] (3.40)

−Tr ∑_{s=3}^{m−1} g_{m}^{−1}g_{m−1}g_{m}^{−1}[g_{m−s+1}, η_{s}] + Tr ∑_{s=3}^{m−1} g_{m}^{−1}[g_{m−s}, η_{s}].

Assume inductively that for some t ≥ 2, one can write Φ as follows:

Φ = Tr(∑_{a=1}^{t}(−1)^{a−1} ∑_{τ_{1}+...+τ_{a}=t} g_{m}^{−1}g_{m−τ_{1}} ⋯ g_{m}^{−1}g_{m−τ_{a}}g_{m}^{−1})[g_{m}, η_{t}] (3.41)

+Tr ∑_{s=t+1}^{m−1} ∑_{ℓ=0}^{t−1}(∑_{a=0}^{ℓ}(−1)^{a} ∑_{τ_{1}+...+τ_{a}=ℓ} g_{m}^{−1}g_{m−τ_{1}} ⋯ g_{m}^{−1}g_{m−τ_{a}}g_{m}^{−1})[g_{m−s+ℓ}, η_{s}].

Applying (3.35) to the first line, and isolating the terms in η_{t+1} and in η_{s}, s ≥ (t + 2), one obtains Φ = F (t + 1) + (3.42)

Tr ∑_{s=t+2}^{m−1} ∑_{ℓ=0}^{t}(∑_{a=0}^{ℓ}(−1)^{a} ∑_{τ_{1}+...+τ_{a}=ℓ} g_{m}^{−1}g_{m−τ_{1}} ⋯ g_{m}^{−1}g_{m−τ_{a}}g_{m}^{−1})[g_{m−s+ℓ}, η_{s}],

with

F (t + 1) = Tr(∑_{a=1}^{t}(−1)^{a} ∑_{τ_{1}+...τ_{a}=t} g_{m}^{−1}g_{m−τ_{1}} ⋯ g_{m}^{−1}g_{m−τ_{a}}g_{m}^{−1})[g_{m−1}, η_{t+1}] (3.43)

+Tr ∑_{ℓ=0}^{t−1}(∑_{a=0}^{ℓ}(−1)^{a} ∑_{τ_{1}+...+τ_{a}=ℓ} g_{m}^{−1}g_{m−τ_{1}} ⋯ g_{m}^{−1}g_{m−τ_{a}}g_{m}^{−1})[g_{m−t−1+ℓ}, η_{t+1}].

It remains to arrange F (t + 1). To this aim, write

F (t + 1) = ∑_{ℓ=0}^{t} ∑_{a=0}^{ℓ} ∑_{τ_{1}+...+τ_{a}=ℓ} Tr (3.44)

((−1)^{a}g_{m−τ_{1}}g_{m}^{−1} ⋯ g_{m−t−1+ℓ}η_{t+1}g_{m}^{−1} − (−1)^{a}g_{m−t−1+ℓ}g_{m}^{−1} ⋯ g_{m−τ_{a}}g_{m}^{−1}η_{t+1}). Now we group those terms differently. To a tuple (τ_{1}, …, τ_{a}), with τ_{1}+…+τ_{a} = ℓ, we associate the tuple (τ′_{1}, …, τ′_{a}) with τ′_{1}+… τ′_{a}+τ_{1} = t + 1, τ′_{a} = t + 1 − ℓ, and otherwise τ_{i} = τ′_{i−1} for i ≥ 2. Using the first relation of (3.35) again, this gives for those 2 terms together

Tr(−1)^{a}g_{m}^{−1}g_{m−τ_{1}}g_{m}^{−1} ⋯ g_{m−t−1+ℓ}g_{m}^{−1}[g_{m}, η_{t+1}]. (3.45)

This shows that the relation (3.41) is true, with t replaced by t + 1. As the last equation of (3.35) for ℓ = m − 1 is [g_{m}, η_{m−1}] = 0, one obtains by induction that Φ vanishes on the variety defined by (3.35).

Remark 3.9. Writing a_{i} = g_{i} and b_{i} = η_{i−1}, the above proposition can be restated as follows. Suppose a(t) = a_{m}t^{m} + … + a_{1}t and b(t) = b_{m}t^{m} + … + b_{1}t are polynomials with matrix coefficients satisfying a(0) = b(0) = 0 and a_{m} invertible. Assume [a(t), b(t)] = c_{m}t^{m} + lower order terms. Then Tr(a_{m}^{−1}c_{m}) = 0.

4. The Gauß-Manin Determinant: Step 1

In this section we begin the computation of the Gauß-Manin determinant appearing in the main theorem 2.8.

