Meromorphic connections on ℙ^{1} and the multiplicity of Abelian integrals

Hossein Movasati ^{1}

Abstract

In this paper we introduce the concept of Abelian integrals in differential equations for an arbitrary vector bundle on ℙ^{1} with a meromorphic connection. In this general context we give an upper bound for the numbers we are looking for.

Let V be a locally free sheaf (vector bundle) of rank α on ℙ^{1} and D = ∑_{i=1}^{r} m_{i}c_{i} be a positive divisor in ℙ^{1}, i.e. all c_{i}’s are positive. We denote by C the set of c_{i}’s. A meromorphic connection ∇ on V with the pole divisor D is a ℂ-linear homomorphism of sheaves

∇ : V → Ω_{ℙ^{1}}^{1}(D) ⊗_{O_{ℙ^{1}}} V

satisfying the Leibniz identity

∇(fω) = df ⊗ ω + f∇ω, f ∈ O_{ℙ^{1}}, ω ∈ V

where Ω_{ℙ^{1}}^{1}(D) is the sheaf of meromorphic 1-forms in ℙ^{1} with poles on D (the pole order of a section of Ω_{ℙ^{1}}^{1}(D) at c_{i} is less than m_{i}). For any two meromorphic connection ∇_{1} and ∇_{2} with the same pole divisor D, ∇_{1} − ∇_{2} is a O_{ℙ^{1}}-linear map.

Let t be the affine coordinate of ℂ = ℙ^{1} − {∞}, where ∞ is the point at infinity in ℙ^{1}. By Leibniz rule and by composing ∇ with the holomorphic vector field \frac{∂}{∂t} we can define:

∇_{\frac{∂}{∂t}} : H^{0}(ℙ^{1}, V (∗∞)) → H^{0}(ℙ^{1}, V (D + ∗∞))

where ∗∞ means that the pole order at ∞ is arbitrary. Since \frac{∂}{∂t} is a holomorphic vector field in ℙ^{1} with a zero of multiplicity two at ∞, if ω ∈ H^{0}(ℙ^{1}, V (∗∞)) has a pole (resp. zero) of order m at ∞ then ∇_{\frac{∂}{∂t}} ω has a pole (resp. zero) of order m − 1 (resp. max{2, m + 1}) at ∞. If there is no confusion we write ∇ = ∇_{\frac{∂}{∂t}}.

1Supported by MPIM-Germany Keywords: Meromorphic connection, Gauss-Manin connection Math. classification: 14F05, 14F43

For any point b ∈ ℙ^{1}\C we can find a frame {e_{1}, e_{2}, …, e_{α}} of holomrphic sections of V in a neighborhood of b such that ∇e_{i} = 0 ∀i and any other solution of ∇ω = 0 is a linear combination of e_{i}’s. Analytic continuations of this frame in ℙ^{1}\C define the monodromy operator

T : π_{1}(ℙ^{1}\C, b) → GL(V_{b})

We say that ∇ is irreducible if the action of monodromy on a non-zero element of V_{b} generates the whole V_{b}. Let V^{∗} be the dual vector bundle of V. There is defined a natural dual connection ∇^{∗} : V^{∗} → Ω_{ℙ^{1}}^{1}(D) ⊗_{O_{ℙ^{1}}} V^{∗} on V^{∗} as follows

< ∇^{∗}δ, ω >= d < δ, ω > − < δ, ∇ω >, δ ∈ V^{∗}ω ∈ V

If {e_{1}, e_{2}, …, e_{α}} is a base of flat sections in a neighborhood of b then we can define the dual of it as follows: < δ_{i}, e_{j} >= 0 if i ≠ j and = 1 if i = j. We can easily check that δ_{i}’s are flat sections. The associated monodromy for ∇^{∗} with respect to this basis is just T^{∗}, where T^{∗} is the composition of T with the transpose operator. We can also define a natural connection on ∧^{k}V = {ω_{1} ∧ ω_{2} ∧ ⋯ ∧ ω_{k} | ω_{i} ∈ V } with the pole divisor D as follows:

∇(ω_{1} ∧ ω_{2} ∧ ⋯ ∧ ω_{k}) = ∑_{i=1}^{k} ω_{1} ∧ ω_{2} ∧ ⋯ { \atop ω_{i}, ∇ω_{i}} ⋯ ∧ ω_{k}

where { \atop ω_{i}, ∇ω_{i}} means that we replace ω_{i} by ∇ω_{i}.

