By Bibhutibhushan Datta

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C Pullback of bundles with connection Let (E, ∇) be a holomorphic bundle with connection on an analytic manifold M and let f : M → M be a holomorphic mapping from an analytic manifold M to M . We know how to deﬁne the pullback E = f ∗ E of the 30 0 The language of ﬁbre bundles bundle E on the manifold M (cf. 4). 9) ⊗ f −1 OM f −1 E . We use the notation E = f ∗ E and E = f ∗ E . When E is the cotangent bundle T ∗ M , the cotangent mapping T ∗f is a morphism of bundles T ∗f : f ∗ T ∗ M −→ T ∗ M deﬁning a morphism of sheaves of OM -modules (cf.

The derivation d has degree one. The interior product ιξ by the vector ﬁeld ξ, deﬁned by the property that, for any p-form ω and any p − 1 vector ﬁelds ξ2 , . . , ξp , (ιξ ω) (ξ2 , . . , ξp ) = ω(ξ, ξ2 , . . , ξp ), has degree −1. These three operators are related by the formulas Lξ = ιξ ◦ d + d ◦ ιξ and ι[ξ,η] = Lξ ιη − ιη Lξ . 5) of rank p of the tangent sheaf ΘM , which is stable by the Lie bracket, that is, such that, for any m ∈ M and all ξ, η ∈ Fm , we have [ξ, η] ∈ Fm . An integral submanifold of the foliation F is a (not necessarily closed) connected analytic submanifold V of M of dimension p such that the tangent map T iV to the inclusion iV (cf.

3 Deﬁnition (Meromorphic bundles, lattices). A meromorphic bundle on M with poles along Z is a locally free sheaf of OM (∗Z)-modules of ﬁnite rank. A lattice of this meromorphic bundle is a locally free OM -submodule of this meromorphic bundle, which has the same rank. In particular, a lattice E of a meromorphic bundle M coincides with M when restricted to M Z. Moreover, we have M = OM (∗Z) ⊗ E . OM A meromorphic bundle M can contain nonisomorphic lattices. It can also contain no lattice at all (see for instance [Mal94, Mal96] where a criterion for the existence of lattices in a meromorphic bundle is also given).