We keep the same notations as in section 3. In particular, E is a trivial bundle on ℙ_{K}^{1} with basis e_{μ}, having at least one point of multiplicity ≥ 2, D = {a_{1}, …, a_{N}}, and H ↪ Γ(E ⊗ ω(∗D)) is the Ksubspace with basis defined in 3.4. We continue to write ∇ = d + γ + η and ∇_{/K} = d + γ_{K} as in definition 3.2.

The Gauß-Manin connection is computed from the diagram

Γ(E(∗D))  Γ(E(∗D))
↓^{∇}  ↓^{∇_{/K}}
Γ(E(∗D)) ⊗ Ω_{K}^{1} ⟶ Γ(E ⊗ Ω^{1}(∗D)) ⟶^{←^{s}} Γ(E ⊗ ω(∗D))
↓^{∇_{/K}⊗1}  ↓^{∇}
Γ(E ⊗ ω(∗D)) ⊗ Ω_{K}^{1} ⟶_{ι}^{≅} Γ(E ⊗ Ω^{2}(∗D)/F^{2})

(4.1)

Here in the central column Ω refers to the Kähler differentials Ω_{ℙ_{K}^{1}/k}, and F^{2} := O_{ℙ^{1}}(∗D) ⊗ Ω_{K}^{2} ⊂ Ω^{2}(∗D). We are interested in the induced map from H_{DR}^{1} = H_{DR/K}^{1}(ℙ^{1} − D, (E, ∇)), which is the cokernel of the right hand column, to H_{DR}^{1} ⊗_{K} Ω_{K}^{1}, which is the cokernel of the left hand column. Let p : Γ(E ⊗ ω(∗D)) → H_{DR}^{1} be the projection. By construction, H ↪ Γ(E ⊗ ω(∗D)) splits p. Let q : H_{DR}^{1} ≅ H denote the splitting. The section s is given on H by

(4.2) s(\frac{e_{μ}dt}{(t − a_{i})^{r}} ) = \frac{e_{μ}d(t − a_{i})}{(t − a_{i})^{r}};

s(e_{μ}dt(\frac{1}{t − a_{i}} − \frac{1}{t − a_{N}} )) = \frac{e_{μ}d(t − a_{i})}{t − a_{i}} − \frac{e_{μ}d(t − a_{N})}{t − a_{N}}.

Lemma 4.1. The Gauß-Manin determinant is the trace of the map

(q ⊗ 1) ∘ (p ⊗ 1) ∘ ι^{−1} ∘ ∇ ∘ s : H → H.

Proof. Straightforward.

Explicitly, this map is obtained by applying the projection (q ⊗ 1) ∘ (p ⊗ 1) to the right hand side in

(4.3) \frac{e_{μ}dt}{(t − a_{i})^{r}} ↦ ∑_{j=1}^{N} ∑_{s=1}^{m_{j}} \frac{g_{s}^{(j)}(e_{μ})d(a_{i} − a_{j}) ∧ dt}{(t − a_{j})^{s}(t − a_{i})^{r}}

+ ∑_{j} ∑_{s=1}^{M_{j}} \frac{η_{s}^{(j)}(e_{μ}) ∧ dt}{(t − a_{j})^{s}(t − a_{i})^{r}}

(4.4) e_{μ}dt(\frac{1}{t − a_{i}} − \frac{1}{t − a_{N}} ) ↦

∑_{j=1}^{N} ∑_{s=1}^{m_{j}} g_{s}^{(j)}(e_{μ})(\frac{d(a_{i} − a_{j}) ∧ dt}{(t − a_{j})^{s}(t − a_{i})} − \frac{d(a_{N} − a_{j}) ∧ dt}{(t − a_{j})^{s}(t − a_{N})})

+ ∑_{j} ∑_{s=1}^{M_{j}} η_{s}^{(j)}(e_{μ}) ∧ dt(\frac{1}{(t − a_{j})^{s}(t − a_{i})} − \frac{1}{(t − a_{j})^{s}(t − a_{N})}).

We leave aside for the moment the terms in the trace involving η and focus on the trace of the map which we rewrite using lemma 3.3 as

(4.5) \frac{e_{μ}dt}{(t − a_{i})^{r}} ↦ ∑_{j=1}^{N} ∑_{s=1}^{m_{j}} g_{s}^{(j)}(e_{μ})d(a_{i} − a_{j}) ∧ dt(\frac{(a_{i} − a_{j})^{−s}}{(t − a_{i})^{r}} + …)

(4.6) e_{μ}dt(\frac{1}{t − a_{i}} − \frac{1}{t − a_{N}} ) ↦

∑_{j=1}^{N} ∑_{s=1}^{m_{j}} g_{s}^{(j)}(e_{μ})[(a_{i} − a_{j})^{−s}d(a_{i} − a_{j}) ∧ dt(\frac{1}{t − a_{i}} − \frac{1}{t − a_{N}} )

+ ((a_{N} − a_{j})^{−s}d(a_{N} − a_{j}) ∧ dt − (a_{i} − a_{j})^{−s}d(a_{i} − a_{j}) ∧ dt)

× (\frac{1}{t − a_{j}} − \frac{1}{t − a_{N}} ) + …].