Proposition 1. If the connection ∇ over V is irreducible then for any global meromorphic non zero section of V with poles at C ∪ {∞}, say ω, we have

1. {∇^{i}ω | i = 0, 1, 2, … } generates each fiber V_{b}, b ∈ ℙ^{1}\C ∪ {∞};

2. {∇^{i}ω | i = 0, 1, 2, …, α − 1} generates a generic fiber V_{b}.

Proof. If there exists a ω ∈ H^{0}(ℙ^{1}, V (∗∞)) such that {∇^{i}ω | i = 0, 1, 2, … } does not generate V_{b} then there is a δ_{b} ∈ V_{b}^{∗} such that

< δ_{b}, ∇^{i}ω >= 0, i = 0, 1, 2, …

Consider the flat section δ passing through δ_{b}. Since \frac{∂^{i}<δ,ω>}{∂^{i}t} |_{b}=< δ_{b}, ∇^{i}ω >= 0, we conclude that < δ, ω > is identically zero. Since ∇ is irreducible, we conclude that ω is the zero section. Now let us prove the second part. Let k be the smallest number such that for all non-zero ω ∈ H^{0}(ℙ^{1}, V (∗∞)) A = ω ∧∇ω ∧⋯∧∇^{k}ω is not identically zero. We want to prove that k = α − 1. Fix a non zero ω with the property A∧∇^{k+1}ω = 0. Let B = ℙ^{1}−C ∪{∞}∪zero(A) and V′ be the vector bundle over B generated by ω, ∇ω, ⋯, ∇^{k}ω. Since A∧∇^{k+1}ω = 0, ∇ induces on V′ a well-defined holomorphic connection. But this means that V′_{b} is invariant under monodromy. ∇ is irreducible and so k + 1 = dim(V′_{b}) = α.

Every line bundle L in ℙ^{1} is of the form L_{a∞}, where a is an integer and L_{a∞} is the line bundle associated to the divisor a∞. We define c(L) = a (Chern class). According to Grothendieck decomposition theorem, every vector bundle V on ℙ^{1} can be written as V = ⊕_{i=1}^{α}L_{i}, where L_{i}’s are line bundles. We define c(V) = ∑_{i=1}^{α} c(L_{i}). In view of Proposition 1 the following definition is natural.

Definition 1. For any meromorphic global section of V with poles at C ∪ {∞} define its degree to be the sum of its pole orders. For any natural number n let H_{∇}(n) be the smallest number such that for all ω of degree n the set {∇^{i}ω | i = 0, 1, 2, …, H_{∇}(n) − 1} generates each fiber V_{b}, b ∈ ℙ^{1}\C ∪ {∞}.

Of course we have

H_{∇}(n) ≥ α

Let V be a line bundle. In this case H_{∇}(n) is the maximum multiplicity of a zero of a ω of degree n minus one and so H_{∇}(n) = n + c(V) + 1. In general case we can only give an upper bound for H_{∇}(n).

Proposition 2. Let ∇ be an irreducible connection then

H_{∇}(n) ≤ (α − 1)(∑ m_{i}) + α(n + 1) − \frac{α(α − 1)}{2} + c(V)

Proof. For any global meromorphic section ω of V with poles at C ∪ {∞} we define A = ω ∧ ∇ω ∧ ⋯ ∧ ∇^{α−1}ω. In Proposition 1 we proved that A is a nonzero global meromorphic section of ∧^{α}V. Let m be the order of the pole of ω at ∞. The sum of pole orders of A at C is at most (α−1)(∑ m_{i})+α(n−m).

