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2) Prove that L(E, E )∇ can be identiﬁed with the subsheaf of the sheaf H omOM (E , E ) consisting of the homomorphisms which commute to the action of ∇ and ∇ (we also say that they are morphisms of bundles with connection), or also which send E ∇ in E ∇ . 12 Exercise. c, show that, if ∇ is integrable, then so is its pullback by f on the manifold M . 13 Corollary (Analytic extension). The homomorphisms of bundles with connection satisfy the principle of analytic extension: if V ⊂ U is the inclusion of connected open sets of M which induces an isomorphism between the fundamental groups of V and U , and if ϕ : E|V → E|V is a homomorphism of bundles with connection, then ϕ can be extended in a unique way as a homomorphism of bundles with connection E|U → E|U .
We will say that E is a logarithmic lattice of the meromorphic bundle with connection (M , ∇) if 1 log Z ⊗ E ∇(E ) ⊂ ΩM OM and more generally that it is a lattice of order r if 1 ∇(E ) ⊂ ΩM (r + 1) log Z ⊗ E OM and of order r if moreover it does not have order r − 1 (logarithmic = order 0). Therefore, E is a logarithmic lattice if, in any local basis of E , the connection matrix of ∇ has forms with logarithmic poles along Z as entries. 2 Exercise (Behaviour of the order by operations on lattices).
If E is a bundle, one sets E (kZ) = E ⊗OM OM (kZ) and one deﬁnes similarly E (k1 Z1 + · · · + kp Zp ). 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 .