Terms are to be dropped if some factor becomes 0. The terms represented by ellipses (…) do not enter into the trace calculation. Also all terms lie in the K-span of the basis (3.19). Let Ψ denote the resulting endomorphism of H. The contribution to Tr(Ψ) from (4.5) is

(4.7) ∑_{i=1}^{N}(m_{i} − 1) ∑_{\begin{matrix}j=1 \\ j≠i \end{matrix}}^{N} ∑_{s=1}^{m_{j}} Tr(g_{s}^{(j)})(a_{i} − a_{j})^{−s}d(a_{i} − a_{j}) ∧ dt

− ∑_{j=1}^{N−1} ∑_{s=1}^{m_{j}} Tr(g_{s}^{(j)})(a_{N} − a_{j})^{−s}d(a_{N} − a_{j}) ∧ dt.

The second sum arises because \frac{e_{μ}dt}{(t−a_{N})^{m_{N}}} is not a basis element for H. The contribution from (4.6) is

(4.8) ∑_{i=1}^{N−1} ∑_{\begin{matrix}j=1 \\ j≠i \end{matrix}}^{N} ∑_{s=1}^{m_{j}} Tr(g_{s}^{(j)})(a_{i} − a_{j})^{−s}d(a_{i} − a_{j}) ∧ dt

+ ∑_{j=1}^{N−1} ∑_{s=1}^{m_{j}} Tr(g_{s}^{(j)})(a_{N} − a_{j})^{−s}d(a_{N} − a_{j}) ∧ dt.

In total, this gives

(4.9) ∑_{i=1}^{N} m_{i} ∑_{\begin{matrix}j=1 \\ j≠i \end{matrix}}^{N} ∑_{s=1}^{m_{j}} Tr(g_{s}^{(j)})(a_{i} − a_{j})^{−s}d(a_{i} − a_{j}) ∧ dt

− ∑_{j=1}^{N−1} ∑_{s=1}^{m_{j}} Tr(g_{s}^{(j)})(a_{N} − a_{j})^{−s}d(a_{N} − a_{j}) ∧ dt.

In fact, the above analysis of Tr(Ψ) omits some terms. On the right in (4.3) taking j = N and s = m_{N} gives a term

\frac{g_{m_{N}}^{(N)}(e_{μ})d(a_{i} − a_{N}) ∧ dt}{(t − a_{N})^{m_{N}} (t − a_{i})^{r}}.

Expanding this by lemma 3.3 yields a term (for 2 ≤ r ≤ m_{i})

(4.10) \frac{g_{m_{N}}^{(N)}(e_{μ})(a_{N} − a_{i})^{−r}d(a_{i} − a_{N}) ∧ dt}{(t − a_{N})^{m_{N}}}

≡ − \begin{matrix}∑ & ′ \\ j,s \end{matrix} \frac{g_{s}^{(j)}(e_{μ})(a_{N} − a_{i})^{−r}d(a_{i} − a_{N}) ∧ dt}{(t − a_{j})^{s}}.

Here ≡ means equivalent in H_{DR}^{1} ⊗ Ω_{K}^{1}. The prime in the sum means omit the pair j = N, s = m_{N}.

Similarly, from (4.4) we get a term

(4.11) \frac{g_{m_{N}}^{(N)}(e_{μ})(a_{N} − a_{i})^{−1}d(a_{i} − a_{N}) ∧ dt}{(t − a_{N})^{m_{n}}}

≡ − \begin{matrix}∑ & ′ \\ j,s \end{matrix} \frac{g_{s}^{(j)}(e_{μ})(a_{N} − a_{i})^{−1}d(a_{i} − a_{N}) ∧ dt}{(t − a_{j})^{s}}.

Of course, in (4.10) and (4.11) the contribution to the trace comes from j = i. These precisely cancel the second double sum on the right in (4.11). Thus, one gets

Tr(Ψ) = ∑_{i=1}^{N} m_{i} ∑_{\begin{matrix}j=1 \\ j≠i \end{matrix}}^{N} ∑_{s=1}^{m_{j}} Tr(g_{s}^{(j)})(a_{i} − a_{j})^{−s}d(a_{i} − a_{j}) ∧ dt. (4.12)

Finally, comparing (4.3), (4.4), and definition 3.5, we have

Proposition 4.2. With notation as above (definition 3.5 and equation (4.12)), the Gauß-Manin trace on H_{DR}^{1} is given by

Tr_{GM}(H_{DR}^{1}) = Tr(Ψ) + Tr(η_{∇})

≡^{thm.3.6} Tr(Ψ) + Tr(η_{γ}) + \frac{1}{2} ∑_{i; m_{i}≥2} m_{i}d log(det(g_{m_{i}}^{(i)})) mod d log K^{×}.