Each ∇^{i}α has a pole (resp. zero if m − i is positive) of order m − i at ∞ and so A has order m + m − 1 + m − 2 + ⋯ + m − (α − 1) = mα − α(α − 1)/2 at infinity. We conclude that the multiplicity of a zero of A in ℙ^{1}\C ∪ {∞} is less than

(α − 1)(∑ m_{i}) + α(n − m) + mα − α(α − 1)/2 + c(V)

If b ∈ ℙ^{1}\C ∪ {∞} be a point such that ω, ∇ω ⋯, ∇^{i}ω, i ≥ α − 1 do not generate V_{b} then A has a zero of multiplicity i − (α − 2). The proposition is proved.

It does not seem to the author this upper bound to be the best one. More precisely for any vector bundle V and divisor D on ℙ^{1} can we find a meromorphic connection ∇ on V with pole divisor D such that H_{∇}(n) is the above number?

Let δ be a flat section of V^{∗} in a small open set U around b and ω be a global meromorphic section of V with poles at C ∪{∞}. From now on we use the notation ∫_{δ} ω instead of < δ, ω >. Let us fix the number n and suppose that the degree of ω is less than n. What is the maximum multiplicity of ∫_{δ} ω at t ∈ U, say H_{∇}(n, δ)? Let S(n) be the vector space of meromorphic sections of V with poles at C ∪ {∞} and degree less than n. Since S(n) is a finite dimensional vector space, H_{∇}(n, δ_{t}) is a finite number.

Proposition 3. If ∇ is irreducible then for all t ∈ U

H_{∇}(n, δ) ≥ dim_{ℂ}S(n) − 1

The equality happens except for a finite number of points in U.

Proof. Let ω_{1}, ω_{2}, …, ω_{b} be a basis for the vector space S(n). Consider the determinant

W_{b}(t) = det[\frac{∂^{i} ∫_{δ} ω_{j}}{∂t^{i}}]_{b×b}

It is enough to prove that W_{b}(t) is not identically zero. Let a ≤ b be the smallest number such that W_{a}(t) is identically zero. There exist holomorphic functions p_{i}, i = 1, 2, …, a in U such that

A_{a} = ∑_{i=1}^{a−1} A_{i}p_{i}/p_{a} = 0

where A_{i} is the i-th column of [^{\frac{∂^{i} ∫_{δ} ω_{j}}{∂t^{i}}}]_{a×a}. This is a equalities. If we act \frac{∂}{∂t} to the i-th equation and subtract the (i + 1)-th equation we conclude that [^{\frac{∂^{i} ∫_{δ} ω_{j}}{∂t^{i}}}]_{b−1×b−1}.[\frac{∂(p_{i}/p_{b})}{∂t}]_{b−1×1} = 0. By hypothesis, this implies that [\frac{∂(p_{i}/p_{b})}{∂t}]_{b_{1}×1} ≡ 0 or equivalently ∑_{i=1}^{a} c_{i} ∫_{δ} ω_{i} = 0, where c_{i}’s are constant. Since ∇^{∗} is irreducible, we have ∑_{i=1}^{a} c_{i}ω_{i} = 0 which is a contradiction.

Proposition 4. If ∇ is irreducible then

H_{∇}(n) = supp{H_{∇}(n, δ)}

where δ runs through all flat sections of V^{∗} in ℙ^{1}\C ∪ {∞}.

Proof. The proof is essentially stated in Proposition 1. If there exists a degree n section ω such that {∇^{i}ω | i = 0, 1, 2, …, p − 1} does not generate V_{b} then there is a δ_{b} ∈ V_{b}^{∗} such that

∫_{δ_{b}} ∇^{i}ω = 0, i = 0, 1, 2, …, p − 1

Consider the flat section δ passing through δ_{b}. We conclude that ∫_{δ} ω has multiplicity p at b and hence H_{∇}(n, δ_{b}) ≤ H_{∇}(n). The proof of the other part is similar.

