Meromorphic connections on P and the multiplicity of Abelian integrals

In this paper we introduce the concept of Abelian integrals in differential equations for an arbitrary vector bundle on P1 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 and D = ∑r i=1 mici be a positive divisor in P , i.e. all ci’s are positive. We denote by C the set of ci’s. A meromorphic connection ∇ on V with the pole divisor D is a C-linear homomorphism of sheaves ∇ : V → Ω P (D)⊗O P1 V satisfying the Leibniz identity ∇(fω) = df ⊗ ω + f∇ω, f ∈ OP1 , ω ∈ V where Ω P (D) is the sheaf of meromorphic 1-forms in P with poles on D (the pole order of a section of Ω P (D) at ci is less than mi). For any two meromorphic connection ∇1 and ∇2 with the same pole divisor D, ∇1 −∇2 is a OP1-linear map. Let t be the affine coordinate of C = P − {∞}, where ∞ is the point at infinity in P. By Leibniz rule and by composing ∇ with the holomorphic vector field ∂ ∂t we can define: ∇ ∂ ∂t : H(P, V (∗∞)) → H(P, V (D + ∗∞)) where ∗∞ means that the pole order at ∞ is arbitrary. Since ∂ ∂t is a holomorphic vector field in P with a zero of multiplicity two at ∞, if ω ∈ H(P, V (∗∞)) has a pole (resp. zero) of order m at ∞ then ∇ ∂ ∂t ω Supported by MPIM-Germany