HURWITZ SPACES AND FROBENIUS MANIFOLDS

MARKUS ROSELLEN

1. Introduction

In this paper we give an exposition of Dubrovin’s theorem [D1, D2, D3] stating that there is a hierarchy of Frobenius structures on Hurwitz spaces.

1.1. A Frobenius manifold is a complex manifold M with a flat metric and a commutative, associative multiplication on the tangent spaces such that its structure constants with respect to flat vector fields are third order derivatives of a function Φ on M; Φ thus satisfies a system of non-linear PDE, the associativity or WDVV-equations.

Besides the A- and B-model of mirror symmetry, i.e. quantum cohomology and the Barannikov-Kontsevich construction, the third large class of Frobenius manifolds comes from Saito frameworks which consist of a family of holomorphic functions F : N → ℂ parametrized by M, N → M, and a relative volume form ω ∈ Ω_{N/M}^{max}, e.g. the versal deformation of an isolated hypersurface singularity and a Saito primitive form. This third construction is also of B-model type in the sense that contrary to the A-model the multiplication is easily obtained from the Milnor rings O_{C} of F, in particular is independent of ω, whereas the flat metric is obtained by pulling back the Grothendieck residue pairing to O_{C} via (f ↦ fω) and finding ω is difficult.

The Hurwitz space H_{g,t}, t = (n_{j}), is the moduli space of meromorphic functions on curves of genus g with poles of order n_{0} + 1, …, n_{M} + 1 and simple branch points over ℂ. Adding rigidifying data to H_{g,t} to obtain a covering H′_{g,t} → H_{g,t}, there is thus the “universal” function F : C^{∘} → ℂ on the (pointed) universal curve C^{∘} → H′_{g,t}.

Dubrovin’s theorem states that if ω is a descendant form, i.e. a multi-valued relative 1-form on C^{∘} → H′_{g,t} with constant principal parts, a-periods, and increments, then (F, ω) is a Saito framework. As descendant forms form a hierarchy, i.e. a filtered local system Ω = ⋃_{k≥0} Ω_{k} on H′_{g,t}, rk Ω_{k} = (k + 1) dim(H_{g,t}), together with surjective maps D : Ω_{k} → Ω_{k−1}, we obtain a hierarchy of (semisimple) Frobenius structures on H′_{g,t}. This hierarchy descends to H_{g,t} and consists exactly of all (finite) flat metrics with the same rotation coefficients. We give an overview in section 2.

1.2. Certain Frobenius structures on Hurwitz spaces may be realized on orbit spaces of their monodromy groups: H_{0,n}, H_{0,(n,m)}, H_{1,n} correspond in this way to the Coxeter group A_{n} [D3], the extended affine Weyl group W̃^{(n+1)}(A_{n+m+1}) [DZ], and the Jacobi group 𝕁(A_{n}) [B], resp.

Krichever showed [K] that there are Frobenius structures on moduli spaces of curves with a marked abelian differential of the second kind with fixed type of pole divisor, vanishing a-periods, and fixed b-periods. Hurwitz spaces are the special case where the b-periods are zero. Except for this case Krichever’s Frobenius manifolds do not have an Euler field.

1.3. I heartily thank my advisor Prof. Yu.I.Manin for support and M.Bhupal, Prof.G.Faltings, C.Hertling, A.Kabanov, T.Kimura, M.Spitzweck, and especially Prof.B.Dubrovin for their help. I am very grateful to the Max-Planck-Institut which provided excellent working conditions.

2. Overview

We first recall some basic statements about Frobenius manifolds and Saito frameworks from [M], Chap.I and Chap.III, §8, resp.

2.1. Definition. A Frobenius manifold is a complex manifold M with a commutative, associative multiplication ∘ : T_{M} ⊗ T_{M} → T_{M} with identity e on the holomorphic tangent sheaf and a non-degenerate, symmetric pairing g : T_{M} ⊗ T_{M} → O_{M} (called metric) which is invariant, i.e. g(X ∘Y, Z) = g(X, Y ∘Z) for any X, Y, Z ∈ T_{M}, such that the Levi-Civita connection of g is flat and locally there exists Φ ∈ O_{M} (called Frobenius potential) such that if T_{M}^{g} denotes the sheaf of g-flat vector fields we have

g(∂ ∘ ∂′, ∂′′) = ∂∂′∂′′ Φ, ∂, ∂′, ∂′′ ∈ T_{M}^{g}.

E ∈ T_{M}(M) is called an Euler field on M if Lie_{E}(∘) = d_{0}∘ and Lie_{E}(g) = Dg for some d_{0}, D ∈ ℂ. An Euler field E is called semisimple if the adjoint action of E on T_{M}^{g} is semisimple.

A Frobenius manifold is called semisimple if ∘ is semisimple, i.e. locally there exists a basis of idempotent vector fields (e_{n}), e_{n} ∘ e_{m} = δ_{nm}e_{n}. A semisimple weak Frobenius manifold is a complex manifold M with a multiplication ∘ such that locally there exist coordinates (u_{n}) (called canonical coordinates) such that e_{n} := \frac{∂}{∂u_{n}} are idempotents. A metric g on a semisimple weak Frobenius manifold M is called compatible if (M, ∘, g) is a Frobenius manifold and admissible if in addition e ∈ T_{M}^{g}.

2.2. Proposition. If (M, ∘, g) is a semisimple Frobenius manifold then (M, ∘) is a semisimple weak Frobenius manifold.

Let (M, ∘) be a semisimple weak Frobenius manifold. A metric g on M is invariant iff g = ∑ η_{n}(du_{n})^{2} is diagonal. A diagonal metric g = ∑ η_{n}(du_{n})^{2} on M is flat if and only if locally there exists η ∈ O_{M} (called metric potential) such that e_{n}η = η_{n} for any n and the rotation coefficients

γ_{nm} := \frac{e_{n}\sqrt{η_{m}}}{\sqrt{η_{n}}} = \frac{1}{2} \frac{e_{n}e_{m} η}{\sqrt{η_{n}η_{m}}}  (2.1)

satisfy the Darboux-Egoroff equations e_{k} γ_{nm} = γ_{nk}γ_{km} and eγ_{nm} = 0 for any n ≠ m ≠ k ≠ n. A diagonal metric g on M is flat if and only if g is compatible. In this case, e = ∑ e_{n} ∈ T_{M}^{g} iff e_{n} γ_{nm} = 0 for any n ≠ m and E := d_{0} ∑ u_{n}e_{n} is an Euler field iff E η_{n} = (D − 2d_{0})η_{n} for any n.

If g = ∑(a_{n})^{2}(du_{n})^{2}, g′ = ∑(a′_{n})^{2}(du_{n})^{2} are metrics on M, g is admissible, and ∂ := ∑ \frac{a′_{n}}{a_{n}} e_{n} (thus g′(X, Y) = g(∂ ∘ X, ∂ ∘ Y) for any X, Y ∈ T_{M}) then g′ is admissible if and only if ∂ ∈ T_{M}^{g} and the rotation coefficients agree, i.e. \frac{e_{n} a_{m}}{a_{n}} = \frac{e_{n} a′_{m}}{a′_{n}} for any n ≠ m.

2.3. Definition. A Saito framework consists of submersions p : N → M, p′ : N′ → M of complex manifolds, morphisms π : N′ → N, F : N → ℂ, and a relative volume form ω ∈ Ω_{N′/M}^{max} such that

(0) p ∘ π = p′ and π : (p′)^{−1}(m) → p^{−1}(m) is a universal covering for any m ∈ M;

(i) if C := Spec(O_{N}/J_{F}), J_{F} := {XF ∈ O_{N} | X ∈ T_{N/M} ⊂ T_{N}}, is the critical set of F then the map s : T_{M} → p_{∗}(O_{C}), X ↦ X̄F |_{C}, where X̄ ∈ T_{N} is a lift of X, is an isomorphism;

(ii) if D_{C} ⊂ C is the zero-set of the Hessian Hess(F) ∈ (Ω_{N/M}^{max})^{⊗2}(C) defined by

Hess(F) := det (\frac{∂^{2} F}{∂z_{i}∂z_{j}} ) (dz_{1} ∧ ⋯ ∧ dz_{max})^{2}|_{C},

z_{1}, …, z_{max} being vertical coordinates, then the discriminant D := p(D_{C}) is a divisor and p : C \ p^{−1}(D) → M \ D is étale;

(iii) if ρ_{1}, …, ρ_{μ} : U → C, U ⊂ M \ D, are sections of p, μ := dim(M), and u_{n} := F ∘ ρ_{n} then

g_{ω} := ∑_{n=1}^{μ} \frac{ω^{2}}{Hess(F)}(ρ_{n}) (du_{n})^{2} = ∑_{n=1}^{μ} res_{ρ_{n}}[ \begin{matrix}& ω \\ \frac{∂F}{∂z_{1}} & … & \frac{∂F}{∂z_{max}} \end{matrix} ] (du_{n})^{2},  (2.2)

z_{1}, …, z_{max} being vertical coordinates such that dz_{1}∧⋯∧dz_{max} = ω (cf.[GH], p.650), is a well-defined (i.e. independent of lifts of ρ_{n} to N′) flat metric on M \ D that extends regularly to M.

It follows that M with multiplication ∘ induced by s : T_{M} ˜ ⟶p_{∗}(O_{C}) and metric g_{ω} is a generically semisimple Frobenius manifold.

2.4. Overview. In section 6 we construct in the category of comlex manifolds the Hurwitz space H_{g,t}, its rigidification H′_{g,t}, the universal curve C → H′_{g,t}, the fibrewise universal covering C̃ → C^{∘} of the pointed universal curve with the fibres over infinity removed, and the universal morphism F ≡ Λ : C^{∘} → ℂ. We thus have all the data of a Saito framework except for ω. Note that the disriminant D is empty.

In order to obtain ω we discuss in sections 3,4,5 multi-valued forms on Riemann surfaces, in particular we define a pairing between multi-valued forms which is used in a reciprocity law and we give a basis of the vector space of multi-valued abelian differentials.

In section 7 we define descendant forms as relative 1-forms on C̃ → H′_{g,t} that are fibrewise multi-valued abelian differentials with poles only at the poles of Λ and with constant principal parts, a-periods, and increments. Let Ω denote the sheaf on H′_{g,t} of descendant forms, ẽ, Ẽ be the (meromorphic) lifts to C̃ of e := ∑ e_{n}, E := ∑ u_{n}e_{n} such that ẽ(Λ) = Ẽ(Λ) = 0, and define D := Lieẽ. We show that Ω_{k} := ker(D^{k+1} : Ω → Ω) is a local system of rank (k + 1) dim(H_{g,t}) and D : Ω_{k} → Ω_{k−1} is surjective. ω is called primary if ω ∈ Ω_{0} and ω is called eigenprimary if in addition ω is an eigenvector of LieẼ. We show that ω ∈ Ω is primary if and only if the order of poles of ω is smaller or equal to one plus the order of Λ at the respective poles and the inrements of ω are at most linear in Λ. We also show that the natural basis of primary forms consists of eigenprimary forms.

In section 8 we prove Dubrovin’s theorem: if H_{ω} ⊂ H′_{g,t}, ω ∈ Ω, denotes the open subset where g_{ω} is non-degenerate, i.e. all residues of (2.2) are non-zero, then (H_{ω}, ∘, g_{ω}) is a Frobenius manifold. In Theorem 8.1 we give flat coordinates, calculate the metric in flat coordinates, show that there is a natural isomorphism between Ω_{0} and g_{ω}-flat vector fields which respects the pairings, characterize primary forms by the flatness of the identity vector field, give the metric potential and the Frobenius potential, prove that the hierarchy of flat metrics descends to H_{g,t} and consists exactly of all (finite) metrics with the same rotation coefficients, prove that the Frobenius structures defined by eigenprimary forms have a semisimple Euler field and compute its spectrum.

The distinguished feature of Saito frameworks of Hurwitz spaces as compared to those of isolated hypersurface singularities and primitive forms consists in the global geometry of the fibres of C → H′_{g,t}.

3. Multi-valued forms on Riemann surfaces

In this section we define multi-valued abelian differentials on a Riemann surface with local coordinates at marked points and prove that they are in one-to-one correspondence with multi-valued Laurent tails and a-periods.

Let C be a compact Riemann surface of genus g, Q_{0}, …, Q_{M} ∈ C, and π : C̃ → C^{∘} be the universal covering of C^{∘} := C \ {Q_{0}, …, Q_{M}}. Let z_{j} be a local coordinate at Q_{j} defined on a small disk D_{j} such that z_{j}(Q_{j}) = 0, j = 0, …, M. Put D_{j}^{∘} := D_{j} \ {Q_{j}}.

Denote by M_{S} and M_{S}^{1} the sheaves of meromorphic functions and meromorphic 1-forms on a Riemann surface S and M(S) := M_{S}(S), M^{1}(S) := M_{S}^{1}(S).

π_{1}(C^{∘}) acts on M1 ; C̃(π^{−1}(U)), U ⊂ C^{∘}. The difference incr_{γ}(ω) := γ^{∗}ω − ω, γ ∈ π_{1}(C^{∘}), is called the increment of ω along γ. The map incr(ω) : π_{1}(C^{∘}) → M1 ; C̃(π^{−1}(U)), γ ↦ incr_{γ}(ω), is a crossed homomorphism.

As a substitute for the notion of “a point in C̃ over Q_{j}” we introduce the notion of a ray over Q_{j}. Everything explicit of what follows depends on choices of rays.