5. The Higgs Trace

The purpose of this section and of the next one is to rewrite the Higgs trace Tr(η_{γ}) as a sum of terms which are in some sense local, associated to the singularities a_{i} of the connection. In the next section we will compute these local traces.

It will be convenient to write

V := Γ(E(∗D)); W := Γ(E ⊗ ω(∗D)) ⊂ Vdt; (5.1)

γ_{K} = ∑_{j=1}^{N} ∑_{r=1}^{m_{j}} \frac{g_{r}^{(j)}dt}{(t − a_{j})^{r}} : V → W.

(See equations (3.5) and definition 3.2).

We identify H ↪ W with basis (3.8). As in section 3, there is a splitting H ≅ W/γ_{K}V. We write

η = η_{0} + ∑_{j=1}^{N} ∑_{s=1}^{M_{j}} \frac{η_{s}^{(j)}}{(t − a_{j})^{s}} = η_{0} + ∑_{j} η^{(j)}. (5.2)

The η_{s}^{(j)} are matrices with entries in Ω_{K}^{1}. We view these objects as linear maps H → H ⊗_{K} Ω_{K}^{1}:

H ↪ W ⟶^{η^{(i)}} W ⊗ Ω_{K}^{1} ↠ H ⊗ Ω_{K}^{1}

Now fix an i. It will be convenient to put a_{i} at ∞, so we set

u := \frac{1}{t − a_{i}} . (5.3)

Define

(5.4) R := Γ(ℙ^{1} − D, O) = K[\frac{1}{t − a_{1}} , …, \frac{1}{t − a_{N}}]

= K[\frac{u}{1 − (a_{1} − a_{i})u}, …, u, …, \frac{u}{1 − (a_{N} − a_{i})u}].

In the u coordinates,

γ_{K} = ∑ g_{s}^{(j)}(\frac{u}{1 − (a_{j} − a_{i})u})^{s}. (5.5)

The basis of H is

e_{μ}dt(\frac{u}{1 − (a_{j} − a_{i})u})^{r}; 2 ≤ r ≤ m_{j} (resp. 2 ≤ r ≤ m_{N} − 1), (5.6)

e_{μ}dt(\frac{u}{1 − (a_{j} − a_{i})u} − \frac{u}{1 − (a_{N} − a_{i})u})

= e_{μ}dt\frac{(a_{j} − a_{N})u^{2}}{(1 − (a_{j} − a_{i})u)(1 − (a_{N} − a_{i})u)}; j ≠ N.

A FORMULA FOR GAUSS-MANIN DETERMINANTS 29

Define

θ := ∏_{j≠i}(1 − (a_{j} − a_{i})u)^{m_{j}}, (5.7)

θ_{1} := θ/(1 − (a_{N} − a_{i})u),

g = θ · γ_{K} = ∑_{j,s} g_{s}^{(j)}u^{s}.

Note that θ is a unit in R. We can write (5.8) g = ∏_{j≠i}(a_{i} − a_{j})^{m_{j}}g_{m_{i}}^{(i)}u^{m} + lower order terms

= u^{2} + higher order terms,

where m = ∑ m_{j}. Let V = ⊕_{μ}Ke_{μ} = Γ(ℙ^{1}, E) and write H′ := V u^{2}dt ⊕ … ⊕ V u^{m−1}dt ⊂ V [u]dt. (5.9) As a consequence of (5.9) we have θ_{1}H = H′. (5.10) We are interested in the trace of η^{(i)} = ∑_{s=1}^{M_{i}} η_{s}^{(i)}u^{s}. Consider the diagram

H ⟶^{⊂} W ⟶^{η^{(i)}} W ⊗ Ω_{K}^{1} ⟶W/γ_{K}V ⊗ Ω_{K}^{1} ⟵^{≅} H ⊗ Ω_{K}^{1}
↓≅  ^{θ_{1}}↓≅  ^{θ_{1}^{−1}}↑≅  ^{θ_{1}^{−1}}↑≅  ↑≅
H′ ⟶^{⊂} W ⟶^{η^{(i)}} W ⊗ Ω_{K}^{1} ⟶W/γ_{K}V ⊗ Ω_{K}^{1} ⟵^{≅} H′ ⊗ Ω_{K}^{1}
∥  ^{∪}↑  ^{∪}↑  ≅↑^{a}  ∥
H′ ⟶^{⊂} u^{2}V [u]dt ⟶^{η^{(i)}} u^{2}V [u]dt ⊗ Ω_{K}^{1} ⟶ \frac{u^{2}V [u]}{gV [u]}dt ⊗ Ω_{K}^{1} ⟵^{≅} H′ ⊗ Ω_{K}^{1}

(5.11)

Proposition 5.1. The following maps have the same trace η^{(i)} : H → H ⊗ Ω_{K}^{1}

η^{(i)} : H′ → H′ ⊗ Ω_{K}^{1}

η^{(i)} : \frac{V [u]}{gV [u]}dt → \frac{V [u]}{gV [u]}dt ⊗ Ω_{K}^{1}.