Regular Connections and Linear Equations: Consider the connection ∇^{∗} on V^{∗} as before and fix a trivialization map for V^{∗} around a singular point c_{i}. ∇^{∗} is called regular at c_{i} if each flat section of V^{∗} in a sector with the vertex c_{i} has at most a polynomial growth near c_{i} (see [Ku] p. 36 or [AB] p. 8). ∇ is called regular if it is regular in all c_{i}’s.

Let ω_{1}, ω_{2}, …, ω_{α} be global meromorphic sections of V with poles at C∪{∞}. Let also δ_{1}, …, δ_{α} be a base of flat sections of V^{∗} in a neighborhood of b. The Wornskian function is defined as follows

W (t) = W (ω_{1}, ⋯, ω_{α})(t) = det[∫_{δ_{j}} ω_{i}]_{α×α}

The division of two such functions is a one valued meromorphic function in ℙ^{1}\C and by regularity of ∇^{∗} we conclude that it extends meromorphically to the whole ℙ^{1}. Fix an ω. By a similar argument as stated in Proposition 3 and by irreducibility of ∇ we know that W (ω, ∇ω, …, ∇^{α−1}ω) is not identically zero. The set {∫_{δ} ω | δ is a flat section of V^{∗}} is a base for the space of solutions of the following linear equation

ψ : | \begin{matrix}Y & ∫_{δ} ω \\ Y′ & ∫_{δ} ∇ω \\ ⋮ & ⋮ \\ Y^{(α)} & ∫_{δ} ∇^{α}ω \end{matrix} | (1)

writing in other form

ψ : Y^{(α)} + ∑_{i=1}^{α}(−1)^{i}P_{i}Y^{(i)} = 0 (2)

where

P_{i} = \frac{W (ω, ∇ω, …, { \atop ∇^{α−i}ω}, …, ∇^{α}ω)}{W}, W = W (ω, ∇ω, …, ∇^{α−1}ω)

Since ∫_{δ_{i}} ω̃ has polynomial growth at the points of C, ψ is regular therefore it must be Fuchsian i.e. P_{i} has poles of order at most i(see [AB]). The union of poles of P_{i}’s is the singular set of the Picard-Fuchs equation ψ. It has three type of singularities:

1. C; in a c_{i} ∈ C the solutions of (2) branch.

2. Z the zeros of W; In these singularities like regular points we have a space of solutions of dimension α. Note that P_{1} = \frac{\frac{∂W}{∂t}}{W} and so neither of these points is regular. For this reason in [AB] these are called apparent singularities. For a zero b of W we can find a flat section δ of V^{∗} such that ∫_{δ} ω has multiplicity greater than α at b.

3. ∞; Let m be the order of the pole of ω at ∞. The solutions of (2) in a neighborhood of ∞ are meromorphic functions with poles of order at most m at ∞.

Since P_{1} = \frac{\frac{∂W}{∂t}}{W} we have

Res(P_{1}dt, t = c) = mul(W, t = c), c ∈ C ∪ Z

Now Consider a regular linear equation ψ with singularities at C ∪ Z ∪ {∞} and suppose that it has apparent singularities in Z and a singularity of type 3 in ∞. Furthermore assume that ψ has the same monodromy representation like as of ∇^{∗}.

Proposition 5. ψ is obtained by a meromorphic global section of V with poles at C ∪ {∞}.

Proof. Consider in a neighborhood of b a base of flat sections δ_{i} of V^{∗} and a base e_{i}, i = 1, 2, …, α for solutions of ψ such that the monodromy representation of the both ∇^{∗} and ψ with respect to these bases is the same Define a section of V = (V^{∗})^{∗} as follows

ω(δ_{t}) = e_{i}(t)

this is a one valued holomorphic section of V in ℙ^{1}\C ∪ {∞}. Since ψ and ∇^{∗} are regular, ω extends meromorphically to C.