3.1. Definition. A ray over Q_{j} (with respect to z_{j}) is a connected subset Q̃_{j} ⊂ C̃ such that there exists a representative z′_{j} : D′_{j} → ℂ of the germ z_{j}|_{Q_{j}} ∈ O_{C,Q_{j}} such that π(Q̃_{j}) ⊂ D′_{j}, z′_{j}(π(Q̃_{j})) = (0, r) ⊂ ℝ, r > 0, is an interval and such that Q̃_{j} is maximal (wrt inclusion) among all connected subsets of C̃ with this property.

For any ray Q̃_{j} over Q_{j} we define a logarithm logQ̃_{j} of z_{j}. Assume g > 0 or M > 0 (if g = M = 0 the map π : C̃ → C^{∘} is an isomorphism and there is only one ray Q̃_{j} over Q_{j}; we define TQ̃_{j} := π^{−1}(D_{j}^{∘}) and logQ̃_{j} := z_{j} : TQ̃_{j} → ℂ). Choose P̃_{0} ∈ Q̃_{j} such that P_{0} := π(P̃_{0}) ∈ D_{j}. Because C̃ is the universal covering of C^{∘} there exists a commutative diagram

\xymatrix{(D̃_{j}^{∘}, P̃_{0}) \ar@{^{(}->}[r]^{ιQ̃_{j}} \ar[d] & (C̃, P̃_{0}) \ar[d]^{π} \\ (D_{j}^{∘}, P_{0}) \ar@{^{(}->}[r] & (C^{∘}, P_{0})}

where D̃_{j}^{∘} is the universal covering of D_{j}^{∘}. Put TQ̃_{j} := ιQ̃_{j}(D̃_{j}^{∘}) and define logQ̃_{j} : TQ̃_{j} → ℂ to be the unique holomorphic function such that exp ∘ logQ̃_{j} = z_{j} and logQ̃_{j}(P) = log(z_{j}(P)) for any P ∈ Q̃_{j} ∩ TQ̃_{j}.

The set Q_{j} of rays over Q_{j} is a π_{1}(C^{∘})-torsor. Endow C̄ := C̃ ⊔ ⨆ Q_{j} with the unique (non-Hausdorff) topology such that C̃ ⊂ C̄ is an open subspace and a neighborhood basis of a ray Q̃_{j} over Q_{j} is given by {Q̃_{j}} ∪ (π^{−1}(U \ {Q_{j}}) ∩ TQ̃_{j}) where U is a neighborhood of Q_{j}. Wethe towers in TQ̃_{j} over neighborhoods of Q_{j}. Extend π to a continuous map π̄ : C̄ → C by mapping a ray over Q_{j} to Q_{j} and write i : C̃ ↪ C̄ for the inclusion.

3.2. Definition. (i) A 1-form ω ∈ i_{∗}M1 ; C̃(Ū), Ū ⊂ C̄, is called logarithmic at infinity if for any ray Q̃_{j} ∈ Ū over Q_{j} there are ϕ, ψ ∈ M_{C}^{1}(U) for some neighborhood U of Q_{j} such that ω = ϕ + ψ logQ̃_{j}.

(ii) A 1-form ω ∈ i_{∗}M1 ; C̃(π̄^{−1}(U)), U ⊂ C, is called a multi-valued abelian differential on (U, C̄) if ω is logarithmic at infinity and im(incr_{γ}(ω)) ⊂ dM_{C}(U).

Denote by M1 ; C̄ the sheaf on C of multi-valued abelian differentials on C̄ and M^{1}(C̄) :=

M1 ; C̄(C).

The condition on ω ∈ M1 ; C̄ to be logarithmic at infinity is imposed to ensure that the following remark holds true for ω.

3.3. Remark. If ω ∈ M1 ; C̄(U), U ⊂ C, and incr(ω) = 0 then ω ∈ M_{C}^{1}(U). In fact, because ω is π_{1}(C^{∘})-invariant ω ∈ M_{C}^{1}(U ∩ C^{∘}) and because ω is logarithmic at infinity it follows that ω ∈ M_{C}^{1}(U).

The condition on the increments of ω ∈ M^{1}(C̄) is made in order that there is a notion of integral of ω over a homology class in H_{1}(C^{∘}, ℤ) provided that ω has no residues. This is done as follows.

3.4. Lemma. Let ω ∈ M^{1}(C̄) have no residues on C̃ and P_{0} ∈ C^{∘} be such that ω is holomorphic at any P̃_{0} ∈ π^{−1}(P_{0}). For a closed path γ in C^{∘} at P_{0} we denote by γP̃_{0} the lift of γ to P̃_{0} ∈ π^{−1}(P_{0}). Then the integral

∫_{γ} ω := ∫_{γP̃_{0}} ω  (3.1)

only depends on P_{0} and the homology class of γ.

Proof. For any δ ∈ π_{1}(C^{∘}, P_{0}) we have

∫_{γ_{δ∗P̃_{0}}} ω = ∫_{δ_{∗}γP̃_{0}} ω = ∫_{γP̃_{0}} δ^{∗}ω = ∫_{γP̃_{0}} ω + ∫_{γP̃_{0}} incr_{δ}(ω) = ∫_{γP̃_{0}} ω.

Thus the integral is independent of the choice of P̃_{0}, in particular the integral is additive for closed paths at P_{0}. Because homotopic paths in C^{∘} lift to homotopic paths in C̃ and ω has no residues on C̃ the integral is well-defined on π_{1}(C^{∘}, P_{0}), hence on H_{1}(C^{∘}, ℤ).

We shall always assume that we have chosen a point P_{0} ∈ C^{∘} by which integrals are defined.

Let ω ∈ M^{1}(C̄). Because the action of π_{1}(C^{∘}) on dM(C) is trivial the crossed homomorphism incr(ω) : π_{1}(C^{∘}) → dM(C) is a homomorphism and we get incr(ω) : H_{1}(C^{∘}, ℤ) → dM(C).

Let a_{i}, b_{i} ∈ H_{1}(C^{∘}, ℤ), i = 1, …, g, be such that their images in H_{1}(C, ℤ) form a symplectic basis of H_{1}(C, ℤ). We call (a_{i}, b_{i})_{i=1,...,g} a symplectic basis of H_{1}(C^{∘}, ℤ). ω ∈ M^{1}(C̃) without residues on C̃ is called normalized if all a-periods of ω vanish.

Let d_{j} ∈ H_{1}(C^{∘}, ℤ) be the homology class of a small, counterclockwise oriented circle around Q_{j}, j = 0, …, M. Because a_{i}, b_{i}, d_{j} ∈ H_{1}(C^{∘}, ℤ), i = 1, …, g, j = 1, …, M, form a basis of H_{1}(C^{∘}, ℤ), giving a homomorphism ι : π_{1}(C^{∘}) → dM(C) is equivalent to giving a map ι : B := {a_{i}, b_{i}, d_{j} | i = 1, …, g, j = 1, …, M} → dM(C). The 1-forms ι(γ), γ ∈ B, are called the basic increments of ι.

We call P_{0} ∈ C^{∘} an ω-base point if P_{0} is not a pole of incr_{d_{j}}(ω), j = 0, …, M. In a neighborhood of a ray Q̃_{j} over Q_{j} we write

ω = ∑_{k≥−K} c_{kj}^{ω}z_{j}^{k}dz_{j} + d(r_{j}^{ω} logQ̃_{j})  (3.2)

where c_{kj}^{ω} ∈ ℂ and r_{j}^{ω} ∈ M(C) is uniquely determined by 2πi dr_{j}^{ω} = incr_{d_{j}}(ω) and r_{j}^{ω}(P_{0}) = 0. We call ∑_{k≥−K} c_{kj}^{ω}z_{j}^{k}, ∑_{k=−K}^{−1} c_{kj}^{ω}z_{j}^{k}, and c_{−1,j}^{ω} the meromorphic part, the Laurent tail, and the residue of ω at Q̃_{j} (with respect to z_{j} and P_{0}) and write

res_{Q̃_{j}}(ω) := c_{−1,j}^{ω}.

3.5. Definition. A 1-form ω ∈ M1 ; C̄(U), U ⊂ C, is called

(i) holomorphic at a ray Q̃_{j} ∈ π̄^{−1}(U) if the Laurent tail of ω at Q̃_{j} is zero; (ii) holomorphic at P ∈ U if ω is holomorphic at some P̃ ∈ π̄^{−1}(P); (iii) a multi-valued holomorphic 1-form on (U, C̄) if ω is holomorphic at any P ∈ U.

Denote by Ω1 ; C̄ the sheaf on C of multi-valued holomorphic 1-forms on C̄ and Ω^{1}(C̄) :=

Ω1 ; C̄(C).

By Mittag-Leffler there is an isomorphism between the space of abelian differentials on C and the space of Laurent tail divisors with vanishing sum of residues and a-periods (see for example [N], p.54). We want to extend this statement to multi-valued abelian differentials on C̄.

A pre-Laurent tail divisor 𝔏 = (ι, (ω_{i})) on C̃ consists of a homomorphism ι : π_{1}(C^{∘}) → dM(C) and a Čech 0-cochain (ω_{i} ∈ Γ(U_{i}, π̄_{∗}M1 ; C̄))_{i∈I} such that

(i) incr(ω_{i}) = ι for any i ∈ I; (ii) the Čech differential of (ω_{i}) is a Čech 1-cochain of Ω_{C}^{1}. Let 𝔏 = (ι, (ω_{i})_{i∈I}), 𝔏′ = (ι′, (ω′_{j})_{j∈J}) be pre-Laurent tail divisors on C̃. 𝔏 is called finer than 𝔏′ if ι = ι′ and the Čech 0-cochain (ω_{i}) is finer than (ω′_{j}). 𝔏, 𝔏′ are called equivalent if they have a common refinement. A Laurent tail divisor on C̃ is an equivalence class of pre-Laurent tail divisors.

The residue at P ∈ C of a Laurent tail divisor (ι, (ω_{i})) is defined to be the residue of ω_{i} at some P̃ ∈ π̄^{−1}(P) for some i ∈ I such that P ∈ U_{i}. The Laurent tail divisor of ω ∈ M^{1}(C̄) is given by (incr(ω), ω).

3.6. Theorem. Let (a_{i}, b_{i})_{i=1,...,g} be a symplectic basis of H_{1}(C^{∘}, ℤ). Let 𝔏 be a Laurent tail divisor on C̃ which has at most residues at Q_{0}, …, Q_{M} and such that the sum of the residues is zero. For any I ∈ ℂ^{g} there exists a unique ω ∈ M^{1}(C̄) with Laurent tail divisor 𝔏 and a-periods I.

Proof. Let ω, ω′ ∈ M^{1}(C̄) satisfying the above properties. By Remark 3.3 ω − ω′ is an abelian differential on C. By Mittag-Leffler ω − ω′ is zero.

Let 𝔏 = (ι, (ω_{i})). By Lemma 3.7 below there exists ω ∈ Ω^{1}(C̄) such that incr(ω) = ι. By Remark 3.3 (ω_{i} − ω) is a Laurent tail divisor on C. By Mittag-Leffler there exists ω′ ∈ M^{1}(C) such that ω + ω′ has Laurent tail divisor 𝔏 and a-periods I_{1}, …, I_{g}.

3.7. Lemma. Let ι : π_{1}(C^{∘}) → dM(C) be a homomorphism and R_{0}, …, R_{K} ∈ C be the poles of the basic increments of ι. Let R̃_{0}, …, R̃_{K} ∈ C̄ be lifts of R_{0}, …, R_{K}. Then there exists a unique normalized 1-form ω ∈ Ω^{1}(C̄) which is holomorphic at R̃_{0}, …, R̃_{K} and satisfies incr(ω) = ι.

Proof. Uniqueness follows in the same way as in the proof of the Theorem. The proof of existence is an adaptation of the proof of Satz (28.4) of [F].

We may assume that R_{j} = Q_{j}, j = 0, …, M, and R_{M+1}, …, R_{K} ∈ C^{∘}. Let (U_{l})_{l=0,...,L}, L ≥ N, be an open covering of C such that U_{j} ⊂ D_{j}, j = 0, …, M, we have R_{k} ∈ U_{k} \ ⋃_{l≠k} U_{l}, k = 0, …, K, and such that there are homeomorphisms ϕ_{l} : π^{−1}(U_{l}) ˜ ⟶U_{l} × π_{1}(C^{∘}), l = M + 1, …, L, which are compatible with π, π_{1}(C^{∘})-equivariant, and satisfy ϕ_{k}(R̃_{k}) = (R_{k}, e), k = M + 1, …, K, where e ∈ π_{1}(C^{∘}) is the identity.

Consider the 1-form on U_{l} × π_{1}(C^{∘}) which equals ι(γ) on U_{l} × {γ}, γ ∈ π_{1}(C^{∘}), l = M + 1, …, L. Let ω_{l} be its pull-back via ϕ_{l} to π^{−1}(U_{l}).

Define Q̃_{j} := R̃_{j}, j = 0, …, M. Choose a set of representatives Γ_{j} of π_{1}(C^{∘})/π_{1}(D_{j}^{∘}) such that e ∈ Γ_{j}. For any γ ∈ Γ_{j} define 1-forms

ω_{0} := −\frac{1}{2πi} ( ∑_{j′=1}^{M} ι(d_{j′})) log_{γ_{∗}Q̃_{0}}  ω_{j} := \frac{1}{2πi} ι(d_{j}) log_{γ_{∗}Q̃_{j}}  j = 1, …, M,

on the tower in γ_{∗}TQ̃_{j} over U_{j} \ {Q_{j}}, j = 0, …, M. This gives ω_{j} ∈ Ω1 ; C̄(π̄^{−1}(U_{j})).