Here the first two maps are given by horizontal rows in (5.11). The third is given by embedding

\frac{V [u]}{gV [u]}dt ↪ V [u]dt

via the basis V ⊕ V u ⊕ … ⊕ V u^{m−1} and then proceeding as in (5.11).

Proof. The first two traces are equal by the diagram. For the third, note that since g = u^{2} + higher, it follows that one has an exact sequence compatible with the endomorphism multiplication by u

0 → H′ → V [u]/gV [u]dt → V [u]/u^{2}V [u]dt → 0.

Since η^{(i)} has no constant term in u, it acts nilpotently on the right.

6. The Higgs trace: local calculation

In this section we give a formula for the trace of η^{(i)} as in proposition 5.1, involving residues. As already mentioned in the introduction, the method here is reminiscent of the classical residue calculation for the trace of an element in a field extension.

To simplify, we write h in place of η^{(i)}. We also suppress the Ω_{K}^{1} and treat h(u) as a polynomial with matrix coefficients. The case of matrices with coefficients in Ω_{K}^{1} follows immediately by applying arbitrary derivations Ω_{K}^{1} → K to the entries.

To avoid confusion we write

ϕ(h) : V [u]/gV [u] ↪ V [u] ⟶^{h·} V [u] ↠ V [u]/gV [u], (6.1)

where V [u]/gV [u] ↪ V [u] is defined as in proposition 5.1 via the invertibility of the leading coefficient of g

V ⊕ V u ⊕ … ⊕ V u^{m−1} ↪ V [u]. (6.2)

By (4.9) and admissibility, the leading coefficient of g(u) is invertible so g^{−1} ∈ End(V)((u^{−1})). Write dg = \frac{dg}{du}du and let

res_{u=∞} : End(V)((u^{−1})) → End(V)

be the evident extension of the residue map.

One has

Proposition 6.1. The notations being as above, one has

Tr_{V [u]/gV [u]}(ϕ(h)) = −Tr_{V} res_{u=∞}(dgg^{−1}h).

Proof. Write g = a_{0}u^{m} + a_{1}u^{m−1} + … + a_{m}. Note that neither side of the identity changes if we replace g by ga_{0}^{−1} so we may assume a_{0} = 1.

Also, by linearity, we may assume h = cu^{p}. The matrix for the action of u on V [u]/gV [u], the entries of which are themselves matrices, is

M = (\begin{matrix}0 & 0 & … & −a_{n} \\ 1 & 0 & … & −a_{n−1} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 1 & −a_{1} \end{matrix})

The matrix for u^{p} is M^{p}. We write TrM^{p} ∈ End(V) for the naive trace, i.e. the sum of the diagonal elements. E.g. TrM = −a_{1} ∈ End(V). Then

Tr(ϕ(cu^{p})) = Tr_{V} (Tr(M^{p})c).

Also

Tr_{V} res_{u=∞}(dgg^{−1}cu^{p}) = Tr_{V} (res (dgg^{−1}u^{p}) · c).

(The residue is computed in the ring End(V)((u^{−1})). Since u is in the center of this ring, we may move c past u^{p} under the residue.) It will therefore suffice to show

Tr(M^{p}) = −res_{u=∞}(dgg^{−1}u^{p}). (6.3)

Let z = u^{−1} and write g = u^{m}G(z) with G(z) = I + a_{1}z + … + a_{m}z^{m}. The assertion becomes

dGG^{−1} = (Tr(M) + Tr(M^{2})z + …)dz. (6.4)

Lemma 6.2. Let X = (x_{ij}) be an m × m matrix. Then

Tr(XM^{p}) = ∑_{q=0}^{p}(−1)^{q} ∑_{\begin{matrix}1≤m_{1},...,m_{q}≤m \\ 1≤i,i_{1}≤m \\ ∑ m_{k}=p+i−i_{1} \\ m_{1}≥m−i_{1}+1 \end{matrix}} x_{i,i_{1}}a_{m_{1}} … a_{m_{q}}. (6.5)