Let ϕ_{b} be the maximum multiplicity of solutions of ψ at b. If b is a regular point of ψ then ϕ_{b} = α − 1 and if it is an apparent singularity of ψ then ϕ_{b} ≥ α. In the last case by definition of W we can see that W has a zero of order at least ϕ_{b} − (α − 1) at b and by P_{1} = \frac{\frac{∂W}{∂t}}{W} we have

ϕ_{b} ≤ Res(P_{1}dt, t = b) + (α − 1)

Remark: Let us choose a trivialization of V in a small disk D around a singular point c_{i} of the connection ∇, V |_{D}≅ D × ℂ^{α}, and a coordinate z in D. In this coordinate we can write ∇v = \frac{∂v}{∂z} + ∑_{j=1}^{m_{i}} \frac{C_{j}}{z^{j}} v + A(z)v, where v is a holomorphic vector in D, C_{j}, 1 ≤ j ≤ m_{i} (resp. A(z)) is a constant (resp. holomorphic in z) matrix. C_{1} is called the residue of the connection at c_{i}. Now we can apply the Levelt’s theory (see [AB] Section 1, 2.2) to understand the local theory of this connection.

Lefschetz Pencil: Let M be a projective compact complex manifold of dimension two, {M_{t}}_{t∈ℙ^{1}} a pencil of hyperplane sections of M and f the meromorphic function on M whose level sets are M_{t}’s (see [La]). We set R the indeterminacy points of f, L_{t} = M_{t} − R, C = {c_{1}, c_{2}, c_{3}, …, c_{r}} the set of critical values of f, β = dim(H^{1}(L_{t}, ℂ)) for a t ∈ ℙ^{1} − C and ℂ[t] the ring of polynomials in t. Since f |_{M−R} is a C^{∞} fibration over ℙ^{1} − C (see [La]), β is independent of t. We assume that

1. The axis of the pencil intersects M transversally. This is equivalent to this fact that in a coordinate system (x, y) around each indeterminacy point of f we can write f = \frac{x}{y};

2. The critical points of f are isolated; 3. The pole divisor D = M_{∞} of f is a regular fiber, i.e. ∞ ∉ C.

We define Ω^{i}(∗D) to be the set of meromorphic i-forms in M with poles of arbitrary order along D. The set H̃ = ∪_{t∈B}H^{1}(L_{t}, ℂ), where B = ℙ^{1} − C, has a natural structure of a complex manifold and the natural projection H̃^{1} → B is a holomorphic vector bundle which is called the cohomology vector bundle. The sheaf of holomorphic sections of H̃ is also denoted by H̃. In what follows when we consider f as a holomorphic function we mean its restriction to M −R. Let ℂ_{M−R} be the sheaf of constant functions in M −R and R^{1}f_{∗}ℂ_{M−R} be the 1-th direct image of the sheaf ℂ_{M−R} (see [GrRe]). Any element of R^{1}f_{∗}ℂ_{M−R}(U), U being an open set in B, is a holomorphic section of the cohomology fiber bundle map. It is easy to verify that

H̃ ≅ R^{1}f_{∗}ℂ_{M−R} ⊗_{ℂ} O_{ℙ^{1}}, in B

Now let us introduce the Gauss-Manin Connection on H̃. Consider a holomorphic coordinate (t, 0) in U, a small open disk in ℙ^{1}. The Gauss-Manin connection is defined as follows:

∇ : H̃(U) → Ω_{ℙ^{1}}^{1} ⊗_{O_{U}} H̃(U)
∇(g ⊗ c) = dg ⊗ c, c ∈ R^{1}f_{∗}ℂ_{M−R}(U), g ∈ O_{ℙ^{1}}(U)

The sheaf of flat sections of ∇ is R^{1}f_{∗}ℂ_{M−R}. Let \frac{∂}{∂t} be a vector field in U. We write ∇_{\frac{∂}{∂t}} = \frac{∂}{∂t} ∘ ∇. (∇_{\frac{∂}{∂t}} (g ⊗ c) = \frac{∂g}{∂t} ⊗ c).