Then incr(ω_{l}) = ι, l = 0, …, L, and therefore the 1-form ρ_{ll′} := ω_{l} − ω_{l′} on π^{−1}(U_{l}) ∩ π^{−1}(U_{l′}) is π_{1}(C^{∘})-invariant, more precisely ρ_{ll′} ∈ Γ(U_{l} ∩ U_{l′}, Ω_{C}^{1}). These elements define a Čech-1-cocycle. Because H^{1}(C, Ω_{C}^{1}(R_{0})) = H^{0}(C, O_{C}(−R_{0})) = 0 there are ρ_{l} ∈ Γ(U_{l}, Ω_{C}^{1}(∗ ∑ R_{k})) such that ρ_{l} − ρ_{l′} = ρ_{ll′} = ω_{l} − ω_{l′} on U_{l} ∩ U_{l′}. The 1-form ω on C̃ given on π^{−1}(U_{l} ∩ C^{∘}) by ω_{l} − ρ_{l} is thus globally defined and satisfies incr(ω) = ι.

The Čech 0-cochain (ρ_{l}) defines a Laurent tail divisor 𝔏. By Mittag-Leffler there exists ω′ ∈ M^{1}(C) with Laurent tail divisor 𝔏 and the same a-periods as −ω. By construction ω′′ := ω + ω′ ∈ Ω^{1}(C̄) is normalized, holomorphic at R̃_{0}, …, R̃_{K} and satisfies incr(ω′′) = ι.

Let ω ∈ M^{1}(C̄). A subset S ⊂ C̄ is called a set of poles of ω if ω is holomorphic on C̄ \ (π_{1}(C^{∘})S) and π̄|_{S} is injective.

3.8. Corollary. If S is a set of poles of ω then ∑_{P ∈S} res_{P}(ω) = 0.

Proof. By Lemma 3.7 there exists ω′ ∈ Ω^{1}(C̄) such that incr(ω′) = incr(ω). Then ω−ω′ ∈

M_{C}^{1}(C) and ∑_{P ∈S} res_{P}(ω) = ∑_{P ∈S} res_{P}(ω − ω′) = 0.

4. A pairing and a reciprocity law

In this section we define a pairing between multi-valued abelian differentials and prove a reciprocity law from them. Reciprocity laws for single-valued abelian differentials are discussed for example in [GH].

Let P_{1}, …, P_{N} ∈ C^{∘} and C^{⋄} := C^{∘} \{P_{1}, …, P_{N}}. The pairing is defined on a subspace of Ω1 ; C̄(C^{⋄}) and depends on a cutting of C^{⋄} into a simply connected domain Č.

Let P_{0} ∈ C^{⋄}. Choose closed smooth paths a_{i}, b_{i} at P_{0} in C^{∘} and smooth paths c_{j} : [0, 1] → C from P_{0} to Q_{j} in such a way that for ϵ > 0 sufficiently small

(i) a_{i}, b_{i}, c_{j} only meet at P_{0}, are not self-intersecting, and do not pass through P_{1}, …, P_{N}; (ii) a_{i}, b_{i} have homology classes a_{i}, b_{i} ∈ H_{1}(C^{∘}, ℤ); (iii) c_{j}([0, 1)) ⊂ C^{∘} and z_{j}(c_{j}(1 − ϵ)) ∈ ℝ_{>0}.

Define c_{jϵ} : [0, 1] → C^{∘}, t ↦ c_{j}((1 − ϵ)t), and s_{jϵ} : [0, 1] → C^{∘} be a small clockwise loop around Q_{j} at c_{jϵ}(1).

Choose a lift P̃_{0} ∈ C̃ of P_{0} and let ã_{1} be the lift of a_{1} to P̃_{0}. By induction on i ≥ 1 we define b̃_{i}, ã_{i}^{−}, b̃_{i}^{−}, and ã_{i+1} to be the lifts of b_{i}, a_{i}^{−1}, b_{i}^{−1}, and a_{i+1} to the end-points of ã_{i}, b̃_{i}, ã_{i}^{−}, and b̃_{i}^{−}, resp. It follows that ã_{i}^{−} = (b_{i})_{∗}ã_{i}^{−1}, b̃_{i}^{−} = (a_{i}^{−1})_{∗}b̃_{i}^{−1}.

Let c̃_{0ϵ} be the lift of c_{0ϵ} to the end-point of b̃_{g}^{−}. By induction on j ≥ 1 we define s̃_{jϵ}, c̃_{jϵ}^{−} and c̃_{j+1,ϵ} to be the lifts of s_{jϵ}, c_{jϵ}^{−1}, and c_{j+1,ϵ} to the end-points of c̃_{jϵ}, s̃_{jϵ}, and c̃_{j,ϵ}^{−}, resp. We denote P̃_{0j} the starting-point of c̃_{jϵ} and Q̃_{j} the unique ray over Q_{j} that is contained in c_{j}([0, 1]). Let t_{j} ∈ π_{1}(C^{∘}, P_{0}) be such that (t_{j})_{∗}(s̃_{jϵ}(0)) = s̃_{jϵ}(1). It follows that c̃_{jϵ}^{−} = (t_{j})_{∗}c̃_{jϵ}^{−1}.

Let Č be the simply connected region in C̃ whose oriented boundary is

ã_{1}b̃_{1}ã_{1}^{−}b̃_{1}^{−} … ã_{g}b̃_{g}ã_{g}^{−}b̃_{g}^{−} c̃_{0ϵ}s̃_{0ϵ}c̃_{0ϵ}^{−} … c̃_{Mϵ}s̃_{Mϵ}c̃_{Mϵ}^{−}.

(We have to choose the paths a_{i}, b_{i}, c_{j} in such a way that the above order is possible, namely the tangents of these paths at P_{0} have to respect the same order.) Č is called a basic region over (C, (z_{j}), (a_{i}, b_{i}), (P_{n})_{n=0,...,N}).

In addition, the pairing involves the notion of principal value. Let ω ∈ M^{1}(C̄) be such that ω has no residues on C̃. For any ray Q̃_{j} we define a primitive of ω, called the principal value,

p.v. ∫_{Q̃_{j}} ω : C̃ → ℂℙ^{1}

by giving its values in a pointed neighborhood of Q̃_{j}:

p.v. ∫_{Q̃_{j}}^{P} ω := ∑_{\begin{matrix}k≥−K \\ k≠−1 \end{matrix}} \frac{c_{kj}^{ω}}{k + 1}z_{j}(P)^{k+1} + (c_{−1,j}^{ω} + r_{j}^{ω}) logQ̃_{j}(P)  (4.1)

where we used (3.2). We also define for P ∈ C̃

p.v. ∫_{P}^{Q̃_{j}}ω := −p.v. ∫_{Q̃_{j}}^{P} ω, p.v. ∫_{Q̃_{j}}^{Q̃_{k}}ω := p.v. ∫_{Q̃_{j}}^{P} ω + p.v. ∫_{P}^{Q̃_{k}}ω

where the second number does not depend on P.

Let ω, ω′ ∈ Ω1 ; C̄(C^{⋄}) be such that P_{0} is an ω- and ω′-base point. Let ∑_{k≥−K} c_{kj}z_{j}^{k}dz_{j} be the meromorphic part of ω at Q̃_{j} and incr_{a_{i}}(ω) = dp_{i}, incr_{b_{i}}(ω) = dq_{i}, incr_{d_{j}}(ω) = 2πi dr_{j} ∈ dM(C), r_{j}(P_{0}) = 0. Define c′_{kj} ∈ ℂ and p′_{i}, q′_{i}, r′_{j} ∈ M(C) in the same way for ω′.

4.1. Definition. Let Č be a basic region over (C, (z_{j}), (a_{i}, b_{i}), (P_{n})). Define on the subspace of Ω1 ; C̄(C^{⋄}) consisting of ω such that P_{0} is an ω-base point a bilinear pairing ⟨., .⟩Č by the formula

⟨ω, ω′⟩Č := −\frac{1}{2πi} ∑_{i=1}^{g} ( ∫_{a_{i}} q_{i}ω′ − ∫_{b_{i}} p_{i}ω′ + ∫_{b_{i}} ω ∫_{a_{i}} ω′ − ∫_{a_{i}} ω ∫_{b_{i}} ω′)

+ ∑_{j=0}^{M} ( ∑_{k=0}^{K−2} (c_{−k−2,j} \frac{c′_{kj}}{k + 1}) + c_{−1,j}p.v. ∫_{P̃_{0j}}^{Q̃_{j}}ω′ + p.v. ∫_{P̃_{0j}}^{Q̃_{j}}r_{j}ω′ + πi res_{Q_{j}}(r_{j}ω′)). (4.2)

4.2. Proposition. Let ω ∈ Ω1 ; C̄(C^{⋄}) and ω′ be an abelian differential of the second kind on C with at most poles at P_{1}, …, P_{N} of order two. Assume that incr_{d_{j}}(ω) vanishes at P_{1}, …, P_{N}, j = 0, …, M. Let Č be a basic region over (C, (z_{j}), (a_{i}, b_{i}), (P_{n})) such that P_{0} is an ω- and ω′-base point. Let P̃_{1}, …, P̃_{N} ∈ Č be the points over P_{1}, …, P_{N}. Then

∑_{n=1}^{N} res_{P̃_{n}}(ω ∫_{P̃_{0}} ω′) = −⟨ω, ω′⟩Č.

Proof. Put f := ∫_{P̃_{0}} ω′. Because P̃_{1}, …, P̃_{N} are the only poles of ωf in Č the residue

theorem gives us

∑_{n=1}^{N} res_{P̃_{n}}(ωf) = \frac{1}{2πi} ∫_{∂Č} ωf = \frac{1}{2πi} ∑_{i=1}^{g} ∫_{ã_{i}b̃_{i}ã_{i}^{−}b̃_{i}^{−}} ωf + \frac{1}{2πi} ∑_{j=0}^{M} ∫_{c̃_{jϵ}s̃_{jϵ}c̃_{jϵ}^{−}} ωf.

The integral over ã_{i}b̃_{i}ã_{i}^{−}b̃_{i}^{−} is

∫_{ã_{i}b̃_{i}ã_{i}^{−}b̃_{i}^{−}} ωf = ∫_{ã_{i}} ωf + ∫_{b̃_{i}} ωf − ∫_{ã_{i}} b_{i}^{∗}ω(f + ∫_{b_{i}} ω′) − ∫_{b̃_{i}} (a_{i}^{−1})^{∗}ω(f − ∫_{a_{i}} ω′)

= − ∫_{ã_{i}} dq_{i} f + ∫_{b̃_{i}} dp_{i} f − ∫_{ã_{i}} b_{i}^{∗}ω ∫_{b_{i}} ω′ + ∫_{b̃_{i}} (a_{i}^{−1})^{∗}ω ∫_{a_{i}} ω′

= ∫_{a_{i}} q_{i}ω′ − ∫_{b_{i}} p_{i}ω′ − ∫_{a_{i}} ω ∫_{b_{i}} ω′ + ∫_{b_{i}} ω ∫_{a_{i}} ω′.

We compute the integral over c̃_{jϵ}s̃_{jϵ}c̃_{jϵ}^{−} by splitting it into four parts

∫_{c̃_{jϵ}s̃_{jϵ}c̃_{jϵ}^{−}} ωf = ∫_{c̃_{jϵ}} ωf − ∫_{c̃_{jϵ}} t_{j}^{∗}(ωf) + ∫_{s̃_{jϵ}} ωf =  (4.3)

= 2πi ∫_{c̃_{jϵ}} dr_{j} f + ∫_{s_{jϵ}} ∑_{\begin{matrix}l≥−K \\ k≥0 \end{matrix}} c_{lj} \frac{c′_{kj}}{k + 1}z_{j}^{l+k+1}dz_{j} + ∫_{s_{jϵ}} ∑_{l≥−K} c_{lj}z_{j}^{l}dz_{j}(p.v. ∫_{P̃_{0j}}^{Q̃_{j}}ω′) + ∫_{s̃_{jϵ}} d(r_{j} logQ̃_{j})f

where we used ω − t_{j}^{∗}ω = 2πi dr_{j} (s_{jϵ} is clockwise oriented), used (3.2) for ω, (4.1) for ω′, and applied distributivity. By the residue theorem the second and third integrals of the very RHS of (4.3) give the fifth and sixth term of −⟨ω, ω′⟩Č.