In particular, taking X = I it follows that

Tr(M^{p}) = ∑_{q=1}^{p}(−1)^{q} ∑_{\begin{matrix}1≤m_{1},...,m_{q}≤m \\ 1≤i≤m \\ ∑ m_{k}=p \\ m_{1}≥m−i+1 \end{matrix}} a_{m_{1}} … a_{m_{q}}. (6.6)

proof of lemma. Write M = (M_{ij})_{1≤i,j≤m}. We have M_{i+1,i} = 1, M_{i,m} = −a_{m+1−i}, and M_{ij} = 0 otherwise. Thus

(6.7) Tr(XM^{p}) = ∑_{i,i_{1},...,i_{p}} x_{i,i_{1}}M_{i_{1},i_{2}} ⋯ M_{i_{p},i} = ∑_{i,i_{1},q,j_{1},...,j_{q}} x_{i,i_{1}}M_{j_{1},m} ⋯ M_{j_{q},m}

= ∑_{i,i_{1},q,j_{1},...,j_{q}}(−1)^{q}x_{i,i_{1}}a_{m−j_{1}+1} ⋯ a_{m−j_{q}+1}

The conditions on the tuples {i, i_{1}, q, j_{1}, …, j_{q}} over which the right hand sums are taken become

0 ≤ q ≤ p; j_{1} ≤ i_{1}; (6.8)

(i_{1} − j_{1}) + 1 + (m − j_{2}) + 1 + … + (m − j_{q}) + 1 + (m − i) = p Replacing j_{k} by m_{k} := m − j_{k} + 1, these become

0 ≤ q ≤ p; m_{1} ≥ m − i_{1} + 1; ∑ m_{k} = p + i − i_{1}, (6.9)

proving the lemma.

Write c_{p} = Tr(M^{p}). We must show

(6.10) − (c_{1} + c_{2}z + c_{3}z^{2} + …)(1 + a_{1}z + … + a_{m}z^{m})

= a_{1} + 2a_{2}z + … + ma_{m}z^{m−1}.

This amounts to

c_{p} + c_{p−1}a_{1} + … + c_{1}a_{p−1} = \begin{cases}−pa_{p} & p < m \\ 0 & else \end{cases} (6.11)

Suppose first p < m. With reference to (6.6), one can isolate the terms in c_{p} ending in a_{k} for 1 ≤ k ≤ p − 1 and write

c_{p} = c_{p,1}a_{1} + c_{p,2}a_{2} + … + c_{p,p−1}a_{p−1} + R_{p}. (6.12)

Here

c_{p,k} = − ∑_{r=1}^{p−k}(−1)^{r} ∑_{\begin{matrix}1≤m_{1},...,m_{r}≤m \\ 1≤i≤m \\ ∑ m_{k}=p−k \\ m_{1}≥m−i+1 \end{matrix}} a_{m_{1}} … a_{m_{r}} = −c_{p−k}. (6.13)

(Notice that since each m_{j} ≥ 1, terms with r > p − k are impossible. Also, since k < p, necessarily r ≥ 1.) The remainder R_{p} is given by the terms a_{p} in c_{p}. In the sum for c_{p} these terms arise when q = 1 and p ≥ m − i + 1, i.e. m − p + 1 ≤ i ≤ m. There are p such terms:

R_{p} = −a_{p}. (6.14)

Finally, in (6.11) we consider the terms with p ≥ m. Writing t_{k} = Tr(XM^{k}) and replacing X with M^{j} for some j, it suffices to show

t_{0}a_{m} + … + t_{m−1}a_{1} + t_{m} = 0. (6.15)

We start with

t_{m} = ∑_{q=1}^{p}(−1)^{q} ∑_{\begin{matrix}1≤m_{1},...,m_{q}≤m \\ 1≤i,i_{1}≤m \\ ∑ m_{k}=n+i−i_{1} \\ m_{1}≥n−i_{1}+1 \end{matrix}} x_{i,i_{1}}a_{m_{1}} … a_{m_{q}}. (6.16)

Note q = 0 is not possible because ∑ m_{k} = m + i − i_{1} ≥ 1. Again, by grouping together the terms ending with a_{k} we get

t_{m} = −t_{m−1}a_{1} − … − t_{0}a_{m},

which is the desired equation.

Corollary 6.3. We have with notation as in definition 3.5

Tr(η_{∇}) ≡ (m − 2)Tr(η_{0}) − ∑_{i} res _{t=a_{i}}Tr(dgg^{−1}η^{(i)})

+ \frac{1}{2} ∑_{i; m_{i}≥2} d log(det(g_{m_{i}}^{(i)})) mod d log(K^{×}).