In the same way we can define the cohomology fiber bundle H̃_{c} of compact fibers M_{t}. Since H̃_{c} is a ∇-invariant sub-vector bundle of H̃, we have the restriction of ∇ to H̃_{c} which we denote it again by ∇.

Let ω be a meromorphic 1-form in M with poles along some fibers of f. Let also {δ_{t}}_{t∈ℙ^{1}−C}, δ_{t} ⊂ L_{t} be a a continuous family of cycles. The Abelian integral ∫_{δ_{t}} ω appears in the deformation df + ϵω of df inside holomorphic foliations (differential equations) and it is related to the number of limit cycles which born from the cycles δ_{t} (see [Ho]). The pair (H̃, ∇) is defined in ℙ^{1} −C and in order to be in the context of this paper we may be interested to prove:

Proposition 6. Under the assumptions 1,2,3, there is a vector bundle V, a sub-vector bundle V̄ ⊂ V and a meromorphic connection on V with poles in C

∇ : V → Ω_{ℙ^{1}}^{1}(D) ⊗_{O_{ℙ^{1}}} V, D = ∑ m_{i}c_{i}

such that:

1. V̄ is ∇-invariant.

2. (V, ∇) (resp. (V̄ , ∇)) coincides with (H̃, ∇)( resp. (H̃_{c}, ∇)) in ℙ^{1} − C;

3. The Brieskorn lattices ( Petrov module in the context of differential equations) of f H′ = \frac{Ω^{1}(∗D)}{df∧Ω^{0}(∗D)+dΩ^{0}(∗D)} is ℂ[t]-isomorphism to the module of global sections of V with poles of arbitrary order at ∞.

This is a task which is done in details in [Ho1]. If the singularities of f are non-degenerated, i.e. in a holomorphic coordinate (x, y) around a singularity p_{i} we can write f = f(p_{i}) + x^{2} + y^{2}, then all the m_{i}’s are equal to one. In other words ∇ is logarithmic.

The pair (V, ∇) is not irreducible but if H^{1}(M, ℂ) = 0 and f satisfies 1, 2, 3 and has non-degenerated singularities with distinct images then (V̄ , ∇) is irreducible (see [La] 7.3). The following proposition justifies the use of (V̄ , ∇) instead of (V, ∇).

Proposition 7. For an ω a meromorphic 1-form in M with poles of order at most n along D, the integral ∫_{δ_{t}} ω is a polynomial of degree n. ∇i_{\frac{∂}{∂t}} ω, i > n restricted to each fiber has not residues in R and hence is a meromorphic section of V̄.

Proof. We have p(t) := ∫_{δ_{t}} ω = t^{n} ∫_{δ_{t}} \frac{ω}{f^{n}}. Since the 1-form \frac{ω}{f^{n}} has not pole along D, \frac{p(t)}{t^{n}} has finite growth at t = ∞. Since p(t) is holomorphic in ℂ (even in the points of C), we conclude that p(t) is a polynomial of degree at most n. The second part is a direct consequence of the first one and the fact that \frac{∂}{∂t} ∫_{δ_{t}} ω = ∫_{δ_{t}} ∇_{\frac{∂}{∂t}} ω.

I tried to study the maximum multiplicity of Abelian integrals in the context of meromorphic connections. My motives were the paper [Ma] and also a paper of mine, where the extension of cohomology vector bundles and their connections to the critical values of a meromorphic function is discussed. The upper bound obtained in Proposition 2 seems to be far from the best one (at least for Guass-Manin connections). Some works in Differential equations (see [HoIl]) suggest that the number H_{∇}(n) must be very sensitive with respect to ∇.

Here I would like to express my thanks to Max-Planck institute for hospitality. Thanks go also to C. Hertling, Y. Holla and S. Archava for many useful conversations.

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