By assumption dr_{j} f has no residues on C̃. We write dr_{j} f = ∑_{k≥−K} h_{kj}z_{j}^{k}dz_{j} in a neighborhood of Q̃_{j}. By definition of the principal value, the first integral of the very RHS of (4.3) is

∫_{c̃_{jϵ}} dr_{j} f = ∑_{\begin{matrix}k≥−K \\ k≠−1 \end{matrix}} (\frac{h_{k}}{k + 1}z_{j}^{k+1}(c̃_{jϵ}(1))) + h_{−1} logQ̃_{j}(c̃_{jϵ}(1)) + p.v. ∫_{P̃_{0j}}^{Q̃_{j}}dr_{j} f =

= ∑_{\begin{matrix}k≥−K \\ k≠−1 \end{matrix}} (\frac{h_{k}}{k + 1}z_{j}(c̃_{jϵ}(1))^{k+1}) + h_{−1} logQ̃_{j}(c̃_{jϵ}(1)) − p.v. ∫_{P̃_{0j}}^{Q̃_{j}}r_{j}ω′ + res_{Q̃_{j}}(\frac{r_{j}f}{z_{j}} ) − r_{j}(P_{0})f(P̃_{0j})

(4.4)

where we used integration by parts and took the constant terms of developments in z_{j} in order to rewrite the principal value. Because r_{j}(P_{0}) = 0 the last term vanishes. The third term of (4.4) is the seventh term of −⟨ω, ω′⟩Č. Finally, the fourth integral of the very RHS of (4.3) is

∫_{s̃_{jϵ}} d(r_{j} logQ̃_{j})f = ∫_{s̃_{jϵ}} dr_{j} f logQ̃_{j} + ∫_{s_{jϵ}} \frac{r_{j}f}{z_{j}} dz_{j} = − ∫_{s_{jϵ}} ( ∑_{\begin{matrix}k≥−K \\ k≠−1 \end{matrix}} \frac{h_{k}}{k + 1}z_{j}^{k+1} \frac{dz_{j}}{z_{j}} )+ (4.5)

+[ ∑_{\begin{matrix}k≥−K \\ k≠−1 \end{matrix}} \frac{h_{k}}{k + 1}z_{j}^{k+1}logQ̃_{j}]_{s̃_{jϵ}(0)}^{s̃_{jϵ}(1)} + h_{−1} ∫_{s_{jϵ}} \frac{logQ̃_{j}}{z_{j}} dz_{j} − 2πi res_{Q̃_{j}}(\frac{r_{j}f}{z_{j}} )

= −2πi ∑_{\begin{matrix}k≥−K \\ k≠−1 \end{matrix}} (\frac{h_{k}}{k + 1}z_{j}(s_{jϵ}(0))^{k+1}) + h_{−1}(−2πi logQ̃_{j}(s̃_{jϵ}(0)) − 2π^{2}) − 2πi res_{Q̃_{j}}(\frac{r_{j}f}{z_{j}} ).

We applied integration by parts to dr_{j} f − h_{−1} \frac{dz_{j}}{z_{j}} and logQ̃_{j} and used the identities

∫_{s̃_{jϵ}} \frac{logQ̃_{j}}{z_{j}} dz_{j} = [\frac{1}{2} log2 ; Q̃_{j} ]_{s̃_{jϵ}(0)}^{s̃_{jϵ}(1)} = \frac{1}{2}(−4πi logQ̃_{j}(s̃_{jϵ}(0)) + (2πi)^{2}).

Because c̃_{jϵ}(1) = s̃_{jϵ}(0) three terms of (4.4) cancel with three terms of the very RHS of (4.5). We obtain the eighth term of −⟨ω, ω′⟩Č because h_{−1} = res_{Q̃_{j}}(dr_{j} f) = −res_{Q_{j}}(r_{j}ω′).

5. Multi-valued abelian differentials on \underline{λ}

In this section we define multi-valued abelian differentials on a Riemann surface with a marked meromorphic function and write down a basis of the vector space they form.

Let λ ∈ M(C) with divisor of poles ∑_{j=0}^{M}(n_{j} + 1)Q_{j}, n_{j} ∈ ℤ_{≥0}. Let z_{j} be a branch of the (n_{j} + 1)-st root of λ^{−1}, z_{i}^{−(n_{j}+1)} = λ, defined on a small disk D_{j} around Q_{j}. It follows that z_{j} is a local coordinate at Q_{j}. Let P_{1}, …, P_{N} ∈ λ^{−1}(ℂ) be the finite branch points of λ.

Define \underline{λ} := (λ, (Q_{j}), (z_{j}|_{Q_{j}}), (a_{i}, b_{i})) where z_{j}|_{Q_{j}} ∈ O_{C,Q_{j}} is the germ of z_{j} at Q_{j} and (a_{i}, b_{i})_{i=1,...,g} is a symplectic basis of H_{1}(C, ℤ). A basic region over (C, (z_{j}), (a′_{i}, b′_{i}), (P_{n})) is called a basic region over \underline{λ} if a′_{i}, b′_{i} ∈ H_{1}(C^{∘}, ℤ) map to a_{i}, b_{i}.

5.1. Definition. A 1-form ω ∈ M1 ; C̄(U) is called a multi-valued abelian differential on (U, \underline{λ}) if im(incr_{γ}(ω)) ⊂ dℂ[λ]. If furthermore ω ∈ Ω1 ; C̄(U) then ω is called a multivalued holomorphic 1-form on (U, \underline{λ}).

Denote by M1 ; \underline{λ}, Ω1 ; \underline{λ} the sheaves on C of multi-valued abelian differentials on \underline{λ} and of multi-valued holomorphic 1-forms on \underline{λ}. Put M^{1}(\underline{λ}) := M1 ; \underline{λ}(C).

The condition on the increments is made in order that for any ω, ω′ ∈ Ω1 ; \underline{λ}(C^{∘}) the residue

res_{P̃} \frac{ωω′}{dλ} ,  P̃ ∈ C̃,

only depends on P := π(P̃) ∈ C^{∘} as one easily shows.

In the following any zero P_{0} ∈ C^{∘} of λ will be taken as ω-base point for any ω ∈ M^{1}(\underline{λ}) and will be used to define integrals (3.1). Both is independent of the choice of the zero P_{0}.

5.2. Lemma. Let ω ∈ Ω1 ; \underline{λ}(C^{∘}) and γ ∈ π_{1}(C^{∘}, P_{0}). If incr_{γ}(ω) = dr, r ∈ λℂ[λ], then

p.v. ∫_{Q̃_{j}}^{γ_{∗}P}ω = p.v. ∫_{Q̃_{j}}^{P} ω + ∫_{γ} ω + r(P).  (5.1)

Proof. Let γ̃ be the lift to P̃_{0} ∈ π^{−1}(P_{0}) of a representative of γ ∈ π_{1}(C^{∘}, P_{0}) and α be a path from P̃_{0} to P. Then we have

∫_{P}^{γ_{∗}P}ω = ∫_{α^{−1}} ω + ∫_{γ̃} ω + ∫_{γ_{∗}(α)} ω = ∫_{γ} ω + ∫_{α} incr_{γ}(ω) = ∫_{γ} ω + r(P).

Introduce a set of labels

N := {s^{i}, r^{i}, w^{j} | i = 1, …, g, j = 1, …, M} ∪ {t^{kj} | 1 ≤ k ≤ n_{j} + 1, j = 0, …, M}.

As a special case of Theorem 3.6 we obtain a basis of Ω1 ; \underline{λ}(C^{∘}).

5.3. Proposition. Let Q̃_{0}, …, Q̃_{M} be rays over Q_{0}, …, Q_{M}. The 1-forms ω_{A,k}, A ∈ N , k ≥ 0, form a basis of the vector space Ω1 ; \underline{λ}(C^{∘}) where

(i) ω_{s^{i},0} is the abelian differential of the first kind on C with a-periods

∫_{a_{j}} ω_{s^{i},0} = δ_{ij} i, j = 1, …, g.

(ii) ω_{s^{i},k}, k ≥ 1, and ω_{r^{i},k}, k ≥ 0, are the normalized multi-valued holomorphic 1-forms on (C, λ) which are holomorphic at Q̃_{0}, …, Q̃_{M} and have all basic increments equal to zero except for

incr_{a_{i}}(ω_{s^{i},k+1}) = incr_{b_{i}}(ω_{r^{i},k}) = −\frac{dλ^{k+1}}{(k + 1)!}, k ≥ 0.

(iii) ω_{t^{kj},k′} is the normalized abelian differential of the second kind on C with only a pole at Q_{j} with Laurent tail equal to the Laurent tail of

−\frac{(n_{j} + 1)^{k′}dz_{j}^{−(k+k′(n_{j}+1))}}{∏_{l=0}^{k′}(k + l(n_{j} + 1))} ;

(iv) ω_{w^{j},0} is the normalized abelian differential of the third kind on C with only poles at Q_{0} and Q_{j} with residues −1 and 1, resp.;

(v) ω_{w^{j},k} is the multi-valued holomorphic 1-form on (C \ {Q_{0}, Q_{j}}, λ), k ≥ 1, which is holomorphic at Q̃_{j′}, j′ ≠ 0, j, has Laurent tail at Q̃_{0}, Q̃_{j} equal to the Laurent tail at Q̃_{0}, Q̃_{j} of

−\frac{1}{n_{0} + 1}( ∑_{n=1}^{k} \frac{1}{n})\frac{dλ^{k}}{k!}  and  \frac{1}{n_{j} + 1}( ∑_{n=1}^{k} \frac{1}{n})\frac{dλ^{k}}{k!} ,

and has all basic increments equal to zero except for

incr_{d_{j}}(ω_{w^{j},k}) = 2πi\frac{dλ^{k}}{k!} .

Note that the development of ω_{w^{j},k} at Q̃_{j} is given by

ω_{w^{j},k} = d(\frac{λ^{k}}{k!} ( logQ̃_{j} +\frac{1}{n_{j} + 1} ∑_{n=1}^{k} \frac{1}{n})) + (reg)  (5.2)

and mutatis mutandis for Q̃_{0}.

6. Hurwitz spaces

In this section we construct the Hurwitz spaces H_{g,t}, H′_{g,t} as complex manifolds together with the universal curve C → H′_{g,t} and the fibrewise universal covering C̃ → C^{∘} of the pointed universal curve.

6.1. Definition. Let g, M ∈ ℤ_{≥0}, t = (n_{0}, …, n_{M}) ∈ ℤ_{≥0}^{M+1} and d := ∑(n_{j} + 1). Define H_{g,t} and H′_{g,t} to be the sets of equivalence classes of

λ and \underline{λ} := (λ, (Q_{0}, …, Q_{M}), (z_{0}, …, z_{M}), (a_{i}, b_{i})_{i=1,...,g})

where

(i) λ is a meromorphic function on a curve C of genus g with poles of order n_{0} + 1, …, n_{M} + 1 and with only simple branch points over ℂ, i.e. λ^{−1}(z) contains at least d − 1 elements for any z ∈ ℂ;

(ii) Q_{j} ∈ C is a pole of λ of order n_{j} + 1; (iii) z_{j} ∈ O_{C,Q_{j}} is the germ of a branch of the n_{j} + 1st root of λ^{−1}, z_{j}^{−(n_{j}+1)} = λ; (iv) (a_{i}, b_{i})_{i=1,...,g} is a symplectic basis of H_{1}(C, ℤ); and where λ, λ′ (resp. \underline{λ}, \underline{λ}′) are equivalent if there is an isomorphism ι : C ˜ ⟶C′ such that λ′ = λ ∘ ι (and ι(Q_{j}) = Q′_{j}, ι^{∗}(z′_{j}) = z_{j}, ι_{∗}(a_{i}) = a′_{i}, ι_{∗}(b_{i}) = b′_{i}).

We will use the notations C(\underline{λ}) := C, C^{∘}(\underline{λ}) := C^{∘}, Q_{j}(\underline{λ}) := Q_{j}, etc.

The Riemann-Hurwitz formula says that the number of branch points of λ is b := 2(g + d) − 2. Let P_{n}(\underline{λ}) ∈ C(\underline{λ}) be the branch points of λ with finite branch values u_{n}(λ) := λ(P_{n}(\underline{λ})) ∈ ℂ, n = 1, …, N := b − ∑ n_{j}. Assigning to a covering its branching locus gives a map Π from H_{g,t} to the N-th symmetric power of ℂℙ^{1}, Π : H_{g,t} → ℂℙ^{N}, \underline{λ} ↦ {u_{1}(λ), …, u_{N}(λ)}.

Define C := ⨆_{\underline{λ}∈H′_{g,t}} C(\underline{λ}) and C^{∘} := ⨆_{\underline{λ}∈H′_{g,t}} C^{∘}(\underline{λ}).

6.2. Lemma. The sets H_{g,t}, H′_{g,t} and C carry unique structures of complex manifolds such that

(i) Π : H_{g,t} → ℂℙ^{N} and ϕ : H′_{g,t} → H_{g,t}, \underline{λ} ↦ λ, are unramified coverings, i.e. u_{1}, …, u_{N} are local coordinates on H_{g,t}, H′_{g,t};

(ii) κ : C → H′_{g,t}, P ↦ \underline{λ} if P ∈ C(\underline{λ}), and Λ : C → ℂℙ^{1}, Λ|_{C(\underline{λ})} = λ, are holomorphic; (iii) there exists a diagram

[figure: C̃; π; κ̃; C^{∘}; κ; H′_{g,t}; H′_{g,t}]

such that π : C̃(\underline{λ}) → C^{∘}(\underline{λ}) is the universal covering for any \underline{λ} ∈ H′_{g,t}.

Proof. (i) (cf.[Fu, MPR]) Let B = {u_{1}, …, u_{N}} ∈ Π(H_{g,t}). Let Δ_{n} ⊂ ℂ be small disks around u_{n} and Δ ⊂ ℂℙ^{N} be their symmetric product considered as an open neighborhood of B. We construct a natural bijection β_{B} : Π^{−1}(Δ) ˜ ⟶Π^{−1}(B) × Δ as follows.