Proof. By theorem 3.6, we may replace η_{∇} with η_{γ}. By proposition 4.2 Tr(η_{γ} − η_{0}) is the sum of the Tr(η^{(i)}) on V [u]/gV [u]dt. By proposition 5.1 this is the same as − ∑_{i} Tr_{V} res _{t=a_{i}}(dgg^{−1}η^{(i)}). Note the factor m − 2 on the right is because as a matrix η_{0} acts on V = Γ(ℙ^{1}, E) while the trace one wants is the action on H ≅ V^{⊕m−2}.

7. The proof of the main Theorem

In this section we deduce the main theorem 2.8 from the equality of the Higgs and de Rham traces (theorem 3.6) and from the shape of the Higgs trace (proposition 6.1) via residues.

We start with an admissible connection (E, ∇) on ℙ_{K}^{1}. Let V ⊂ U be a Zariski open subset, with complement Z. By localization, one obtains

(7.1) det H_{DR}(U, ∇_{/K}) = det H_{DR}(V, ∇_{/K}) + det ∇|_{Z}.

On the other hand, at a special pseudo-logarithmic point the local factor Tr(dgg^{−1}A) = 0. Indeed, writing the connection as g_{1} \frac{dz}{z} +g_{0}dz + … + \frac{η_{1}}{z} + \frac{η_{0}}{+} …, the local factor is Trg_{0}g_{1}^{−1}η_{1}. The special pseudologarithmic points have, in the notations of the definition 2.2, a local matrix of the shape

( \begin{matrix}A + m\frac{dz}{z} & zB \\ \frac{C}{z} & D + n\frac{dz}{z} \end{matrix} ). (7.2)

Thus in particular

η_{1} = ( \begin{matrix}0 & 0 \\ γ_{0} & 0 \end{matrix} ), (7.3)

where C = cdz + γ_{0} + γ_{1}z + …, while g_{0} and g_{1} are both of the shape

( \begin{matrix}∗ & 0 \\ ∗ & ∗ \end{matrix} ). (7.4)

Thus Trg_{0}g_{1}^{−1}η_{1} = 0.

Thus the difference of the right hand side of the theorem 2.8 for U and V is the difference of the global factors, which is det ∇|_{Z}, as one

34 SPENCER BLOCH AND HÉLÈNE ESNAULT

sees taking a trivializing section which is good for V. Thus by theorem 2.3, we may assume that E = ⊕_{1}^{r}O_{ℙ_{K}^{1}}.

Let G = γ_{K} · ∏_{j}(t − a_{j})^{m_{j}}. Note G = u^{−m}g(u) with u as in (5.3). Write the absolute connection as ∇ = d + A with A = γ + η as in definition 3.2.

Proposition 7.1. We have

(7.5) ∑_{i} Tr res_{t=a_{i}}dGG^{−1}A = − ∑_{\begin{matrix}i,j,r \\ j≠i \end{matrix}} Tr(g_{r}^{(i)})m_{j}(a_{j} − a_{i})^{−r}d(a_{j} − a_{i})

+ ∑_{i} Tr res_{t=a_{i}}(dGG^{−1}η^{(i)}).

Proof. Define absolute forms s, s(i) and η(i):

s = \frac{dt}{∏_{j}(t − a_{j})^{m_{j}}}; s(i) = \frac{d(t − a_{i})}{∏_{j}(t − a_{j})^{m_{j}}}; ∇ = d + G · s(i) + η(i). (7.6)

The local term at t = a_{i} is

Tr res_{t=a_{i}}dGG^{−1}A = Tr res_{t=a_{i}}dG · s(i) + Tr res_{t=a_{i}}dGG^{−1}η(i).

Applying trace to dA = A ∧ A yields

Tr(dG · s(i) + G · ds(i) + dη(i)) = 0.

(Note here that s(i) is not closed as an absolute form!) Also, modulo Ω_{K}^{2}, we have res Tr(dη(i)) = 0 because the residue of an exact form vanishes. The local term thus becomes

Tr res_{t=a_{i}}dGG^{−1}A = −Tr res_{t=a_{i}}G · ds(i) + Tr res_{t=a_{i}}dGG^{−1}η(i). (7.7)

We have

(7.8) G · ds(i) =

( ∏_{j}(t − a_{j})^{m_{j}} ∑_{r,k} \frac{g_{r}^{(k)}}{(t − a_{k})^{r}} )\frac{∑_{j} m_{j} ∏_{k≠j}(t − a_{k})d(t − a_{j}) ∧ d(t − a_{i})}{∏_{j}(t − a_{j})^{m_{j}+1}}

= ( ∑_{r,k} \frac{g_{r}^{(k)}}{(t − a_{k})^{r}} ) ∑_{j} m_{j} \frac{d(a_{i} − a_{j}) ∧ dt}{t − a_{j}}.