Fix u_{0} ∈ ℂ \ S, S := ⋃ Δ_{n} ⊂ ℂℙ^{1}. For λ ∈ Π^{−1}(Δ) giving the branched covering λ is equivalent to giving {u_{1}(λ), …, u_{N}(λ)} ∈ Δ plus a representation π_{1}(ℂℙ^{1} \ {u_{1}(λ), …, u_{N}(λ), ∞}, u_{0}) → S_{d} up to conjugacy such that the permutation corresponding to a loop around ∞ decomposes into (n_{0}+1)–, …, (n_{M} +1)–cycles. Because ℂℙ^{1}\(S∪ {∞}) is a deformation retract of ℂℙ^{1} \ {u_{1}(λ), …, u_{N}(λ), ∞} we get a natural isomorphism between π_{1}(ℂℙ^{1} \ {u_{1}(λ), …, u_{N}(λ), ∞}, u_{0}) and π_{1}(ℂℙ^{1} \ {u_{1}, …, u_{N}, ∞}, u_{0}). Let λ′ : C′ → ℂℙ^{1} be the covering given by {u_{1}, …, u_{N}} and the representation of π_{1}(ℂℙ^{1} \ {u_{1}, …, u_{N}, ∞}, u_{0}) corresponding to that of λ via this isomorphism of fundamental groups. We define β_{B}(λ) := (λ′, Π(λ)).

The structure of a complex manifold on H_{g,t} is defined by the condition that Π : H_{g,t} → Π(H_{g,t}) ⊂ ℂℙ^{N} becomes a covering of complex manifolds which is locally trivialized by the bijections β_{B} for B ∈ Π(H_{g,t}).

Let U be a small neighborhood of λ′ ∈ H_{g,t}. Again it suffices to construct a natural bijection β_{λ} : ϕ^{−1}(U) ˜ ⟶ϕ^{−1}(λ′) × U.

For \underline{λ} ∈ ϕ^{−1}(U) there is a (non-canonical) isomorphism ι = ι(\underline{λ}, λ′) : C(\underline{λ}) \ λ^{−1}(S) ˜ ⟶ C′ \ (λ′)^{−1}(S) of Riemann surfaces over ℂℙ^{1} \ S. Define Q′_{j} := ι(Q_{j}) and z′_{j} := (ι^{∗})^{−1}(z_{j}). Because C \ λ^{−1}(S), C′ \ (λ′)^{−1}(S) are deformation retracts of C^{∘}(\underline{λ}), (C′)^{∘}, we obtain from a_{i}, b_{i} ∈ H_{1}(C(\underline{λ}), ℤ) via ι classes a′_{i}, b′_{i} ∈ H_{1}(C′, ℤ). Altogether we constructed \underline{λ}′ := (λ′, (Q′_{j}), (z′_{j}), (a′_{i}, b′_{i})) ∈ ϕ^{−1}(λ′) ⊂ H′_{g,t} and the isomorphism ι(\underline{λ}, \underline{λ}′) := ι(\underline{λ}, λ′) compatible with the extra data is now canonical. We define β_{λ}(\underline{λ}) := (\underline{λ}′, λ).

(ii) Let U be a small neighborhood of \underline{λ} ∈ H′_{g,t} and z be a local coordinate defined on V ⊂ C(\underline{λ}) \ λ^{−1}(S). Then we define a map

⨆_{\underline{λ}′∈U} ι(\underline{λ}, \underline{λ}′)(V) ↪ ℂ^{N+1}, P ↦ (z(ι(\underline{λ}, \underline{λ}′)^{−1}(P)), u_{1}(κ(P)), …, u_{N}(κ(P))).

Because P_{n}(\underline{λ}′) is a simple branch point of λ′ there is a local coordinate τ_{n}(λ′) : V_{n}(λ′) → ℂ such that λ′ = u_{n}(λ′)+τ_{n}(λ′)^{2}, τ_{n}(λ′)(P_{n}(\underline{λ}′)) = 0. For any small neighborhood D of 0 ∈ ℂ we define a map

⨆_{\underline{λ}′∈U} τ_{n}(λ′)^{−1}(D) ↪ ℂ^{N+1}, P ↦ (τ_{n}(λ′)(P), u_{1}(κ(P)), …, u_{N}(κ(P))).

These two types of coordinate charts define on C the structure of a complex manifold.

(iii) Let P(\underline{λ}) := \{p : (0, 1] → C^{∘}(\underline{λ}) | ∃ϵ > 0 z_{0}(p((0, ϵ)) ⊂ ℝ\} and P := ⨆ P(\underline{λ}). Define maps π : P → C^{∘}, p ↦ p(1), and κ̃ : P → H′_{g,t}, p ↦ \underline{λ} if p ∈ P(\underline{λ}). Let C̃(\underline{λ}) be the set of homotopy classes of paths p ∈ P(\underline{λ}) with homotopies relative to \{1\} ⊂ (0, 1] and inside P(\underline{λ}). Put C̃ := ⨆ C̃(\underline{λ}) and let q : P → C̃ be the quotient map, π : C̃ → C^{∘} and κ̃ : C̃ → H′_{g,t} be the induced maps. It follows that C̃(\underline{λ}) has a natural topology such that π(\underline{λ}) : C̃(\underline{λ}) → C^{∘}(\underline{λ}) is the universal covering for any \underline{λ} ∈ H′_{g,t}.

Let p ∈ C̃(\underline{λ}) and U ⊂ C^{∘} be an open neighborhood of π(p) such that U(\underline{λ}) := U ∩ C^{∘}(\underline{λ}) is a connected and simply connected neighborhood of π(p). Let Ũ(\underline{λ}) ⊂ C̃(\underline{λ}) be the connected neighborhood of p such that π(Ũ(\underline{λ})) = U(\underline{λ}). Define for any open neighborhood V ⊂ κ(U) of \underline{λ} the set of V -parametrized paths

S_{U,V} := \{s : (0, 1] × V → C^{∘} | \begin{matrix}∀ \underline{λ}′ ∈ V s(., \underline{λ}′) ∈ P(\underline{λ}′); q(s(., \underline{λ})) ∈ Ũ(\underline{λ}) \\ ∃ ϵ > 0 s((1 − ϵ, 1] × V) ⊂ U \end{matrix} \}

and Ũ := q\{p′ ∈ P | ∃ open V ⊂ κ(U) κ̃(p′), \underline{λ} ∈ V ∃ s ∈ S_{U,V} s(., κ̃(p′)) = p′\} ⊂ C̃.

The sets B(p) := {Ũ | U ⊂ C^{∘} open, p ∈ U, U(κ̃(p)) connected simply connected}, p ∈ C̃, form a system of neighborhood bases for C̃, i.e.

(i) Ũ_{1}, Ũ_{2} ∈ B(p) ⇒ ∃ Ũ ∈ B(p) Ũ ⊂ Ũ_{1} ∩ Ũ_{2}; (ii) Ũ ∈ B(p) ⇒ ∃ Ũ′ ∈ B(p) ∀ p′ ∈ Ũ′ Ũ ∈ B(p′); and thus we obtain a topology on C̃. The map π : C̃ → C^{∘} becomes an unramified covering of topological spaces. We endow C̃ with the structure of a complex manifold induced by π.

6.3. Definition. Define a multiplication on the tangent sheaves of H_{g,t}, H′_{g,t} by

e_{n} ∘ e_{m} := δ_{nm}e_{n}, e_{n} := \frac{∂}{∂u_{n}} .

By definition, (H_{g,t}, ∘), (H′_{g,t}, ∘) are semisimple weak Frobenius manifolds.

Note that objects on the curve C(\underline{λ}) extend to a small neighborhood U of \underline{λ}, e.g. the point P_{n}(\underline{λ}) ∈ C(\underline{λ}) extends to a section P_{n} : U → C, \underline{λ}′ ↦ P_{n}(\underline{λ}′).

The functions u_{1}, …, u_{N}, Λ form locally a coordinate system on C^{⋄} := ⨆ C^{⋄}(\underline{λ}). Let ẽ_{1}, …, ẽ_{N}, ∂_{λ} be the corresponding vector fields. Because (Λ = u_{n} + τ_{n}^{2} at P_{n})

ẽ_{m}τ_{n} = −\frac{δ_{mn}}{2τ_{n}}  ∂_{λ}τ_{n} = \frac{1}{2τ_{n}}  ẽ_{m}z_{j} = 0 ∂_{λ}z_{j} = −\frac{1}{n_{j} + 2}z_{j}^{n_{j}+1}  (6.1)

ẽ_{1}, …, ẽ_{N} are locally defined and ẽ:= ∑ ẽ_{n}, Ẽ := ∑ u_{n}ẽ_{n}, ∂_{λ} are globally defined meromorphic vector fields on C with simple poles only at the finite branch points P_{1}, …, P_{N}.

7. A hierarchy of primitive forms

In this section we define descendant, primary, and eigenprimary forms, show that descendant forms form a hierarchy (Ω, D), and compute the e-derivative of the pairing ⟨ω_{1}, ω_{2}⟩Č for primary forms ω_{1}, ω_{2}.

7.1. Definition. A 1-form ω ∈ κ̃_{∗}Ω_{C̃/H′_{g,t}}^{1}(U), U ⊂ H′_{g,t}, is called a descendant form on U if

(i) ω(\underline{λ}) := ω|_{C̃(\underline{λ})} ∈ Ω1 ; \underline{λ}(C^{∘}(\underline{λ})) for any \underline{λ} ∈ U; (ii) the Laurent tails at Q_{j} with respect to z_{j}, j = 0, …, M, the a-periods, and the

increments of ω are constant, i.e. incr_{γ}(ω) ∈ ℂ[Λ]dΛ for any γ ∈ (R^{1}κ_{∗}\underline{ℤ})^{∗} (where incr_{γ}(ω)(\underline{λ}) := incr_{γ(\underline{λ})}(ω(\underline{λ}))).

Denote by Ω ⊂ κ̃_{∗}Ω_{C̃/H′_{g,t}}^{1} the sheaf of descendant forms and Ω_{O} := Ω ⊗\underline{ℂ} O_{H′_{g,t}}. For ω ∈ Ω(U) let H_{ω} be the set of \underline{λ} ∈ U such that ω(\underline{λ}) is not vanishing at P_{1}(\underline{λ}), …, P_{N}(\underline{λ}).

Note that the condition that the Laurent tails are constant is independent of the choice of rays because the increments are constant. Also the condition that the a-periods are constant is independent of the choice of pre-images of (a_{i}, b_{i}) in H_{1}(C^{∘}, ℤ) because the residues at Q_{0}, …, Q_{M} are constant.

Define Lie derivatives D := Lie_{ẽ+∂_{λ}} and D_{E} := Lie_{Ẽ+Λ∂_{λ}} that act on κ̃_{∗}Ω_{C̃/H′_{g,t}}^{1} and

Ω_{k} := ker(D^{k+1}) ∩ Ω, k ≥ 0.

7.2. Definition. A primary form is a section of Ω_{0}. A primary form ω is called eigenprimary if D_{E}ω = [ω]ω for some [ω] ∈ ℂ.

Expand ω in a neighborhood of P̃_{n} ∈ π^{−1}(P_{n}(\underline{λ})) with respect to τ_{n}, Λ = u_{n} + τ_{n}^{2},

ω = ∑_{k≥0} a_{kn}^{ω}τ_{n}^{k}dτ_{n}.  (7.1)

Note that the function a_{0n}^{ω} does not depend on the choice of P̃_{n} because the increments r(Λ)dΛ ∈ ℂ[Λ]dΛ of ω vanish at P_{n}.

7.3. Lemma. (i) Ω_{k} is a local system of rank (k +1)N and D : Ω_{k} → Ω_{k−1} is surjective. Ω is the union of Ω_{k}, k ≥ 0.

(ii) The action of D_{E} on Ω_{0} is diagonalizable.

(iii) Let E be the sheaf of eigenprimary forms ω ∈ Ω_{0}(U), U ⊂ H′_{g,t}, which satisfy H_{ω} = U. The family (H_{ω})_{ω∈E} is a covering of H′_{g,t}.

Proof. Let U ⊂ H′_{g,t} be a small neighborhood of \underline{λ} ∈ H′_{g,t}. Choose rays Q̃_{j}(\underline{λ}) over Q_{j}(\underline{λ}), j = 0, …, M.

(i) Let ω_{A,k} ∈ Ω(U), A ∈ N , k ≥ 0, be the relative 1-form such that ω_{A,k}(\underline{λ}′) ∈ Ω_{\underline{λ}′}^{1}(C^{∘}(\underline{λ}′)) is the 1-form defined in Proposition 5.3 using Q̃_{0}(\underline{λ}′), …, Q̃_{M}(\underline{λ}′). It follows that (ω_{A,k}) is a local basis of Ω. So it remains to prove

Dω_{A,k} = ω_{A,k−1},  Dω_{A,0} = 0,  for any A ∈ N , k ≥ 1.  (7.2)

We have incr(Dω_{A,k}) = D incr(ω_{A,k}) = ∂_{λ} incr(ω_{A,k}) = incr(ω_{A,k−1}) because Lieẽ(incr (ω_{A,k})) = 0 and Lie_{∂_{λ}}(\frac{dΛ^{k}}{k!} ) = \frac{dΛ^{k−1}}{(k−1)!}, k ≥ 1, Lie_{∂_{λ}}(dΛ) = 0.

Put C̃^{⋄} := π^{−1}(C^{⋄}). The 1-form Dω_{A,k} is holomorphic on κ̃^{−1}(U)∩C̃^{⋄} because ẽ, ∂_{λ}, and ω_{A,k} are holomorphic there. From (6.1) we obtain D(τ_{n}) = (ẽ + ∂_{λ})τ_{n} = −\frac{1}{2τ_{n}} + \frac{1}{2τ_{n}} = 0 and therefore

Dω = ∑_{k≥0} (D(a_{kn}^{ω})τ_{n}^{k}dτ_{n} + a_{kn}^{ω}D(τ_{n}^{k}dτ_{n})) = ∑_{k≥0} e(a_{kn}^{ω})τ_{n}^{k}dτ_{n}.  (7.3)

Hence Dω is also holomorphic on π^{−1}(P_{n}^{−1}(U)), n = 1, …, N. The Laurent tail of Dω_{A,k}(\underline{λ}) at Q̃_{j}(\underline{λ}) is equal to the Laurent tail of Lie_{∂_{λ}}(ω_{A,k})(\underline{λ}) at Q̃_{j}(\underline{λ}) because Lieẽ(dz_{j}^{k}) = 0. We have

Lie_{∂_{λ}}(ω_{t^{kj},k′}) = −\frac{1}{n_{j} + 1}Lie_{z_{j}^{n_{j} +2} \frac{∂}{∂z_{j}}} ( − \frac{(n_{j} + 1)^{k′}dz_{j}^{−(k+k′(n_{j}+1))}}{∏_{l=0}^{k′}(k + l(n_{j} + 1))} ) + (reg)

which gives the same Laurent tail at Q̃_{j}(\underline{λ}) as that of ω_{t^{kj},k′−1}(\underline{λ}) and using (5.2)

Lie_{∂_{λ}}(ω_{w^{j},k}) = d(\frac{Λ^{k−1}}{(k − 1)!}( logQ̃_{j} +\frac{1}{n_{j} + 1} ∑_{n=1}^{k} \frac{1}{n}) + \frac{Λ^{k}}{k!} ( − \frac{z_{j}^{n_{j}+2}}{n_{j} + 1})\frac{1}{z_{j}} ) + (reg)

which gives the same Laurent tail at Q̃_{j}(\underline{λ}) as that of ω_{w^{j},k−1}(\underline{λ}). Thus the Laurent tails of Lie_{∂_{λ}}(ω_{A,k}) and ω_{A,k−1} are the same.

Finally, the a-periods of Dω_{A,k} are

∫_{a_{i}} Dω_{A,k} = e ∫_{a_{i}} ω_{A,k} + ∫_{a_{i}} Lie_{∂_{λ}}(ω_{A,k}) = (\frac{a_{i}^{∗}ω_{A,k} − ω_{A,k}}{dΛ} )|_{P_{0}} = δ_{A,s^{i}}δ_{k1}

because Lie_{∂_{λ}}(ω) = ∂_{λ}(f)dΛ = d(\frac{ω}{dΛ}) if ω = fdΛ, and \frac{dΛ^{k}}{dΛ} = kΛ^{k−1}. Theorem 3.6 now implies (7.2).

(ii) Introduce a new set of labels

N_{p} := {s^{i}, r^{i}, v^{j}, w^{j} | i = 1, …, g, j = 1, …, M}
∪ {t^{kj} | k = 1, …, n_{j}, j = 0, …, M}.  (7.4)

Define ω_{A} := ω_{A,0} for A ∈ N_{p} \ {v^{1}, …, v^{M}} ⊂ N and ω_{v^{j}} := (n_{j} + 1)ω_{t^{n_{j} +1,j},0}, j = 1, …, M. Because (ω_{A})_{A∈N_{p}} is a local basis of Ω_{0} it suffices to prove that D_{E}ω_{A} = [ω_{A}]ω_{A} where

[ω_{t^{kj}}] = \frac{k}{n_{j} + 1},  [ω_{v^{j}}] = [ω_{r^{i}}] = 1,  [ω_{w^{j}}] = [ω_{s^{i}}] = 0.  (7.5)

The proof is analogous to that of (7.2).

We have incr(D_{E}ω_{A}) = D_{E} incr(ω_{A}) = Lie_{Λ∂_{λ}}(incr(ω_{A})) = incr(ω_{A}) because LieẼ(incr (ω_{A})) = 0 and Lie_{Λ∂_{λ}}(dΛ) = dΛ.

The 1-form D_{E}ω_{A} is holomorphic on κ̃^{−1}(U) ∩ C̃^{⋄} because Ẽ, Λ∂_{λ}, and ω_{A} are holomorphic there. Because Ẽτ_{n} = ∑ u^{m}ẽ_{m}τ_{n} = −\frac{u^{n}}{2τ_{n}} and Λ∂_{λ}(τ_{n}) = (u^{n} + τ_{n}^{2})\frac{1}{2τ_{n}} we have D_{E}τ_{n} = \frac{1}{2}τ_{n}. We obtain for any ω ∈ Ω_{O}

D_{E}ω = ∑_{k≥0} (E(a_{kn}^{ω})τ_{n}^{k}dτ_{n} + a_{kn}^{ω} \frac{k}{2}τ_{n}^{k}dτ_{n} + a_{kn}^{ω}τ_{n}^{k} \frac{1}{2}dτ_{n}) = ∑_{k≥0} (E(a_{kn}^{ω}) + a_{kn}^{ω} \frac{k + 1}{2})τ_{n}^{k}dτ_{n}. (7.6)

Hence D_{E}ω_{A} is also holomorphic on π^{−1}(P_{n}^{−1}(U)), n = 1, …, N. The Laurent tail of D_{E}ω_{A}(\underline{λ}) at Q̃_{j}(\underline{λ}) is equal to the Laurent tail of Lie_{Λ∂_{λ}}(ω_{A})(\underline{λ}) at Q̃_{j}(\underline{λ}) because LieẼ(dz_{j}^{k}) = 0. From Lie_{Λ∂_{λ}}(dΛ) = dΛ,

Lie_{Λ∂_{λ}}( − \frac{dz_{j}^{−k}}{k} ) = \frac{1}{n_{j} + 1}Lie_{z_{j} \frac{∂}{∂z_{j}}} (\frac{dz_{j}^{−k}}{k} ) = \frac{k}{n_{j} + 1}( − \frac{dz_{j}^{−k}}{k} )

and

Lie_{Λ∂_{λ}}(\frac{dz_{j}}{z_{j}} ) = −\frac{1}{n_{j} + 1}Lie_{z_{j} \frac{∂}{∂z_{j}}} (\frac{dz_{j}}{z_{j}} ) = −\frac{1}{n_{j} + 1}(\frac{dz_{j}}{z_{j}} − \frac{dz_{j}}{z_{j}} ) = 0

we see that D_{E}ω_{A} and [ω_{A}]ω_{A} have the same Laurent tails at Q̃_{0}, …, Q̃_{M}.

Finally, the a-periods of D_{E}ω_{A} are

∫_{a_{i}} D_{E}ω_{A} = E ∫_{a_{i}} ω_{A} + ∫_{a_{i}} Λd(\frac{ω_{A}}{dΛ}) + ∫_{a_{i}} ω_{A} = − ∫_{a_{i}} ω_{A} + [Λ\frac{ω_{A}}{dΛ}](a_{i})_{∗}P̃_{0} ; P̃_{0} + ∫_{a_{i}} ω_{A} = 0

because Lie_{Λ∂_{λ}}(ω) = Λ∂_{λ}(f)dΛ + fdΛ = Λd(\frac{ω}{dΛ}) + ω if ω = fdΛ, and Λ ∘ P_{0} = 0. By Theorem 3.6 this implies D_{E}ω_{A} = [ω_{A}]ω_{A}.

(iii) First assume g > 0. For any \underline{λ} ∈ H′_{g,t} one has H^{0}(C(\underline{λ}), K) ⊋ H^{0}(C(\underline{λ}), K −P_{n}(\underline{λ})) because the canonical divisor K is base-point-free. Thus there is a holomorphic 1-form ω′ on C(\underline{λ}) which is not vanishing at P_{1}(\underline{λ}), …, P_{N}(\underline{λ}). If ω ∈ Ω satisfies ω(\underline{λ}) = ω′ then ω ∈ E and \underline{λ} ∈ H_{ω}.

If g = 0 and M > 0 then ω_{w^{1}} ∈ E has no zeros and if g = M = 0 and n_{0} > 0 then ω_{t^{1,0}} has no zeros.

Let U be a small neighborhood of \underline{λ} ∈ H′_{g,t} and ω ∈ Ω(U). Choosing a ray Q̃_{0} over Q_{0}(\underline{λ}), we define a local primitive z_{ω} : κ̃^{−1}(U) → ℂℙ^{1}, U ⊂ H′_{g,t}, of ω by

z_{ω} := p.v. ∫_{Q̃_{0}} ω = ∑_{k=−(n_{j}+2)}^{−1} \frac{c_{kj}^{ω}}{k + 1}z_{j}^{k+1} + c_{−1,j}^{ω} logQ̃_{j} +p.v. ∫_{Q̃_{0}}^{Q̃_{j}}ω + ∑_{k≥0} \frac{c_{kj}^{ω}}{k + 1}z_{j}^{k+1}(7.7)

where we also wrote down the expansion of z_{ω} at a ray Q̃_{j} using (3.2).

For a basic domain Č over \underline{λ} ∈ H′_{g,t} the pairing ⟨., .⟩Č of (4.2) extends over an open neighborhood U ⊂ H′_{g,t} of \underline{λ} to a pairing ⟨., .⟩Č : Ω_{O}|_{U} ⊗ Ω_{O}|_{U} → O_{U}.

7.4. Lemma. Let U be small neighborhood of \underline{λ} ∈ H′_{g,t}, ω_{1}, ω_{2} ∈ Ω_{0}(U), and Č be a basic

region over \underline{λ}. Write c_{kj}^{(α)} := c_{kj}^{ω_{α}}, incr_{b_{i}}(ω_{α}) = q_{i}^{(α)}dΛ, q_{i}^{(α)} ∈ ℂ, and A_{i}^{(α)} := ∫_{a_{i}} ω_{α} where

α = 1, 2. Then we have

e⟨ω_{1}, ω_{2}⟩Č = − ∑_{j=0}^{M} (\frac{1}{n_{j} + 1} ∑_{k=0}^{n_{j}−1} c_{−k−2,j}^{(1)}c_{k−n_{j}−1,j}^{(2)})

− ∑_{j=1}^{M} \frac{1}{n_{j} + 1}(c_{−1,j}^{(1)}c_{−n_{j}−2,j}^{(2)} + c_{−1,j}^{(2)}c_{−n_{j}−2,j}^{(1)}) − \frac{1}{2πi} ∑_{i=1}^{g} (q_{i}^{(1)}A_{i}^{(2)} + q_{i}^{(2)}A_{i}^{(1)}).

Proof. By continuity we may assume that λ has a simple zero P_{0}(\underline{λ}).

Let ω ∈ Ω_{0}(U). Because d(Dz_{ω}) = D(dz_{ω}) = D(ω) = 0 we have Dz_{ω} = c ∈ O_{H′_{g,t}}. For any sections P, P′ : U → C̃ of κ̃ such that Λ ∘ P, Λ ∘ P′ are constant we obtain

e(z_{ω} ∘ P) = ((∂_{λ}z_{ω}) ∘ P)e(Λ ∘ P) + (ẽz_{ω}) ∘ P = (Dz_{ω} − ∂_{λ}z_{ω}) ∘ P = c − \frac{ω}{dΛ} ∘ P (7.8)

and

e ∫_{P}^{P′}ω = e([z_{ω}]_{P}^{P′}) = −[\frac{ω}{dΛ}]_{P}^{P′},  (7.9)

in particular for any a ∈ {a_{i}, b_{i}|i = 1, …, g} we have

e ∫_{a} ω = e ∫_{ã} ω = −\frac{a^{∗}ω}{dΛ} ∘ P̃_{0} + \frac{ω}{dΛ} ∘ P̃_{0}.  (7.10)

For Λω ∈ Ω_{O} we have analogously D(Λω) = ω, D(z_{Λω}) = z_{ω} + c′, c′ ∈ O_{H′_{g,t}}, e(z_{Λω} ∘ P) = z_{ω} ∘ P + c′ − Λ\frac{ω}{dΛ} ∘ P and

e ∫_{a} Λω = e ∫_{ã} Λω = [z_{ω} − Λ\frac{ω}{dΛ}]_{ã(0)}^{ã(1)} = ∫_{a} ω  (7.11)

because Λ ∘ ã(0) = Λ ∘ ã(1) = Λ ∘ P_{0} = 0.

Applying (7.10) and (7.11) we get

∑_{i=1}^{g} −q_{i}^{(1)} e ∫_{a_{i}} Λω_{2} + A_{i}^{(1)} e ∫_{b_{i}} ω_{2}

= ∑_{i=1}^{g} −q_{i}^{(1)} ∫_{a_{i}} ω_{2} + A_{i}^{(1)}( − \frac{b_{i}^{∗}ω_{2}}{dΛ} ∘ P̃_{0} + \frac{ω_{2}}{dΛ} ∘ P̃_{0}) = − ∑_{i=1}^{g} q_{i}^{(1)}A_{i}^{(2)} + A_{i}^{(1)}q_{i}^{(2)}.