Thus

(7.9) res_{t=a_{i}} Tr(G · ds(i)) = res_{t=a_{i}}( ∑_{\begin{matrix}j,k,r \\ j≠i \end{matrix}} \frac{Tr(g_{r}^{(k)})m_{j}d(a_{i} − a_{j}) ∧ dt}{(t − a_{k})^{r}(t − a_{j})}).

A FORMULA FOR GAUSS-MANIN DETERMINANTS 35

Expanding

\frac{1}{(t − a_{j})} = −(a_{j} − a_{i})^{−1} ∑_{n=0}^{∞} (\frac{t − a_{i}}{a_{j} − a_{i}} )^{n}

and substituting on the right in (7.9)

res_{t=a_{i}} Tr(G · ds(i)) = ∑_{\begin{matrix}j,r \\ j≠i \end{matrix}} Tr(g_{r}^{(i)})m_{j}(a_{j} − a_{i})^{−r}d(a_{j} − a_{i}) (7.10)

Thus, (7.9) becomes after summing over i

(7.11) ∑_{i} Tr res_{t=a_{i}}dGG^{−1}A = − ∑_{\begin{matrix}i,j,r \\ j≠i \end{matrix}} Tr(g_{r}^{(i)})m_{j}(a_{j} − a_{i})^{−r}d(a_{j} − a_{i})

+ ∑_{i} Tr res_{t=a_{i}}(dGG^{−1}η(i)).

It remains to compare η^{(i)} and η(i). Recall γ_{K} = G · s. We have

γ = γ_{K} − ∑_{j,r} \frac{g_{r}^{(j)}da_{j}}{(t − a_{j})^{r}} (7.12)

G · s(i) = γ_{K} − ∑_{j,r} \frac{g_{r}^{(j)}da_{i}}{(t − a_{j})^{r}} (7.13)

∇ = d + G · s(i) + η(i) = d + γ + η_{0} + ∑_{j} η^{(j)}. (7.14)

From these equations it follows that

η(i) − η^{(i)} = ∑_{j≠i} \frac{k_{r}^{(j)}}{(t − a_{j})^{r}} .

Since this difference has no pole at a_{i}, and since G(a_{i}) is invertible, we find

res_{t=a_{i}}(dGG^{−1}η(i)) = res_{t=a_{i}}(dGG^{−1}η^{(i)})

Making this substitution in (7.11) proves the proposition.

Our calculations to this point have been on H_{DR}^{1} which introduces a minus sign in the final formula. (Note, admissibility forces H_{DR}^{0} = (0).)

We find therefore

(7.15) − Tr_{GM}(H_{DR}^{∗}) = Tr(Ψ) + Tr(η_{∇}) (proposition 4.2)

≡ Tr(Ψ) + (m − 2)Tr(η_{0}) − ∑_{i} Tr res_{t=a_{i}}(dgg^{−1}η^{(i)})

+ \frac{1}{2} ∑_{i; m_{i}≥2} d log(det(g_{m_{i}}^{(i)})) mod d log K^{×} (corollary 6.3)

≡ Tr(Ψ) + (m − 2)Tr(η_{0}) − ∑_{i} Tr res_{t=a_{i}}(dGG^{−1}η^{(i)})

+ \frac{1}{2} ∑_{i; m_{i}≥2} d log(det(g_{m_{i}}^{(i)})) mod d log K^{×}

≡ (m − 2)Tr(η_{0}) − ∑_{i} Tr res_{t=a_{i}}(dGG^{−1}A)

+ \frac{1}{2} ∑_{i; m_{i}≥2} d log(det(g_{m_{i}}^{(i)})) mod d log K^{×} (proposition 7.1).

Note here we can replace g by G = u^{−m}g because η^{(i)} only involves terms of degrees ≥ 1 in u, so res (\frac{du}{u}η^{(i)}) = 0.

View s = \frac{dt}{∏_{j}(t−a_{j})^{m_{j}}} as a relative form, i.e. as a meromorphic section of ω(∑ m_{j}(a_{j})). As such, it has a zero of order m − 2 at infinity,

(s) = (m − 2)∞. Note det(E, ∇)|_{∞} = (K, Tr(η_{0})). Since the relative connection is given by ∇_{/K} = G · s, the desired formula is

(7.16) Tr_{GM}(H_{DR}^{∗}) ≡ − det(E, ∇)|_{(s)} + ∑_{i} res_{t=a_{i}}Tr(dGG^{−1}A)

+ \frac{1}{2} ∑_{i; m_{i}≥2} d log(det(g_{m_{i}}^{(i)})) mod d log K^{×}.

Finally, comparing (7.15) and (7.16) we see that theorem 2.8 holds for pseudo-admissible connections on bundles on ℙ^{1}.

References

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

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

Mathematik, Universität Essen, Essen, Germany

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