From (7.9) and (3.2) we obtain for P (\underline{λ}) in a neighborhood of Q̃_{j}(\underline{λ})

e ∫_{P̃_{0}}^{P} ω = −[\frac{ω}{dΛ}]P ; P̃_{0} = ∑_{k≥−(n_{j}+2)} \frac{c_{kj}^{ω}}{n_{j} + 1}(z_{j} ∘ P)^{k+n_{j}+2} + \frac{ω}{dΛ} ∘ P̃_{0}.

On the other hand because e(c_{kj}^{ω}) = 0 for k < 0

e ∫_{P̃_{0}}^{P} ω = ∑_{k≥0} \frac{e(c_{kj}^{ω})}{k + 1}(z_{j} ∘ P)^{k+1} + e p.v. ∫_{P̃_{0}}^{Q̃_{j}}ω

and therefore by comparison of coefficients we obtain e(c_{ki}^{ω}) = \frac{k+1}{n_{j}+1}c_{k−n_{j}−1,j}^{ω} for k ≥ 0, and

e p.v. ∫_{P̃_{0}}^{Q̃_{j}}ω = \frac{c_{−n_{j}−2}^{ω}}{n_{j} + 1} + \frac{ω}{dΛ} ∘ P̃_{0}.

Applying these results the assertion follows from

∑_{j=0}^{M} ( ∑_{k=0}^{n_{j}} c_{−k−2,j}^{(1)} \frac{e(c_{kj}^{(2)})}{k + 1}) + c_{−1,j}^{(1)} e p.v. ∫_{P̃_{0j}}^{Q̃_{j}}ω_{2}

= ∑_{j=0}^{M} ( ∑_{k=0}^{n_{j}} c_{−k−2,j}^{(1)} \frac{c_{k−n_{j}−1,j}^{(2)}}{n_{j} + 1} ) + c_{−1,j}^{(1)}(\frac{c_{−n_{j}−2,j}^{(2)}}{n_{j} + 1} + \frac{ω_{2}}{dΛ} ∘ P̃_{0})

= ∑_{j=0}^{M} \frac{1}{n_{j} + 1} ∑_{k=0}^{n_{j}−1} c_{−k−2,j}^{(1)}c_{k−n_{j}−1,j}^{(2)} + ∑_{j=1}^{M} \frac{1}{n_{j} + 1}(c_{−n_{j}−2,j}^{(1)}c_{−1,j}^{(2)} + c_{−1,j}^{(1)}c_{−n_{j}−2,j}^{(2)})

because c_{−n_{j}−2,0}^{(2)} = 0 and ∑_{j=0}^{M} c_{−1,j}^{(1)} = 0 by Corollary 3.8.

8. A hierarchy of Frobenius manifolds

In the case of Saito frameworks with one-dimensional fibres the pairing (2.2) is given by (see (7.1))

g_{ω} := ∑_{n=1}^{N} (res_{P_{n}} \frac{ω^{2}}{dΛ}) (du_{n})^{2} = ∑_{n=1}^{N} \frac{(a_{0n}^{ω})^{2}}{2} (du_{n})^{2},  ω ∈ Ω.  (8.1)

8.1. Theorem. (H_{ω}, ∘, g_{ω}) is a Frobenius manifold for any ω ∈ Ω. More explicitly, (i) the functions (see (7.7))

x_{ω}^{t^{kj}} := −\frac{(n_{j} + 1)c_{n_{j}−k,j}^{ω}}{n_{j} − k + 1}  x_{ω}^{v^{j}} := −p.v. ∫_{Q̃_{0}}^{Q̃_{j}}ω  x_{ω}^{w^{j}} := −c_{n_{j},j}^{ω}

x_{ω}^{r^{i}} := − ∫_{b_{i}} ω  x_{ω}^{s^{i}} := ∫_{a_{i}} z_{ω}dΛ = − ∫_{a_{i}} Λω  (8.2)

are local g_{ω}-flat coordinates and if ∂_{A}^{ω} := \frac{∂}{∂x_{ω}^{A}} , A ∈ N_{p} (cf. (7.4)), then g_{ω}(∂_{A}^{ω}, ∂_{B}^{ω}) = 0 except for

g_{ω}(∂_{t^{kj}}^{ω}, ∂_{t^{n_{j} +1−k,j}}^{ω}) = g_{ω}(∂_{v^{j}}^{ω}, ∂_{w^{j}}^{ω}) = \frac{1}{n_{j} + 1}  g_{ω}(∂_{s^{i}}^{ω}, ∂_{r^{i}}^{ω}) = \frac{1}{2πi};  (8.3)

(ii) the morphism

∂^{ω} : Ω|_{H_{ω}} → T_{H_{ω}}, ω′ ↦ ∂_{ω′}^{ω} := − ∑_{n=1}^{N} \frac{a_{0n}^{ω′}}{a_{0n}^{ω}} e_{n},

induces an isomorphism ∂^{ω} : Ω_{0}|_{H_{ω}} ˜ ⟶T_{H_{ω}}^{g_{ω}} such that ∂_{ω_{A}}^{ω} = ∂_{A}^{ω}, A ∈ N_{p} and for any ω′ ∈ Ω

g_{ω′}(X, Y) = g_{ω}(∂_{ω′}^{ω} ∘ X, ∂_{ω′}^{ω} ∘ Y), X, Y ∈ T_{H_{ω}};  (8.4)

for a basic region Č over \underline{λ} ∈ H_{ω} and ω_{1}, ω_{2} ∈ Ω_{0} we have

g_{ω}(∂_{ω_{1}}^{ω}, ∂_{ω_{2}}^{ω}) = e ⟨ω_{1}, ω_{2}⟩Č;  (8.5)

if ∇ := 1 ⊗ d is the canonical connection on Ω_{O} = Ω ⊗ O_{H′_{g,t}} then z_{ω}dΛ ∈ Ω_{O} and

∇_{∂_{ω′}^{ω}}(z_{ω}dΛ) = ω′;  (8.6)

e ∈ T_{M}^{g_{ω}} if and only if ω ∈ Ω_{0}; (iii) for any basic region Č over \underline{λ} ∈ H_{ω} a metric potential of g_{ω} is given by

η^{ω} := ⟨ω, ω⟩Č

and a Frobenius potential is given by

Φ_{ω} := \frac{1}{2}⟨z_{ω}dΛ, z_{ω}dΛ⟩Č;

(iv) the rotation coefficients of the metrics g_{ω}

γ_{mn} := γ_{mn}^{ω} := \frac{e_{m}(a_{0n}^{ω})}{a_{0m}^{ω}}, m ≠ n,

are independent of ω and for any metric g = ∑ a_{n}^{2}(du_{n})^{2} on H ⊂ H_{g,t} such that \frac{e_{m}(a_{n})}{a_{m}} = γ_{mn}, m ≠ n, e^{k+1}(a_{n}) = 0, and any small open U ⊂ ϕ^{−1}(H) ⊂ H′_{g,t} there exists ω ∈ Ω(U) such that g = g_{ω}.

(v) if ω is eigenprimary then the vector field E is a semisimple Euler field on H_{ω} with spectrum d_{0} = 1, D = 1 + 2[ω] and

d_{t^{kj}} = 1 + [ω] − [ω_{t^{kj}}]  d_{v^{j}} = d_{r^{i}} = [ω]  d_{w^{j}} = d_{s^{i}} = 1 + [ω]

where [∂_{A}^{ω}, E] = d_{A}∂_{A}^{ω}, A ∈ N_{p};

Proof. Let U be a neighborhood of \underline{λ} ∈ H_{ω} and Q̃_{0}, …, Q̃_{M} be rays over Q_{0}(\underline{λ}), …, Q_{M}(\underline{λ}). By continuity we may assume that λ has a simple zero P_{0}(\underline{λ}).

(i), (ii) Using (7.7) we see that z_{ω}dΛ has the following expansion at Q̃_{j}

z_{ω}dΛ = − ∑_{k=−K}^{−1} \frac{(n_{j} + 1)c_{kj}^{ω}}{k + 1} z_{j}^{k−n_{j}−1}dz_{j} + c_{−1,j}^{ω} logQ̃_{j} dΛ + (p.v. ∫_{Q̃_{0}}^{Q̃_{j}}ω)dΛ

− ∑_{k≥0}^{n_{j}−1} \frac{(n_{j} + 1)c_{kj}^{ω}}{k + 1} \frac{dz_{j}^{k−n_{j}}}{k − n_{j}} − c_{n_{j},j}^{ω} \frac{dz_{j}}{z_{j}} − ∑_{k>n_{j}} \frac{(n_{j} + 1)c_{kj}^{ω}}{k + 1} z_{j}^{k−n_{j}−1}dz_{j},

(8.7)

in particular z_{ω}dΛ is logarithmic at infinity. As dΛ is invariant the basic increments of z_{ω}dΛ are determined by the basic increments of z_{ω}. If incr_{a_{i}}(ω) = dp_{i}^{ω}, incr_{b_{i}}(ω) = q_{i}^{ω}, incr_{d_{j}}(ω) = dr_{j}^{ω}, p_{i}^{ω}, q_{i}^{ω}, r_{j}^{ω} ∈ Λℂ[Λ], then we have by (5.1) and (8.7)

incr_{a_{i}}(z_{ω}) = ∫_{a_{i}} ω + p_{i}^{ω}  incr_{b_{i}}(z_{ω}) = −x_{ω}^{r^{i}} + q_{i}^{ω}  incr_{d_{j}}(z_{ω}) = 2πi c_{−1} + r_{j}^{ω}. (8.8)

From (8.7) and (8.8) we see that z_{ω}dΛ ∈ Ω_{O} and that expressing z_{ω}dΛ in the local basis ω_{A,k} ∈ Ω_{O} gives

z_{ω}dΛ = ψ^{ω} + ∑_{A∈N_{p}} x_{ω}^{A}ω_{A},  ψ^{ω} ∈ span_{ℂ}{ω_{A,k} | A ∈ N , k ≥ 1}.  (8.9)

We calculate the expansion of ẽ_{m}(z_{ω}dΛ) = ẽ_{m}(z_{ω})dΛ in a neighborhood of P̃_{n} ∈ π^{−1}(P_{n}(\underline{λ})) using (6.1) and (7.7)

ẽ_{m}(z_{ω}dΛ) = (e_{m} p.v. ∫_{Q̃_{0}}^{P̃_{n}}ω + ∑_{k≥0} \frac{e_{m}a_{kn}^{ω}}{k + 1} τ_{n}^{k+1} + ∑_{k≥0} \frac{a_{kn}^{ω}}{k + 1}( − \frac{(k + 1)δ_{nm}}{2} )τ_{n}^{k−1})2τ_{n}dτ_{n}. (8.10)

This gives ẽ_{m}(z_{ω}dΛ) = (−δ_{nm}a_{0n}^{ω} + …)dτ_{n}. On the other hand from (8.9) we obtain

ẽ_{m}(z_{ω}dΛ) = ∑_{A} \frac{∂x_{ω}^{A}}{∂u^{m}} ω_{A} + ∑_{A} x_{ω}^{A}ẽ_{m}(ω_{A}) + ẽ_{m}(ψ^{ω})  (8.11)

Put ρ := ∑_{A} x_{ω}^{A}ẽ_{m}(ω_{A}) + ẽ_{m}(ψ^{ω}). For any ω′ ∈ Ω we have

⟨ω′, ρ⟩Č = ∑_{A} x_{ω}^{A}⟨ω′, ẽ_{m}(ω_{A})⟩Č + ⟨ω′, ẽ_{m}(ψ^{ω})⟩Č

= ∑_{A} x_{ω}^{A}e_{m}⟨ω′, ω_{A}⟩Č + e_{m}⟨ω′, ψ^{ω}⟩Č = \frac{a_{0m}^{ω′}a_{0m}^{z_{ω}dΛ}}{2} ≡ 0.

Let ω_{k} ∈ Ω, k ≥ 0, such that ω_{k}(\underline{λ}) for any \underline{λ} ∈ U is the normalized abelian differential of the second kind on C(\underline{λ}) with only a pole at Q_{0}(\underline{λ}) with Laurent tail equal to the Laurent tail of (k + 1)z_{0}^{−k−2}dz_{0}. Then 0 ≡ ⟨ω_{k}, ρ⟩Č = c_{k0}^{ρ} for any k ≥ 0. Because ρ is holomorphic on Q_{0}(U) we have ρ = 0.

Therefore putting (8.10) and (8.11) together we obtain

−δ_{nm}a_{0n}^{ω} = ∑_{A} \frac{∂x_{ω}^{A}}{∂u^{m}} a_{0n}^{A}.

We thus see that (x_{ω}^{A})_{A} are local coordinates and

∂_{A}^{ω}(x_{ω}^{A}) = −\frac{a_{0n}^{A}}{a_{0n}^{ω}} .  (8.12)

This shows that ∂_{ω_{A}}^{ω} = ∂_{A}^{ω}, A ∈ N_{p} which in turn implies that ∂^{ω} is an isomorphism, and that (8.4) holds; (8.6) follows from (8.9).

Let Č be a basic region over \underline{λ}, ω_{1}, ω_{2} ∈ Ω(U) and \underline{λ}′ ∈ U. From (7.1) and dΛ = 2τ_{n}dτ_{n} we get

res_{P_{n}(\underline{λ}′)} \frac{ω_{1}(\underline{λ}′)ω_{2}(\underline{λ}′)}{dλ′} = \frac{a_{0n}^{ω_{1}}(\underline{λ}′)a_{0n}^{ω_{2}}(\underline{λ}′)}{2}.  (8.13)

On the other hand, from

\begin{matrix}ẽ_{m}ω(\underline{λ}) = ∑_{k≥0} (e_{m}(a_{kn}^{ω})τ_{n}^{k}dτ_{n} + a_{kn}^{ω}kτ_{n}^{k−1}( − \frac{δ_{mn}}{2τ_{n}} )dτ_{n} + a_{kn}^{ω}τ_{n}^{k} \frac{δ_{mn}}{2τ_{n}^{2}} dτ_{n}) \\ = ∑_{k≥−2} (e_{m}(a_{kn}^{ω}) − δ_{mn} \frac{(k + 1)a_{k+2,n}^{ω}}{2} )τ_{n}^{k}dτ_{n} \\ = (δ_{mn} \frac{a_{0n}^{ω}}{2} τ_{n}^{−2} + e_{m}(a_{0n}^{ω}) − δ_{mn} \frac{a_{2n}^{ω}}{2} + …)dτ_{n} \end{matrix} (8.14)

we obtain for P in a neighborhood of P̃_{n}(\underline{λ}′) ∈ Č that ∫_{P_{0}(\underline{λ}′)}^{P} ẽ_{n}ω_{2}(\underline{λ}′) = −\frac{a_{0n}^{ω_{2}}(\underline{λ}′)}{2τ_{n}(P)} + (reg)

and therefore

− \frac{a_{0n}^{ω_{1}}(\underline{λ})a_{0n}^{ω_{2}}(\underline{λ})}{2} = res_{P̃_{n}(\underline{λ}′)}(ω_{1}(\underline{λ}′) ∫_{P_{0}(\underline{λ}′)}^{P} ẽ_{n}ω_{2}(\underline{λ}′)).  (8.15)

Because ω_{1}(\underline{λ}′) is holomorphic on C̃(\underline{λ}′) and P̃_{n}(\underline{λ}′) is the only pole of ẽ_{n}ω_{2}(\underline{λ}′) in Č(\underline{λ}′) the reciprocity law together with (8.13) and (8.15) implies

res_{P_{n}(\underline{λ}′)} \frac{ω_{1}(\underline{λ}′)ω_{2}(\underline{λ}′)}{dλ′} = ⟨ω_{1}(\underline{λ}′), ẽ_{n}ω_{2}(\underline{λ}′)⟩Č.

Because ω_{1} is a descendant form one obtains from (4.2) ⟨ω_{1}(\underline{λ}′), ẽ_{n}ω_{2}(\underline{λ}′)⟩Č = e_{n}⟨ω_{1}(\underline{λ}′), ω_{2}(\underline{λ}′)⟩Č and thus

res_{P_{n}(\underline{λ}′)} \frac{ω_{1}(\underline{λ}′)ω_{2}(\underline{λ}′)}{dλ′} = e_{n}⟨ω_{1}, ω_{2}⟩Č(\underline{λ}′).  (8.16)

This implies (8.5) and from Lemma 7.4 we obtain (8.3). It remains to prove the statement about the flat identity.

By (7.3) Dω = 0 implies ea_{0n}^{ω} = 0 which is equivalent to eη_{n}^{ω} = 0 by (8.1) which is equivalent to the flatness of e by Proposition 2.2. Conversely, assume ea_{0n}^{ω} = 0, n = 1, …, N. We may assume that Dω ∈ Ω_{0}. The matrix (a_{0n}^{A})_{A,n} is invertible by (8.12). Therefore a_{0n}^{Dω} = ea_{0n}^{ω} = 0, n = 1, …, N, (see (7.3)) implies Dω = 0.

(iii) That η^{ω} is a metric potential follows from (8.16). Because z_{ω}dΛ is vanishing on P_{1}(U), …, P_{N}(U) we have for any ω′ ∈ Ω, ∂ ∈ T_{U}

∂⟨ω′, ψ^{ω}⟩ + ∑_{A} x_{ω}^{A}∂⟨ω′, ω_{A}⟩ = ∑_{n} ∂(u^{n})e_{n}⟨ω′, ψ^{ω}⟩ + ∑_{A} x_{ω}^{A}∂(u^{n})e_{n}⟨ω′, ω_{A}⟩

= ∑_{n} ∂(u^{n})\frac{a_{0n}^{ω′}}{2} (a_{0n}^{ψ^{ω}} + ∑_{A} x_{ω}^{A}a_{0n}^{A}) = 0.

Using this we obtain from (8.9) by linearity

∂_{A}^{ω}⟨z_{ω}dΛ, z_{ω}dΛ⟩ = ⟨ψ^{ω}, ω_{A}⟩ + ⟨ω_{A}, ψ^{ω}⟩ + ∑_{C} x_{ω}^{C}(⟨ω_{A}, ω_{C}⟩ + ⟨ω_{C}, ω_{A}⟩)

and

∂_{B}^{ω}∂_{A}^{ω}⟨z_{ω}dΛ, z_{ω}dΛ⟩ = ⟨ω_{A}, ω_{B}⟩ + ⟨ω_{B}, ω_{A}⟩.

The assertion now follows from

∂_{A}^{ω}⟨ω_{B}, ω_{C}⟩ = ∑_{n=1}^{N} \frac{∂u^{n}}{∂t_{ω}^{A}} e_{n}⟨ω_{B}, ω_{C}⟩ = ∑_{n=1}^{N} ( − \frac{a_{0n}^{A}}{a_{0n}^{ω}} )\frac{a_{0n}^{B}a_{0n}^{C}}{2}

= − ∑_{n=1}^{N} res_{P_{n}} \frac{ω_{A}ω_{B}ω_{C}}{dΛ ω} = g_{ω}(∂_{A}^{ω} ∘ ∂_{B}^{ω}, ∂_{C}^{ω}).

(iv) Define two operations on the space of (possibly degenerate) metrics g = ∑(a_{n})^{2} (du_{n})^{2}, g_{+ω} := ∑(a_{n} + a_{0n}^{ω})^{2}(du_{n})^{2} and Dg := ∑(e(a_{n}))^{2}(du_{n})^{2}.

The second Darboux-Egoroff equation eγ_{mn}^{ω} = 0, m ≠ n, implies e_{m}(e(a_{n}^{ω}))a_{m}^{ω} − e_{m}(a_{n}^{ω}) e(a_{m}^{ω}) = 0, i.e. γ_{mn}^{ω} = γ_{mn}^{Dω} on H_{ω} ∩ H_{Dω} because e(a_{m}^{ω}) = a_{m}^{Dω} by (7.3). Proposition 2.2 says that (8.4) implies that γ_{mn}^{ω} = γ_{mn}^{ω′}, m ≠ n, for any ω, ω′ ∈ Ω_{0}. Thus γ_{mn}^{ω} = γ_{mn}^{ω′}, m ≠ n, for any ω, ω′ ∈ Ω.

The converse statement is proven by induction on k. The case k = 0 follows from Proposition of 2.2 and (8.4) because k = 0 means that e is flat. If g = ∑(a_{n})^{2}(du_{n})^{2} is such that \frac{e_{m}(a_{n})}{a_{m}} = γ_{mn}, m ≠ n, and e^{k+1}(a_{n}) = 0 then g_{+ω}, ω ∈ Ω_{k}, satisfies the same properties provided that g_{+ω} is non-degenerate. There exists ω_{1} ∈ Ω_{k} such that D(g_{+ω_{1}}) is non-degenerate and satisfies these properties for k − 1. By induction there is ω_{2} = ∑ c_{A,l}ω_{A,l} ∈ Ω_{k−1}, c_{A,l} ∈ ℂ, such that D(g_{+ω_{1}}) = g_{ω_{2}}. Thus g_{+(ω_{1}−D^{−1}ω_{2})}, D^{−1}ω_{2} := ∑ c_{A,l}ω_{A,l+1} ∈ Ω_{k}, also satisfies these properties for k − 1. Again by induction there is ω_{3} ∈ Ω_{k−1} such that g_{+(ω_{1}−D^{−1}ω_{2})} = g_{ω_{3}}, thus g = g_{ω} for ω := −ω_{1} + D^{−1}ω_{2} + ω_{3} ∈ Ω_{k}.

(v) By (7.6) D_{E}ω = [ω]ω implies Ea_{0n}^{ω} = ([ω] − \frac{1}{2})a_{0n}^{ω}. By (8.1) this implies Eη_{n}^{ω} = (2[ω] − 1)η_{n}^{ω}. From Proposition 2.2 we obtain Lie_{E}(g_{ω}) = D_{ω}g_{ω} where D_{ω} = (2[ω] − 1) + 2d_{0} = 2[ω] + 1. The rest of the proof is analogous to that of Lemma 7.4 with e replaced by E.

Since d(D_{E}z_{ω}) = D_{E}(dz_{ω}) = D_{E}ω = [ω]ω we have D_{E}z_{ω} = [ω]z_{ω} +c for some c ∈ O_{H′_{g,t}}. As in (7.8) we obtain for any sections P, P′ : U → C̃ of κ̃ such that Λ∘P, Λ∘P′ are constant

E(z_{ω} ∘ P) = (Ẽz_{ω}) ∘ P = ((D_{E} − Lie_{Λ∂_{λ}})z_{ω}) ∘ P = [ω]z_{ω} ∘ P + c − Λ\frac{ω}{dΛ} ∘ P

and

E ∫_{P}^{P′}ω = [ω] ∫_{P}^{P′}ω − [Λ\frac{ω}{dΛ}]_{P}^{P′},  (8.17)

in particular

E(x_{ω}^{r^{i}}) = −E ∫_{b_{i}} ω = −[ω] ∫_{b_{i}} ω + [Λ\frac{ω}{dΛ}]_{b_{i}(0)}^{b_{i}(1)} = [ω]x_{ω}^{r^{i}}.

For Λω ∈ Ω_{O} we have analogously D_{E}(Λω) = ([ω]+1)Λω, D_{E}(z_{Λω}) = ([ω]+1)z_{Λω} +c′, c′ ∈ O_{H′_{g,t}}, E(z_{Λω} ∘ P) = ([ω] + 1)z_{Λω} ∘ P + c − Λ^{2} \frac{ω}{dΛ} ∘ P and

E(x_{ω}^{s^{i}}) = −E ∫_{ã_{i}} Λω = [([ω] + 1)z_{Λω} − Λ^{2} \frac{ω}{dΛ}]_{ã_{i}(0)}^{ã_{i}(1)} = ([ω] + 1)x_{ω}^{s^{i}}. (8.18)

From (8.17) and the development of ω at Q̃_{j} we obtain

E ∫_{P̃_{0}}^{P} ω = [ω] ∫_{P̃_{0}}^{P} ω − [Λ\frac{ω}{dΛ}]P ; P̃_{0}

= ∑_{\begin{matrix}k≥−K \\ k≠−1 \end{matrix}}[ω]\frac{c_{kj}^{ω}}{k + 1}(z_{j} ∘ P)^{k+1} + [ω]c_{−1,j}^{ω} logQ̃_{j} ∘P + [ω] p.v. ∫_{P̃_{0}}^{Q̃_{j}}ω + ∑_{k≥−K} \frac{c_{kj}^{ω}}{n_{j} + 1}(z_{j} ∘ P)^{k+1}

On the other hand

E ∫_{P̃_{0}}^{P} ω = ∑_{\begin{matrix}k≥−K \\ k≠−1 \end{matrix}} \frac{E(c_{kj}^{ω})}{k + 1}(z_{j} ∘ P)^{k+1} + E(c_{−1,j}^{ω}) logQ̃_{j} ∘P + E(p.v. ∫_{P̃_{0}}^{Q̃_{j}}ω).

Comparing coefficients we get

E(c_{kj}^{ω}) = ([ω] + \frac{k + 1}{n_{j} + 1})c_{kj}^{ω}, E (p.v. ∫_{P̃_{0}}^{Q̃_{j}}ω) = [ω] p.v. ∫_{P̃_{0}}^{Q̃_{j}}ω + \frac{c_{−1,j}^{ω}}{n_{j} + 1}.

From this we obtain

E(x_{ω}^{v^{j}}) = E(p.v. ∫_{Q̃_{0}}^{Q̃_{j}}ω) = [ω]p.v. ∫_{Q̃_{0}}^{Q̃_{j}}ω + \frac{c_{−1,j}^{ω}}{n_{j} + 1} − \frac{c_{−1,0}^{ω}}{n_{0} + 1} = [ω]x_{ω}^{v^{j}} + \frac{c_{−1,j}^{ω}}{n_{j} + 1} − \frac{c_{−1,0}^{ω}}{n_{0} + 1},

E(x_{ω}^{t^{kj}}) = −\frac{n_{j} + 1}{n_{j} + 1 − k} E(c_{n_{j}−k,j}) = −\frac{n_{j} + 1}{n_{j} + 1 − k} ([ω] + \frac{n_{j} − k + 1}{n_{j} + 1})c_{n_{j}−k,j}

= ([ω] + 1 − \frac{k}{n_{j} + 1})x_{ω}^{t^{kj}},

and

E(x_{ω}^{w^{j}}) = −E(c_{n_{j}j}^{ω}) = −([ω] + \frac{n_{j} + 1}{n_{j} + 1})c_{n_{j}j}^{ω} = ([ω] + 1)x_{ω}^{w^{j}}.

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Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

  E-mail address: rosellen@mpim-bonn.mpg.de